-
Notifications
You must be signed in to change notification settings - Fork 0
/
RegressionTheory.html
1047 lines (998 loc) · 255 KB
/
RegressionTheory.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
<meta charset="utf-8">
<meta name="generator" content="quarto-1.4.358">
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<meta name="author" content="ហាស់ សុធា">
<meta name="dcterms.date" content="2023-11-28">
<title>ទ្រឹស្តីតម្រែតម្រង់តម្លៃ (Regression Theory)</title>
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
/* CSS for syntax highlighting */
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
}
pre.numberSource { margin-left: 3em; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
</style>
<script src="RegressionTheory_files/libs/clipboard/clipboard.min.js"></script>
<script src="RegressionTheory_files/libs/quarto-html/quarto.js"></script>
<script src="RegressionTheory_files/libs/quarto-html/popper.min.js"></script>
<script src="RegressionTheory_files/libs/quarto-html/tippy.umd.min.js"></script>
<script src="RegressionTheory_files/libs/quarto-html/anchor.min.js"></script>
<link href="RegressionTheory_files/libs/quarto-html/tippy.css" rel="stylesheet">
<link href="RegressionTheory_files/libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles">
<script src="RegressionTheory_files/libs/bootstrap/bootstrap.min.js"></script>
<link href="RegressionTheory_files/libs/bootstrap/bootstrap-icons.css" rel="stylesheet">
<link href="RegressionTheory_files/libs/bootstrap/bootstrap.min.css" rel="stylesheet" id="quarto-bootstrap" data-mode="light">
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" integrity="sha512-c3Nl8+7g4LMSTdrm621y7kf9v3SDPnhxLNhcjFJbKECVnmZHTdo+IRO05sNLTH/D3vA6u1X32ehoLC7WFVdheg==" crossorigin="anonymous"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.5.1/jquery.min.js" integrity="sha512-bLT0Qm9VnAYZDflyKcBaQ2gg0hSYNQrJ8RilYldYQ1FxQYoCLtUjuuRuZo+fjqhx/qtq/1itJ0C2ejDxltZVFg==" crossorigin="anonymous"></script>
<script type="application/javascript">define('jquery', [],function() {return window.jQuery;})</script>
<script type="text/javascript">
window.PlotlyConfig = {MathJaxConfig: 'local'};
if (window.MathJax && window.MathJax.Hub && window.MathJax.Hub.Config) {window.MathJax.Hub.Config({SVG: {font: "STIX-Web"}});}
if (typeof require !== 'undefined') {
require.undef("plotly");
requirejs.config({
paths: {
'plotly': ['https://cdn.plot.ly/plotly-2.27.0.min']
}
});
require(['plotly'], function(Plotly) {
window._Plotly = Plotly;
});
}
</script>
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml-full.js" type="text/javascript"></script>
<link rel="stylesheet" href="style.css">
</head>
<body>
<header id="title-block-header" class="quarto-title-block default page-columns page-full">
<div class="quarto-title-banner page-columns page-full">
<div class="quarto-title column-body">
<h1 class="title">ទ្រឹស្តីតម្រែតម្រង់តម្លៃ (Regression Theory)</h1>
</div>
<div class="quarto-title-meta-container">
<div class="quarto-title-meta-column-start">
<div class="quarto-title-meta">
<div>
<div class="quarto-title-meta-heading">ដោយ៖</div>
<div class="quarto-title-meta-contents">
<p><a href="https://hassothea.github.io/">ហាស់ សុធា</a> </p>
</div>
</div>
<div>
<div class="quarto-title-meta-heading">កាលបរិច្ឆេទ</div>
<div class="quarto-title-meta-contents">
<p class="date">ថ្ងៃទី ២៨ ខែ វិច្ឆិកា ឆ្នាំ ២០២៣</p>
</div>
</div>
</div>
</div>
<div class="quarto-title-meta-column-end quarto-other-formats-target">
</div>
</div>
<div class="quarto-other-links-text-target">
</div> </div>
</header><div id="quarto-content" class="page-columns page-rows-contents page-layout-article">
<div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
<nav id="TOC" role="doc-toc" class="toc-active">
<h2 id="toc-title">Table of contents</h2>
<ul>
<li><a href="#សចកតផតម" id="toc-សចកតផតម" class="nav-link active" data-scroll-target="#សចកតផតម">១. សេចក្តីផ្តើម</a></li>
<li><a href="#ទនននយ" id="toc-ទនននយ" class="nav-link" data-scroll-target="#ទនននយ">២. ទិន្នន័យ</a></li>
<li><a href="#រងវសនកហសនង-បរមមដល" id="toc-រងវសនកហសនង-បរមមដល" class="nav-link" data-scroll-target="#រងវសនកហសនង-បរមមដល">៣. រង្វាស់នៃកំហុសនិង បរមាម៉ូដែល</a></li>
<li><a href="#សងឃមគណតមនលកខខណឌនង-បរមមដល-eta" id="toc-សងឃមគណតមនលកខខណឌនង-បរមមដល-eta" class="nav-link" data-scroll-target="#សងឃមគណតមនលកខខណឌនង-បរមមដល-eta">៤. សង្ឃឹមគណិតមានលក្ខខណ្ឌនិង បរមាម៉ូដែល <span class="math inline">\(\eta\)</span></a>
<ul class="collapse">
<li><a href="#វចទរចដនយ-random-vectors" id="toc-វចទរចដនយ-random-vectors" class="nav-link" data-scroll-target="#វចទរចដនយ-random-vectors">៤.១. វុិចទ័រចៃដន្យ (Random vectors)</a></li>
<li><a href="#សងឃមគណតមនលកខខណឌ-conditional-expectation" id="toc-សងឃមគណតមនលកខខណឌ-conditional-expectation" class="nav-link" data-scroll-target="#សងឃមគណតមនលកខខណឌ-conditional-expectation">៤.២. សង្ឃឹមគណិតមានលក្ខខណ្ឌ (conditional expectation)</a></li>
</ul></li>
</ul>
</nav>
</div>
<main class="content quarto-banner-title-block page-columns page-full" id="quarto-document-content">
<div class="no-row-height column-margin column-container"><div class="">
<div class="float-left">
<p><a href="https://hassothea.github.io/MLcourses/index.html">👈 មេរៀនមុន៖ “សិក្ខាម៉ាសុីន”</a></p>
</div>
<div class="float-right">
<p><a href="https://hassothea.github.io/MLcourses/LinearRegression.html">👉 មេរៀនបន្ទាប់៖ “តម្រែតម្រង់លីនេអ៊ែរ” </a></p>
</div>
</div></div><section id="សចកតផតម" class="level2">
<h2 class="anchored" data-anchor-id="សចកតផតម">១. សេចក្តីផ្តើម</h2>
<p><strong>តម្រែតម្រង់តម្លៃ ឬ Regression</strong> សំដៅដល់កិច្ចការទាំងឡាយណាដែលយើងចង់ព្យាករណ៍តម្លៃនៃ output ជាចំនួនពិត (<span class="math inline">\(y\in\mathbb{R}\)</span>)។ មានកិច្ចការជាច្រើនដែលអាចចាត់ចូលជាប្រភេទតម្រែតម្រង់តម្លៃ ហើយលើសពីនេះទៅទៀត វាក៏មានលក្ខណៈទូទៅជាងបញ្ហាដែលមានទម្រង់ជា <strong>ការធ្វើចំណែកថ្នាក់បែបមានការណែនាំ (supervised classification)</strong> ផងដែរ ដោយសារគេអាចបម្លែងតម្លៃជាចំនួនពិតទៅជាប្រូបាបប៊ីលីតេនិង នាំទៅដល់ការធ្វើចំណែកថ្នាក់បានដោយងាយ ឬនិយាយម្យ៉ាងទៀតគឺថា ការធ្វើចំណែកថ្នាក់បែបមានការណែនាំជាទូទៅកើតចេញពីការកែច្នៃតម្រែតម្រង់តម្លៃ។ ហេតុនេះហើយ យើងចាប់ផ្តើមនិយាយពីទ្រឹស្តីសំខាន់ៗនៃតម្រែតម្រង់តម្លៃមុននឹងឈានទៅដល់ការសិក្សាម៉ូដែលផ្សេងៗសម្រាប់ដោះស្រាយបញ្ហបែបនេះ។</p>
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="./figures/reg_vs_clas.png" class="img-fluid figure-img"></p>
<figcaption class="figure-caption">រូបទី១៖ តម្រែតម្រង់តម្លៃ (regression) និង ការធ្វើចំណែកថ្នាក់ដោយមានការណែនាំ (supervised classification) ។</figcaption>
</figure>
</div>
<p>ក្នុងអត្ថបទនេះ យើងនឹងសិក្សាពីប្រភេទរង្វាស់នៃកំហុសមួយប្រភេទដែលគេនិយមប្រើក្នុងការសិក្សាតម្រែតម្រង់តម្លៃហៅថា <strong>មធ្យមនៃលម្អៀងការ៉េ (Mean Squared Error)</strong> ហើយនិងម៉ូដែលដែលល្អបំផុតសម្រាប់ប្រើប្រាស់ជាមួយនឹងរង្វាស់កំហុសប្រភេទនេះ។ បើដឹងថាម៉ូដែលដែលល្អបំផុតជានរណាទៅហើយ ហេតុអ្វីបញ្ហាតម្រែតម្រង់តម្លៃនៅតែមានភាពស្មុគស្មាញទៀត? ក្រៅពីឆ្លើយសំនួរនេះ គោលដៅចម្បងគឺយើងចង់បង្ហាញពីបរមាភាពនៃម៉ូដែលដែលយើងហៅថាជាបរមាម៉ូដែលនេះតាមរយៈទ្រឹស្តីនិង ការគណនាផងដែរ។ យើងក៏នឹងស្គាល់ផងដែរពីធាតុសំខាន់ៗជាច្រើនដូចជា វុិចទ័រចៃដន្យ (random vector) និង សង្ឃឹមគណិតមានលក្ខខណ្ឌ (conditional expectation) នៃរបាយប្រូបាបមានវិមាត្រលើសពីមួយ។</p>
<blockquote class="blockquote">
<p><strong>សម្រាប់មិត្តអ្នកអាន</strong>៖ ប្រសិនបើមិត្តអ្នកអានមិនមែនជាអ្នកចូលចិត្តស្វែងយល់ពីមូលហេតុជាទ្រឹស្តីគណិតវិទ្យានៃបញ្ហាផ្សេងៗនោះទេ អត្ថបទនេះមិនមែនសម្រាប់ប្រិយមិត្តទេ។ ប្រិយមិត្តអាចអានត្រឹមផ្នែកទី២ដែលនិយាយអំពីការតាងទិន្នន័យហើយ រំលងទៅអានផ្នែកបន្ទាប់ដែលនិយាយអំពីម៉ូដែលតម្រែតម្រង់លីនេអ៊ែរតែម្តង។</p>
</blockquote>
<hr>
</section>
<section id="ទនននយ" class="level2">
<h2 class="anchored" data-anchor-id="ទនននយ">២. ទិន្នន័យ</h2>
<p>មុននឹងចូលដល់ការស្វែងយល់ពីមូលដ្ឋាននៃតម្រែតម្រង់តម្លៃនិង ម៉ូដែលនានាព្រមទាំងការសិក្សាសុីជម្រៅក្នុងសិក្ខាម៉ាសុីន យើងនឹងកំណត់តាងធាតុសំខាន់ៗមួយចំនួនដែលនឹងត្រូវប្រើប្រាស់ជាទូទៅសម្រាប់ការសិក្សានេះ។</p>
<p>យើងបានកំណត់តាងទិន្នន័យសម្រាប់បង្វឹកម៉ូដែលសិក្ខាម៉ាសុីនដោយ៖ <span class="math inline">\({\cal T}=\{(x_1,y_1),\dots, (x_n,y_n)\}\)</span> ដែលមានផ្ទុក <span class="math inline">\(n\geq 1\)</span> ទិន្នន័យហើយក្នុងនោះ គូ input-output <span class="math inline">\((x_i,y_i)\in\mathbb{R}^d\times{\cal Y}\)</span> ចំពោះគ្រប់ <span class="math inline">\(i=1,...,n\)</span> ។ ក្នុងករណីតម្រែតម្រង់តម្លៃយើងមាន <span class="math inline">\({\cal Y}=\mathbb{R}\)</span> ហើយក្នុងករណីកិច្ចការធ្វើចំណែកថ្នាក់បែបមានការណែនាំយើងបាន <span class="math inline">\(\cal Y\)</span> រាប់អស់និង មានចំនួនធាតុតូចបង្គួរ ។ យើងបានទិន្នន័យសម្រាប់បង្វឹកម៉ូដែល៖</p>
<p><span class="math display">\[{\cal T}=\left[\begin{array}
\ x_{11} & x_{12} & \dots & x_{1d} & y_{1}\\
x_{21} & x_{22} & \dots & x_{1d} & y_{2}\\
\vdots & \vdots & \vdots & \vdots & \vdots\\
x_{n1} & x_{n2} & \dots & x_{nd} & y_{n}
\end{array}\right]\]</span> ដែលក្នុងនោះ ជួរឈរនីមួយៗហៅថា<strong>អថេរ</strong>ឬ <strong>អញ្ញាត</strong> (variable) ហើយជាវុិចទ័រមានប្រវែង <span class="math inline">\(n\)</span> កំណត់ដោយ៖ <span class="math display">\[
X_j=\left(\begin{array}
\ x_{1j} \\
x_{2j} \\
\vdots \\
x_{nj}
\end{array}\right)\in\mathbb{R}^n, j=1,...,d\]</span> ចំណែកជួរដេកនីមួយៗហៅថា <strong>ឧទាហរណ៍</strong> ឬ <strong>ករណី</strong> (example ឬ individual) ហើយជាវុិចទ័រដែលមានប្រវែង <span class="math inline">\(d\geq 1\)</span> ៖ <span class="math display">\[
x_i=\left(\begin{array}
\ x_{i1} \\
x_{i2} \\
\vdots \\
x_{id}
\end{array}\right)\in\mathbb{R}^d, i=1,...,n\ \text{។}\]</span></p>
<p>ក្នុងករណីនៃតម្រែតម្រង់តម្លៃ output <span class="math inline">\(y_i\)</span> ទាំងអស់បង្ករបានជាវុិចទ័រដែលមានប្រវែង <span class="math inline">\(n\)</span> តាងដោយ៖ <span class="math display">\[
y=\left(\begin{array}
\ y_{1} \\
y_{2} \\
\vdots \\
y_{n}
\end{array}\right)\in\mathbb{R}^n\text{ ។}\]</span> រាល់វុិចទ័រក្នុងការសិក្សានេះមានទម្រង់ជាជួរឈហើយអក្សរតូចនិង ធំ (small និង capital) មានន័យខុសគ្នា។</p>
<p>ក្នុងការសិក្សាសិក្ខាម៉ាសុីន ទិន្នន័យតាមជួរដេកទាំងអស់ត្រូវបានគេសន្មតថា មិនអាស្រ័យគ្នានិង មានរបាយដូចៗគ្នាពេលគេស្រង់យកវាមកពីសកល <a href="https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables">(independent and identically distributed ឬ iid)</a> ហេតុនេះគេអាចនិយាយថាមានអថេរចៃដន្យ <span class="math inline">\((X,Y)\sim {\cal P}_{X,Y}\)</span> ណាមួយដែលធម្មជាតិនៃការកើតឡើងរបស់វាដូចគ្នានឹងធម្មជាតិនៃទិន្នន័យ <span class="math inline">\((x_i,y_i)\)</span> និងកំណត់សរសេរដោយ៖ <span class="math display">\[(x_i,y_i)\sim (X,Y)\overset{iid}{\sim} {\cal P}_{X,Y},\ \forall i=1,...,n \text{ ។}\]</span></p>
<p>យើងមិនដាក់លក្ខខណ្ឌនៃភាពមិនអាស្រ័យគ្នារវាងអថេរនៃទិន្នន័យទេ (មានន័យថារវាងចំណោម <span class="math inline">\(X_j\)</span> ទាំងអស់ និងរវាង <span class="math inline">\(Y\)</span>) ដោយសារធម្មជាតិនៃអថេរទាំងអស់ត្រូវបានពណ៌នាយ៉ាងពេញលេញដោយ <span class="math inline">\({\cal P}_{X,Y}\)</span> រួចទៅហើយ ។ ក្នុងកិច្ចការជាក់ស្តែងយើងមិនអាចស្គាល់ <span class="math inline">\({\cal P}_{X,Y}\)</span> បានទេ ក៏ប៉ុន្តែដោយប្រើប្រាស់លក្ខណៈ <span class="math inline">\(iid\)</span> នៃទិន្នន័យ រាល់តម្លៃទ្រឹស្តី (សង្ឃឹមមគណិត វ៉ារ្យង់ … ) ដែលទាក់ទងនឹងរបាយ <span class="math inline">\({\cal P}_{X,Y}\)</span> អាចត្រូវបានប៉ាន់ស្មានដោយប្រើច្បាប់ប្រូបាបប៊ីលីតេ។ ការប៉ានស្មាន<strong>របាយទ្រឹស្តី</strong>ដោយប្រើប្រាស់<strong>របាយទិន្នន័យ</strong>ត្រូវបានសិក្សាសុីជ្រៅក្នុងផ្នែកមួយដែលហៅថា <a href="http://www.stat.columbia.edu/~bodhi/Talks/Emp-Proc-Lecture-Notes.pdf">ទ្រឹស្តីនៃ Empirical Process</a> ហើយភាពរួមនៃរបាយទិន្នន័យទៅរករបាយទ្រឹស្តីត្រូវបានបកស្រាយនៅក្នុងទ្រឹស្តីបទដ៏ល្បីល្បាញមួយនៃស្ថិតិទ្រឹស្តីគឺ <a href="https://www.wiley.com/en-us/Statistical+Learning+Theory-p-9780471030034">ទ្រឹស្តីបទ Vapnik Chervonenkis</a> ។ ដោយសារទ្រឹស្តីទាំងនេះ ទាមទារបច្ចេកទេសផ្នែកទ្រឹស្តីសុីជម្រៅ ហេតុនេះយើងនឹងមិនសិក្សាពីវានៅក្នុងផ្នែកនេះទេ។ តែទោះជាយ៉ាងណាក៏ដោយ យើងគួរតែដឹងថា រាល់ការគណនាដោយប្រើប្រាស់ទិន្នន័យក្នុងការសិក្សាសិក្ខាម៉ាសុីន មានបាតគ្រឹះចេញពីទ្រឹស្តីស្ថិតិនិង ប្រូបាបប៊ីលីតេ។</p>
<hr>
</section>
<section id="រងវសនកហសនង-បរមមដល" class="level2">
<h2 class="anchored" data-anchor-id="រងវសនកហសនង-បរមមដល">៣. រង្វាស់នៃកំហុសនិង បរមាម៉ូដែល</h2>
<p>ឥលូវយើងមានឧបរករណ៍គ្រប់គ្រាន់សម្រាប់ពណ៌នាពីសញ្ញាណដ៏សំខាន់នៃសិក្ខាម៉ាសុីនគឺ <strong>រង្វាស់នៃកំហុស (Loss)</strong> ។ ក្នុងការសិក្សាជាទូទៅចំពោះកិច្ចការបែបតម្រែតម្រង់តម្លៃ គេនិយមប្រើមធ្យមនៃលម្អៀងការ៉េ ឬ Mean Squared Error (MSE) ជារង្វាស់នៃកំហុសនិង កំណត់គុណភាពនៃម៉ូដែល។ ចំពោះម៉ូដែល <span class="math inline">\(f:\mathbb{R}^d\to\mathbb{R}\)</span> ណាមួយដែលប្រើសម្រាប់ព្យាករណ៍តម្លៃនៃ output <span class="math inline">\(Y\)</span> ផ្ទៀងផ្ទាត់ <span class="math inline">\(\mathbb{E}[|f(X)|^2]<+\infty\)</span> គេកំណត់ MSE នៃ <span class="math inline">\(f\)</span> ដោយ៖</p>
<p><span id="eq-loss"><span class="math display">\[
\text{MSE}(f)=\mathbb{E}_{X,Y}[(Y-f(X))^2]
\tag{1}\]</span></span></p>
<p>ដែល <span class="math inline">\(\mathbb{E}_{X,Y}\)</span> ជាតម្លៃមធ្យមតាមទ្រឹស្តីឬ សង្ឃឹមមគណិតធៀបនឹងរបាយនៃទិន្នន័យ <span class="math inline">\((X,Y)\)</span> គឺ <span class="math inline">\({\cal P}_{X,Y}\)</span>។ តើរង្វាស់ខាងលើមានន័យដូចម្តេច? វាជា<strong>តម្លៃមធ្យម</strong>នៃគម្លាតការ៉េរវាងអ្វីដែលព្យាករណ៍ដោយម៉ូដែល ពោលគឺ <span class="math inline">\(f(X)\)</span> ទៅនឹងតម្លៃ output ពិតប្រាកដ <span class="math inline">\(Y\)</span> ។ <strong>យើងថាម៉ូដែល <span class="math inline">\(f\)</span> ល្អជាងម៉ូដែល <span class="math inline">\(g\)</span> បើ <span class="math inline">\(\text{MSE}(f)<\text{MSE}(g)\)</span></strong> ។ នេះមានន័យថា ម៉ូដែលដែលល្អជាងគេគឺជាម៉ូដែលដែលមានតម្លៃ MSE អប្បរមា។</p>
<blockquote class="blockquote">
<p><strong>🤔 តើយើងគួរជ្រើសយកម៉ូដែល <span class="math inline">\(f\)</span> បែបណាដើម្បីឲ្យ <a href="#eq-loss" class="quarto-xref">Equation 1</a> មានតម្លៃអប្បរមា?</strong></p>
</blockquote>
<p>បើយើងអាចឆ្លើយនឹងសំនួរនេះបាន នោះមានន័យថាយើងនឹងស្គាល់ម៉ូដែលល្អបំផុតសម្រាប់ធ្វើកិច្ចការបែបតម្រែតម្រង់តម្លៃដែលមាន MSE ជារង្វាស់នៃកំហុស។ ជារឿងល្អនោះគឺ យើងដឹងថាម៉ូដែលដែលល្អជាងគេនោះគឺ <span class="math inline">\(\eta:\mathbb{R}^d\to\mathbb{R}\)</span> ដែលកំណត់ដោយ៖</p>
<p><span id="eq-eta"><span class="math display">\[\eta(x)=\mathbb{E}_{Y|X}(Y|X=x) \text{ ។} \tag{2}\]</span></span></p>
<p>យើងអាចស្រាយបានដោយងាយ (នៅផ្នែកចុងក្រោយនៃអត្ថបទ) ថា៖</p>
<p><span id="eq-minimal-loss"><span class="math display">\[
\begin{align}
\text{MSE}(\eta)&=\min_{f\in{\cal M}}\text{MSE}(f)\\
&=\min_{f\in{\cal M}}\mathbb{E}_{X,Y}[(Y-f(X))^2]
\end{align}
\tag{3}\]</span></span></p>
<p>ដែល <span class="math inline">\({\cal M}=\{f:\mathbb{R}^d\to\mathbb{R},\mathbb{E}[|f(X)|^2]<+\infty\}\)</span> ហើយសមភាពខាងលើមានន័យថា <span class="math inline">\(\eta\)</span> ជាម៉ូដែលដែលផ្តល់តម្លៃ MSE តូចបំផុត។</p>
<p>ម៉ូដែល <span class="math inline">\(\eta\)</span> ដែលកំណត់ក្នុង <a href="#eq-eta" class="quarto-xref">Equation 2</a> ខាងលើត្រូវបានហៅថា <strong>អនុគមន៍តម្រែតម្រង់តម្លៃ ឬ ការព្យាករណ៍បែប Bayes (regression function ឬ Bayes estimator)</strong> ។</p>
<p>រូបមន្តនៃបរមាម៉ូដែល <span class="math inline">\(\eta\)</span> ខាងលើមានន័យថា តម្លៃព្យាករណ៍ <span class="math inline">\(\hat{y}=\eta(x)\)</span> នៃ output របស់ <span class="math inline">\(x\)</span> គឺជាតម្លៃមធ្យមឬ សង្ឃឹមគណិតមានលក្ខខណ្ឌ (conditional expectation) នៃអថេរចៃដន្យ <span class="math inline">\(Y\)</span> ដោយដឹងថា input <span class="math inline">\(X\)</span> យកតម្លៃស្មើនឹង <span class="math inline">\(x\)</span> ។ យើងអាចគិតពីតម្លៃនេះតាមលំនាំដូចតទៅ៖ បើគេដឹងថា input <span class="math inline">\(X\)</span> យកតម្លៃស្មើនឹង <span class="math inline">\(x\)</span> ណាមួយ តើ output <span class="math inline">\(Y\)</span> នៃ input ដទៃទៀតដែលស្រដៀងៗនឹង <span class="math inline">\(x\)</span> មានតម្លៃជាមធ្យមប៉ុន្មាន? ពាក្យថា <strong>ស្រដៀងនឹង <span class="math inline">\(x\)</span></strong> ចង់សំដៅដល់របាយនៃទិន្នន័យក្រោមលក្ខខណ្ឌ <span class="math inline">\(X=x\)</span> ហើយអ្វីដែលយើងចាប់អារម្មណ៍គឺតម្លៃមធ្យមនៃ output <span class="math inline">\(Y\)</span> នៃទិន្នន័យទាំងនោះ។</p>
<blockquote class="blockquote">
<p><strong>🗝️ ជារួម រូបមន្ត <span class="math inline">\(\eta\)</span> បញ្ជាក់ថាបើយើងចង់ព្យាករណ៍តម្លៃនៃវត្ថុណាមួយ យើងគួរពិនិត្យមើលតម្លៃនៃវត្ថុដែលមានលក្ខណៈប្រហាក់ប្រហែលនឹងវត្ថុនោះ។</strong></p>
</blockquote>
<hr>
<blockquote class="blockquote">
<p><strong>🤔 បើយើងស្គាល់រូបមន្តម៉ូដែលល្អបំផុតទៅហើយ តើបញ្ហាតម្រែត្រង់តម្លៃនៅមានអ្វីជាភាពស្មុគស្មាញទៀត?</strong></p>
</blockquote>
<blockquote class="blockquote">
<p><strong>😣 ចម្លើយគឺព្រោះថា ក្នុងការងារជាក់ស្តែង យើងមិនអាចគណនា <span class="math inline">\(\eta\)</span> បានទេដោយសារយើងមិនស្គាល់របាយនៃអថេរចៃដន្យមានលក្ខខណ្ឌ <span class="math inline">\(Y|X\)</span> ។</strong></p>
</blockquote>
<hr>
<p>នៅផ្នែកបន្តបន្ទាប់ទៀត យើងនឹងឃើញថាមានម៉ូដែលតម្រែតម្រង់តម្លៃជាច្រើនប្រភេទដែលសុទ្ធតែជាការប៉ាន់ស្មានតម្លៃនៃ <span class="math inline">\(\eta\)</span> ពិសេសប្រភេទម៉ូដែលបែបអប៉ារ៉ាម៉ែត្រ (nonparametric models) ។</p>
<p><strong>ឧទារហណ៍.១.</strong> ពិនិត្យមើលឧទាហរណ៍ខាងក្រោម៖</p>
<p><strong>១.</strong> ដើម្បីព្យារករណ៍តម្លៃនៃផ្ទះមួយ យើងពិនិត្យមើលផ្ទះចំនួន <span class="math inline">\(K\)</span> ដែលមានលក្ខណៈស្រដៀងនឹងវាបំផុតក្នុងចំណោមផ្ទះដែលមានទាំងអស់ (ស្រដៀងក្នុងន័យ input) ហើយគណនាតម្លៃមធ្យមនៃតម្លៃផ្ទះទាំងនោះ។ វិធីសាសស្រ្តបែបនេះហៅថា <span class="math inline">\(K\)</span>-Nearest Neighbor (<span class="math inline">\(K\)</span>NN) សម្រាប់តម្រែតម្រង់តម្លៃដែលជាវិធីសាស្រមួយដែលមានគោលដៅប៉ាន់ស្មានដោយផ្ទាល់ទៅលើតម្លៃនៃ <span class="math inline">\(\eta\)</span> ។</p>
<p><strong>២.</strong> បើយើងសន្មត់ថាបរមាម៉ូដែល <span class="math inline">\(\eta\)</span> មានរាងជាអនុគមន៍លីនេអ៊ែរនៃ input នោះតម្លៃព្យាករណ៍ <span class="math inline">\(\hat{y}\)</span> កំណត់ដោយ៖</p>
<p><span id="eq-linear-model"><span class="math display">\[\hat{y}=\beta_0+\beta_1X_1+\beta_2X_2+\dots+\beta_dX_d \tag{4}\]</span></span></p>
<p>ដែល <span class="math inline">\(\beta=(\beta_0,\beta_1,\dots,\beta_d)^t\in\mathbb{R}^{d+1}\)</span> ជាមេគុណឬ ប៉ារ៉ាម៉ែត្រគន្លឹះនៃម៉ូដែលដែលយើងត្រូវកំណត់រក។ បញ្ហាប្រភេទនេះហៅថា <strong>តម្រែតម្រង់លីនេអ៊ែរ (linear regression)</strong> ដែលជាប្រភេទម៉ូដែលមូលដ្ឋានគ្រឹះដ៏សំខាន់និង ពេញនិយមមួយហើយក៏ជាប្រធានបទនៃអត្ថបទបន្ទាប់ផងដែរ។</p>
<hr>
</section>
<section id="សងឃមគណតមនលកខខណឌនង-បរមមដល-eta" class="level2">
<h2 class="anchored" data-anchor-id="សងឃមគណតមនលកខខណឌនង-បរមមដល-eta">៤. សង្ឃឹមគណិតមានលក្ខខណ្ឌនិង បរមាម៉ូដែល <span class="math inline">\(\eta\)</span></h2>
<p>យើងនឹងបកស្រាយពីលទ្ធផលជាទ្រឹស្តីដែលបានបង្ហាញក្នុងផ្នែកមុន ជាពិសេសគឺ ភាពជាម៉ូដែលល្អបំផុតធៀបនឹង MSE នៃអនុគមន៍តម្រែតម្រង់តម្លៃ <span class="math inline">\(\eta\)</span> កំណត់ក្នុង <a href="#eq-eta" class="quarto-xref">Equation 2</a> ។</p>
<section id="វចទរចដនយ-random-vectors" class="level3">
<h3 class="anchored" data-anchor-id="វចទរចដនយ-random-vectors">៤.១. វុិចទ័រចៃដន្យ (Random vectors)</h3>
<p>ដោយទិន្នន័យត្រូវបានប្រដូចនឹងឧបករណ៍គណិតវិទ្យាហៅថា <strong>អថេរនិង វុិចទ័រចៃដន្យ (random variable និង random vector)</strong> នោះយើងនឹងសិក្សានិយមន័យខ្លះៗនិងលក្ខណៈរបស់<strong>វុិចទ័រចៃដន្យ</strong>ពិសេសគឺ សង្ឃឹមគណិតមានលក្ខខណ្ឌដែលជានិយមន័យនៃបរមាម៉ូដែល <span class="math inline">\(\eta\)</span> ។</p>
<hr>
<blockquote class="blockquote">
<p><strong>និយមន័យ.១.</strong> បើ <span class="math inline">\(X=(X_1,...,X_d)\)</span> ជាវុិចទ័រចៃដន្យមានអនុគមន៍ដង់សុីតេ <span class="math inline">\(f_X\)</span> (ធៀបនឹងរង្វាស់ Lebesgue) កំណត់លើ <span class="math inline">\(\mathbb{R}^d\)</span> នោះសង្ឃឹមគណិតនៃ <span class="math inline">\(X\)</span> តាងដោយ <span class="math inline">\(\mu=\mathbb{E}(X)=(\mu_1, ..., \mu_d)\in\mathbb{R}^d\)</span> ជាតម្លៃមធ្យមទ្រឹស្តីនៃ <span class="math inline">\(X\)</span> កំណត់ដោយ៖ <span class="math display">\[
\begin{align}
\mu&=\int_{\mathbb{R}^d}xf_X(x)dx
\end{align}\]</span></p>
</blockquote>
<p>ខុសពីអថេរចៃដន្យមានមួយវិមាត្រ និយមន័យខាងលើមានន័យថា សង្ឃឹមគណិតនៃវុិចទ័រចៃដន្យមួយជាវុិចទ័រនៃសង្ឃឹមគណិតនៃកូអរដោនេររបស់វាគឺ <span class="math inline">\(\mu_j=\mathbb{E}(X_j)\)</span> ចំពោះ <span class="math inline">\(j=1,...,d\)</span> ។</p>
<p>លើសពីនេះទៅទៀតដើម្បីពណ៌នាពីភាពរាយប៉ាយនៃ <span class="math inline">\(X\)</span> ក្នុងករណីវិមាត្រមួយគេប្រើប្រាស់ វ៉ារ្យង់ឬ គម្លាតស្តង់ដារ (variance ឬ standard deviation) តែក្នុងករណីនៃវុិចទ័រចៃដន្យយើងមិនត្រឹមតែត្រូវពណ៌នាពីកម្រិតនៃភាពរាយប៉ាយនៃ កូរអដោនេនីមួយៗរបស់វានោះទេ យើងត្រូវគិតពីទំនាក់ទំនងរវាងកូរអដោនេនីមួយៗរបស់វាថែមទៀងផង។ ទំនាក់ទំនងនិង កម្រិតរាយប៉ាយនេះត្រូវបានពណ៌នាដោយ <strong>ម៉ាទ្រីសកូវ៉ារ្យង់</strong> កំណត់ដូចខាងក្រោម៖</p>
<hr>
<blockquote class="blockquote">
<p><strong>និយមន័យ.២.</strong> ក្នុងករណីខាងលើ ម៉ាទ្រីសកូវ៉ារ្យង់នៃ <span class="math inline">\(X\)</span> តាងដោយ <span class="math inline">\(\Sigma\in\mathbb{R}^{d\times d}\)</span> កំណត់ដោយ៖ <span class="math display">\[
\begin{align}
\Sigma&=\mathbb{E}[(X-\mathbb{E}(X))(X-\mathbb{E}(X))^t]\\
&=\int_{\mathbb{R}^d}(x-\mu)(x-\mu)^tf_X(x)dx
\end{align}\]</span> ហើយយើងមាន៖ <span class="math display">\[\begin{align}
\Sigma_{ij}&=\mathbb{E}[(X_i-\mu_i)(X_j-\mu_j)]
\end{align}\]</span> ហៅថាកូវ៉ារ្យង់នៃកូអរដោនេ <span class="math inline">\(i\)</span> និង <span class="math inline">\(j\)</span> នៃវុិចទ័រចៃដន្យ <span class="math inline">\(X\)</span> ហើយ ចំពោះ <span class="math inline">\(i=j\)</span> យើងបានធាតុនៃអង្កត់ទ្រូង <span class="math inline">\(\Sigma_{jj}=\mathbb{V}(X_j)\)</span> ជាវ៉ារ្យង់នៃកូអរដោនេទី <span class="math inline">\(j\)</span> របស់វា។</p>
</blockquote>
<p>អថេរចៃដន្យដែលមានសារៈសំខាន់និង ត្រូវបានប្រើប្រាស់ច្រើនជាងគេក្នុងការសិក្សាទិន្នន័យគឺ របាយណរម៉ាល់ឬ Gaussian (Normal ឬ Gaussian variable) ។ ក្នុងករណីវិមាត្រច្រើនជាង១ របាយនៃវុិចទ័រណរម៉ាល់នៅតែមានទម្រង់ល្អ (យើងអាចសរសេររូបមន្តនៃអនុគមន៍ដង់សុីតេ) និង មានសារៈសខាន់ជាខ្លាំងក្នុងការសិក្សានិង បង្ហាញពីទំនាក់ទំនងនៃកូអរដោនេរបស់វុិចទ័រចៃដន្យ។ ខាងក្រោមនេះជានិយមន័យនៃវុិចទ័រចៃដន្យណរម៉ាល់ក្នុងលំហវិមាត្រ <span class="math inline">\(d\geq 2\)</span> ។</p>
<hr>
<blockquote class="blockquote">
<p><strong>និយមន័យ.៣.</strong> <span class="math inline">\(X=(X_1,...,X_d)\)</span> ហៅថា <strong>វុិចទ័រណរម៉ាល់</strong>ឬ <strong>វុិចទ័រ Guassian</strong> កាលណា គ្រប់បន្សំលីនេអ៊ែរនៃកូអរដោយនេរបស់វា ជាអថេរណរម៉ាល់ មានន័យថា <span class="math inline">\(\sum_{j=1}^d\alpha_jX_j\)</span> ជាអថេរចៃដន្យណរម៉ាល់ចំពោះគ្រប់តម្លៃមេគុណ <span class="math inline">\(\alpha_j\in\mathbb{R}\)</span> ដែល <span class="math inline">\(j=1,...,d\)</span> ។ បើគេស្គាល់មធ្យម <span class="math inline">\(\mu\in\mathbb{R}^d\)</span> និង ម៉ាទ្រីសកូវ៉ារ្យង់ <span class="math inline">\(\Sigma\in\mathbb{R}^{d\times d}\)</span> នោះអនុគមន៍ដង់សុីតេរបស់វាកំណត់ដោយ៖ <span class="math display">\[f_X(x)=\frac{1}{(2\pi)^{d/2}\sqrt{|\Sigma|}}e^{-\frac{1}{2}(x-\mu)^t\Sigma^{-1}(x-\mu)}\ \text{។}\]</span> ក្នុងរូបមន្តខាងលើ <span class="math inline">\(|\Sigma|\)</span> និង <span class="math inline">\(\Sigma^{-1}\)</span> ជាដេទែមីណង់និង ចម្រាស់នៃ <span class="math inline">\(\Sigma\)</span> រាងគ្នា។ គេកំណត់តាងវុិចទ័រណរម៉ាល់ដែលមានមធ្យម <span class="math inline">\(\mu\)</span> និង ម៉ាទ្រីសកូវ៉ារ្យង់ <span class="math inline">\(\Sigma\)</span> ដោយ <span class="math inline">\(X\sim{\cal N}(\mu, \Sigma)\)</span> ។</p>
</blockquote>
<p><strong>ឧទាហរណ៍.២.</strong> ខាងក្រោមយើងមានចំណុចចំនួន <span class="math inline">\(n\)</span> នៃវុិចទ័រណរម៉ាល់ក្នុងប្លង់ដែលកំណត់ដោយមធ្យមនិង ម៉ាទ្រីសកូវ៉ារ្យង់ផ្សេងៗគ្នាតាមករណីនីមួយៗ៖</p>
<p>១. <span class="math inline">\(n=200, X\sim{\cal N}\left(\left(\begin{array}\ 0 \\ 0\end{array}\right), \left(\begin{array}\ 1 & 0\\ 0 & 2\end{array}\right)\right)\)</span></p>
<p>២. <span class="math inline">\(n=200, X\sim{\cal N}\left(\left(\begin{array}\ 5 \\ 5\end{array}\right), \left(\begin{array}\ 2 & -1\\ -1 & 2\end{array}\right)\right)\)</span></p>
<p>៣. <span class="math inline">\(n=200, X\sim{\cal N}\left(\left(\begin{array}\ -1 \\ 6\end{array}\right), \left(\begin{array}\ 2 & 1.2\\ 1.2 & 1\end{array}\right)\right)\)</span></p>
<p>៤. <span class="math inline">\(n=200, X\sim{\cal N}\left(\left(\begin{array}\ 6 \\ 0\end{array}\right), \left(\begin{array}\ 3 & 0.1\\ 0.1 & 0.25\end{array}\right)\right)\)</span></p>
<div id="448f5dae" class="cell" data-execution_count="1">
<details>
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> plotly.graph_objs <span class="im">as</span> go</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>mu1, Sigma1 <span class="op">=</span> [<span class="dv">0</span>, <span class="dv">0</span>], [[<span class="dv">1</span>, <span class="dv">0</span>], [<span class="dv">0</span>, <span class="dv">3</span>]]</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>mu2, Sigma2 <span class="op">=</span> [<span class="dv">5</span>, <span class="dv">5</span>], [[<span class="dv">2</span>, <span class="op">-</span><span class="dv">1</span>], [<span class="op">-</span><span class="dv">1</span>, <span class="dv">2</span>]]</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>mu3, Sigma3 <span class="op">=</span> [<span class="op">-</span><span class="dv">1</span>, <span class="dv">6</span>], [[<span class="dv">2</span>, <span class="fl">1.2</span>], [<span class="fl">1.2</span>, <span class="dv">1</span>]]</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>mu4, Sigma4 <span class="op">=</span> [<span class="dv">6</span>, <span class="dv">0</span>], [[<span class="dv">3</span>, <span class="fl">0.1</span>], [<span class="fl">0.1</span>, <span class="fl">0.25</span>]]</span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a>x1 <span class="op">=</span> np.random.multivariate_normal(mean<span class="op">=</span>mu1, cov<span class="op">=</span>Sigma1, size<span class="op">=</span><span class="dv">100</span>)</span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a>x2 <span class="op">=</span> np.random.multivariate_normal(mean<span class="op">=</span>mu2, cov<span class="op">=</span>Sigma2, size<span class="op">=</span><span class="dv">100</span>)</span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a>x3 <span class="op">=</span> np.random.multivariate_normal(mean<span class="op">=</span>mu3, cov<span class="op">=</span>Sigma3, size<span class="op">=</span><span class="dv">100</span>)</span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a>x4 <span class="op">=</span> np.random.multivariate_normal(mean<span class="op">=</span>mu4, cov<span class="op">=</span>Sigma4, size<span class="op">=</span><span class="dv">100</span>)</span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a>fig <span class="op">=</span> go.Figure(</span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a> go.Scatter(x<span class="op">=</span>x1[:,<span class="dv">0</span>], y <span class="op">=</span> x1[:,<span class="dv">1</span>],</span>
<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a> mode <span class="op">=</span> <span class="st">"markers"</span>,</span>
<span id="cb1-17"><a href="#cb1-17" aria-hidden="true" tabindex="-1"></a> name <span class="op">=</span> <span class="st">"ករណីទី១"</span>,</span>
<span id="cb1-18"><a href="#cb1-18" aria-hidden="true" tabindex="-1"></a> showlegend <span class="op">=</span> <span class="va">True</span>,</span>
<span id="cb1-19"><a href="#cb1-19" aria-hidden="true" tabindex="-1"></a> marker <span class="op">=</span> <span class="bu">dict</span>(size <span class="op">=</span> <span class="dv">10</span>)))</span>
<span id="cb1-20"><a href="#cb1-20" aria-hidden="true" tabindex="-1"></a>fig.add_trace(</span>
<span id="cb1-21"><a href="#cb1-21" aria-hidden="true" tabindex="-1"></a> go.Scatter(x<span class="op">=</span>x2[:,<span class="dv">0</span>], y <span class="op">=</span> x2[:,<span class="dv">1</span>],</span>
<span id="cb1-22"><a href="#cb1-22" aria-hidden="true" tabindex="-1"></a> mode <span class="op">=</span> <span class="st">"markers"</span>,</span>
<span id="cb1-23"><a href="#cb1-23" aria-hidden="true" tabindex="-1"></a> name <span class="op">=</span> <span class="st">"ករណីទី២"</span>,</span>
<span id="cb1-24"><a href="#cb1-24" aria-hidden="true" tabindex="-1"></a> showlegend <span class="op">=</span> <span class="va">True</span>,</span>
<span id="cb1-25"><a href="#cb1-25" aria-hidden="true" tabindex="-1"></a> marker <span class="op">=</span> <span class="bu">dict</span>(size <span class="op">=</span> <span class="dv">10</span>)))</span>
<span id="cb1-26"><a href="#cb1-26" aria-hidden="true" tabindex="-1"></a>fig.add_trace(</span>
<span id="cb1-27"><a href="#cb1-27" aria-hidden="true" tabindex="-1"></a> go.Scatter(x<span class="op">=</span>x3[:,<span class="dv">0</span>], y <span class="op">=</span> x3[:,<span class="dv">1</span>],</span>
<span id="cb1-28"><a href="#cb1-28" aria-hidden="true" tabindex="-1"></a> mode <span class="op">=</span> <span class="st">"markers"</span>,</span>
<span id="cb1-29"><a href="#cb1-29" aria-hidden="true" tabindex="-1"></a> name <span class="op">=</span> <span class="st">"ករណីទី៣"</span>,</span>
<span id="cb1-30"><a href="#cb1-30" aria-hidden="true" tabindex="-1"></a> showlegend <span class="op">=</span> <span class="va">True</span>,</span>
<span id="cb1-31"><a href="#cb1-31" aria-hidden="true" tabindex="-1"></a> marker <span class="op">=</span> <span class="bu">dict</span>(size <span class="op">=</span> <span class="dv">10</span>)))</span>
<span id="cb1-32"><a href="#cb1-32" aria-hidden="true" tabindex="-1"></a>fig.add_trace(</span>
<span id="cb1-33"><a href="#cb1-33" aria-hidden="true" tabindex="-1"></a> go.Scatter(x<span class="op">=</span>x4[:,<span class="dv">0</span>], y <span class="op">=</span> x4[:,<span class="dv">1</span>],</span>
<span id="cb1-34"><a href="#cb1-34" aria-hidden="true" tabindex="-1"></a> mode <span class="op">=</span> <span class="st">"markers"</span>,</span>
<span id="cb1-35"><a href="#cb1-35" aria-hidden="true" tabindex="-1"></a> name <span class="op">=</span> <span class="st">"ករណីទី៤"</span>,</span>
<span id="cb1-36"><a href="#cb1-36" aria-hidden="true" tabindex="-1"></a> showlegend <span class="op">=</span> <span class="va">True</span>,</span>
<span id="cb1-37"><a href="#cb1-37" aria-hidden="true" tabindex="-1"></a> marker <span class="op">=</span> <span class="bu">dict</span>(size <span class="op">=</span> <span class="dv">10</span>)))</span>
<span id="cb1-38"><a href="#cb1-38" aria-hidden="true" tabindex="-1"></a>fig.update_layout(title <span class="op">=</span> <span class="st">"វុិចទ័រណរម៉ាល់វិមាត្រ២"</span>,</span>
<span id="cb1-39"><a href="#cb1-39" aria-hidden="true" tabindex="-1"></a> width <span class="op">=</span> <span class="dv">600</span>,</span>
<span id="cb1-40"><a href="#cb1-40" aria-hidden="true" tabindex="-1"></a> height <span class="op">=</span> <span class="dv">600</span>)</span>
<span id="cb1-41"><a href="#cb1-41" aria-hidden="true" tabindex="-1"></a>fig.show()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-display">
<div> <div id="1747c359-48ad-42a7-9b28-650dffe063e5" class="plotly-graph-div" style="height:600px; width:600px;"></div> <script type="text/javascript"> require(["plotly"], function(Plotly) { window.PLOTLYENV=window.PLOTLYENV || {}; if (document.getElementById("1747c359-48ad-42a7-9b28-650dffe063e5")) { Plotly.newPlot( "1747c359-48ad-42a7-9b28-650dffe063e5", [{"marker":{"size":10},"mode":"markers","name":"\u1780\u179a\u178e\u17b8\u1791\u17b8\u17e1","showlegend":true,"x":[-0.7953984474454445,0.8418800291541689,-0.38951993119806605,-0.8798793275443674,-0.16165651813477158,-0.49016375073094276,-1.1292213468317522,-0.08356407104162308,-0.07608695256772437,0.33977155362948025,-0.8222110564125261,-0.4580293094759987,0.8124495265963859,-0.16577961056932058,-0.9435753392868891,0.9102122529059751,1.0097096037003617,-0.38859163206423025,1.0184920673110143,-0.47388910063472584,-0.8935326506817948,-0.5454572814334516,-0.3712516472002274,0.20167288052505547,-0.5920830986301294,-0.6723064488217639,0.1414146092937489,0.7345693151786084,-1.7754957411405559,-0.8326983573316206,0.06566718389927115,0.5471842085688542,-2.2035803674655146,0.6100390240208439,0.5761266302628091,-0.025971587283073687,-0.38332638710210853,1.8721134501181138,-0.8698757923864182,-0.4781498976159309,-0.16509471042244517,1.1396374036169399,-0.13018204638431558,0.46264476401562366,0.7241493257931877,-0.33221971441512743,0.703235019084534,0.24325199195824387,0.15695315951532768,-0.003910982527061235,0.022107916360531003,0.009834744754098984,-1.0377523234409518,-1.4965666720273343,0.2707589778563335,-0.8651260992964422,-0.2576880886371723,0.5724903786861074,-0.2381739982082649,2.190702379686309,-0.1344243150769005,-0.10209564011929338,-0.9492903680894209,-0.7612023647898152,0.5243656437480837,0.20578688622259175,-1.181643413311679,-1.0281871057499379,-0.34488425702272485,-0.30871200454863595,1.5948019292313227,-0.009455077709107184,0.7720895891751111,0.6148226249931685,0.1984410708427845,-1.394674153120621,0.8343385343212807,1.8179475016438753,-0.7788637169558912,0.14398762887746042,0.7284594496051099,-1.6252713388291378,0.33039638654075687,1.5955221352575233,1.0696613226191516,0.004569140556302963,-0.14481833328447039,-0.4341774540413823,0.5910576531068105,-0.44711340343714623,0.801150888349948,0.47986440101884187,-0.6382780904985798,-0.441330367106781,-0.8706959944086682,1.321056438529566,1.6593377358672867,2.2780866782751676,-0.12202310921323252,-0.2770884257302011],"y":[-0.6284399072491871,-1.8007620968562454,1.39893644394564,-1.0320531667395485,0.09315888637463206,1.5802479314869025,-0.34866994254885453,1.5838620875934997,1.9845293729121216,0.09963863217358729,-0.8754445350040143,2.797783977815979,-2.5038954060666714,-1.1069537264864366,-1.155849946237066,-1.1129049675999583,-0.6388820597903871,2.6408179251623327,-2.585138525744197,0.7329854652035315,-1.045792012349405,1.119063540679252,0.8555562908776034,-0.8804927488725037,0.9548004151972646,0.8288125861402471,-0.6270803692608281,1.0466679379545387,0.11752094329390475,-0.9024164909088951,0.41701425684330873,0.029954032216717816,1.7239093985430352,1.179481696430455,0.45715874219332747,0.4772302447211735,-0.8713377815301835,-0.4823364585520813,-2.2210062601549914,-0.18274570940444623,-0.46940314707801645,-1.074621892388543,2.151418300289886,-1.898174879331462,0.04373198784104527,0.3562612429841098,-0.33866266226641656,-3.7526406865422097,1.4312292288684545,-0.9429345035444537,-1.3138576191621065,-1.2452987725153666,0.1033349330253965,-2.577154224967211,-0.006716621327592223,0.6056761912189932,2.6733332411818638,-0.5659775487565628,0.6186602265213033,-0.3236584385164235,0.21687206711867574,0.5630338620487948,-1.64975207877442,-0.007901506238496612,0.2712278526355756,-1.2602071261836123,0.6885057325046422,-3.2652370232728414,1.941661261991139,-3.842595400989269,-2.6461017629412757,3.5322496820031537,-1.7140944415049109,0.7063368712831839,0.17483907053315784,-1.8137438506880017,0.2123921809907718,-3.243759576422954,1.2271260440787288,-0.8179573381009517,0.4071907894381872,-0.6791380653123269,-1.0883164961894123,3.213657332673319,1.6288736485342334,0.7811698318763182,2.5218702044294985,-5.181477632716801,-1.1916018562494424,-0.06025170396654949,-0.4784614595240955,0.7827521605683873,7.017582356423781,0.14476286739578526,1.5538281952044584,2.008932837951299,-3.0332477758929697,0.37901387617284316,1.0885170316324821,1.5068981873188743],"type":"scatter"},{"marker":{"size":10},"mode":"markers","name":"\u1780\u179a\u178e\u17b8\u1791\u17b8\u17e2","showlegend":true,"x":[6.09512501220593,5.717809457016952,3.8799979168751935,6.515739588892175,4.374999106424835,5.74548974322213,4.923306484329329,5.308426954747936,7.4645908028029195,5.263091838760994,5.600282605151987,5.163717798637608,7.274203617596716,5.771958572266471,4.70047034837351,6.884681584025734,6.457639077213153,5.630558861482912,4.499278122688359,5.839621193525219,6.971047489191695,5.91277441344599,3.2267444204951112,3.7939884442624745,5.748834770744768,4.046500174582232,2.1666810539891452,5.619995881173428,4.7346114535907144,3.3664676728830765,2.3242955844392816,7.437352147757464,5.948478510890508,3.58006117756098,2.3538004707945315,7.473290167465356,3.5981880493300764,3.2428357297751793,3.7531381356107376,4.731835320824369,7.811510181728205,5.577087759509759,3.694267634675791,3.7161247729862215,4.370359952290038,3.963437552055807,5.392661214957211,6.029659258159191,7.860330304312022,4.760244206233865,7.560365478388087,5.6457954626439415,3.589878274577617,5.111187980816262,4.991838363609099,4.955268593471879,5.9102795891157385,6.1930231811067395,6.060765458239064,3.8953619298575193,4.711389889293882,7.926089521710415,3.229573802314632,4.8276269086811565,4.635615937082159,5.996243704269943,3.5554549187263085,4.016259992964434,4.640119573337463,5.262303619005966,3.5131169033118956,4.37970688976729,4.082886669144432,3.5289488056798857,6.349375323950401,4.924309456500052,3.1699229183015403,4.4237892607771085,5.494850340095168,5.87985869989515,6.794082780099318,5.234562259002815,5.048435141142333,2.347607592689852,4.979603004977072,7.180008678586356,3.589151468850826,5.754318776508209,2.714296548037565,3.5911081864072827,5.3884356549461145,5.908467015578715,8.18766004206534,2.5215097812302214,3.5682254475293718,6.948442835607681,6.334518900283025,5.58041838805347,4.875854188008243,4.312690095106781],"y":[1.5917550622069014,5.176126805715324,4.572368492971409,4.416495194555354,5.570835026215098,4.004934403799499,3.7270350443235496,6.053470626836036,5.887396306929695,5.10536240194546,3.361044052867533,5.292887778808502,4.676124952194224,4.266246956955627,5.334078387384445,4.337912683256824,1.42155945718036,5.507203467305689,4.0640957122508805,3.3138176051893256,2.243923346877296,1.7789866790173279,7.211946480148159,4.207269797440617,4.764507084795307,7.227994762399381,5.303318038520123,6.666446324081942,6.0595764035634865,4.954944649522129,4.756854625852578,6.077623511923277,3.79636886487517,4.669879254238577,6.2404824333961075,5.513318475047275,4.157200620079508,4.75776712461987,6.1203652759472105,2.1903663240423255,3.2528548327900593,5.464106020692787,6.34370098022354,5.751999903581808,5.204973597583972,6.726620939974165,3.1024799106038747,3.561413921279431,1.9891459671508165,4.307779660804071,3.1233531812663005,3.158982168816867,3.6417306588638594,2.668367366959388,3.8913750450922455,4.050122766350659,5.550852853317376,5.411958376397827,5.266370378733747,6.15432443329735,3.8921025041656385,4.212502178749233,5.946037758293198,4.376891365234306,6.433483233025207,6.457668218238146,6.307933247304448,3.672600482411098,6.226814557311185,5.480400692098802,5.677426391571959,0.8470713533380287,6.423771516608836,5.7786113492196405,3.3806305705703785,5.730744905759879,5.7975802225670865,3.4626222776209996,5.940829806842141,4.022511834309091,4.979212927097323,3.321604267571643,4.4842104801994145,4.743766879614385,4.320305727221476,4.262883882660994,5.706964628679344,4.533055519620809,7.712820667916844,4.943248067793658,5.195646686858724,3.597145521288776,5.103178361912736,6.8525849440542626,3.886262160835154,4.913953747564661,3.9241386766516997,4.883844219319733,4.890813686113537,4.353185620331174],"type":"scatter"},{"marker":{"size":10},"mode":"markers","name":"\u1780\u179a\u178e\u17b8\u1791\u17b8\u17e3","showlegend":true,"x":[-3.0369860556972275,-0.1438327711453331,-1.898777395841407,-0.8015582806371777,-1.5222222931700653,-0.17805465291234956,-3.167929587209125,-1.3856799648555804,-1.127530609855148,-1.0787793321200934,-1.8229908762369023,-0.5574478841167756,0.9241124197584101,-1.0341189396921153,-2.3344970260908022,-1.834987836701921,-2.0770249015967623,-2.5068499778935447,-2.8071537284704933,3.106396394908683,0.564354733192691,-0.5476283575095642,-3.0501148659826627,-2.1778390031609254,-2.508771240420243,-0.972102993366218,-0.8143836113249214,-1.5158596875893076,0.68003265817729,-1.14721279645241,-0.9092522640597125,-0.6075742076970527,-1.5438139938803053,-0.683244687485691,-1.8420237019062973,-2.2116836278685654,0.7712578184408803,-1.0301137311935362,-2.38638514940616,0.8470932960886535,0.7415228590490108,-3.743762667892733,-1.783162000988598,-1.7987385752158085,-0.7367505067001039,0.3882164289330854,-1.3778534165140586,-2.9990403635583203,-0.20686892365543486,1.0405609722726132,-3.424415104097187,-0.5057057879494157,-1.206830651894622,-3.505978147834631,0.3904572634503154,1.079871355492363,-0.6232167482166212,-2.1246158987540085,-2.8449807327749,-2.131546581838784,-0.463506356947165,-0.18845035267005783,0.32679579853036156,0.4824685314252082,-1.290973481839683,-0.68056189755311,-0.7144769936731473,-0.9885234774825236,1.2300982281479165,-0.24504067861161127,-1.3022415303151673,-0.18289003498421785,-1.1286529281729494,1.0925338481173452,1.2699844382135899,-1.8661467770098537,0.5103955227504997,-2.193080454819139,0.07358287919761519,-4.171763285558045,-2.5630727531679085,-1.9869641434013547,-0.30449829617078683,-0.5810123596942354,-2.0530684012310916,-3.241015283899727,-0.7494889509184445,-0.6493674244003693,-0.0894035822017053,2.497669890056554,-2.9107716202170084,-2.428542253872961,-0.7063407329688047,-2.3320592148616197,1.5391341631235473,-2.0028374986876587,-1.435960331587943,2.2154434822192823,0.5047413106223702,-1.2227589519713384],"y":[5.404034437687137,6.496109294682272,5.628287988391393,5.455778778098557,5.678287100980391,6.880481668255336,4.7255182744847035,6.368318592630888,5.8032871670389135,5.7758299569049685,5.81669122531141,5.729754283436307,6.938729479424984,6.480236847216761,4.95566957590715,5.818548365298049,5.971156690369533,6.131603233252475,4.895723884546907,8.782857501955528,6.443610213420372,6.765095810231493,4.430321178648541,5.339378095935034,4.945240028402628,5.799858391865887,6.008173596766771,5.465972450980112,6.767112507109102,5.814459279261009,5.233892100294818,6.247155597599782,6.434386285103867,6.827835520255458,5.456329195054868,6.107703717779218,7.418988330511363,6.741516057661687,4.162492123958051,7.244709435732283,7.595293740478304,4.341456220275342,5.287599841471522,6.8103743525770915,6.266234650383033,6.303278162027603,5.541433167111085,4.703941022235179,6.300564295954796,6.941439339462931,5.478463142576805,6.219955520284135,5.664629251924105,3.910437528566701,6.561275436816684,6.99394065666821,5.35621608230052,4.407339089354779,4.787134446335856,5.523094332119194,6.900752209056505,6.844678272518145,6.543641095523806,7.099194483235125,5.833509992891258,6.291068502483238,5.515954940306237,6.341784274001587,8.335307864272421,6.3596548305835015,5.847984757462413,6.37847281255608,7.117812526242515,7.301079517964839,7.563926024778321,5.610430695194503,7.120310941465081,4.760911731797733,7.5059430535617215,4.2421004080881275,5.348383957227,4.614147750400661,6.727658944743169,4.706351127201999,5.455399316585787,5.1153094165838375,5.92442023235577,6.312805433460076,6.1314904561798445,7.489935945612351,4.161382676082844,5.371414833299714,7.001308187844904,5.093376094654373,7.72505793626822,4.61725494367222,6.330033628118339,7.387253577557915,7.595539390759826,6.631683310222618],"type":"scatter"},{"marker":{"size":10},"mode":"markers","name":"\u1780\u179a\u178e\u17b8\u1791\u17b8\u17e4","showlegend":true,"x":[6.272814081234496,7.3365052781621225,6.882825753711822,3.1645670733876545,8.081157932711562,6.39497807028433,7.153897815837494,6.194136471246923,9.049770780092622,4.483434318587108,5.089824852808976,4.860440814616595,6.486305803727596,6.912185831233814,7.069471771392333,7.2907848061754486,5.011062831240059,4.887469697699576,7.00044991134692,5.071426376211953,2.5351278769880525,5.499014749086212,4.893449323756066,7.942328884937404,6.910793447604793,7.652978018848325,11.133603286712097,5.0399044751084165,2.7705536501032313,5.615011492903019,4.757558738894636,2.2553472429671118,6.581655188244433,3.3237673698313537,7.025553727425035,4.49648878204175,7.083695334402219,6.293754990345231,8.222828960980237,3.5718812417877444,7.18768306100071,7.4924245128914455,3.798150608810263,4.178977057622517,3.48788849560874,3.7255550308760284,5.741873175729555,5.480028268440094,4.916127820556816,7.048511964712055,7.019322428925755,3.0347475432393245,3.3087141448010944,3.830367872410996,7.318999118581715,7.401420067793052,5.294108666676221,6.971561598858203,4.793833754751526,3.484109127247222,7.68466022239065,7.018588861513554,5.279978247040695,6.192253568548841,5.771410603115028,6.639113884886737,6.955592726501263,6.214865407534983,5.676197080321424,6.131476776939439,7.595355228357855,5.240774979188924,5.422642107258586,1.5978766779904152,2.92661616980623,3.607435763843521,7.607659955669795,8.121615261474748,7.013994122785498,6.617381284601316,6.204278646345697,6.125744710043678,7.300694333758843,9.343702192882883,4.938209910067041,4.860644607673601,7.190683707205711,7.3015109200033885,6.537088169124397,8.012837083281948,6.339464557029557,6.310624485811087,8.132394872039084,4.535726789198613,7.443973925167061,10.396426536056012,6.416719959293396,6.9872901838459365,5.537202117713643,5.189768276328413],"y":[-0.31947547469602117,-0.1460528937968336,-0.11812535730631586,0.07806336511059228,-0.3656332254303303,0.20301597532407714,0.39452046736571533,-0.10986414966732619,-0.5291893040602468,0.08869857812021512,0.23706613443944136,-0.17976494859888154,-0.4435574014844299,0.6555508207691856,1.1023017730680609,0.09739326146432108,0.8419422775358013,0.7390832630251545,-0.07199484097841942,0.3292330816748284,-0.13256341978849973,-0.16310937732897557,-0.10253682553505343,0.11426345009983246,-0.06869076142455756,0.24751267806647084,0.42508550359666425,0.05793800266548716,0.09031084872240193,-0.5360962533740886,-0.4398995829526246,-0.13784309143887719,0.8618328955461854,0.34578802611561177,-0.5343219602003381,0.03870897952593879,-0.177028896771163,0.2290889286754899,-0.38469671356181334,0.4592452979262429,0.340050779312739,-0.4795006140979925,-1.2258233659364925,0.3672154495500722,-0.6934666722358477,-0.15332330920973228,-0.30602263526498164,0.11757718719482496,0.6473939306131056,-0.19814549581007668,-0.0828834524314723,0.023889534712683843,0.3291730562923228,0.057040983359950635,0.02304626768008648,-0.019843347795960385,-0.4256124069108012,-0.24041938478722,-0.15943361421678484,0.511562838224083,-0.7232314948046153,0.3857542517669334,-0.28277037752029777,-0.8800163188540315,-1.3488119726430485,-0.2752810225777483,-0.4000008745240387,0.42819332796449305,-0.049530222678224774,0.23414739677947932,-0.0027454082294152966,0.6338878705194814,-0.13644364440227313,-1.3279582410769448,-0.3070558617875188,-1.06599553717791,-0.17490461475264898,-0.176252169307345,0.2786708491480686,-0.23285137414782928,0.6091925119391075,0.14301382577680197,0.14771024519418974,0.16778155218477445,0.48537464806162345,0.7319537836495058,0.09283811798958717,-0.053935234349238706,-0.18638045975041193,0.3822711825950118,0.5403638005070365,-0.4694096226777123,0.2183051081695643,0.2312644145432604,-0.7168351428900035,0.23948355675303035,0.441883487810039,0.1802390752011337,0.29802269178006724,-0.3439787897223176],"type":"scatter"}], {"template":{"data":{"histogram2dcontour":[{"type":"histogram2dcontour","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"choropleth":[{"type":"choropleth","colorbar":{"outlinewidth":0,"ticks":""}}],"histogram2d":[{"type":"histogram2d","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"heatmap":[{"type":"heatmap","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"heatmapgl":[{"type":"heatmapgl","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"contourcarpet":[{"type":"contourcarpet","colorbar":{"outlinewidth":0,"ticks":""}}],"contour":[{"type":"contour","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"surface":[{"type":"surface","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"mesh3d":[{"type":"mesh3d","colorbar":{"outlinewidth":0,"ticks":""}}],"scatter":[{"fillpattern":{"fillmode":"overlay","size":10,"solidity":0.2},"type":"scatter"}],"parcoords":[{"type":"parcoords","line":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterpolargl":[{"type":"scatterpolargl","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"bar":[{"error_x":{"color":"#2a3f5f"},"error_y":{"color":"#2a3f5f"},"marker":{"line":{"color":"#E5ECF6","width":0.5},"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"bar"}],"scattergeo":[{"type":"scattergeo","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterpolar":[{"type":"scatterpolar","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"histogram":[{"marker":{"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"histogram"}],"scattergl":[{"type":"scattergl","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatter3d":[{"type":"scatter3d","line":{"colorbar":{"outlinewidth":0,"ticks":""}},"marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scattermapbox":[{"type":"scattermapbox","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterternary":[{"type":"scatterternary","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scattercarpet":[{"type":"scattercarpet","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"carpet":[{"aaxis":{"endlinecolor":"#2a3f5f","gridcolor":"white","linecolor":"white","minorgridcolor":"white","startlinecolor":"#2a3f5f"},"baxis":{"endlinecolor":"#2a3f5f","gridcolor":"white","linecolor":"white","minorgridcolor":"white","startlinecolor":"#2a3f5f"},"type":"carpet"}],"table":[{"cells":{"fill":{"color":"#EBF0F8"},"line":{"color":"white"}},"header":{"fill":{"color":"#C8D4E3"},"line":{"color":"white"}},"type":"table"}],"barpolar":[{"marker":{"line":{"color":"#E5ECF6","width":0.5},"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"barpolar"}],"pie":[{"automargin":true,"type":"pie"}]},"layout":{"autotypenumbers":"strict","colorway":["#636efa","#EF553B","#00cc96","#ab63fa","#FFA15A","#19d3f3","#FF6692","#B6E880","#FF97FF","#FECB52"],"font":{"color":"#2a3f5f"},"hovermode":"closest","hoverlabel":{"align":"left"},"paper_bgcolor":"white","plot_bgcolor":"#E5ECF6","polar":{"bgcolor":"#E5ECF6","angularaxis":{"gridcolor":"white","linecolor":"white","ticks":""},"radialaxis":{"gridcolor":"white","linecolor":"white","ticks":""}},"ternary":{"bgcolor":"#E5ECF6","aaxis":{"gridcolor":"white","linecolor":"white","ticks":""},"baxis":{"gridcolor":"white","linecolor":"white","ticks":""},"caxis":{"gridcolor":"white","linecolor":"white","ticks":""}},"coloraxis":{"colorbar":{"outlinewidth":0,"ticks":""}},"colorscale":{"sequential":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]],"sequentialminus":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]],"diverging":[[0,"#8e0152"],[0.1,"#c51b7d"],[0.2,"#de77ae"],[0.3,"#f1b6da"],[0.4,"#fde0ef"],[0.5,"#f7f7f7"],[0.6,"#e6f5d0"],[0.7,"#b8e186"],[0.8,"#7fbc41"],[0.9,"#4d9221"],[1,"#276419"]]},"xaxis":{"gridcolor":"white","linecolor":"white","ticks":"","title":{"standoff":15},"zerolinecolor":"white","automargin":true,"zerolinewidth":2},"yaxis":{"gridcolor":"white","linecolor":"white","ticks":"","title":{"standoff":15},"zerolinecolor":"white","automargin":true,"zerolinewidth":2},"scene":{"xaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2},"yaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2},"zaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2}},"shapedefaults":{"line":{"color":"#2a3f5f"}},"annotationdefaults":{"arrowcolor":"#2a3f5f","arrowhead":0,"arrowwidth":1},"geo":{"bgcolor":"white","landcolor":"#E5ECF6","subunitcolor":"white","showland":true,"showlakes":true,"lakecolor":"white"},"title":{"x":0.05},"mapbox":{"style":"light"}}},"title":{"text":"\u179c\u17bb\u17b7\u1785\u1791\u17d0\u179a\u178e\u179a\u1798\u17c9\u17b6\u179b\u17cb\u179c\u17b7\u1798\u17b6\u178f\u17d2\u179a\u17e2"},"width":600,"height":600}, {"responsive": true} ).then(function(){
var gd = document.getElementById('1747c359-48ad-42a7-9b28-650dffe063e5');
var x = new MutationObserver(function (mutations, observer) {{
var display = window.getComputedStyle(gd).display;
if (!display || display === 'none') {{
console.log([gd, 'removed!']);
Plotly.purge(gd);
observer.disconnect();
}}
}});
// Listen for the removal of the full notebook cells
var notebookContainer = gd.closest('#notebook-container');
if (notebookContainer) {{
x.observe(notebookContainer, {childList: true});
}}
// Listen for the clearing of the current output cell
var outputEl = gd.closest('.output');
if (outputEl) {{
x.observe(outputEl, {childList: true});
}}
}) }; }); </script> </div>
<p>រូបទី២៖ របាយនៃវុិចទ័រណរម៉ាល់២វិមាត្រអាស្រ័យទៅតាមមធ្យមនិង ម៉ាទ្រីសកូវ៉ារ្យង់របស់វា។</p>
</div>
</div>
<p>ក្នុងឧទាហរណ៍ខាងលើ របាយចំណុចក្នុងក្រុមនីមួយៗនៅរាយប៉ាយព័ទ្ធជុំវិញមធ្យម <span class="math inline">\(\mu\)</span> ហើយទ្រង់ទ្រាយនៃរបាយចំណុចអាស្រ័យនឹងម៉ាទ្រីសកូវ៉ារ្យង់របស់វា។ បើយើងសង្កេតមើលក្រុមទី២ (ពណ៌ក្រហម) របាយចំណុចនៅរាយប៉ាយជុំវិញមធ្យមរបស់វាគឺ <span class="math inline">\(\mu=\left(\begin{array}\ 5\\5\end{array}\right)\)</span> ហើយបើនិយាយពីទ្រង់ទ្រាយនៃចំណុចវិញ ពេលដែលតម្លៃលើអ័ក្ស <span class="math inline">\(x\)</span> កើនយើងឃើញថាតម្លៃលើអ័ក្ស <span class="math inline">\(y\)</span> មានទំនោរធ្លាក់ចុះដោយសារម៉ាទ្រីសកូវ៉ារ្យង់របស់វាមានកូវ៉ារ្យង់អវិជ្ជមានស្មើនឹង <span class="math inline">\(-1\)</span> (ធាតុលើអង្កត់ទ្រូងច្រាស់នៃម៉ាទ្រីសកូវ៉ារ្យង់) ហើយកម្រិតនៃការរាយប៉ាយតាមទិស <span class="math inline">\(x\)</span> និង <span class="math inline">\(y\)</span> មានតម្លៃប្រហែលគ្នាដោយសារវ៉ារ្យង់នៃកូអរដោនេទាំងពីរមានតម្លៃស្មើគ្នាស្មើនឹង <span class="math inline">\(2\)</span> (ធាតុនៃអង្កត់ទ្រូងនៃម៉ាទ្រីសកូវ៉ារ្យង់) ។</p>
</section>
<section id="សងឃមគណតមនលកខខណឌ-conditional-expectation" class="level3">
<h3 class="anchored" data-anchor-id="សងឃមគណតមនលកខខណឌ-conditional-expectation">៤.២. សង្ឃឹមគណិតមានលក្ខខណ្ឌ (conditional expectation)</h3>
<p>ចំពោះវុិចទ័រចៃដន្យជាប់ (continuous random vector) ដើម្បីយល់ពីសង្ឃឹមគណិតមានលក្ខខណ្ឌយើងត្រូវស្គាល់ដង់សុីតេមានលក្ខខណ្ឌជាមុនសិន។</p>
<hr>
<blockquote class="blockquote">
<p><strong>និយមន័យ.៤.</strong> បើ <span class="math inline">\((X,Y)\)</span> ជាវុិចទ័រចៃដន្យរួមនៃវុិចទ័រចៃដន្យ <span class="math inline">\(X\)</span> និង <span class="math inline">\(Y\)</span> កំណត់លើ <span class="math inline">\(\mathbb{R}^{d_1}\)</span> និង <span class="math inline">\(\mathbb{R}^{d_2}\)</span> រាងគ្នាហើយមានដង់សុីតេរួម (joint density) <span class="math inline">\(f_{X,Y}\)</span> នោះដង់សុីតេមានលក្ខខណ្ឌនៃ <span class="math inline">\(Y\)</span> ដោយដឹងថា <span class="math inline">\(X=x\in\mathbb{R}^{d_1}\)</span> កំណត់តែចំពោះ <span class="math inline">\(x\)</span> ដែល <span class="math inline">\(f_X(x)>0\)</span> ដោយ៖ <span class="math display">\[f_{Y|X}(y|X=x)=\frac{f_{X,Y}(x,y)}{f_{X}(x)}\]</span> ដែល <span class="math inline">\(f_X(x)=\int_{\mathbb{R}^{d_2}}f_{X,Y}(x,y)dy\)</span> ហៅថាដង់សុីតេដោយផ្នែក (marginal density) នៃ <span class="math inline">\(X\)</span> ចេញពីដង់សុីតេរួម <span class="math inline">\(f_{X,Y}\)</span> ។</p>
</blockquote>
<hr>
<p><strong>ឧទាហរណ៍.៣.</strong> តាង <span class="math inline">\(\mathbb{R}^*=\mathbb{R}\setminus \{0\}\)</span> និង <span class="math inline">\(\Omega=(\mathbb{R}^*)^2\times[0,1]\)</span> ។ គណនាដង់សុីតេមានលក្ខខណ្ឌ <span class="math inline">\(f_{Y|X}(y|X=x)\)</span> ចំពោះ <span class="math inline">\(x\neq(0,0)\)</span> ដោយដឹងថា <span class="math display">\[f_{X,Y}(x,y)=\begin{cases}Cye^{-\|x\|^2y}, & (x,y)\in \Omega\\ 0, &(x,y)\notin \Omega\end{cases}\]</span> ចំពោះចំនួនពិត <span class="math inline">\(C>0\)</span> ។</p>
<div id="c14829d6" class="cell" data-execution_count="2">
<details>
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> plotly.graph_objects <span class="im">as</span> go</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>C <span class="op">=</span> <span class="dv">1</span><span class="op">/</span>np.pi</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> f_XY(x,y):</span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> C <span class="op">*</span> y <span class="op">*</span> np.exp(<span class="op">-</span>np.dot(x,x)<span class="op">*</span>y)</span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a>N <span class="op">=</span> <span class="dv">30</span></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> grid(y):</span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a> x <span class="op">=</span> np.linspace(<span class="op">-</span><span class="dv">3</span>,<span class="dv">3</span>,N)</span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a> X <span class="op">=</span> np.array([[i,j] <span class="cf">for</span> i <span class="kw">in</span> x <span class="cf">for</span> j <span class="kw">in</span> x])</span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="bu">len</span>(y)):</span>
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> i <span class="op">==</span> <span class="dv">0</span>:</span>
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a> res <span class="op">=</span> np.column_stack([X, [f_XY(X[j], y[i]) <span class="cf">for</span> j <span class="kw">in</span> <span class="bu">range</span>(X.shape[<span class="dv">0</span>])]])</span>
<span id="cb2-12"><a href="#cb2-12" aria-hidden="true" tabindex="-1"></a> <span class="cf">else</span>:</span>
<span id="cb2-13"><a href="#cb2-13" aria-hidden="true" tabindex="-1"></a> Y <span class="op">=</span> np.column_stack([X, [f_XY(X[j], y[i]) <span class="cf">for</span> j <span class="kw">in</span> <span class="bu">range</span>(X.shape[<span class="dv">0</span>])]])</span>
<span id="cb2-14"><a href="#cb2-14" aria-hidden="true" tabindex="-1"></a> res <span class="op">=</span> np.row_stack([res, Y])</span>
<span id="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> res</span>
<span id="cb2-16"><a href="#cb2-16" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-17"><a href="#cb2-17" aria-hidden="true" tabindex="-1"></a>y <span class="op">=</span> np.linspace(<span class="dv">0</span>,<span class="dv">1</span>,<span class="dv">5</span>)</span>
<span id="cb2-18"><a href="#cb2-18" aria-hidden="true" tabindex="-1"></a>data <span class="op">=</span> grid(y)</span>
<span id="cb2-19"><a href="#cb2-19" aria-hidden="true" tabindex="-1"></a>layout <span class="op">=</span> go.Layout(</span>
<span id="cb2-20"><a href="#cb2-20" aria-hidden="true" tabindex="-1"></a> title <span class="op">=</span> <span class="st">'ដង់សុីតេរួមនៃចំពោះតម្លៃខ្លះៗនៃ y'</span>,</span>
<span id="cb2-21"><a href="#cb2-21" aria-hidden="true" tabindex="-1"></a> width <span class="op">=</span> <span class="dv">650</span>,</span>
<span id="cb2-22"><a href="#cb2-22" aria-hidden="true" tabindex="-1"></a> height <span class="op">=</span> <span class="dv">400</span></span>
<span id="cb2-23"><a href="#cb2-23" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb2-24"><a href="#cb2-24" aria-hidden="true" tabindex="-1"></a>fig <span class="op">=</span> go.Figure(layout<span class="op">=</span>layout)</span>
<span id="cb2-25"><a href="#cb2-25" aria-hidden="true" tabindex="-1"></a>frames <span class="op">=</span> []</span>
<span id="cb2-26"><a href="#cb2-26" aria-hidden="true" tabindex="-1"></a>start, end <span class="op">=</span> <span class="dv">0</span>, <span class="dv">900</span></span>
<span id="cb2-27"><a href="#cb2-27" aria-hidden="true" tabindex="-1"></a>x_, y_ <span class="op">=</span> np.linspace(<span class="op">-</span><span class="dv">3</span>,<span class="dv">3</span>,N), np.linspace(<span class="op">-</span><span class="dv">3</span>,<span class="dv">3</span>,N)</span>
<span id="cb2-28"><a href="#cb2-28" aria-hidden="true" tabindex="-1"></a>op <span class="op">=</span> [<span class="dv">1</span><span class="op">-</span><span class="fl">0.15</span><span class="op">*</span>i <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="dv">1</span>, <span class="dv">6</span>)]</span>
<span id="cb2-29"><a href="#cb2-29" aria-hidden="true" tabindex="-1"></a><span class="co"># col = ['Viridis', 'RdBu', 'Inferno', 'Bluered_r', 'Cividis_r']</span></span>
<span id="cb2-30"><a href="#cb2-30" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="bu">len</span>(y)):</span>
<span id="cb2-31"><a href="#cb2-31" aria-hidden="true" tabindex="-1"></a> <span class="bu">id</span> <span class="op">=</span> <span class="bu">range</span>(start, end)</span>
<span id="cb2-32"><a href="#cb2-32" aria-hidden="true" tabindex="-1"></a> z <span class="op">=</span> data[<span class="bu">id</span>,<span class="op">-</span><span class="dv">1</span>].reshape(N,N)</span>
<span id="cb2-33"><a href="#cb2-33" aria-hidden="true" tabindex="-1"></a> trace <span class="op">=</span> go.Surface(</span>
<span id="cb2-34"><a href="#cb2-34" aria-hidden="true" tabindex="-1"></a> z<span class="op">=</span>z,</span>
<span id="cb2-35"><a href="#cb2-35" aria-hidden="true" tabindex="-1"></a> x<span class="op">=</span>x_,</span>
<span id="cb2-36"><a href="#cb2-36" aria-hidden="true" tabindex="-1"></a> y<span class="op">=</span>y_,</span>
<span id="cb2-37"><a href="#cb2-37" aria-hidden="true" tabindex="-1"></a> colorbar_x <span class="op">=</span> <span class="op">-</span><span class="fl">0.01</span>,</span>
<span id="cb2-38"><a href="#cb2-38" aria-hidden="true" tabindex="-1"></a> opacity<span class="op">=</span>op[i],</span>
<span id="cb2-39"><a href="#cb2-39" aria-hidden="true" tabindex="-1"></a> name <span class="op">=</span> <span class="st">'y = </span><span class="sc">{}</span><span class="st">'</span>.<span class="bu">format</span>(np.<span class="bu">round</span>(y[i],<span class="dv">3</span>)),</span>
<span id="cb2-40"><a href="#cb2-40" aria-hidden="true" tabindex="-1"></a> showlegend <span class="op">=</span> <span class="va">True</span>,</span>
<span id="cb2-41"><a href="#cb2-41" aria-hidden="true" tabindex="-1"></a> colorscale<span class="op">=</span> <span class="st">'RdBu'</span>,</span>
<span id="cb2-42"><a href="#cb2-42" aria-hidden="true" tabindex="-1"></a> colorbar<span class="op">=</span><span class="bu">dict</span>(title<span class="op">=</span><span class="st">"ដង់សុីតេ"</span>),</span>
<span id="cb2-43"><a href="#cb2-43" aria-hidden="true" tabindex="-1"></a> cmin<span class="op">=</span><span class="dv">0</span>,</span>
<span id="cb2-44"><a href="#cb2-44" aria-hidden="true" tabindex="-1"></a> cmax<span class="op">=</span><span class="fl">0.325</span></span>
<span id="cb2-45"><a href="#cb2-45" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb2-46"><a href="#cb2-46" aria-hidden="true" tabindex="-1"></a> fig.add_trace(trace)</span>
<span id="cb2-47"><a href="#cb2-47" aria-hidden="true" tabindex="-1"></a> start, end <span class="op">=</span> end, end <span class="op">+</span> <span class="dv">900</span></span>
<span id="cb2-48"><a href="#cb2-48" aria-hidden="true" tabindex="-1"></a>fig.update_layout(scene <span class="op">=</span> <span class="bu">dict</span>(xaxis_title <span class="op">=</span> <span class="st">'x1'</span>,</span>
<span id="cb2-49"><a href="#cb2-49" aria-hidden="true" tabindex="-1"></a> yaxis_title <span class="op">=</span><span class="st">'x2'</span>,</span>
<span id="cb2-50"><a href="#cb2-50" aria-hidden="true" tabindex="-1"></a> zaxis_title<span class="op">=</span><span class="st">'f_(x1,x2,y)'</span>))</span>
<span id="cb2-51"><a href="#cb2-51" aria-hidden="true" tabindex="-1"></a>fig.show()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-display">
<div> <div id="c5b31e9f-e107-47aa-b9ca-ec4659f23ee4" class="plotly-graph-div" style="height:400px; width:650px;"></div> <script type="text/javascript"> require(["plotly"], function(Plotly) { window.PLOTLYENV=window.PLOTLYENV || {}; if (document.getElementById("c5b31e9f-e107-47aa-b9ca-ec4659f23ee4")) { Plotly.newPlot( "c5b31e9f-e107-47aa-b9ca-ec4659f23ee4", [{"cmax":0.325,"cmin":0,"colorbar":{"title":{"text":"\u178a\u1784\u17cb\u179f\u17bb\u17b8\u178f\u17c1"},"x":-0.01},"colorscale":[[0.0,"rgb(103,0,31)"],[0.1,"rgb(178,24,43)"],[0.2,"rgb(214,96,77)"],[0.3,"rgb(244,165,130)"],[0.4,"rgb(253,219,199)"],[0.5,"rgb(247,247,247)"],[0.6,"rgb(209,229,240)"],[0.7,"rgb(146,197,222)"],[0.8,"rgb(67,147,195)"],[0.9,"rgb(33,102,172)"],[1.0,"rgb(5,48,97)"]],"name":"y = 0.0","opacity":0.85,"showlegend":true,"x":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"y":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"z":[[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0]],"type":"surface"},{"cmax":0.325,"cmin":0,"colorbar":{"title":{"text":"\u178a\u1784\u17cb\u179f\u17bb\u17b8\u178f\u17c1"},"x":-0.01},"colorscale":[[0.0,"rgb(103,0,31)"],[0.1,"rgb(178,24,43)"],[0.2,"rgb(214,96,77)"],[0.3,"rgb(244,165,130)"],[0.4,"rgb(253,219,199)"],[0.5,"rgb(247,247,247)"],[0.6,"rgb(209,229,240)"],[0.7,"rgb(146,197,222)"],[0.8,"rgb(67,147,195)"],[0.9,"rgb(33,102,172)"],[1.0,"rgb(5,48,97)"]],"name":"y = 0.25","opacity":0.7,"showlegend":true,"x":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"y":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"z":[[0.0008840258559260083,0.0011928844882429471,0.0015755658917099723,0.002036945949975808,0.002577669277683294,0.0031928582714325594,0.0038711218528552803,0.004594082590234108,0.005336610549331791,0.006067879919817759,0.00675325580442451,0.007356888581079765,0.00784476453358481,0.008187859873498201,0.008364994237599,0.008364994237599,0.008187859873498201,0.00784476453358481,0.007356888581079765,0.00675325580442451,0.006067879919817757,0.005336610549331789,0.004594082590234108,0.0038711218528552803,0.0031928582714325594,0.002577669277683294,0.002036945949975808,0.0015755658917099723,0.0011928844882429471,0.0008840258559260083],[0.0011928844882429471,0.0016096513385347629,0.0021260329659213073,0.0027486087774776704,0.003478248601618012,0.004308370710673959,0.005223604241226928,0.006199151102720615,0.007201101530478754,0.008187859873498201,0.009112690585046542,0.009927218996223851,0.010585547767977259,0.011048512856868706,0.01128753395998937,0.01128753395998937,0.011048512856868706,0.010585547767977259,0.009927218996223851,0.009112690585046542,0.008187859873498198,0.007201101530478751,0.006199151102720615,0.005223604241226928,0.004308370710673959,0.003478248601618012,0.0027486087774776704,0.0021260329659213073,0.0016096513385347629,0.0011928844882429471],[0.001575565891709973,0.0021260329659213073,0.0028080715767295557,0.0036303718273906116,0.004594082590234108,0.005690510696956645,0.00689935425884465,0.008187859873498201,0.009511239408330768,0.01081455318593859,0.012036072737146524,0.0131119046346423,0.01398142751680651,0.014592913381782634,0.014908613268223871,0.014908613268223871,0.014592913381782634,0.01398142751680651,0.0131119046346423,0.012036072737146524,0.01081455318593859,0.009511239408330768,0.008187859873498198,0.006899354258844646,0.005690510696956642,0.004594082590234108,0.0036303718273906116,0.0028080715767295557,0.0021260329659213073,0.001575565891709973],[0.002036945949975809,0.0027486087774776704,0.0036303718273906116,0.004693469965057365,0.00593938849226772,0.007356888581079765,0.00891972324924442,0.010585547767977259,0.012296458494048167,0.013981427516806507,0.015560650141351164,0.016951522740199097,0.018075671925298183,0.018866221950964678,0.019274369657435002,0.019274369657435002,0.018866221950964678,0.018075671925298183,0.016951522740199097,0.015560650141351164,0.013981427516806507,0.012296458494048163,0.010585547767977254,0.008919723249244414,0.007356888581079761,0.00593938849226772,0.004693469965057365,0.0036303718273906116,0.0027486087774776704,0.002036945949975809],[0.002577669277683295,0.003478248601618012,0.004594082590234108,0.00593938849226772,0.007516045894554064,0.0093098325338547,0.01128753395998937,0.013395564702778686,0.015560650141351164,0.017692907447375315,0.019691347141840197,0.021451437814479016,0.022874001244793423,0.023874408994275665,0.024390902720453304,0.024390902720453304,0.023874408994275665,0.022874001244793423,0.021451437814479016,0.019691347141840197,0.017692907447375315,0.01556065014135116,0.013395564702778685,0.011287533959989368,0.009309832533854697,0.007516045894554064,0.00593938849226772,0.004594082590234108,0.003478248601618012,0.002577669277683295],[0.0031928582714325594,0.004308370710673959,0.005690510696956645,0.007356888581079765,0.0093098325338547,0.011531725993214125,0.01398142751680651,0.016592562875334053,0.019274369657435005,0.02191551351374867,0.024390902720453307,0.026571058301799615,0.028333132069152358,0.029572298080631686,0.03021205868919643,0.03021205868919643,0.029572298080631686,0.028333132069152358,0.026571058301799615,0.024390902720453307,0.021915513513748666,0.019274369657435002,0.01659256287533405,0.013981427516806507,0.011531725993214122,0.0093098325338547,0.007356888581079765,0.005690510696956645,0.004308370710673959,0.0031928582714325594],[0.0038711218528552803,0.005223604241226928,0.00689935425884465,0.00891972324924442,0.011287533959989368,0.01398142751680651,0.016951522740199097,0.02011734542565887,0.023368852369206515,0.026571058301799615,0.029572298080631686,0.03221558732058515,0.03435198101152144,0.035854384882458325,0.03663005077861468,0.03663005077861468,0.035854384882458325,0.03435198101152144,0.03221558732058515,0.029572298080631686,0.026571058301799608,0.023368852369206508,0.020117345425658868,0.016951522740199093,0.013981427516806507,0.011287533959989368,0.00891972324924442,0.00689935425884465,0.005223604241226928,0.0038711218528552803],[0.004594082590234108,0.006199151102720615,0.008187859873498198,0.010585547767977254,0.013395564702778686,0.016592562875334053,0.02011734542565887,0.023874408994275665,0.027733158992124415,0.03153340064931213,0.03509514423195878,0.038232087355892355,0.0407674684248652,0.04255045736949905,0.04347098462873319,0.04347098462873319,0.04255045736949905,0.0407674684248652,0.038232087355892355,0.03509514423195878,0.03153340064931212,0.027733158992124404,0.023874408994275658,0.020117345425658868,0.01659256287533405,0.013395564702778686,0.010585547767977254,0.008187859873498198,0.006199151102720615,0.004594082590234108],[0.005336610549331791,0.007201101530478751,0.009511239408330768,0.012296458494048167,0.015560650141351164,0.019274369657435005,0.023368852369206515,0.027733158992124415,0.03221558732058515,0.03663005077861467,0.0407674684248652,0.04441142637273149,0.04735659357282031,0.049427761738473616,0.05049707109155733,0.05049707109155733,0.049427761738473616,0.04735659357282031,0.04441142637273149,0.0407674684248652,0.03663005077861467,0.032215587320585146,0.027733158992124404,0.023368852369206508,0.019274369657435002,0.015560650141351164,0.012296458494048167,0.009511239408330768,0.007201101530478751,0.005336610549331791],[0.006067879919817759,0.008187859873498198,0.01081455318593859,0.013981427516806507,0.017692907447375315,0.02191551351374867,0.026571058301799615,0.031533400649312135,0.03663005077861468,0.041649422892455844,0.04635378593778369,0.05049707109155733,0.053845811035895697,0.05620078890185729,0.05741662443860306,0.05741662443860306,0.05620078890185729,0.053845811035895697,0.05049707109155733,0.04635378593778369,0.041649422892455844,0.03663005077861467,0.03153340064931212,0.026571058301799608,0.021915513513748666,0.017692907447375315,0.013981427516806507,0.01081455318593859,0.008187859873498198,0.006067879919817759],[0.00675325580442451,0.009112690585046539,0.012036072737146524,0.015560650141351164,0.019691347141840197,0.024390902720453307,0.02957229808063169,0.03509514423195878,0.0407674684248652,0.04635378593778369,0.05158951364858101,0.05620078890185729,0.059927773905754746,0.06254874995550394,0.06390191588895248,0.06390191588895248,0.06254874995550394,0.059927773905754746,0.05620078890185729,0.05158951364858101,0.046353785937783686,0.04076746842486519,0.03509514423195877,0.029572298080631686,0.024390902720453304,0.019691347141840197,0.015560650141351164,0.012036072737146524,0.009112690585046539,0.00675325580442451],[0.007356888581079765,0.009927218996223851,0.0131119046346423,0.016951522740199097,0.021451437814479012,0.026571058301799615,0.03221558732058515,0.038232087355892355,0.04441142637273149,0.05049707109155733,0.05620078890185729,0.061224238218380696,0.06528435591732298,0.06813960519709264,0.06961372246621364,0.06961372246621364,0.06813960519709264,0.06528435591732298,0.061224238218380696,0.05620078890185729,0.05049707109155733,0.04441142637273149,0.03823208735589235,0.032215587320585146,0.026571058301799608,0.021451437814479012,0.016951522740199097,0.0131119046346423,0.009927218996223851,0.007356888581079765],[0.00784476453358481,0.010585547767977259,0.01398142751680651,0.018075671925298183,0.02287400124479342,0.028333132069152358,0.03435198101152144,0.0407674684248652,0.04735659357282031,0.05384581103589569,0.059927773905754746,0.06528435591732298,0.06961372246621364,0.0726583191102467,0.07423019324476678,0.07423019324476678,0.0726583191102467,0.06961372246621364,0.06528435591732298,0.059927773905754746,0.05384581103589569,0.04735659357282031,0.04076746842486519,0.034351981011521426,0.028333132069152354,0.02287400124479342,0.018075671925298183,0.01398142751680651,0.010585547767977259,0.00784476453358481],[0.008187859873498201,0.011048512856868706,0.014592913381782634,0.018866221950964678,0.023874408994275665,0.02957229808063169,0.035854384882458325,0.04255045736949905,0.049427761738473616,0.05620078890185729,0.06254874995550394,0.06813960519709264,0.0726583191102467,0.07583607295944655,0.07747669392354645,0.07747669392354645,0.07583607295944655,0.0726583191102467,0.06813960519709264,0.06254874995550394,0.056200788901857285,0.04942776173847361,0.04255045736949904,0.03585438488245832,0.029572298080631686,0.023874408994275665,0.018866221950964678,0.014592913381782634,0.011048512856868706,0.008187859873498201],[0.008364994237599,0.01128753395998937,0.014908613268223871,0.019274369657435002,0.024390902720453304,0.03021205868919643,0.03663005077861468,0.04347098462873319,0.05049707109155733,0.05741662443860306,0.06390191588895248,0.06961372246621364,0.07423019324476678,0.07747669392354645,0.07915280772163426,0.07915280772163426,0.07747669392354645,0.07423019324476678,0.06961372246621364,0.06390191588895248,0.05741662443860304,0.050497071091557315,0.04347098462873319,0.03663005077861467,0.030212058689196422,0.024390902720453304,0.019274369657435002,0.014908613268223871,0.01128753395998937,0.008364994237599],[0.008364994237599,0.01128753395998937,0.014908613268223871,0.019274369657435002,0.024390902720453304,0.03021205868919643,0.03663005077861468,0.04347098462873319,0.05049707109155733,0.05741662443860306,0.06390191588895248,0.06961372246621364,0.07423019324476678,0.07747669392354645,0.07915280772163426,0.07915280772163426,0.07747669392354645,0.07423019324476678,0.06961372246621364,0.06390191588895248,0.05741662443860304,0.050497071091557315,0.04347098462873319,0.03663005077861467,0.030212058689196422,0.024390902720453304,0.019274369657435002,0.014908613268223871,0.01128753395998937,0.008364994237599],[0.008187859873498201,0.011048512856868706,0.014592913381782634,0.018866221950964678,0.023874408994275665,0.02957229808063169,0.035854384882458325,0.04255045736949905,0.049427761738473616,0.05620078890185729,0.06254874995550394,0.06813960519709264,0.0726583191102467,0.07583607295944655,0.07747669392354645,0.07747669392354645,0.07583607295944655,0.0726583191102467,0.06813960519709264,0.06254874995550394,0.056200788901857285,0.04942776173847361,0.04255045736949904,0.03585438488245832,0.029572298080631686,0.023874408994275665,0.018866221950964678,0.014592913381782634,0.011048512856868706,0.008187859873498201],[0.00784476453358481,0.010585547767977259,0.01398142751680651,0.018075671925298183,0.02287400124479342,0.028333132069152358,0.03435198101152144,0.0407674684248652,0.04735659357282031,0.05384581103589569,0.059927773905754746,0.06528435591732298,0.06961372246621364,0.0726583191102467,0.07423019324476678,0.07423019324476678,0.0726583191102467,0.06961372246621364,0.06528435591732298,0.059927773905754746,0.05384581103589569,0.04735659357282031,0.04076746842486519,0.034351981011521426,0.028333132069152354,0.02287400124479342,0.018075671925298183,0.01398142751680651,0.010585547767977259,0.00784476453358481],[0.007356888581079765,0.009927218996223851,0.0131119046346423,0.016951522740199097,0.021451437814479012,0.026571058301799615,0.03221558732058515,0.038232087355892355,0.04441142637273149,0.05049707109155733,0.05620078890185729,0.061224238218380696,0.06528435591732298,0.06813960519709264,0.06961372246621364,0.06961372246621364,0.06813960519709264,0.06528435591732298,0.061224238218380696,0.05620078890185729,0.05049707109155733,0.04441142637273149,0.03823208735589235,0.032215587320585146,0.026571058301799608,0.021451437814479012,0.016951522740199097,0.0131119046346423,0.009927218996223851,0.007356888581079765],[0.00675325580442451,0.009112690585046539,0.012036072737146524,0.015560650141351164,0.019691347141840197,0.024390902720453307,0.02957229808063169,0.03509514423195878,0.0407674684248652,0.04635378593778369,0.05158951364858101,0.05620078890185729,0.059927773905754746,0.06254874995550394,0.06390191588895248,0.06390191588895248,0.06254874995550394,0.059927773905754746,0.05620078890185729,0.05158951364858101,0.046353785937783686,0.04076746842486519,0.03509514423195877,0.029572298080631686,0.024390902720453304,0.019691347141840197,0.015560650141351164,0.012036072737146524,0.009112690585046539,0.00675325580442451],[0.006067879919817757,0.008187859873498198,0.010814553185938588,0.013981427516806507,0.017692907447375308,0.021915513513748666,0.026571058301799608,0.03153340064931213,0.03663005077861467,0.041649422892455844,0.046353785937783686,0.05049707109155733,0.05384581103589569,0.056200788901857285,0.05741662443860305,0.05741662443860305,0.056200788901857285,0.05384581103589569,0.05049707109155733,0.046353785937783686,0.04164942289245584,0.03663005077861466,0.03153340064931212,0.026571058301799608,0.02191551351374866,0.017692907447375308,0.013981427516806507,0.010814553185938588,0.008187859873498198,0.006067879919817757],[0.005336610549331789,0.007201101530478751,0.009511239408330768,0.012296458494048163,0.01556065014135116,0.019274369657435002,0.023368852369206508,0.027733158992124415,0.032215587320585146,0.03663005077861467,0.04076746842486519,0.04441142637273149,0.04735659357282031,0.049427761738473616,0.05049707109155733,0.05049707109155733,0.049427761738473616,0.04735659357282031,0.04441142637273149,0.04076746842486519,0.03663005077861466,0.032215587320585146,0.027733158992124404,0.023368852369206505,0.019274369657435,0.01556065014135116,0.012296458494048163,0.009511239408330768,0.007201101530478751,0.005336610549331789],[0.004594082590234108,0.006199151102720615,0.008187859873498198,0.010585547767977254,0.013395564702778685,0.01659256287533405,0.020117345425658868,0.023874408994275665,0.027733158992124415,0.03153340064931212,0.03509514423195877,0.03823208735589235,0.04076746842486519,0.04255045736949904,0.04347098462873319,0.04347098462873319,0.04255045736949904,0.04076746842486519,0.03823208735589235,0.03509514423195877,0.03153340064931212,0.027733158992124404,0.023874408994275658,0.02011734542565886,0.016592562875334043,0.013395564702778685,0.010585547767977254,0.008187859873498198,0.006199151102720615,0.004594082590234108],[0.0038711218528552803,0.005223604241226928,0.006899354258844646,0.008919723249244414,0.011287533959989368,0.01398142751680651,0.016951522740199097,0.020117345425658868,0.023368852369206508,0.026571058301799608,0.029572298080631686,0.03221558732058515,0.03435198101152143,0.03585438488245832,0.03663005077861467,0.03663005077861467,0.03585438488245832,0.03435198101152143,0.03221558732058515,0.029572298080631686,0.026571058301799608,0.023368852369206505,0.02011734542565886,0.016951522740199093,0.013981427516806507,0.011287533959989368,0.008919723249244414,0.006899354258844646,0.005223604241226928,0.0038711218528552803],[0.0031928582714325594,0.004308370710673959,0.005690510696956642,0.007356888581079761,0.009309832533854697,0.011531725993214122,0.013981427516806507,0.01659256287533405,0.019274369657435002,0.021915513513748666,0.024390902720453304,0.026571058301799608,0.028333132069152354,0.029572298080631682,0.030212058689196422,0.030212058689196422,0.029572298080631682,0.028333132069152354,0.026571058301799608,0.024390902720453304,0.02191551351374866,0.019274369657435,0.016592562875334043,0.013981427516806503,0.01153172599321412,0.009309832533854697,0.007356888581079761,0.005690510696956642,0.004308370710673959,0.0031928582714325594],[0.002577669277683295,0.003478248601618012,0.004594082590234108,0.00593938849226772,0.007516045894554064,0.0093098325338547,0.01128753395998937,0.013395564702778686,0.015560650141351164,0.017692907447375315,0.019691347141840197,0.021451437814479016,0.022874001244793423,0.023874408994275665,0.024390902720453304,0.024390902720453304,0.023874408994275665,0.022874001244793423,0.021451437814479016,0.019691347141840197,0.017692907447375315,0.01556065014135116,0.013395564702778685,0.011287533959989368,0.009309832533854697,0.007516045894554064,0.00593938849226772,0.004594082590234108,0.003478248601618012,0.002577669277683295],[0.002036945949975809,0.0027486087774776704,0.0036303718273906116,0.004693469965057365,0.00593938849226772,0.007356888581079765,0.00891972324924442,0.010585547767977259,0.012296458494048167,0.013981427516806507,0.015560650141351164,0.016951522740199097,0.018075671925298183,0.018866221950964678,0.019274369657435002,0.019274369657435002,0.018866221950964678,0.018075671925298183,0.016951522740199097,0.015560650141351164,0.013981427516806507,0.012296458494048163,0.010585547767977254,0.008919723249244414,0.007356888581079761,0.00593938849226772,0.004693469965057365,0.0036303718273906116,0.0027486087774776704,0.002036945949975809],[0.001575565891709973,0.0021260329659213073,0.0028080715767295557,0.0036303718273906116,0.004594082590234108,0.005690510696956645,0.00689935425884465,0.008187859873498201,0.009511239408330768,0.01081455318593859,0.012036072737146524,0.0131119046346423,0.01398142751680651,0.014592913381782634,0.014908613268223871,0.014908613268223871,0.014592913381782634,0.01398142751680651,0.0131119046346423,0.012036072737146524,0.01081455318593859,0.009511239408330768,0.008187859873498198,0.006899354258844646,0.005690510696956642,0.004594082590234108,0.0036303718273906116,0.0028080715767295557,0.0021260329659213073,0.001575565891709973],[0.0011928844882429471,0.0016096513385347629,0.0021260329659213073,0.0027486087774776704,0.003478248601618012,0.004308370710673959,0.005223604241226928,0.006199151102720615,0.007201101530478754,0.008187859873498201,0.009112690585046542,0.009927218996223851,0.010585547767977259,0.011048512856868706,0.01128753395998937,0.01128753395998937,0.011048512856868706,0.010585547767977259,0.009927218996223851,0.009112690585046542,0.008187859873498198,0.007201101530478751,0.006199151102720615,0.005223604241226928,0.004308370710673959,0.003478248601618012,0.0027486087774776704,0.0021260329659213073,0.0016096513385347629,0.0011928844882429471],[0.0008840258559260083,0.0011928844882429471,0.0015755658917099723,0.002036945949975808,0.002577669277683294,0.0031928582714325594,0.0038711218528552803,0.004594082590234108,0.005336610549331791,0.006067879919817759,0.00675325580442451,0.007356888581079765,0.00784476453358481,0.008187859873498201,0.008364994237599,0.008364994237599,0.008187859873498201,0.00784476453358481,0.007356888581079765,0.00675325580442451,0.006067879919817757,0.005336610549331789,0.004594082590234108,0.0038711218528552803,0.0031928582714325594,0.002577669277683294,0.002036945949975808,0.0015755658917099723,0.0011928844882429471,0.0008840258559260083]],"type":"surface"},{"cmax":0.325,"cmin":0,"colorbar":{"title":{"text":"\u178a\u1784\u17cb\u179f\u17bb\u17b8\u178f\u17c1"},"x":-0.01},"colorscale":[[0.0,"rgb(103,0,31)"],[0.1,"rgb(178,24,43)"],[0.2,"rgb(214,96,77)"],[0.3,"rgb(244,165,130)"],[0.4,"rgb(253,219,199)"],[0.5,"rgb(247,247,247)"],[0.6,"rgb(209,229,240)"],[0.7,"rgb(146,197,222)"],[0.8,"rgb(67,147,195)"],[0.9,"rgb(33,102,172)"],[1.0,"rgb(5,48,97)"]],"name":"y = 0.5","opacity":0.55,"showlegend":true,"x":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"y":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"z":[[1.964128034639744e-05,3.576322229511952e-05,6.238971485005047e-05,0.00010427948318833298,0.00016699145564774272,0.00025621180827714905,0.00037662881487761493,0.0005304414537399787,0.0007157656960436467,0.0009253665894641641,0.0011462154570633818,0.0013602797008490269,0.001546677203782656,0.0016849253439607662,0.0017586165339435879,0.0017586165339435879,0.0016849253439607662,0.001546677203782656,0.0013602797008490269,0.0011462154570633818,0.0009253665894641633,0.000715765696043646,0.0005304414537399787,0.00037662881487761493,0.00025621180827714905,0.00016699145564774272,0.00010427948318833298,6.238971485005047e-05,3.576322229511952e-05,1.964128034639744e-05],[3.576322229511952e-05,6.511836531902715e-05,0.00011360039680512626,0.0001898740953905568,0.00030406126506978047,0.0004665154049252642,0.0006857730143690002,0.0009658380355091629,0.0013032799923109423,0.0016849253439607662,0.0020870512240602746,0.002476823530189334,0.002816219700658092,0.0030679445822279143,0.0032021229230523684,0.0032021229230523684,0.0030679445822279143,0.002816219700658092,0.002476823530189334,0.0020870512240602746,0.0016849253439607647,0.0013032799923109412,0.0009658380355091629,0.0006857730143690002,0.0004665154049252642,0.00030406126506978047,0.0001898740953905568,0.00011360039680512626,6.511836531902715e-05,3.576322229511952e-05],[6.238971485005053e-05,0.00011360039680512626,0.00019817834939587134,0.00033123946637338456,0.0005304414537399787,0.000813846214590541,0.0011963458567932296,0.0016849253439607662,0.002273600136449445,0.0029393887073671473,0.003640906003157847,0.004320872221950083,0.004912956182450325,0.005352095683141995,0.005586172700979197,0.005586172700979197,0.005352095683141995,0.004912956182450325,0.004320872221950083,0.003640906003157847,0.0029393887073671473,0.002273600136449445,0.0016849253439607647,0.0011963458567932285,0.0008138462145905402,0.0005304414537399787,0.00033123946637338456,0.00019817834939587134,0.00011360039680512626,6.238971485005053e-05],[0.00010427948318833308,0.0001898740953905568,0.00033123946637338456,0.0005536406192593425,0.0008865910156925211,0.0013602797008490269,0.0019995976574142087,0.002816219700658092,0.003800143144999158,0.0049129561824503235,0.006085486963021312,0.007221996819679552,0.008211618419224605,0.008945605427977102,0.009336846684729054,0.009336846684729054,0.008945605427977102,0.008211618419224605,0.007221996819679552,0.006085486963021312,0.0049129561824503235,0.0038001431449991565,0.0028162197006580897,0.001999597657414207,0.0013602797008490258,0.0008865910156925211,0.0005536406192593425,0.00033123946637338456,0.0001898740953905568,0.00010427948318833308],[0.00016699145564774289,0.00030406126506978047,0.0005304414537399787,0.0008865910156925211,0.001419772324794848,0.0021783296233124183,0.0032021229230523684,0.0045098480818838715,0.006085486963021312,0.00786752752657243,0.009745199105417005,0.011565187366948787,0.013149951180459567,0.014325345949119345,0.01495187328677852,0.01495187328677852,0.014325345949119345,0.013149951180459567,0.011565187366948787,0.009745199105417005,0.00786752752657243,0.006085486963021309,0.004509848081883869,0.0032021229230523667,0.002178329623312416,0.001419772324794848,0.0008865910156925211,0.0005304414537399787,0.00030406126506978047,0.00016699145564774289],[0.00025621180827714905,0.0004665154049252642,0.000813846214590541,0.0013602797008490269,0.0021783296233124183,0.003342169631659831,0.004912956182450325,0.006919373974151699,0.00933684668472906,0.012070997563524834,0.014951873286778527,0.017744246595469696,0.020175719515906364,0.021979105311602878,0.02294037426691601,0.02294037426691601,0.021979105311602878,0.020175719515906364,0.017744246595469696,0.014951873286778527,0.01207099756352483,0.009336846684729054,0.0069193739741516954,0.0049129561824503235,0.0033421696316598298,0.0021783296233124183,0.0013602797008490269,0.000813846214590541,0.0004665154049252642,0.00025621180827714905],[0.00037662881487761493,0.0006857730143690002,0.0011963458567932296,0.0019995976574142087,0.0032021229230523667,0.004912956182450325,0.007221996819679552,0.01017141105674868,0.013725071944223595,0.017744246595469696,0.021979105311602878,0.026083866356849512,0.029658107413844354,0.03230906663218604,0.03372212245444776,0.03372212245444776,0.03230906663218604,0.029658107413844354,0.026083866356849512,0.021979105311602878,0.017744246595469686,0.013725071944223592,0.010171411056748675,0.007221996819679549,0.0049129561824503235,0.0032021229230523667,0.0019995976574142087,0.0011963458567932296,0.0006857730143690002,0.00037662881487761493],[0.0005304414537399787,0.0009658380355091629,0.0016849253439607647,0.0028162197006580897,0.0045098480818838715,0.006919373974151699,0.01017141105674868,0.014325345949119345,0.01933029770211686,0.024990875864556773,0.030955221992717608,0.03673633944864934,0.04177027617201327,0.04550386905188757,0.04749400723295003,0.04749400723295003,0.04550386905188757,0.04177027617201327,0.03673633944864934,0.030955221992717608,0.02499087586455677,0.019330297702116854,0.014325345949119338,0.010171411056748675,0.0069193739741516954,0.0045098480818838715,0.0028162197006580897,0.0016849253439607647,0.0009658380355091629,0.0005304414537399787],[0.0007157656960436467,0.0013032799923109412,0.002273600136449445,0.003800143144999158,0.006085486963021312,0.00933684668472906,0.013725071944223595,0.01933029770211686,0.02608386635684951,0.03372212245444775,0.04177027617201327,0.04957118526495805,0.05636386558289658,0.061401891339678466,0.0640873387728451,0.0640873387728451,0.061401891339678466,0.05636386558289658,0.04957118526495805,0.04177027617201327,0.033722122454447744,0.026083866356849502,0.019330297702116854,0.013725071944223592,0.009336846684729054,0.006085486963021312,0.003800143144999158,0.002273600136449445,0.0013032799923109412,0.0007157656960436467],[0.0009253665894641641,0.0016849253439607647,0.0029393887073671473,0.0049129561824503235,0.00786752752657243,0.012070997563524834,0.017744246595469696,0.02499087586455678,0.03372212245444776,0.04359712349676836,0.05400205432577081,0.0640873387728451,0.07286915027048335,0.07938248380679964,0.08285432291022486,0.08285432291022486,0.07938248380679964,0.07286915027048335,0.0640873387728451,0.05400205432577081,0.04359712349676834,0.033722122454447744,0.02499087586455677,0.017744246595469686,0.01207099756352483,0.00786752752657243,0.0049129561824503235,0.0029393887073671473,0.0016849253439607647,0.0009253665894641641],[0.0011462154570633818,0.002087051224060273,0.003640906003157847,0.006085486963021312,0.009745199105417005,0.014951873286778527,0.02197910531160288,0.030955221992717618,0.04177027617201327,0.05400205432577081,0.06689023581153622,0.07938248380679966,0.09026017076266708,0.09832798265617632,0.10262841417174182,0.10262841417174182,0.09832798265617632,0.09026017076266708,0.07938248380679966,0.06689023581153622,0.0540020543257708,0.04177027617201326,0.0309552219927176,0.02197910531160287,0.01495187328677852,0.009745199105417005,0.006085486963021312,0.003640906003157847,0.002087051224060273,0.0011462154570633818],[0.0013602797008490269,0.002476823530189334,0.004320872221950083,0.007221996819679552,0.011565187366948781,0.017744246595469696,0.026083866356849512,0.03673633944864934,0.04957118526495805,0.0640873387728451,0.07938248380679966,0.09420775183229373,0.10711692756105771,0.11669146320475397,0.12179503222353374,0.12179503222353374,0.11669146320475397,0.10711692756105771,0.09420775183229373,0.07938248380679966,0.06408733877284509,0.049571185264958045,0.03673633944864933,0.026083866356849502,0.017744246595469686,0.011565187366948781,0.007221996819679552,0.004320872221950083,0.002476823530189334,0.0013602797008490269],[0.001546677203782656,0.002816219700658092,0.004912956182450325,0.008211618419224605,0.013149951180459562,0.020175719515906364,0.029658107413844354,0.04177027617201327,0.05636386558289658,0.07286915027048334,0.09026017076266708,0.10711692756105771,0.12179503222353374,0.1326815550523803,0.13848446003901746,0.13848446003901746,0.1326815550523803,0.12179503222353374,0.10711692756105771,0.09026017076266708,0.07286915027048332,0.05636386558289657,0.04177027617201326,0.02965810741384434,0.020175719515906354,0.013149951180459562,0.008211618419224605,0.004912956182450325,0.002816219700658092,0.001546677203782656],[0.0016849253439607662,0.0030679445822279143,0.005352095683141995,0.008945605427977102,0.014325345949119345,0.02197910531160288,0.032309066632186044,0.04550386905188758,0.061401891339678466,0.07938248380679964,0.09832798265617632,0.11669146320475397,0.1326815550523803,0.14454115845060084,0.15086275009019365,0.15086275009019365,0.14454115845060084,0.1326815550523803,0.11669146320475397,0.09832798265617632,0.07938248380679963,0.061401891339678445,0.04550386905188756,0.03230906663218603,0.02197910531160287,0.014325345949119345,0.008945605427977102,0.005352095683141995,0.0030679445822279143,0.0016849253439607662],[0.0017586165339435879,0.0032021229230523684,0.005586172700979197,0.009336846684729054,0.01495187328677852,0.02294037426691601,0.03372212245444776,0.04749400723295003,0.0640873387728451,0.08285432291022486,0.10262841417174182,0.12179503222353374,0.13848446003901746,0.15086275009019365,0.1574608202172024,0.1574608202172024,0.15086275009019365,0.13848446003901746,0.12179503222353374,0.10262841417174182,0.08285432291022483,0.06408733877284507,0.04749400723295002,0.03372212245444775,0.022940374266916,0.01495187328677852,0.009336846684729054,0.005586172700979197,0.0032021229230523684,0.0017586165339435879],[0.0017586165339435879,0.0032021229230523684,0.005586172700979197,0.009336846684729054,0.01495187328677852,0.02294037426691601,0.03372212245444776,0.04749400723295003,0.0640873387728451,0.08285432291022486,0.10262841417174182,0.12179503222353374,0.13848446003901746,0.15086275009019365,0.1574608202172024,0.1574608202172024,0.15086275009019365,0.13848446003901746,0.12179503222353374,0.10262841417174182,0.08285432291022483,0.06408733877284507,0.04749400723295002,0.03372212245444775,0.022940374266916,0.01495187328677852,0.009336846684729054,0.005586172700979197,0.0032021229230523684,0.0017586165339435879],[0.0016849253439607662,0.0030679445822279143,0.005352095683141995,0.008945605427977102,0.014325345949119345,0.02197910531160288,0.032309066632186044,0.04550386905188758,0.061401891339678466,0.07938248380679964,0.09832798265617632,0.11669146320475397,0.1326815550523803,0.14454115845060084,0.15086275009019365,0.15086275009019365,0.14454115845060084,0.1326815550523803,0.11669146320475397,0.09832798265617632,0.07938248380679963,0.061401891339678445,0.04550386905188756,0.03230906663218603,0.02197910531160287,0.014325345949119345,0.008945605427977102,0.005352095683141995,0.0030679445822279143,0.0016849253439607662],[0.001546677203782656,0.002816219700658092,0.004912956182450325,0.008211618419224605,0.013149951180459562,0.020175719515906364,0.029658107413844354,0.04177027617201327,0.05636386558289658,0.07286915027048334,0.09026017076266708,0.10711692756105771,0.12179503222353374,0.1326815550523803,0.13848446003901746,0.13848446003901746,0.1326815550523803,0.12179503222353374,0.10711692756105771,0.09026017076266708,0.07286915027048332,0.05636386558289657,0.04177027617201326,0.02965810741384434,0.020175719515906354,0.013149951180459562,0.008211618419224605,0.004912956182450325,0.002816219700658092,0.001546677203782656],[0.0013602797008490269,0.002476823530189334,0.004320872221950083,0.007221996819679552,0.011565187366948781,0.017744246595469696,0.026083866356849512,0.03673633944864934,0.04957118526495805,0.0640873387728451,0.07938248380679966,0.09420775183229373,0.10711692756105771,0.11669146320475397,0.12179503222353374,0.12179503222353374,0.11669146320475397,0.10711692756105771,0.09420775183229373,0.07938248380679966,0.06408733877284509,0.049571185264958045,0.03673633944864933,0.026083866356849502,0.017744246595469686,0.011565187366948781,0.007221996819679552,0.004320872221950083,0.002476823530189334,0.0013602797008490269],[0.0011462154570633818,0.002087051224060273,0.003640906003157847,0.006085486963021312,0.009745199105417005,0.014951873286778527,0.02197910531160288,0.030955221992717618,0.04177027617201327,0.05400205432577081,0.06689023581153622,0.07938248380679966,0.09026017076266708,0.09832798265617632,0.10262841417174182,0.10262841417174182,0.09832798265617632,0.09026017076266708,0.07938248380679966,0.06689023581153622,0.0540020543257708,0.04177027617201326,0.0309552219927176,0.02197910531160287,0.01495187328677852,0.009745199105417005,0.006085486963021312,0.003640906003157847,0.002087051224060273,0.0011462154570633818],[0.0009253665894641633,0.0016849253439607647,0.0029393887073671456,0.0049129561824503235,0.007867527526572427,0.01207099756352483,0.017744246595469686,0.024990875864556773,0.03372212245444775,0.04359712349676834,0.0540020543257708,0.06408733877284509,0.07286915027048332,0.07938248380679963,0.08285432291022485,0.08285432291022485,0.07938248380679963,0.07286915027048332,0.06408733877284509,0.0540020543257708,0.043597123496768336,0.03372212245444773,0.024990875864556762,0.017744246595469686,0.012070997563524824,0.007867527526572427,0.0049129561824503235,0.0029393887073671456,0.0016849253439607647,0.0009253665894641633],[0.000715765696043646,0.0013032799923109412,0.002273600136449445,0.0038001431449991565,0.006085486963021309,0.009336846684729054,0.013725071944223592,0.01933029770211686,0.026083866356849502,0.033722122454447744,0.04177027617201326,0.049571185264958045,0.05636386558289657,0.06140189133967845,0.06408733877284509,0.06408733877284509,0.06140189133967845,0.05636386558289657,0.049571185264958045,0.04177027617201326,0.03372212245444773,0.026083866356849495,0.019330297702116854,0.013725071944223585,0.00933684668472905,0.006085486963021309,0.0038001431449991565,0.002273600136449445,0.0013032799923109412,0.000715765696043646],[0.0005304414537399787,0.0009658380355091629,0.0016849253439607647,0.0028162197006580897,0.004509848081883869,0.0069193739741516954,0.010171411056748675,0.014325345949119345,0.01933029770211686,0.02499087586455677,0.0309552219927176,0.03673633944864933,0.04177027617201326,0.04550386905188756,0.04749400723295002,0.04749400723295002,0.04550386905188756,0.04177027617201326,0.03673633944864933,0.0309552219927176,0.024990875864556762,0.019330297702116854,0.014325345949119338,0.010171411056748672,0.006919373974151693,0.004509848081883869,0.0028162197006580897,0.0016849253439607647,0.0009658380355091629,0.0005304414537399787],[0.00037662881487761493,0.0006857730143690002,0.0011963458567932285,0.001999597657414207,0.0032021229230523667,0.004912956182450325,0.007221996819679552,0.010171411056748675,0.013725071944223592,0.017744246595469686,0.02197910531160287,0.02608386635684951,0.029658107413844344,0.03230906663218603,0.03372212245444775,0.03372212245444775,0.03230906663218603,0.029658107413844344,0.02608386635684951,0.02197910531160287,0.017744246595469686,0.013725071944223585,0.010171411056748672,0.007221996819679549,0.0049129561824503235,0.0032021229230523667,0.001999597657414207,0.0011963458567932285,0.0006857730143690002,0.00037662881487761493],[0.00025621180827714905,0.0004665154049252642,0.0008138462145905402,0.0013602797008490258,0.002178329623312416,0.0033421696316598298,0.0049129561824503235,0.0069193739741516954,0.009336846684729054,0.01207099756352483,0.01495187328677852,0.017744246595469686,0.020175719515906354,0.021979105311602867,0.022940374266916,0.022940374266916,0.021979105311602867,0.020175719515906354,0.017744246595469686,0.01495187328677852,0.012070997563524824,0.00933684668472905,0.006919373974151693,0.004912956182450322,0.003342169631659828,0.002178329623312416,0.0013602797008490258,0.0008138462145905402,0.0004665154049252642,0.00025621180827714905],[0.00016699145564774289,0.00030406126506978047,0.0005304414537399787,0.0008865910156925211,0.001419772324794848,0.0021783296233124183,0.0032021229230523684,0.0045098480818838715,0.006085486963021312,0.00786752752657243,0.009745199105417005,0.011565187366948787,0.013149951180459567,0.014325345949119345,0.01495187328677852,0.01495187328677852,0.014325345949119345,0.013149951180459567,0.011565187366948787,0.009745199105417005,0.00786752752657243,0.006085486963021309,0.004509848081883869,0.0032021229230523667,0.002178329623312416,0.001419772324794848,0.0008865910156925211,0.0005304414537399787,0.00030406126506978047,0.00016699145564774289],[0.00010427948318833308,0.0001898740953905568,0.00033123946637338456,0.0005536406192593425,0.0008865910156925211,0.0013602797008490269,0.0019995976574142087,0.002816219700658092,0.003800143144999158,0.0049129561824503235,0.006085486963021312,0.007221996819679552,0.008211618419224605,0.008945605427977102,0.009336846684729054,0.009336846684729054,0.008945605427977102,0.008211618419224605,0.007221996819679552,0.006085486963021312,0.0049129561824503235,0.0038001431449991565,0.0028162197006580897,0.001999597657414207,0.0013602797008490258,0.0008865910156925211,0.0005536406192593425,0.00033123946637338456,0.0001898740953905568,0.00010427948318833308],[6.238971485005053e-05,0.00011360039680512626,0.00019817834939587134,0.00033123946637338456,0.0005304414537399787,0.000813846214590541,0.0011963458567932296,0.0016849253439607662,0.002273600136449445,0.0029393887073671473,0.003640906003157847,0.004320872221950083,0.004912956182450325,0.005352095683141995,0.005586172700979197,0.005586172700979197,0.005352095683141995,0.004912956182450325,0.004320872221950083,0.003640906003157847,0.0029393887073671473,0.002273600136449445,0.0016849253439607647,0.0011963458567932285,0.0008138462145905402,0.0005304414537399787,0.00033123946637338456,0.00019817834939587134,0.00011360039680512626,6.238971485005053e-05],[3.576322229511952e-05,6.511836531902715e-05,0.00011360039680512626,0.0001898740953905568,0.00030406126506978047,0.0004665154049252642,0.0006857730143690002,0.0009658380355091629,0.0013032799923109423,0.0016849253439607662,0.0020870512240602746,0.002476823530189334,0.002816219700658092,0.0030679445822279143,0.0032021229230523684,0.0032021229230523684,0.0030679445822279143,0.002816219700658092,0.002476823530189334,0.0020870512240602746,0.0016849253439607647,0.0013032799923109412,0.0009658380355091629,0.0006857730143690002,0.0004665154049252642,0.00030406126506978047,0.0001898740953905568,0.00011360039680512626,6.511836531902715e-05,3.576322229511952e-05],[1.964128034639744e-05,3.576322229511952e-05,6.238971485005047e-05,0.00010427948318833298,0.00016699145564774272,0.00025621180827714905,0.00037662881487761493,0.0005304414537399787,0.0007157656960436467,0.0009253665894641641,0.0011462154570633818,0.0013602797008490269,0.001546677203782656,0.0016849253439607662,0.0017586165339435879,0.0017586165339435879,0.0016849253439607662,0.001546677203782656,0.0013602797008490269,0.0011462154570633818,0.0009253665894641633,0.000715765696043646,0.0005304414537399787,0.00037662881487761493,0.00025621180827714905,0.00016699145564774272,0.00010427948318833298,6.238971485005047e-05,3.576322229511952e-05,1.964128034639744e-05]],"type":"surface"},{"cmax":0.325,"cmin":0,"colorbar":{"title":{"text":"\u178a\u1784\u17cb\u179f\u17bb\u17b8\u178f\u17c1"},"x":-0.01},"colorscale":[[0.0,"rgb(103,0,31)"],[0.1,"rgb(178,24,43)"],[0.2,"rgb(214,96,77)"],[0.3,"rgb(244,165,130)"],[0.4,"rgb(253,219,199)"],[0.5,"rgb(247,247,247)"],[0.6,"rgb(209,229,240)"],[0.7,"rgb(146,197,222)"],[0.8,"rgb(67,147,195)"],[0.9,"rgb(33,102,172)"],[1.0,"rgb(5,48,97)"]],"name":"y = 0.75","opacity":0.4,"showlegend":true,"x":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"y":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"z":[[3.272923730621637e-07,8.041483154085872e-07,1.852894508991026e-06,4.003865669884971e-06,8.1137676875721e-06,1.541984135844373e-05,2.7482200816959082e-05,4.593432915919405e-05,7.200083183374619e-05,0.00010584050807842824,0.00014590849718186142,0.00018863553975460043,0.0002287076650732598,0.00026004720329089883,0.00027729237094616316,0.00027729237094616316,0.00026004720329089883,0.0002287076650732598,0.00018863553975460043,0.00014590849718186142,0.00010584050807842805,7.200083183374619e-05,4.593432915919405e-05,2.7482200816959082e-05,1.541984135844373e-05,8.1137676875721e-06,4.003865669884971e-06,1.852894508991026e-06,8.041483154085872e-07,3.272923730621637e-07],[8.041483154085872e-07,1.9757701871398326e-06,4.552510601131406e-06,9.837387298202492e-06,1.993530297248473e-05,3.788612406774621e-05,6.7523007896309e-05,0.00011285937728152853,0.00017690405396683987,0.00026004720329089883,0.00035849314518034104,0.0004634723079570225,0.0005619284124129653,0.0006389287916995628,0.0006812996920330261,0.0006812996920330261,0.0006389287916995628,0.0005619284124129653,0.0004634723079570225,0.00035849314518034104,0.00026004720329089845,0.00017690405396683968,0.00011285937728152853,6.7523007896309e-05,3.788612406774621e-05,1.993530297248473e-05,9.837387298202492e-06,4.552510601131406e-06,1.9757701871398326e-06,8.041483154085872e-07],[1.8528945089910295e-06,4.552510601131406e-06,1.0489758833448325e-05,2.266701373165966e-05,4.593432915919405e-05,8.729607450139534e-05,0.0001555844962481314,0.00026004720329089883,0.00040761703274459017,0.0005991929919188553,0.0008260291882574801,0.0010679190368590476,0.0012947786494796002,0.001472200621566766,0.0015698303834708123,0.0015698303834708123,0.001472200621566766,0.0012947786494796002,0.0010679190368590476,0.0008260291882574801,0.0005991929919188553,0.00040761703274459017,0.00026004720329089845,0.00015558449624813127,8.729607450139527e-05,4.593432915919405e-05,2.266701373165966e-05,1.0489758833448325e-05,4.552510601131406e-06,1.8528945089910295e-06],[4.003865669884971e-06,9.837387298202492e-06,2.266701373165966e-05,4.89804884620352e-05,9.925815133957201e-05,0.00018863553975460043,0.00033619799738812727,0.0005619284124129653,0.0008808077502238131,0.0012947786494796002,0.0017849423661944115,0.0023076350807607,0.0027978493970357518,0.003181235358662849,0.003392200661949064,0.003392200661949064,0.003181235358662849,0.0027978493970357518,0.0023076350807607,0.0017849423661944115,0.0012947786494796002,0.0008808077502238131,0.0005619284124129642,0.00033619799738812694,0.0001886355397546003,9.925815133957201e-05,4.89804884620352e-05,2.266701373165966e-05,9.837387298202492e-06,4.003865669884971e-06],[8.113767687572115e-06,1.993530297248473e-05,4.593432915919405e-05,9.925815133957201e-05,0.00020114500521949287,0.00038226680747572916,0.0006812996920330261,0.0011387386519128625,0.0017849423661944115,0.0026238475600525433,0.0036171562407643866,0.004676384398660888,0.005669795618546303,0.006446720941170343,0.006874238645841918,0.006874238645841918,0.006446720941170343,0.005669795618546303,0.004676384398660888,0.0036171562407643866,0.0026238475600525433,0.0017849423661944115,0.0011387386519128618,0.0006812996920330254,0.00038226680747572883,0.00020114500521949287,9.925815133957201e-05,4.593432915919405e-05,1.993530297248473e-05,8.113767687572115e-06],[1.541984135844373e-05,3.788612406774621e-05,8.729607450139534e-05,0.00018863553975460043,0.00038226680747572916,0.0007264804410043835,0.0012947786494796002,0.002164120299884835,0.003392200661949064,0.004986501300341838,0.006874238645841919,0.008887252918135923,0.01077518513460722,0.012251695885656987,0.01306417357022997,0.01306417357022997,0.012251695885656987,0.01077518513460722,0.008887252918135923,0.006874238645841919,0.0049865013003418356,0.003392200661949064,0.0021641202998848334,0.0012947786494796002,0.0007264804410043827,0.00038226680747572916,0.00018863553975460043,8.729607450139534e-05,3.788612406774621e-05,1.541984135844373e-05],[2.7482200816959082e-05,6.7523007896309e-05,0.0001555844962481314,0.00033619799738812727,0.0006812996920330254,0.0012947786494796002,0.0023076350807607,0.003857029867620851,0.006045791110039971,0.008887252918135923,0.012251695885656987,0.01583941518785947,0.0192042054665519,0.021835734807580345,0.023283783095995828,0.023283783095995828,0.021835734807580345,0.0192042054665519,0.01583941518785947,0.012251695885656987,0.008887252918135915,0.006045791110039965,0.0038570298676208472,0.0023076350807606978,0.0012947786494796002,0.0006812996920330254,0.00033619799738812727,0.0001555844962481314,6.7523007896309e-05,2.7482200816959082e-05],[4.593432915919405e-05,0.00011285937728152853,0.00026004720329089845,0.0005619284124129642,0.0011387386519128625,0.002164120299884835,0.003857029867620851,0.006446720941170343,0.010105062572169716,0.014854341673056524,0.02047774249662087,0.026474332087672324,0.03209831341444208,0.03649670696916369,0.038917005371081996,0.038917005371081996,0.03649670696916369,0.03209831341444208,0.026474332087672324,0.02047774249662087,0.014854341673056517,0.010105062572169707,0.006446720941170337,0.0038570298676208472,0.0021641202998848334,0.0011387386519128625,0.0005619284124129642,0.00026004720329089845,0.00011285937728152853,4.593432915919405e-05],[7.200083183374619e-05,0.00017690405396683968,0.00040761703274459017,0.0008808077502238131,0.0017849423661944115,0.003392200661949064,0.006045791110039971,0.010105062572169716,0.015839415187859463,0.023283783095995818,0.03209831341444208,0.0414978071400377,0.050313247381725185,0.05720761158533862,0.06100136456734171,0.06100136456734171,0.05720761158533862,0.050313247381725185,0.0414978071400377,0.03209831341444208,0.023283783095995807,0.015839415187859453,0.010105062572169707,0.006045791110039965,0.003392200661949064,0.0017849423661944115,0.0008808077502238131,0.00040761703274459017,0.00017690405396683968,7.200083183374619e-05],[0.00010584050807842824,0.00026004720329089845,0.0005991929919188553,0.0012947786494796002,0.0026238475600525433,0.004986501300341838,0.008887252918135923,0.014854341673056524,0.023283783095995828,0.03422693002434295,0.047184202094910935,0.06100136456734171,0.07395997421604195,0.08409462115835918,0.08967140038316619,0.08967140038316619,0.08409462115835918,0.07395997421604195,0.06100136456734171,0.047184202094910935,0.03422693002434293,0.023283783095995807,0.014854341673056517,0.008887252918135915,0.0049865013003418356,0.0026238475600525433,0.0012947786494796002,0.0005991929919188553,0.00026004720329089845,0.00010584050807842824],[0.00014590849718186142,0.00035849314518034066,0.0008260291882574801,0.0017849423661944115,0.0036171562407643866,0.006874238645841919,0.012251695885656992,0.02047774249662087,0.03209831341444208,0.047184202094910935,0.06504670228238316,0.08409462115835921,0.10195896529025951,0.11593028054251162,0.12361825833645476,0.12361825833645476,0.11593028054251162,0.10195896529025951,0.08409462115835921,0.06504670228238316,0.047184202094910914,0.03209831341444207,0.02047774249662086,0.012251695885656982,0.006874238645841918,0.0036171562407643866,0.0017849423661944115,0.0008260291882574801,0.00035849314518034066,0.00014590849718186142],[0.00018863553975460043,0.0004634723079570225,0.0010679190368590476,0.0023076350807607,0.004676384398660883,0.008887252918135923,0.01583941518785947,0.026474332087672324,0.0414978071400377,0.06100136456734171,0.08409462115835921,0.10872042793295097,0.13181606843895072,0.14987866687970755,0.15981794984671854,0.15981794984671854,0.14987866687970755,0.13181606843895072,0.10872042793295097,0.08409462115835921,0.0610013645673417,0.041497807140037694,0.02647433208767231,0.015839415187859453,0.008887252918135915,0.004676384398660883,0.0023076350807607,0.0010679190368590476,0.0004634723079570225,0.00018863553975460043],[0.0002287076650732598,0.0005619284124129653,0.0012947786494796002,0.0027978493970357518,0.005669795618546301,0.01077518513460722,0.0192042054665519,0.03209831341444208,0.050313247381725185,0.07395997421604192,0.10195896529025951,0.13181606843895072,0.15981794984671854,0.18171761265636494,0.19376831213136705,0.19376831213136705,0.18171761265636494,0.15981794984671854,0.13181606843895072,0.10195896529025951,0.0739599742160419,0.05031324738172517,0.03209831341444207,0.019204205466551886,0.010775185134607216,0.005669795618546301,0.0027978493970357518,0.0012947786494796002,0.0005619284124129653,0.0002287076650732598],[0.00026004720329089883,0.0006389287916995628,0.001472200621566766,0.003181235358662849,0.006446720941170343,0.012251695885656992,0.021835734807580356,0.0364967069691637,0.05720761158533862,0.08409462115835918,0.11593028054251162,0.14987866687970755,0.18171761265636494,0.20661816010779396,0.22032015254066498,0.22032015254066498,0.20661816010779396,0.18171761265636494,0.14987866687970755,0.11593028054251162,0.08409462115835914,0.05720761158533859,0.036496706969163684,0.021835734807580345,0.012251695885656982,0.006446720941170343,0.003181235358662849,0.001472200621566766,0.0006389287916995628,0.00026004720329089883],[0.00027729237094616316,0.0006812996920330261,0.0015698303834708123,0.003392200661949064,0.006874238645841918,0.01306417357022997,0.023283783095995828,0.038917005371081996,0.06100136456734171,0.08967140038316619,0.12361825833645479,0.15981794984671854,0.19376831213136705,0.22032015254066498,0.2349307998397516,0.2349307998397516,0.22032015254066498,0.19376831213136705,0.15981794984671854,0.12361825833645479,0.08967140038316614,0.06100136456734168,0.03891700537108198,0.023283783095995818,0.013064173570229959,0.006874238645841918,0.003392200661949064,0.0015698303834708123,0.0006812996920330261,0.00027729237094616316],[0.00027729237094616316,0.0006812996920330261,0.0015698303834708123,0.003392200661949064,0.006874238645841918,0.01306417357022997,0.023283783095995828,0.038917005371081996,0.06100136456734171,0.08967140038316619,0.12361825833645479,0.15981794984671854,0.19376831213136705,0.22032015254066498,0.2349307998397516,0.2349307998397516,0.22032015254066498,0.19376831213136705,0.15981794984671854,0.12361825833645479,0.08967140038316614,0.06100136456734168,0.03891700537108198,0.023283783095995818,0.013064173570229959,0.006874238645841918,0.003392200661949064,0.0015698303834708123,0.0006812996920330261,0.00027729237094616316],[0.00026004720329089883,0.0006389287916995628,0.001472200621566766,0.003181235358662849,0.006446720941170343,0.012251695885656992,0.021835734807580356,0.0364967069691637,0.05720761158533862,0.08409462115835918,0.11593028054251162,0.14987866687970755,0.18171761265636494,0.20661816010779396,0.22032015254066498,0.22032015254066498,0.20661816010779396,0.18171761265636494,0.14987866687970755,0.11593028054251162,0.08409462115835914,0.05720761158533859,0.036496706969163684,0.021835734807580345,0.012251695885656982,0.006446720941170343,0.003181235358662849,0.001472200621566766,0.0006389287916995628,0.00026004720329089883],[0.0002287076650732598,0.0005619284124129653,0.0012947786494796002,0.0027978493970357518,0.005669795618546301,0.01077518513460722,0.0192042054665519,0.03209831341444208,0.050313247381725185,0.07395997421604192,0.10195896529025951,0.13181606843895072,0.15981794984671854,0.18171761265636494,0.19376831213136705,0.19376831213136705,0.18171761265636494,0.15981794984671854,0.13181606843895072,0.10195896529025951,0.0739599742160419,0.05031324738172517,0.03209831341444207,0.019204205466551886,0.010775185134607216,0.005669795618546301,0.0027978493970357518,0.0012947786494796002,0.0005619284124129653,0.0002287076650732598],[0.00018863553975460043,0.0004634723079570225,0.0010679190368590476,0.0023076350807607,0.004676384398660883,0.008887252918135923,0.01583941518785947,0.026474332087672324,0.0414978071400377,0.06100136456734171,0.08409462115835921,0.10872042793295097,0.13181606843895072,0.14987866687970755,0.15981794984671854,0.15981794984671854,0.14987866687970755,0.13181606843895072,0.10872042793295097,0.08409462115835921,0.0610013645673417,0.041497807140037694,0.02647433208767231,0.015839415187859453,0.008887252918135915,0.004676384398660883,0.0023076350807607,0.0010679190368590476,0.0004634723079570225,0.00018863553975460043],[0.00014590849718186142,0.00035849314518034066,0.0008260291882574801,0.0017849423661944115,0.0036171562407643866,0.006874238645841919,0.012251695885656992,0.02047774249662087,0.03209831341444208,0.047184202094910935,0.06504670228238316,0.08409462115835921,0.10195896529025951,0.11593028054251162,0.12361825833645476,0.12361825833645476,0.11593028054251162,0.10195896529025951,0.08409462115835921,0.06504670228238316,0.047184202094910914,0.03209831341444207,0.02047774249662086,0.012251695885656982,0.006874238645841918,0.0036171562407643866,0.0017849423661944115,0.0008260291882574801,0.00035849314518034066,0.00014590849718186142],[0.00010584050807842805,0.00026004720329089845,0.0005991929919188548,0.0012947786494796002,0.002623847560052541,0.0049865013003418356,0.008887252918135915,0.014854341673056524,0.023283783095995818,0.03422693002434293,0.047184202094910914,0.0610013645673417,0.0739599742160419,0.08409462115835914,0.08967140038316615,0.08967140038316615,0.08409462115835914,0.0739599742160419,0.0610013645673417,0.047184202094910914,0.034226930024342926,0.023283783095995807,0.01485434167305651,0.008887252918135915,0.00498650130034183,0.002623847560052541,0.0012947786494796002,0.0005991929919188548,0.00026004720329089845,0.00010584050807842805],[7.200083183374619e-05,0.00017690405396683968,0.00040761703274459017,0.0008808077502238131,0.0017849423661944115,0.003392200661949064,0.006045791110039965,0.010105062572169716,0.015839415187859453,0.023283783095995807,0.03209831341444207,0.041497807140037694,0.05031324738172517,0.0572076115853386,0.0610013645673417,0.0610013645673417,0.0572076115853386,0.05031324738172517,0.041497807140037694,0.03209831341444207,0.023283783095995807,0.015839415187859453,0.010105062572169707,0.006045791110039962,0.0033922006619490604,0.0017849423661944115,0.0008808077502238131,0.00040761703274459017,0.00017690405396683968,7.200083183374619e-05],[4.593432915919405e-05,0.00011285937728152853,0.00026004720329089845,0.0005619284124129642,0.0011387386519128618,0.0021641202998848334,0.0038570298676208472,0.006446720941170343,0.010105062572169716,0.014854341673056517,0.02047774249662086,0.02647433208767231,0.03209831341444207,0.036496706969163684,0.03891700537108198,0.03891700537108198,0.036496706969163684,0.03209831341444207,0.02647433208767231,0.02047774249662086,0.01485434167305651,0.010105062572169707,0.006446720941170337,0.0038570298676208472,0.0021641202998848312,0.0011387386519128618,0.0005619284124129642,0.00026004720329089845,0.00011285937728152853,4.593432915919405e-05],[2.7482200816959082e-05,6.7523007896309e-05,0.00015558449624813127,0.00033619799738812694,0.0006812996920330254,0.0012947786494796002,0.0023076350807607,0.0038570298676208472,0.006045791110039965,0.008887252918135915,0.012251695885656982,0.015839415187859463,0.019204205466551893,0.021835734807580345,0.023283783095995818,0.023283783095995818,0.021835734807580345,0.019204205466551893,0.015839415187859463,0.012251695885656982,0.008887252918135915,0.006045791110039962,0.0038570298676208472,0.0023076350807606978,0.0012947786494796002,0.0006812996920330254,0.00033619799738812694,0.00015558449624813127,6.7523007896309e-05,2.7482200816959082e-05],[1.541984135844373e-05,3.788612406774621e-05,8.729607450139527e-05,0.0001886355397546003,0.00038226680747572883,0.0007264804410043827,0.0012947786494796002,0.0021641202998848334,0.003392200661949064,0.0049865013003418356,0.006874238645841918,0.008887252918135915,0.010775185134607216,0.012251695885656982,0.013064173570229959,0.013064173570229959,0.012251695885656982,0.010775185134607216,0.008887252918135915,0.006874238645841918,0.00498650130034183,0.0033922006619490604,0.0021641202998848312,0.0012947786494795993,0.0007264804410043822,0.00038226680747572883,0.0001886355397546003,8.729607450139527e-05,3.788612406774621e-05,1.541984135844373e-05],[8.113767687572115e-06,1.993530297248473e-05,4.593432915919405e-05,9.925815133957201e-05,0.00020114500521949287,0.00038226680747572916,0.0006812996920330261,0.0011387386519128625,0.0017849423661944115,0.0026238475600525433,0.0036171562407643866,0.004676384398660888,0.005669795618546303,0.006446720941170343,0.006874238645841918,0.006874238645841918,0.006446720941170343,0.005669795618546303,0.004676384398660888,0.0036171562407643866,0.0026238475600525433,0.0017849423661944115,0.0011387386519128618,0.0006812996920330254,0.00038226680747572883,0.00020114500521949287,9.925815133957201e-05,4.593432915919405e-05,1.993530297248473e-05,8.113767687572115e-06],[4.003865669884971e-06,9.837387298202492e-06,2.266701373165966e-05,4.89804884620352e-05,9.925815133957201e-05,0.00018863553975460043,0.00033619799738812727,0.0005619284124129653,0.0008808077502238131,0.0012947786494796002,0.0017849423661944115,0.0023076350807607,0.0027978493970357518,0.003181235358662849,0.003392200661949064,0.003392200661949064,0.003181235358662849,0.0027978493970357518,0.0023076350807607,0.0017849423661944115,0.0012947786494796002,0.0008808077502238131,0.0005619284124129642,0.00033619799738812694,0.0001886355397546003,9.925815133957201e-05,4.89804884620352e-05,2.266701373165966e-05,9.837387298202492e-06,4.003865669884971e-06],[1.8528945089910295e-06,4.552510601131406e-06,1.0489758833448325e-05,2.266701373165966e-05,4.593432915919405e-05,8.729607450139534e-05,0.0001555844962481314,0.00026004720329089883,0.00040761703274459017,0.0005991929919188553,0.0008260291882574801,0.0010679190368590476,0.0012947786494796002,0.001472200621566766,0.0015698303834708123,0.0015698303834708123,0.001472200621566766,0.0012947786494796002,0.0010679190368590476,0.0008260291882574801,0.0005991929919188553,0.00040761703274459017,0.00026004720329089845,0.00015558449624813127,8.729607450139527e-05,4.593432915919405e-05,2.266701373165966e-05,1.0489758833448325e-05,4.552510601131406e-06,1.8528945089910295e-06],[8.041483154085872e-07,1.9757701871398326e-06,4.552510601131406e-06,9.837387298202492e-06,1.993530297248473e-05,3.788612406774621e-05,6.7523007896309e-05,0.00011285937728152853,0.00017690405396683987,0.00026004720329089883,0.00035849314518034104,0.0004634723079570225,0.0005619284124129653,0.0006389287916995628,0.0006812996920330261,0.0006812996920330261,0.0006389287916995628,0.0005619284124129653,0.0004634723079570225,0.00035849314518034104,0.00026004720329089845,0.00017690405396683968,0.00011285937728152853,6.7523007896309e-05,3.788612406774621e-05,1.993530297248473e-05,9.837387298202492e-06,4.552510601131406e-06,1.9757701871398326e-06,8.041483154085872e-07],[3.272923730621637e-07,8.041483154085872e-07,1.852894508991026e-06,4.003865669884971e-06,8.1137676875721e-06,1.541984135844373e-05,2.7482200816959082e-05,4.593432915919405e-05,7.200083183374619e-05,0.00010584050807842824,0.00014590849718186142,0.00018863553975460043,0.0002287076650732598,0.00026004720329089883,0.00027729237094616316,0.00027729237094616316,0.00026004720329089883,0.0002287076650732598,0.00018863553975460043,0.00014590849718186142,0.00010584050807842805,7.200083183374619e-05,4.593432915919405e-05,2.7482200816959082e-05,1.541984135844373e-05,8.1137676875721e-06,4.003865669884971e-06,1.852894508991026e-06,8.041483154085872e-07,3.272923730621637e-07]],"type":"surface"},{"cmax":0.325,"cmin":0,"colorbar":{"title":{"text":"\u178a\u1784\u17cb\u179f\u17bb\u17b8\u178f\u17c1"},"x":-0.01},"colorscale":[[0.0,"rgb(103,0,31)"],[0.1,"rgb(178,24,43)"],[0.2,"rgb(214,96,77)"],[0.3,"rgb(244,165,130)"],[0.4,"rgb(253,219,199)"],[0.5,"rgb(247,247,247)"],[0.6,"rgb(209,229,240)"],[0.7,"rgb(146,197,222)"],[0.8,"rgb(67,147,195)"],[0.9,"rgb(33,102,172)"],[1.0,"rgb(5,48,97)"]],"name":"y = 1.0","opacity":0.25,"showlegend":true,"x":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"y":[-3.0,-2.793103448275862,-2.586206896551724,-2.3793103448275863,-2.1724137931034484,-1.9655172413793103,-1.7586206896551724,-1.5517241379310345,-1.3448275862068966,-1.1379310344827587,-0.9310344827586206,-0.7241379310344827,-0.5172413793103448,-0.31034482758620685,-0.10344827586206895,0.10344827586206895,0.31034482758620685,0.5172413793103448,0.7241379310344827,0.9310344827586206,1.137931034482759,1.3448275862068968,1.5517241379310347,1.7586206896551726,1.9655172413793105,2.1724137931034484,2.3793103448275863,2.586206896551724,2.793103448275862,3.0],"z":[[4.8478531191209144e-09,1.6072489412931907e-08,4.891430254633197e-08,1.3664936071444667e-07,3.5042764890124325e-07,8.249129989351791e-07,1.7825304252625519e-06,3.535776274109454e-06,6.438009673843865e-06,1.076062493895792e-05,1.6509821793626974e-05,2.3252320394139795e-05,3.006140213087425e-05,3.5675592093700735e-05,3.886441794866564e-05,3.886441794866564e-05,3.5675592093700735e-05,3.006140213087425e-05,2.3252320394139795e-05,1.6509821793626974e-05,1.0760624938957901e-05,6.438009673843854e-06,3.535776274109454e-06,1.7825304252625519e-06,8.249129989351791e-07,3.5042764890124325e-07,1.3664936071444667e-07,4.891430254633197e-08,1.6072489412931907e-08,4.8478531191209144e-09],[1.6072489412931907e-08,5.328645682558397e-08,1.6216964303560241e-07,4.5304495606606313e-07,1.1618018406434693e-06,2.734902464285018e-06,5.909770920092691e-06,1.1722452255819409e-05,2.1344467289052794e-05,3.5675592093700735e-05,5.473638112938134e-05,7.709034580418192e-05,9.966505907134003e-05,0.00011827824856419322,0.00012885042732750142,0.00012885042732750142,0.00011827824856419322,9.966505907134003e-05,7.709034580418192e-05,5.473638112938134e-05,3.567559209370067e-05,2.1344467289052756e-05,1.1722452255819409e-05,5.909770920092691e-06,2.734902464285018e-06,1.1618018406434693e-06,4.5304495606606313e-07,1.6216964303560241e-07,5.328645682558397e-08,1.6072489412931907e-08],[4.8914302546332056e-08,1.6216964303560241e-07,4.935399103073417e-07,1.3787769572443997e-06,3.535776274109454e-06,8.323281050980836e-06,1.798553511768633e-05,3.5675592093700735e-05,6.495880655718281e-05,0.00010857351716696362,0.00016658227845523817,0.0002346133446529513,0.00030331622734130483,0.00035996278400184795,0.0003921376846856388,0.0003921376846856388,0.00035996278400184795,0.00030331622734130483,0.0002346133446529513,0.00016658227845523817,0.00010857351716696362,6.495880655718281e-05,3.567559209370067e-05,1.7985535117686295e-05,8.323281050980821e-06,3.535776274109454e-06,1.3787769572443997e-06,4.935399103073417e-07,1.6216964303560241e-07,4.8914302546332056e-08],[1.3664936071444694e-07,4.5304495606606313e-07,1.3787769572443997e-06,3.851817974850912e-06,9.877715562410625e-06,2.3252320394139795e-05,5.024526054748722e-05,9.966505907134003e-05,0.00018147206290848125,0.00030331622734130456,0.000465372307735608,0.0006554271837268191,0.0008473588787501829,0.001005609438430674,0.0010954947967126664,0.0010954947967126664,0.001005609438430674,0.0008473588787501829,0.0006554271837268191,0.000465372307735608,0.00030331622734130456,0.00018147206290848106,9.966505907133984e-05,5.024526054748713e-05,2.3252320394139754e-05,9.877715562410625e-06,3.851817974850912e-06,1.3787769572443997e-06,4.5304495606606313e-07,1.3664936071444694e-07],[3.5042764890124384e-07,1.1618018406434693e-06,3.535776274109454e-06,9.877715562410625e-06,2.5330704973322516e-05,5.962893587364875e-05,0.00012885042732750142,0.0002555840155078079,0.000465372307735608,0.0007778330748500256,0.001193414464662725,0.001680796791280345,0.0021729920879507464,0.002578814488262039,0.002809319150769944,0.002809319150769944,0.002578814488262039,0.0021729920879507464,0.001680796791280345,0.001193414464662725,0.0007778330748500256,0.0004653723077356076,0.0002555840155078077,0.0001288504273275013,5.9628935873648644e-05,2.5330704973322516e-05,9.877715562410625e-06,3.535776274109454e-06,1.1618018406434693e-06,3.5042764890124384e-07],[8.249129989351791e-07,2.734902464285018e-06,8.323281050980836e-06,2.3252320394139795e-05,5.962893587364875e-05,0.0001403675893414086,0.00030331622734130483,0.0006016493771924317,0.0010954947967126674,0.001831033071897626,0.0028093191507699462,0.003956625928470724,0.005115262524387828,0.006070575766152622,0.006613187894545275,0.006613187894545275,0.006070575766152622,0.005115262524387828,0.003956625928470724,0.0028093191507699462,0.0018310330718976245,0.0010954947967126664,0.0006016493771924312,0.00030331622734130456,0.00014036758934140849,5.962893587364875e-05,2.3252320394139795e-05,8.323281050980836e-06,2.734902464285018e-06,8.249129989351791e-07],[1.7825304252625519e-06,5.909770920092691e-06,1.798553511768633e-05,5.024526054748722e-05,0.0001288504273275013,0.00030331622734130483,0.0006554271837268191,0.0013000865807304943,0.0023672227354615727,0.003956625928470724,0.006070575766152622,0.008549757499258374,0.011053421505900074,0.013117730010312127,0.014290244723045341,0.014290244723045341,0.013117730010312127,0.011053421505900074,0.008549757499258374,0.006070575766152622,0.00395662592847072,0.0023672227354615705,0.0013000865807304932,0.0006554271837268186,0.00030331622734130456,0.0001288504273275013,5.024526054748722e-05,1.798553511768633e-05,5.909770920092691e-06,1.7825304252625519e-06],[3.535776274109454e-06,1.1722452255819409e-05,3.567559209370067e-05,9.966505907133984e-05,0.0002555840155078079,0.0006016493771924317,0.0013000865807304943,0.002578814488262039,0.004695555186579591,0.007848249816747161,0.012041420140688345,0.01695905398687065,0.02192525016931262,0.02601995336702518,0.028345719953430416,0.028345719953430416,0.02601995336702518,0.02192525016931262,0.01695905398687065,0.012041420140688345,0.007848249816747158,0.004695555186579587,0.0025788144882620367,0.0013000865807304932,0.0006016493771924312,0.0002555840155078079,9.966505907133984e-05,3.567559209370067e-05,1.1722452255819409e-05,3.535776274109454e-06],[6.438009673843865e-06,2.1344467289052756e-05,6.495880655718281e-05,0.00018147206290848125,0.000465372307735608,0.0010954947967126674,0.0023672227354615727,0.004695555186579591,0.008549757499258369,0.014290244723045334,0.02192525016931262,0.03087937277768318,0.03992191862507864,0.04737763322767536,0.05161243391119739,0.05161243391119739,0.04737763322767536,0.03992191862507864,0.03087937277768318,0.02192525016931262,0.014290244723045327,0.008549757499258365,0.004695555186579587,0.0023672227354615705,0.0010954947967126664,0.000465372307735608,0.00018147206290848125,6.495880655718281e-05,2.1344467289052756e-05,6.438009673843865e-06],[1.076062493895792e-05,3.567559209370067e-05,0.00010857351716696362,0.00030331622734130456,0.0007778330748500256,0.001831033071897626,0.003956625928470724,0.007848249816747165,0.014290244723045341,0.023885015950714274,0.03664632482975648,0.05161243391119739,0.06672633545634034,0.0791879738438071,0.08626610888168376,0.08626610888168376,0.0791879738438071,0.06672633545634034,0.05161243391119739,0.03664632482975648,0.023885015950714267,0.014290244723045327,0.007848249816747158,0.00395662592847072,0.0018310330718976245,0.0007778330748500256,0.00030331622734130456,0.00010857351716696362,3.567559209370067e-05,1.076062493895792e-05],[1.6509821793626974e-05,5.473638112938125e-05,0.00016658227845523817,0.000465372307735608,0.001193414464662725,0.0028093191507699462,0.006070575766152625,0.012041420140688349,0.02192525016931262,0.03664632482975648,0.05622575786841228,0.07918797384380713,0.10237694497998522,0.12149659929381929,0.13235644700428947,0.13235644700428947,0.12149659929381929,0.10237694497998522,0.07918797384380713,0.05622575786841228,0.03664632482975646,0.021925250169312613,0.012041420140688338,0.006070575766152619,0.002809319150769944,0.001193414464662725,0.000465372307735608,0.00016658227845523817,5.473638112938125e-05,1.6509821793626974e-05],[2.3252320394139795e-05,7.709034580418192e-05,0.0002346133446529513,0.0006554271837268191,0.0016807967912803436,0.003956625928470724,0.008549757499258374,0.01695905398687065,0.03087937277768318,0.05161243391119739,0.07918797384380713,0.11152780218922388,0.14418699095630136,0.1711149816692041,0.186409917105327,0.186409917105327,0.1711149816692041,0.14418699095630136,0.11152780218922388,0.07918797384380713,0.05161243391119736,0.030879372777683167,0.016959053986870642,0.008549757499258365,0.00395662592847072,0.0016807967912803436,0.0006554271837268191,0.0002346133446529513,7.709034580418192e-05,2.3252320394139795e-05],[3.006140213087425e-05,9.966505907134003e-05,0.00030331622734130483,0.0008473588787501829,0.0021729920879507447,0.005115262524387828,0.011053421505900074,0.02192525016931262,0.03992191862507864,0.06672633545634032,0.10237694497998522,0.14418699095630136,0.186409917105327,0.22122335265393708,0.240997172940145,0.240997172940145,0.22122335265393708,0.186409917105327,0.14418699095630136,0.10237694497998522,0.0667263354563403,0.03992191862507862,0.021925250169312613,0.011053421505900066,0.005115262524387824,0.0021729920879507447,0.0008473588787501829,0.00030331622734130483,9.966505907134003e-05,3.006140213087425e-05],[3.5675592093700735e-05,0.00011827824856419322,0.00035996278400184795,0.001005609438430674,0.002578814488262039,0.006070575766152625,0.013117730010312134,0.02601995336702519,0.04737763322767536,0.0791879738438071,0.12149659929381929,0.1711149816692041,0.22122335265393708,0.2625384556755949,0.2860051836609932,0.2860051836609932,0.2625384556755949,0.22122335265393708,0.1711149816692041,0.12149659929381929,0.07918797384380706,0.04737763322767533,0.02601995336702517,0.013117730010312124,0.006070575766152619,0.002578814488262039,0.001005609438430674,0.00035996278400184795,0.00011827824856419322,3.5675592093700735e-05],[3.886441794866564e-05,0.00012885042732750142,0.0003921376846856388,0.0010954947967126664,0.002809319150769944,0.006613187894545275,0.014290244723045341,0.028345719953430416,0.05161243391119739,0.08626610888168376,0.1323564470042895,0.186409917105327,0.240997172940145,0.2860051836609932,0.3115694608260862,0.3115694608260862,0.2860051836609932,0.240997172940145,0.186409917105327,0.1323564470042895,0.0862661088816837,0.05161243391119735,0.028345719953430402,0.014290244723045334,0.006613187894545269,0.002809319150769944,0.0010954947967126664,0.0003921376846856388,0.00012885042732750142,3.886441794866564e-05],[3.886441794866564e-05,0.00012885042732750142,0.0003921376846856388,0.0010954947967126664,0.002809319150769944,0.006613187894545275,0.014290244723045341,0.028345719953430416,0.05161243391119739,0.08626610888168376,0.1323564470042895,0.186409917105327,0.240997172940145,0.2860051836609932,0.3115694608260862,0.3115694608260862,0.2860051836609932,0.240997172940145,0.186409917105327,0.1323564470042895,0.0862661088816837,0.05161243391119735,0.028345719953430402,0.014290244723045334,0.006613187894545269,0.002809319150769944,0.0010954947967126664,0.0003921376846856388,0.00012885042732750142,3.886441794866564e-05],[3.5675592093700735e-05,0.00011827824856419322,0.00035996278400184795,0.001005609438430674,0.002578814488262039,0.006070575766152625,0.013117730010312134,0.02601995336702519,0.04737763322767536,0.0791879738438071,0.12149659929381929,0.1711149816692041,0.22122335265393708,0.2625384556755949,0.2860051836609932,0.2860051836609932,0.2625384556755949,0.22122335265393708,0.1711149816692041,0.12149659929381929,0.07918797384380706,0.04737763322767533,0.02601995336702517,0.013117730010312124,0.006070575766152619,0.002578814488262039,0.001005609438430674,0.00035996278400184795,0.00011827824856419322,3.5675592093700735e-05],[3.006140213087425e-05,9.966505907134003e-05,0.00030331622734130483,0.0008473588787501829,0.0021729920879507447,0.005115262524387828,0.011053421505900074,0.02192525016931262,0.03992191862507864,0.06672633545634032,0.10237694497998522,0.14418699095630136,0.186409917105327,0.22122335265393708,0.240997172940145,0.240997172940145,0.22122335265393708,0.186409917105327,0.14418699095630136,0.10237694497998522,0.0667263354563403,0.03992191862507862,0.021925250169312613,0.011053421505900066,0.005115262524387824,0.0021729920879507447,0.0008473588787501829,0.00030331622734130483,9.966505907134003e-05,3.006140213087425e-05],[2.3252320394139795e-05,7.709034580418192e-05,0.0002346133446529513,0.0006554271837268191,0.0016807967912803436,0.003956625928470724,0.008549757499258374,0.01695905398687065,0.03087937277768318,0.05161243391119739,0.07918797384380713,0.11152780218922388,0.14418699095630136,0.1711149816692041,0.186409917105327,0.186409917105327,0.1711149816692041,0.14418699095630136,0.11152780218922388,0.07918797384380713,0.05161243391119736,0.030879372777683167,0.016959053986870642,0.008549757499258365,0.00395662592847072,0.0016807967912803436,0.0006554271837268191,0.0002346133446529513,7.709034580418192e-05,2.3252320394139795e-05],[1.6509821793626974e-05,5.473638112938125e-05,0.00016658227845523817,0.000465372307735608,0.001193414464662725,0.0028093191507699462,0.006070575766152625,0.012041420140688349,0.02192525016931262,0.03664632482975648,0.05622575786841228,0.07918797384380713,0.10237694497998522,0.12149659929381929,0.13235644700428947,0.13235644700428947,0.12149659929381929,0.10237694497998522,0.07918797384380713,0.05622575786841228,0.03664632482975646,0.021925250169312613,0.012041420140688338,0.006070575766152619,0.002809319150769944,0.001193414464662725,0.000465372307735608,0.00016658227845523817,5.473638112938125e-05,1.6509821793626974e-05],[1.0760624938957901e-05,3.567559209370067e-05,0.00010857351716696353,0.00030331622734130456,0.0007778330748500249,0.0018310330718976245,0.00395662592847072,0.007848249816747161,0.014290244723045334,0.023885015950714267,0.03664632482975646,0.05161243391119736,0.06672633545634031,0.07918797384380706,0.08626610888168372,0.08626610888168372,0.07918797384380706,0.06672633545634031,0.05161243391119736,0.03664632482975646,0.023885015950714253,0.014290244723045324,0.007848249816747154,0.00395662592847072,0.0018310330718976228,0.0007778330748500249,0.00030331622734130456,0.00010857351716696353,3.567559209370067e-05,1.0760624938957901e-05],[6.438009673843854e-06,2.1344467289052756e-05,6.495880655718281e-05,0.00018147206290848106,0.0004653723077356076,0.0010954947967126664,0.0023672227354615705,0.004695555186579591,0.008549757499258365,0.014290244723045327,0.021925250169312613,0.030879372777683167,0.03992191862507862,0.047377633227675346,0.05161243391119736,0.05161243391119736,0.047377633227675346,0.03992191862507862,0.030879372777683167,0.021925250169312613,0.014290244723045324,0.008549757499258362,0.004695555186579587,0.002367222735461568,0.0010954947967126655,0.0004653723077356076,0.00018147206290848106,6.495880655718281e-05,2.1344467289052756e-05,6.438009673843854e-06],[3.535776274109454e-06,1.1722452255819409e-05,3.567559209370067e-05,9.966505907133984e-05,0.0002555840155078077,0.0006016493771924312,0.0013000865807304932,0.002578814488262039,0.004695555186579591,0.007848249816747158,0.012041420140688338,0.016959053986870642,0.021925250169312613,0.02601995336702517,0.028345719953430402,0.028345719953430402,0.02601995336702517,0.021925250169312613,0.016959053986870642,0.012041420140688338,0.007848249816747154,0.004695555186579587,0.0025788144882620367,0.0013000865807304921,0.0006016493771924306,0.0002555840155078077,9.966505907133984e-05,3.567559209370067e-05,1.1722452255819409e-05,3.535776274109454e-06],[1.7825304252625519e-06,5.909770920092691e-06,1.7985535117686295e-05,5.024526054748713e-05,0.0001288504273275013,0.00030331622734130483,0.0006554271837268191,0.0013000865807304932,0.0023672227354615705,0.00395662592847072,0.006070575766152619,0.008549757499258369,0.011053421505900071,0.013117730010312124,0.014290244723045334,0.014290244723045334,0.013117730010312124,0.011053421505900071,0.008549757499258369,0.006070575766152619,0.00395662592847072,0.002367222735461568,0.0013000865807304921,0.0006554271837268186,0.00030331622734130456,0.0001288504273275013,5.024526054748713e-05,1.7985535117686295e-05,5.909770920092691e-06,1.7825304252625519e-06],[8.249129989351791e-07,2.734902464285018e-06,8.323281050980821e-06,2.3252320394139754e-05,5.9628935873648644e-05,0.00014036758934140849,0.00030331622734130456,0.0006016493771924312,0.0010954947967126664,0.0018310330718976245,0.002809319150769944,0.00395662592847072,0.005115262524387824,0.006070575766152617,0.006613187894545269,0.006613187894545269,0.006070575766152617,0.005115262524387824,0.00395662592847072,0.002809319150769944,0.0018310330718976228,0.0010954947967126655,0.0006016493771924306,0.0003033162273413043,0.00014036758934140835,5.9628935873648644e-05,2.3252320394139754e-05,8.323281050980821e-06,2.734902464285018e-06,8.249129989351791e-07],[3.5042764890124384e-07,1.1618018406434693e-06,3.535776274109454e-06,9.877715562410625e-06,2.5330704973322516e-05,5.962893587364875e-05,0.00012885042732750142,0.0002555840155078079,0.000465372307735608,0.0007778330748500256,0.001193414464662725,0.001680796791280345,0.0021729920879507464,0.002578814488262039,0.002809319150769944,0.002809319150769944,0.002578814488262039,0.0021729920879507464,0.001680796791280345,0.001193414464662725,0.0007778330748500256,0.0004653723077356076,0.0002555840155078077,0.0001288504273275013,5.9628935873648644e-05,2.5330704973322516e-05,9.877715562410625e-06,3.535776274109454e-06,1.1618018406434693e-06,3.5042764890124384e-07],[1.3664936071444694e-07,4.5304495606606313e-07,1.3787769572443997e-06,3.851817974850912e-06,9.877715562410625e-06,2.3252320394139795e-05,5.024526054748722e-05,9.966505907134003e-05,0.00018147206290848125,0.00030331622734130456,0.000465372307735608,0.0006554271837268191,0.0008473588787501829,0.001005609438430674,0.0010954947967126664,0.0010954947967126664,0.001005609438430674,0.0008473588787501829,0.0006554271837268191,0.000465372307735608,0.00030331622734130456,0.00018147206290848106,9.966505907133984e-05,5.024526054748713e-05,2.3252320394139754e-05,9.877715562410625e-06,3.851817974850912e-06,1.3787769572443997e-06,4.5304495606606313e-07,1.3664936071444694e-07],[4.8914302546332056e-08,1.6216964303560241e-07,4.935399103073417e-07,1.3787769572443997e-06,3.535776274109454e-06,8.323281050980836e-06,1.798553511768633e-05,3.5675592093700735e-05,6.495880655718281e-05,0.00010857351716696362,0.00016658227845523817,0.0002346133446529513,0.00030331622734130483,0.00035996278400184795,0.0003921376846856388,0.0003921376846856388,0.00035996278400184795,0.00030331622734130483,0.0002346133446529513,0.00016658227845523817,0.00010857351716696362,6.495880655718281e-05,3.567559209370067e-05,1.7985535117686295e-05,8.323281050980821e-06,3.535776274109454e-06,1.3787769572443997e-06,4.935399103073417e-07,1.6216964303560241e-07,4.8914302546332056e-08],[1.6072489412931907e-08,5.328645682558397e-08,1.6216964303560241e-07,4.5304495606606313e-07,1.1618018406434693e-06,2.734902464285018e-06,5.909770920092691e-06,1.1722452255819409e-05,2.1344467289052794e-05,3.5675592093700735e-05,5.473638112938134e-05,7.709034580418192e-05,9.966505907134003e-05,0.00011827824856419322,0.00012885042732750142,0.00012885042732750142,0.00011827824856419322,9.966505907134003e-05,7.709034580418192e-05,5.473638112938134e-05,3.567559209370067e-05,2.1344467289052756e-05,1.1722452255819409e-05,5.909770920092691e-06,2.734902464285018e-06,1.1618018406434693e-06,4.5304495606606313e-07,1.6216964303560241e-07,5.328645682558397e-08,1.6072489412931907e-08],[4.8478531191209144e-09,1.6072489412931907e-08,4.891430254633197e-08,1.3664936071444667e-07,3.5042764890124325e-07,8.249129989351791e-07,1.7825304252625519e-06,3.535776274109454e-06,6.438009673843865e-06,1.076062493895792e-05,1.6509821793626974e-05,2.3252320394139795e-05,3.006140213087425e-05,3.5675592093700735e-05,3.886441794866564e-05,3.886441794866564e-05,3.5675592093700735e-05,3.006140213087425e-05,2.3252320394139795e-05,1.6509821793626974e-05,1.0760624938957901e-05,6.438009673843854e-06,3.535776274109454e-06,1.7825304252625519e-06,8.249129989351791e-07,3.5042764890124325e-07,1.3664936071444667e-07,4.891430254633197e-08,1.6072489412931907e-08,4.8478531191209144e-09]],"type":"surface"}], {"height":400,"title":{"text":"\u178a\u1784\u17cb\u179f\u17bb\u17b8\u178f\u17c1\u179a\u17bd\u1798\u1793\u17c3\u1785\u17c6\u1796\u17c4\u17c7\u178f\u1798\u17d2\u179b\u17c3\u1781\u17d2\u179b\u17c7\u17d7\u1793\u17c3 y"},"width":650,"template":{"data":{"histogram2dcontour":[{"type":"histogram2dcontour","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"choropleth":[{"type":"choropleth","colorbar":{"outlinewidth":0,"ticks":""}}],"histogram2d":[{"type":"histogram2d","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"heatmap":[{"type":"heatmap","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"heatmapgl":[{"type":"heatmapgl","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"contourcarpet":[{"type":"contourcarpet","colorbar":{"outlinewidth":0,"ticks":""}}],"contour":[{"type":"contour","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"surface":[{"type":"surface","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"mesh3d":[{"type":"mesh3d","colorbar":{"outlinewidth":0,"ticks":""}}],"scatter":[{"fillpattern":{"fillmode":"overlay","size":10,"solidity":0.2},"type":"scatter"}],"parcoords":[{"type":"parcoords","line":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterpolargl":[{"type":"scatterpolargl","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"bar":[{"error_x":{"color":"#2a3f5f"},"error_y":{"color":"#2a3f5f"},"marker":{"line":{"color":"#E5ECF6","width":0.5},"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"bar"}],"scattergeo":[{"type":"scattergeo","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterpolar":[{"type":"scatterpolar","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"histogram":[{"marker":{"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"histogram"}],"scattergl":[{"type":"scattergl","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatter3d":[{"type":"scatter3d","line":{"colorbar":{"outlinewidth":0,"ticks":""}},"marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scattermapbox":[{"type":"scattermapbox","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterternary":[{"type":"scatterternary","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scattercarpet":[{"type":"scattercarpet","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"carpet":[{"aaxis":{"endlinecolor":"#2a3f5f","gridcolor":"white","linecolor":"white","minorgridcolor":"white","startlinecolor":"#2a3f5f"},"baxis":{"endlinecolor":"#2a3f5f","gridcolor":"white","linecolor":"white","minorgridcolor":"white","startlinecolor":"#2a3f5f"},"type":"carpet"}],"table":[{"cells":{"fill":{"color":"#EBF0F8"},"line":{"color":"white"}},"header":{"fill":{"color":"#C8D4E3"},"line":{"color":"white"}},"type":"table"}],"barpolar":[{"marker":{"line":{"color":"#E5ECF6","width":0.5},"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"barpolar"}],"pie":[{"automargin":true,"type":"pie"}]},"layout":{"autotypenumbers":"strict","colorway":["#636efa","#EF553B","#00cc96","#ab63fa","#FFA15A","#19d3f3","#FF6692","#B6E880","#FF97FF","#FECB52"],"font":{"color":"#2a3f5f"},"hovermode":"closest","hoverlabel":{"align":"left"},"paper_bgcolor":"white","plot_bgcolor":"#E5ECF6","polar":{"bgcolor":"#E5ECF6","angularaxis":{"gridcolor":"white","linecolor":"white","ticks":""},"radialaxis":{"gridcolor":"white","linecolor":"white","ticks":""}},"ternary":{"bgcolor":"#E5ECF6","aaxis":{"gridcolor":"white","linecolor":"white","ticks":""},"baxis":{"gridcolor":"white","linecolor":"white","ticks":""},"caxis":{"gridcolor":"white","linecolor":"white","ticks":""}},"coloraxis":{"colorbar":{"outlinewidth":0,"ticks":""}},"colorscale":{"sequential":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]],"sequentialminus":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]],"diverging":[[0,"#8e0152"],[0.1,"#c51b7d"],[0.2,"#de77ae"],[0.3,"#f1b6da"],[0.4,"#fde0ef"],[0.5,"#f7f7f7"],[0.6,"#e6f5d0"],[0.7,"#b8e186"],[0.8,"#7fbc41"],[0.9,"#4d9221"],[1,"#276419"]]},"xaxis":{"gridcolor":"white","linecolor":"white","ticks":"","title":{"standoff":15},"zerolinecolor":"white","automargin":true,"zerolinewidth":2},"yaxis":{"gridcolor":"white","linecolor":"white","ticks":"","title":{"standoff":15},"zerolinecolor":"white","automargin":true,"zerolinewidth":2},"scene":{"xaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2},"yaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2},"zaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2}},"shapedefaults":{"line":{"color":"#2a3f5f"}},"annotationdefaults":{"arrowcolor":"#2a3f5f","arrowhead":0,"arrowwidth":1},"geo":{"bgcolor":"white","landcolor":"#E5ECF6","subunitcolor":"white","showland":true,"showlakes":true,"lakecolor":"white"},"title":{"x":0.05},"mapbox":{"style":"light"}}},"scene":{"xaxis":{"title":{"text":"x1"}},"yaxis":{"title":{"text":"x2"}},"zaxis":{"title":{"text":"f_(x1,x2,y)"}}}}, {"responsive": true} ).then(function(){
var gd = document.getElementById('c5b31e9f-e107-47aa-b9ca-ec4659f23ee4');
var x = new MutationObserver(function (mutations, observer) {{
var display = window.getComputedStyle(gd).display;
if (!display || display === 'none') {{
console.log([gd, 'removed!']);
Plotly.purge(gd);
observer.disconnect();
}}
}});
// Listen for the removal of the full notebook cells
var notebookContainer = gd.closest('#notebook-container');
if (notebookContainer) {{
x.observe(notebookContainer, {childList: true});
}}
// Listen for the clearing of the current output cell
var outputEl = gd.closest('.output');
if (outputEl) {{
x.observe(outputEl, {childList: true});
}}
}) }; }); </script> </div>
<p>រូបទី៣៖ ដង់សុីតេរួម <span class="math inline">\(f_{X,Y}\)</span> ចំពោះតម្លៃខ្លះៗនៃ <span class="math inline">\(y\)</span> ។</p>
</div>
</div>
<ul>
<li><p>ជាដំបូងយើងកំណត់ <span class="math inline">\(C\)</span> ដោយប្រើលក្ខណៈនៃអនុគមន៍ដង់សុីតេ <span class="math inline">\(\int_{\Omega}f_{X,Y}(x,y)dxdy=1\)</span> ហើយដោយបម្លែងអាំងតេក្រាលធៀប <span class="math inline">\(x\)</span> ជាទម្រង់ប៉ូលែរ គេបាន៖ <span class="math display">\[
\begin{align}
1&=C\int_{\Omega}ye^{-\|x\|^2y}dxdy\\
&=C\int_0^{+\infty}\int_0^{+\infty}\int_0^{2\pi}yre^{-r^2y}drd\theta dy\\
&=\pi C\int_0^1\int_0^{+\infty}e^{-u}dudy,\ \ (u=r^2y)\\
&=\pi C\int_0^1dy\\
\Rightarrow C&=1/\pi
\end{align}
\]</span></p></li>
<li><p>យើងគណនាដង់សុីតេដោយផ្នែកនៃ <span class="math inline">\(X\)</span> ចំពោះ <span class="math inline">\(x\neq (0,0)\)</span> ដោយអាំងតេក្រាលធៀបនឹង <span class="math inline">\(Y\)</span> ៖ <span class="math display">\[
\begin{align}
f_{X}(x)&=\int_0^1 f_{X,Y}(x,y)dy\\
&=-\frac{1}{\pi\|x\|^2}\int_0^1yd(e^{-\|x\|^2y})\\
&=-\frac{1}{\pi\|x\|^2}\left[e^{-\|x\|^2}-\int_0^1e^{-\|x\|^2y}dy\right]\\
&=-\frac{1}{\pi\|x\|^2}\left[e^{-\|x\|^2}+\frac{1}{\|x\|^2}e^{-\|x\|^2y}\Big|_0^1\right]\\
&=\frac{1}{\pi\|x\|^4}\left[1-(1+\|x\|^2)e^{-\|x\|^2}\right]
\end{align}
\]</span></p></li>
<li><p>ចុងក្រោយ យើងគណនាដង់សុីតេមានលក្ខខណ្ឌតាមនិយមន័យ៖</p></li>
</ul>
<p><span class="math display">\[
\begin{align}
f_{Y|X}(y|X=x)&=\frac{f_{X,Y}(x,y)}{f_{X}(x)}\\
&=\frac{\frac{1}{\pi}ye^{-\|x\|^2y}}{\frac{1}{\pi\|x\|^4}\left[1-(1+\|x\|^2)e^{-\|x\|^2}\right]}\\
&=\frac{e^{\|x\|^2(1-y)}y\|x\|^4}{e^{\|x\|^2}-\|x\|^2-1} \qquad \blacksquare
\end{align}
\]</span></p>
<p>ក្នុងចម្លើយចុងក្រោយនៃដង់សុីតេមានលក្ខខណ្ឌនេះ <span class="math inline">\(x\)</span> ត្រូវបានចាត់ទុកជាចំនួនថេរហើយអញ្ញាតរបស់វាគឺ <span class="math inline">\(y\in[0,1]\)</span> ។</p>
<p>ឥលូវនេះយើងអាចនិយាយពីចំណុចសំខាន់នៃការសិក្សាផ្នែកទ្រឹស្តីនៃតម្រែតម្រង់តម្លៃគឺ សង្ឃឹមគណិតមានលក្ខខណ្ឌ។</p>
<hr>
<blockquote class="blockquote">
<p><strong>និយមន័យ.៥.</strong> បើ <span class="math inline">\((X,Y)\)</span> ជាវុិចទ័រចៃដន្យរួមនៃវុិចទ័រចៃដន្យ <span class="math inline">\(X\)</span> និង <span class="math inline">\(Y\)</span> កំណត់លើ <span class="math inline">\(\mathbb{R}^{d_1}\)</span> និង <span class="math inline">\(\mathbb{R}^{d_2}\)</span> រាងគ្នា ហើយមានដង់សុីតេមានលក្ខខណ្ឌ <span class="math inline">\(f_{Y|X}\)</span> នៃ <span class="math inline">\(Y|X\)</span> នោះសង្ឃឹមគណិតមានលក្ខខណ្ឌនៃ <span class="math inline">\(Y\)</span> ដោយដឹងថា <span class="math inline">\(X=x\in\mathbb{R}^{d_1}\)</span> កំណត់ដោយ៖ <span class="math display">\[\mathbb{E}(Y|X=x)=\int_{\mathbb{R}^{d_2}}yf_{Y|X}(y|X=x)dy\]</span> ដែលជាអនុគមន៍នៃ <span class="math inline">\(x\)</span> ។</p>
</blockquote>
<p><strong>ឧទាហរណ៍.៤.</strong> ចំពោះ <em>ឧទាហរណ៍.៣.</em> ខាងលើ យើងអាចគណនា <span class="math inline">\(\mathbb{E}(Y|X=x)\)</span> ដោយ៖</p>
<p><span class="math display">\[
\begin{align}
\mathbb{E}(Y|X=x)&=\int_0^1yf_{Y|X}(y|X=x)dy\\
&=\frac{\|x\|^4e^{\|x\|^2}}{e^{\|x\|^2}-\|x\|^2-1}\int_0^1y^2e^{-\|x\|^2y}dy\\
&=\frac{\|x\|^4e^{\|x\|^2}}{e^{\|x\|^2}-\|x\|^2-1}\left[-e^{-\|x\|^2y}\left(\frac{y^2}{\|x\|^2}+\frac{2y}{\|x\|^4}+\frac{2}{\|x\|^6}\right)\right]_0^1\\
&=\frac{\|x\|^4e^{\|x\|^2}}{e^{\|x\|^2}-\|x\|^2-1}\left[\frac{2}{\|x\|^6}-e^{-\|x\|^2}\left(\frac{1}{\|x\|^2}+\frac{2}{\|x\|^4}+\frac{2}{\|x\|^6}\right)\right]\\
&=\frac{2(e^{\|x\|^2}-\|x\|^2-1)-\|x\|^4}{(e^{\|x\|^2}-\|x\|^2-1)\|x\|^2}\qquad \blacksquare
\end{align}
\]</span></p>
<p>ប្រសិនបើយើងឧបមាថាទិន្នន័យ <span class="math inline">\((X,Y)\in \Omega\)</span> មានរបាយកំណត់ ក្នុង<em>ឧទាហរណ៍.៣.</em> ខាងលើ នោះយើងនឹងអាចគណនារូបមន្តសម្រាប់ព្យាករណ៍ <span class="math inline">\(Y\)</span> ដែលល្អជាងគេសម្រាប់រង្វាស់កំហុស MSE កំណត់ដោយ៖ <span class="math display">\[\hat{y}=\frac{2(e^{\|x\|^2}-\|x\|^2-1)-\|x\|^4}{(e^{\|x\|^2}-\|x\|^2-1)\|x\|^2}\text{ ។}\]</span></p>
<div id="37a3bb87" class="cell" data-execution_count="3">
<details>
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="co"># របាយដោយផ្នែកធៀប X</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> f_Y_known_X(x):</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a> norm_x <span class="op">=</span> np.dot(x,x)</span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> (np.exp(norm_x<span class="op">*</span>(<span class="dv">1</span><span class="op">-</span>y))<span class="op">*</span>y<span class="op">*</span>norm_x <span class="op">**</span> <span class="dv">2</span>)<span class="op">/</span> (np.exp(norm_x)<span class="op">-</span>norm_x<span class="op">-</span><span class="dv">1</span>)</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> eta(x):</span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> (<span class="dv">2</span><span class="op">*</span>(np.exp(x)<span class="op">-</span>x<span class="op">-</span><span class="dv">1</span>)<span class="op">-</span>x<span class="op">**</span><span class="dv">2</span>) <span class="op">/</span> ((np.exp(x)<span class="op">-</span>x<span class="op">-</span><span class="dv">1</span>)<span class="op">*</span>x)</span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> np.linspace(<span class="fl">0.001</span>, <span class="dv">100</span>, <span class="dv">200</span>)</span>
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a>y <span class="op">=</span> eta(x)</span>
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a>fig <span class="op">=</span> go.Figure([go.Scatter(x <span class="op">=</span> x,</span>
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a> y <span class="op">=</span> y,</span>
<span id="cb3-13"><a href="#cb3-13" aria-hidden="true" tabindex="-1"></a> mode <span class="op">=</span> <span class="st">'lines'</span>,</span>
<span id="cb3-14"><a href="#cb3-14" aria-hidden="true" tabindex="-1"></a> name <span class="op">=</span> <span class="st">"បរមាម៉ូដែល"</span>,</span>
<span id="cb3-15"><a href="#cb3-15" aria-hidden="true" tabindex="-1"></a> showlegend <span class="op">=</span> <span class="va">True</span>)])</span>
<span id="cb3-16"><a href="#cb3-16" aria-hidden="true" tabindex="-1"></a>fig.update_layout(title <span class="op">=</span> <span class="st">"បរមាម៉ូដែលជាអនុគន៍នៃការ៉េរបស់ណម x"</span>,</span>
<span id="cb3-17"><a href="#cb3-17" aria-hidden="true" tabindex="-1"></a> width <span class="op">=</span> <span class="dv">600</span>, height <span class="op">=</span> <span class="dv">400</span>,</span>
<span id="cb3-18"><a href="#cb3-18" aria-hidden="true" tabindex="-1"></a> xaxis_title<span class="op">=</span> <span class="st">"||x||^2"</span>,</span>
<span id="cb3-19"><a href="#cb3-19" aria-hidden="true" tabindex="-1"></a> yaxis_title<span class="op">=</span><span class="st">"តម្លៃ output"</span>)</span>
<span id="cb3-20"><a href="#cb3-20" aria-hidden="true" tabindex="-1"></a>fig.show()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-display">
<div> <div id="89ed12f3-91de-4742-a08d-66a730b29d2d" class="plotly-graph-div" style="height:400px; width:600px;"></div> <script type="text/javascript"> require(["plotly"], function(Plotly) { window.PLOTLYENV=window.PLOTLYENV || {}; if (document.getElementById("89ed12f3-91de-4742-a08d-66a730b29d2d")) { Plotly.newPlot( "89ed12f3-91de-4742-a08d-66a730b29d2d", [{"mode":"lines","name":"\u1794\u179a\u1798\u17b6\u1798\u17c9\u17bc\u178a\u17c2\u179b","showlegend":true,"x":[0.001,0.5035075376884421,1.0060150753768842,1.5085226130653262,2.0110301507537685,2.5135376884422107,3.0160452261306525,3.5185527638190948,4.0210603015075375,4.523567839195979,5.026075376884422,5.528582914572864,6.031090452261306,6.533597989949748,7.03610552763819,7.538613065326633,8.041120603015074,8.543628140703516,9.046135678391957,9.5486432160804,10.051150753768843,10.553658291457284,11.056165829145726,11.558673366834169,12.06118090452261,12.563688442211053,13.066195979899495,13.568703517587938,14.071211055276379,14.573718592964822,15.076226130653264,15.578733668341705,16.08124120603015,16.58374874371859,17.086256281407035,17.588763819095476,18.091271356783917,18.59377889447236,19.096286432160802,19.598793969849243,20.101301507537688,20.60380904522613,21.10631658291457,21.608824120603014,22.111331658291455,22.613839195979896,23.11634673366834,23.61885427135678,24.121361809045222,24.623869346733667,25.126376884422108,25.628884422110552,26.131391959798993,26.633899497487434,27.13640703517588,27.63891457286432,28.14142211055276,28.643929648241205,29.146437185929646,29.648944723618087,30.15145226130653,30.653959798994972,31.156467336683413,31.658974874371857,32.161482412060295,32.66398994974874,33.16649748743718,33.66900502512562,34.171512562814065,34.6740201005025,35.17652763819095,35.67903517587939,36.18154271356783,36.684050251256274,37.18655778894472,37.689065326633155,38.1915728643216,38.694080402010044,39.19658793969848,39.699095477386926,40.20160301507537,40.70411055276381,41.20661809045225,41.7091256281407,42.211633165829134,42.71414070351758,43.21664824120602,43.71915577889446,44.221663316582905,44.72417085427135,45.22667839195979,45.72918592964823,46.231693467336676,46.73420100502511,47.23670854271356,47.739216080402,48.24172361809044,48.744231155778884,49.24673869346733,49.74924623115577,50.25175376884421,50.754261306532655,51.2567688442211,51.75927638190954,52.26178391959798,52.764291457286426,53.26679899497486,53.76930653266331,54.27181407035175,54.77432160804019,55.276829145728634,55.77933668341708,56.281844221105516,56.78435175879396,57.286859296482405,57.78936683417084,58.291874371859286,58.79438190954773,59.29688944723617,59.79939698492461,60.30190452261306,60.804412060301495,61.30691959798994,61.80942713567838,62.31193467336682,62.814442211055265,63.31694974874371,63.81945728643215,64.3219648241206,64.82447236180904,65.32697989949749,65.82948743718593,66.33199497487436,66.8345025125628,67.33701005025125,67.8395175879397,68.34202512562814,68.84453266331658,69.34704020100502,69.84954773869346,70.3520552763819,70.85456281407035,71.3570703517588,71.85957788944724,72.36208542713567,72.86459296482411,73.36710050251256,73.869608040201,74.37211557788945,74.87462311557789,75.37713065326632,75.87963819095476,76.38214572864321,76.88465326633165,77.3871608040201,77.88966834170854,78.39217587939697,78.89468341708542,79.39719095477386,79.8996984924623,80.40220603015075,80.9047135678392,81.40722110552763,81.90972864321607,82.41223618090451,82.91474371859296,83.4172512562814,83.91975879396985,84.42226633165828,84.92477386934672,85.42728140703517,85.92978894472361,86.43229648241206,86.9348040201005,87.43731155778893,87.93981909547738,88.44232663316582,88.94483417085426,89.44734170854271,89.94984924623115,90.45235678391958,90.95486432160803,91.45737185929647,91.95987939698492,92.46238693467336,92.9648944723618,93.46740201005024,93.96990954773868,94.47241708542713,94.97492462311557,95.47743216080401,95.97993969849246,96.48244723618089,96.98495477386933,97.48746231155778,97.98996984924622,98.49247738693467,98.99498492462311,99.49749246231156,100.0],"y":[0.666611719253268,0.637799180883805,0.6074173050352899,0.5758608423978682,0.5436094800581267,0.5111935293603085,0.479151656322233,0.4479873204797668,0.4181315712994545,0.38991796812719587,0.3635720671514635,0.33921447279882716,0.3168740034107847,0.2965065492370281,0.2780155266759344,0.2612709182873122,0.24612517253545052,0.23242533872694393,0.22002156896944378,0.2087725302757865,0.19854841705448262,0.18923223580597692,0.18071993262877867,0.17291980642403548,0.16575152847356486,0.15914498681046865,0.15303909524500137,0.14738065026454542,0.14212328030042495,0.13722650668456957,0.13265492013938177,0.12837746777499504,0.1243668410191517,0.12059895305637347,0.11705249410532026,0.11370855349685943,0.11055029857701103,0.107562701674703,0.10473230758753531,0.10204703516710918,0.09949600759186246,0.09706940678921673,0.0947583482142357,0.09255477282020938,0.09045135358180062,0.08844141436823606,0.08651885932624177,0.08467811123230579,0.08291405752203883,0.08122200290992547,0.07959762768315903,0.07803695089477843,0.07653629779912417,0.07509227097090922,0.07370172463141143,0.07236174177424759,0.07106961374118524,0.06982282194736641,0.06861902149669398,0.06745602646322732,0.0663317966442861,0.06524442561642069,0.06419212994717911,0.06317323943426738,0.06218618825974484,0.061229506960720485,0.06030181512995825,0.05940181477013613,0.058528284234474974,0.05768007269425229,0.05685609508051494,0.05605532745324015,0.05527680275638884,0.054519606921848845,0.05378287528926613,0.053065789312281765,0.05236757352479593,0.05168749274362086,0.05102484948630889,0.050378981585089626,0.04974925997975767,0.049135086674045904,0.04853589284152892,0.04795113706844577,0.04738030372203237,0.04682290143402782,0.046278461689980115,0.04574653751583889,0.04522670225409657,0.04471854842243523,0.04422168664846171,0.0437357446746786,0.043260366428346984,0.04279521115135678,0.04233995258563602,0.04189427821000698,0.04145788852473772,0.04103049638034735,0.040611826347503974,0.04020161412510985,0.03979960598390077,0.03940555824309824,0.039019236777846335,0.038640416555340854,0.03826888119771984,0.03790442256993128,0.03754684039092865,0.03719594186666807,0.03685154134349424,0.03651345998060567,0.03618152544038508,0.035855571595468434,0.03553543825150645,0.03522097088464637,0.03491202039283041,0.03460844286006955,0.03431009933290995,0.03401685560836241,0.03372858203261487,0.03344515330989372,0.03316644832088189,0.03289234995014083,0.03262274492201976,0.032357523644569355,0.032096580061007705,0.03183981150831588,0.03158711858256695,0.03133840501061727,0.031093577527812152,0.030852545761379593,0.030615222119205678,0.030381521683704234,0.030151362110510505,0.029924663531744885,0.0297013484636082,0.029481341718083708,0.029264570318534553,0.029050963418997666,0.028840452226986536,0.028632969929626204,0.028428451622953876,0.028226834244227932,0.02802805650709712,0.02783205883948986,0.027638783324091473,0.027448173641284376,0.02726017501443317,0.027074734157402983,0.026891799224205377,0.026711319760671946,0.02653324665806092,0.026357532108507206,0.02618412956223096,0.026012993686424214,0.02584408032573931,0.025677346464306812,0.025512750189214225,0.025350250655380414,0.02518980805176392,0.02503138356884637,0.02487493936733529,0.02472043854803324,0.02456784512282287,0.02441712398672003,0.02426824089094932,0.02412116241699866,0.023975855951611662,0.02383228966267849,0.023690432475987695,0.02355025405280346,0.02341172476823422,0.023274815690360277,0.02313949856008947,0.023005745771711557,0.022873530354123057,0.022742825952695837,0.02261360681176383,0.022485847757703355,0.022359524182583832,0.022234612028366452,0.022111087771629578,0.02198892840880048,0.021868111441873932,0.02174861486459903,0.021630417149116458,0.02151349723302912,0.02139783450688986,0.021283408802090613,0.02117020037913808,0.021058189916301633,0.02094735849861966,0.020837687607251264,0.020729159109160743,0.020621755247122724,0.020515458630036437,0.020410252223538008,0.0203061193409001,0.02020304363420877,0.02010100908580762,0.02],"type":"scatter"}], {"template":{"data":{"histogram2dcontour":[{"type":"histogram2dcontour","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"choropleth":[{"type":"choropleth","colorbar":{"outlinewidth":0,"ticks":""}}],"histogram2d":[{"type":"histogram2d","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"heatmap":[{"type":"heatmap","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"heatmapgl":[{"type":"heatmapgl","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"contourcarpet":[{"type":"contourcarpet","colorbar":{"outlinewidth":0,"ticks":""}}],"contour":[{"type":"contour","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"surface":[{"type":"surface","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"mesh3d":[{"type":"mesh3d","colorbar":{"outlinewidth":0,"ticks":""}}],"scatter":[{"fillpattern":{"fillmode":"overlay","size":10,"solidity":0.2},"type":"scatter"}],"parcoords":[{"type":"parcoords","line":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterpolargl":[{"type":"scatterpolargl","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"bar":[{"error_x":{"color":"#2a3f5f"},"error_y":{"color":"#2a3f5f"},"marker":{"line":{"color":"#E5ECF6","width":0.5},"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"bar"}],"scattergeo":[{"type":"scattergeo","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterpolar":[{"type":"scatterpolar","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"histogram":[{"marker":{"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"histogram"}],"scattergl":[{"type":"scattergl","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatter3d":[{"type":"scatter3d","line":{"colorbar":{"outlinewidth":0,"ticks":""}},"marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scattermapbox":[{"type":"scattermapbox","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterternary":[{"type":"scatterternary","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scattercarpet":[{"type":"scattercarpet","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"carpet":[{"aaxis":{"endlinecolor":"#2a3f5f","gridcolor":"white","linecolor":"white","minorgridcolor":"white","startlinecolor":"#2a3f5f"},"baxis":{"endlinecolor":"#2a3f5f","gridcolor":"white","linecolor":"white","minorgridcolor":"white","startlinecolor":"#2a3f5f"},"type":"carpet"}],"table":[{"cells":{"fill":{"color":"#EBF0F8"},"line":{"color":"white"}},"header":{"fill":{"color":"#C8D4E3"},"line":{"color":"white"}},"type":"table"}],"barpolar":[{"marker":{"line":{"color":"#E5ECF6","width":0.5},"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"barpolar"}],"pie":[{"automargin":true,"type":"pie"}]},"layout":{"autotypenumbers":"strict","colorway":["#636efa","#EF553B","#00cc96","#ab63fa","#FFA15A","#19d3f3","#FF6692","#B6E880","#FF97FF","#FECB52"],"font":{"color":"#2a3f5f"},"hovermode":"closest","hoverlabel":{"align":"left"},"paper_bgcolor":"white","plot_bgcolor":"#E5ECF6","polar":{"bgcolor":"#E5ECF6","angularaxis":{"gridcolor":"white","linecolor":"white","ticks":""},"radialaxis":{"gridcolor":"white","linecolor":"white","ticks":""}},"ternary":{"bgcolor":"#E5ECF6","aaxis":{"gridcolor":"white","linecolor":"white","ticks":""},"baxis":{"gridcolor":"white","linecolor":"white","ticks":""},"caxis":{"gridcolor":"white","linecolor":"white","ticks":""}},"coloraxis":{"colorbar":{"outlinewidth":0,"ticks":""}},"colorscale":{"sequential":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]],"sequentialminus":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]],"diverging":[[0,"#8e0152"],[0.1,"#c51b7d"],[0.2,"#de77ae"],[0.3,"#f1b6da"],[0.4,"#fde0ef"],[0.5,"#f7f7f7"],[0.6,"#e6f5d0"],[0.7,"#b8e186"],[0.8,"#7fbc41"],[0.9,"#4d9221"],[1,"#276419"]]},"xaxis":{"gridcolor":"white","linecolor":"white","ticks":"","title":{"standoff":15},"zerolinecolor":"white","automargin":true,"zerolinewidth":2},"yaxis":{"gridcolor":"white","linecolor":"white","ticks":"","title":{"standoff":15},"zerolinecolor":"white","automargin":true,"zerolinewidth":2},"scene":{"xaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2},"yaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2},"zaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2}},"shapedefaults":{"line":{"color":"#2a3f5f"}},"annotationdefaults":{"arrowcolor":"#2a3f5f","arrowhead":0,"arrowwidth":1},"geo":{"bgcolor":"white","landcolor":"#E5ECF6","subunitcolor":"white","showland":true,"showlakes":true,"lakecolor":"white"},"title":{"x":0.05},"mapbox":{"style":"light"}}},"title":{"text":"\u1794\u179a\u1798\u17b6\u1798\u17c9\u17bc\u178a\u17c2\u179b\u1787\u17b6\u17a2\u1793\u17bb\u1782\u1793\u17cd\u1793\u17c3\u1780\u17b6\u179a\u17c9\u17c1\u179a\u1794\u179f\u17cb\u178e\u1798 x"},"width":600,"height":400,"xaxis":{"title":{"text":"||x||^2"}},"yaxis":{"title":{"text":"\u178f\u1798\u17d2\u179b\u17c3 output"}}}, {"responsive": true} ).then(function(){
var gd = document.getElementById('89ed12f3-91de-4742-a08d-66a730b29d2d');
var x = new MutationObserver(function (mutations, observer) {{
var display = window.getComputedStyle(gd).display;
if (!display || display === 'none') {{
console.log([gd, 'removed!']);
Plotly.purge(gd);
observer.disconnect();
}}
}});
// Listen for the removal of the full notebook cells
var notebookContainer = gd.closest('#notebook-container');
if (notebookContainer) {{
x.observe(notebookContainer, {childList: true});
}}
// Listen for the clearing of the current output cell
var outputEl = gd.closest('.output');
if (outputEl) {{
x.observe(outputEl, {childList: true});
}}
}) }; }); </script> </div>
<p>រូបទី៤៖ បរមាម៉ូដែលក្នុង ឧទាហរណ៍.៤. ជាអនុគន៍នៃ <span class="math inline">\(x\)</span> ។</p>
</div>
</div>
<p>ជាពិសេសដោយប្រើពន្លាតជាស៊េរីពិតនៃអនុគមន៍អ៊ិស៉្បូណង់ស្យែល <span class="math inline">\(e^{t}=1+t+t^2/2+...=\sum_{k=0}^{+\infty}\frac{t^k}{k!}\)</span> នោះកន្សោមនៃការព្យាករណ៍ខាងលើសមមូលនឹង <span class="math inline">\(\frac{\|x\|^6/3+o(\|x\|^6)}{\|x\|^6/2+o(\|x\|^6)}\)</span> ពេល <span class="math inline">\(\|x\|\to 0\)</span> ដូច្នេះ យើងអាចព្យាករណ៍តម្លៃ <span class="math inline">\(Y\)</span> ត្រង់ <span class="math inline">\(X=(0,0)\)</span> តាមបន្លាយភាពជាប់គឺ <span class="math inline">\(\hat{y}(0,0)=\mathbb{E}(Y|X=(0,0))=2/3\)</span> ។</p>
<p>ឥលូវយើងសាកស្រង់ទិន្នន័យចេញពីរបាយប្រូបាបប៊ីលីតេខាងលើនិង ផ្ទៀតផ្ទាត់មើលថាតើម៉ូដែលដែលយើងបានគណនាពិតជាមានសមត្ថភាពក្នុងការព្យាករណ៍បានល្អឬ យ៉ាងណា?</p>
<div id="29cce1fa" class="cell" data-execution_count="4">
<details>
<summary>Code</summary>
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> f_nX_Y(x,y):</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> y<span class="op">*</span>np.exp(<span class="op">-</span>x<span class="op">*</span>y)</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> d_cauchy(x):</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> <span class="dv">1</span><span class="op">/</span>(<span class="dv">1</span><span class="op">+</span>x)</span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a><span class="co"># បង្កើតទិន្នន័យតាម វិធីសាស្រ Rejection sampling</span></span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a>res <span class="op">=</span> []</span>
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a><span class="cf">while</span> <span class="bu">len</span>(res) <span class="op"><</span> <span class="dv">100</span>:</span>
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a> n_x <span class="op">=</span> np.tan(np.pi<span class="op">*</span>(np.random.uniform(<span class="dv">0</span>,<span class="dv">1</span>) <span class="op">-</span> <span class="dv">1</span><span class="op">/</span><span class="dv">2</span>)) <span class="op">**</span> <span class="dv">2</span></span>
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a> n_y <span class="op">=</span> np.random.uniform(<span class="dv">0</span>,<span class="dv">1</span>)</span>
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> np.random.uniform(<span class="dv">0</span>,<span class="dv">1</span>) <span class="op"><=</span> f_nX_Y(n_x, n_y) <span class="op">/</span> d_cauchy(n_x):</span>
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> <span class="bu">len</span>(res) <span class="op"><</span> <span class="dv">80</span>:</span>
<span id="cb4-14"><a href="#cb4-14" aria-hidden="true" tabindex="-1"></a> res.append([n_x, n_y])</span>
<span id="cb4-15"><a href="#cb4-15" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> <span class="bu">len</span>(res) <span class="op">>=</span> <span class="dv">80</span>:</span>
<span id="cb4-16"><a href="#cb4-16" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> n_x <span class="op">>=</span> <span class="dv">10</span>:</span>
<span id="cb4-17"><a href="#cb4-17" aria-hidden="true" tabindex="-1"></a> res.append([n_x, n_y])</span>
<span id="cb4-18"><a href="#cb4-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb4-19"><a href="#cb4-19" aria-hidden="true" tabindex="-1"></a>res <span class="op">=</span> np.array(res)</span>
<span id="cb4-20"><a href="#cb4-20" aria-hidden="true" tabindex="-1"></a>y_mod <span class="op">=</span> eta(res[:,<span class="dv">0</span>])</span>
<span id="cb4-21"><a href="#cb4-21" aria-hidden="true" tabindex="-1"></a>fig <span class="op">=</span> go.Figure([go.Scatter(x <span class="op">=</span> res[:,<span class="dv">0</span>],</span>
<span id="cb4-22"><a href="#cb4-22" aria-hidden="true" tabindex="-1"></a> y <span class="op">=</span> y_mod,</span>
<span id="cb4-23"><a href="#cb4-23" aria-hidden="true" tabindex="-1"></a> mode <span class="op">=</span> <span class="st">'markers'</span>,</span>
<span id="cb4-24"><a href="#cb4-24" aria-hidden="true" tabindex="-1"></a> name <span class="op">=</span> <span class="st">"បរមាម៉ូដែល"</span>,</span>
<span id="cb4-25"><a href="#cb4-25" aria-hidden="true" tabindex="-1"></a> showlegend <span class="op">=</span> <span class="va">True</span>)])</span>
<span id="cb4-26"><a href="#cb4-26" aria-hidden="true" tabindex="-1"></a>fig.add_trace(go.Scatter(x <span class="op">=</span> res[:,<span class="dv">0</span>],</span>
<span id="cb4-27"><a href="#cb4-27" aria-hidden="true" tabindex="-1"></a> y <span class="op">=</span> res[:,<span class="dv">1</span>],</span>
<span id="cb4-28"><a href="#cb4-28" aria-hidden="true" tabindex="-1"></a> mode <span class="op">=</span> <span class="st">'markers'</span>,</span>
<span id="cb4-29"><a href="#cb4-29" aria-hidden="true" tabindex="-1"></a> name <span class="op">=</span> <span class="st">"ទិន្នន័យដែលស្រង់ពីរបាយប្រូបាប"</span>,</span>
<span id="cb4-30"><a href="#cb4-30" aria-hidden="true" tabindex="-1"></a> showlegend <span class="op">=</span> <span class="va">True</span>,</span>
<span id="cb4-31"><a href="#cb4-31" aria-hidden="true" tabindex="-1"></a> marker <span class="op">=</span> <span class="bu">dict</span>(color <span class="op">=</span> <span class="st">"red"</span>)))</span>
<span id="cb4-32"><a href="#cb4-32" aria-hidden="true" tabindex="-1"></a>fig.update_layout(title <span class="op">=</span> <span class="st">"បរមាម៉ូដែលនិង ទិន្នន័យស្រង់ពីរបាយប្រូបាបជាអនុគន៍នៃការ៉េរបស់ណម x"</span>,</span>
<span id="cb4-33"><a href="#cb4-33" aria-hidden="true" tabindex="-1"></a> width <span class="op">=</span> <span class="dv">650</span>, height <span class="op">=</span> <span class="dv">400</span>,</span>
<span id="cb4-34"><a href="#cb4-34" aria-hidden="true" tabindex="-1"></a> xaxis_title<span class="op">=</span><span class="st">'||x||^2'</span>,</span>
<span id="cb4-35"><a href="#cb4-35" aria-hidden="true" tabindex="-1"></a> yaxis_title<span class="op">=</span><span class="st">"តម្លៃ ouput"</span>)</span>
<span id="cb4-36"><a href="#cb4-36" aria-hidden="true" tabindex="-1"></a>mse <span class="op">=</span> np.mean((res[:,<span class="dv">1</span>] <span class="op">-</span> y_mod <span class="op">**</span> <span class="dv">2</span>))</span>
<span id="cb4-37"><a href="#cb4-37" aria-hidden="true" tabindex="-1"></a>fig.add_annotation(</span>
<span id="cb4-38"><a href="#cb4-38" aria-hidden="true" tabindex="-1"></a> x<span class="op">=</span>np.<span class="bu">max</span>(res[:,<span class="dv">0</span>])<span class="op">/</span><span class="dv">2</span>, </span>
<span id="cb4-39"><a href="#cb4-39" aria-hidden="true" tabindex="-1"></a> y<span class="op">=</span>np.mean(y_mod),</span>
<span id="cb4-40"><a href="#cb4-40" aria-hidden="true" tabindex="-1"></a> text<span class="op">=</span><span class="st">"MSE = </span><span class="sc">{}</span><span class="st">"</span>.<span class="bu">format</span>(np.<span class="bu">round</span>(mse, <span class="dv">3</span>)),</span>
<span id="cb4-41"><a href="#cb4-41" aria-hidden="true" tabindex="-1"></a> showarrow<span class="op">=</span><span class="va">False</span>,</span>
<span id="cb4-42"><a href="#cb4-42" aria-hidden="true" tabindex="-1"></a> arrowhead<span class="op">=</span><span class="dv">1</span>)</span>
<span id="cb4-43"><a href="#cb4-43" aria-hidden="true" tabindex="-1"></a>fig.show()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</details>
<div class="cell-output cell-output-display">
<div> <div id="876335fa-c159-40c7-a860-9a8e35cdd39e" class="plotly-graph-div" style="height:400px; width:650px;"></div> <script type="text/javascript"> require(["plotly"], function(Plotly) { window.PLOTLYENV=window.PLOTLYENV || {}; if (document.getElementById("876335fa-c159-40c7-a860-9a8e35cdd39e")) { Plotly.newPlot( "876335fa-c159-40c7-a860-9a8e35cdd39e", [{"mode":"markers","name":"\u1794\u179a\u1798\u17b6\u1798\u17c9\u17bc\u178a\u17c2\u179b","showlegend":true,"x":[2.4040880138795484,8.459973674770449,3.9239657912979453,2.0664838526386373,3.3451613951838355,3.693496586863031,2.214704815978943,0.16156619005268616,2.9091032066134064,0.6351758184301962,0.5389936763648502,0.42062090824828785,2.0881781613489157,0.5020455388199059,0.3508061880504504,1.1301469488289049,0.5423976067898201,0.29488952682848857,8.325004073661205,0.5372860799906032,0.13405021689240765,0.2186587602100002,0.08423081043435408,1.026777462846671,15.672850578381473,0.12406441501658717,0.5327194886554625,0.665612855844851,0.043656066796130974,0.01029239071270185,0.2178305342842938,0.028409673461199186,2.4140720339346897,2.104615475360733,0.18992117755084595,0.7928143490759075,2.439473900258623e-05,0.10653599414215513,0.17766896708276508,0.05408199511658487,0.4317415071313732,0.057902155565617015,0.23596193224789938,0.034816565082906696,0.59879494624622,0.2883531430176155,0.1291893659384791,1.3553692068003296,0.24591412157997042,0.058983992971682414,0.014441682714857037,0.27981686154548013,0.013269838300462059,0.15443570567966988,0.19909968839339323,1.3893717030228485,1.0021673987355888,0.06374872320933438,1.843462293430947,0.6686744754427955,1.35181110770706,4.229111435128576,3.400114111848467,0.01021365022205415,2.5426508919365456,0.015343427416546257,0.13010461418799596,1.349344478151989,0.011763716668563563,1.786115696438613,0.42181116254535017,0.3953217659610606,1.4769382599552847,5.341285591894664,0.26180406427654557,3.857983780848097,0.05328452117370537,2.0853345208615868,0.05375163782930177,2.6600465755828497,24.91023469202897,10.153239795337406,71.83297135171267,26.57096734938632,10.170046082844886,13.8093814661234,12.916817039480438,13.85918318617565,35.152977125910766,13.658044369671176,13.525218952877,118.17400383518171,33.403115622852,11.351578136703898,47.9323081300085,14.864845930993948,17.31368996957698,32.25213433584577,20.87187158052439,11.76940167647486],"y":[0.51824045824901,0.23461212513031265,0.4237811562378951,0.5400304537874535,0.4586127598883494,0.43742557484376365,0.5304598854565399,0.6575954388876516,0.48591336921750355,0.6299758945967291,0.6357011191888757,0.6426688403322309,0.5386298971151,0.6378854512159524,0.6467360758123072,0.5997139812572029,0.6354994578657435,0.6499703124224379,0.23821773773542917,0.6358022566080457,0.6591536460400805,0.6543452723502239,0.6619610881983234,0.6061336798737097,0.12760675850865125,0.659717798637463,0.6360726381393714,0.628152684774215,0.6642342968563032,0.6660944746907793,0.654392584774667,0.665085369291334,0.5175970401505054,0.5375686218103655,0.6559841073620846,0.6204761371525301,0.6436236414738528,0.6607063462659089,0.656681060659215,0.6636513280061568,0.642018045879757,0.6634375237302618,0.6533557383660954,0.6647279365943227,0.6321479939744002,0.6503470020842885,0.6594283512169445,0.5855756305458647,0.652785653002819,0.663376956795436,0.6658635795574798,0.6508385076276455,0.6659288018100787,0.6579997491387942,0.6554613070085389,0.5834251399971399,0.6076549652513962,0.6631101006260418,0.5544092147591482,0.6279689923904369,0.585800439597218,0.4062358404937684,0.45522927303406946,0.6660988555249541,0.5093218575664016,0.665813383284109,0.6593766399659271,0.5859562618328149,0.6660126145914818,0.5580973613773695,0.6425992230211015,0.6441463972180823,0.5778700848849109,0.3480571082750455,0.6518740233674911,0.42765452690569977,0.663695947072542,0.5388134884773785,0.6636698122318928,0.5017894935055294,0.08028828372351221,0.1965858227429833,0.027842367681095725,0.07527012365277161,0.1962662575686983,0.14481518456343376,0.15480517725441048,0.14429538173577455,0.056894185457912735,0.14641786091978287,0.14785383188024206,0.01692419597451752,0.05987465428608914,0.17605354523410682,0.04172550995406541,0.13454042066264685,0.11551500199725957,0.062011399901894755,0.0958227256624419,0.16984109534895453],"type":"scatter"},{"marker":{"color":"red"},"mode":"markers","name":"\u1791\u17b7\u1793\u17d2\u1793\u1793\u17d0\u1799\u178a\u17c2\u179b\u179f\u17d2\u179a\u1784\u17cb\u1796\u17b8\u179a\u1794\u17b6\u1799\u1794\u17d2\u179a\u17bc\u1794\u17b6\u1794","showlegend":true,"x":[2.4040880138795484,8.459973674770449,3.9239657912979453,2.0664838526386373,3.3451613951838355,3.693496586863031,2.214704815978943,0.16156619005268616,2.9091032066134064,0.6351758184301962,0.5389936763648502,0.42062090824828785,2.0881781613489157,0.5020455388199059,0.3508061880504504,1.1301469488289049,0.5423976067898201,0.29488952682848857,8.325004073661205,0.5372860799906032,0.13405021689240765,0.2186587602100002,0.08423081043435408,1.026777462846671,15.672850578381473,0.12406441501658717,0.5327194886554625,0.665612855844851,0.043656066796130974,0.01029239071270185,0.2178305342842938,0.028409673461199186,2.4140720339346897,2.104615475360733,0.18992117755084595,0.7928143490759075,2.439473900258623e-05,0.10653599414215513,0.17766896708276508,0.05408199511658487,0.4317415071313732,0.057902155565617015,0.23596193224789938,0.034816565082906696,0.59879494624622,0.2883531430176155,0.1291893659384791,1.3553692068003296,0.24591412157997042,0.058983992971682414,0.014441682714857037,0.27981686154548013,0.013269838300462059,0.15443570567966988,0.19909968839339323,1.3893717030228485,1.0021673987355888,0.06374872320933438,1.843462293430947,0.6686744754427955,1.35181110770706,4.229111435128576,3.400114111848467,0.01021365022205415,2.5426508919365456,0.015343427416546257,0.13010461418799596,1.349344478151989,0.011763716668563563,1.786115696438613,0.42181116254535017,0.3953217659610606,1.4769382599552847,5.341285591894664,0.26180406427654557,3.857983780848097,0.05328452117370537,2.0853345208615868,0.05375163782930177,2.6600465755828497,24.91023469202897,10.153239795337406,71.83297135171267,26.57096734938632,10.170046082844886,13.8093814661234,12.916817039480438,13.85918318617565,35.152977125910766,13.658044369671176,13.525218952877,118.17400383518171,33.403115622852,11.351578136703898,47.9323081300085,14.864845930993948,17.31368996957698,32.25213433584577,20.87187158052439,11.76940167647486],"y":[0.43775790940447346,0.2817440986064663,0.3110416802474819,0.8420048093135132,0.39989299002504364,0.2865563287889644,0.5699687292925433,0.8497162300325373,0.9810147185428861,0.9110767709901216,0.5829671439424474,0.7729919882377324,0.48038468515123467,0.46314932351903293,0.30445038573503524,0.8678610831104205,0.802163406261983,0.9278543277996381,0.27261415006072354,0.3271700717295306,0.4686537128816024,0.7416429376748929,0.4180492396402111,0.22791283100902493,0.06695509956036627,0.4011041747901395,0.86223194820648,0.9632777760431692,0.9440876001029803,0.7225867826463183,0.5145433490412279,0.7425710794774661,0.8611352097610181,0.5262927454689976,0.6445772603918531,0.2832630226017251,0.815676706308566,0.79935489476323,0.09246727159248613,0.868855967404736,0.6046525634053395,0.4118129367353026,0.6574725080360071,0.9860608283326432,0.93309547100342,0.7299767543337805,0.9422124792966333,0.9460477380265362,0.5159782857451698,0.7456195497699771,0.9366201416933735,0.5118592448285434,0.7597663454669733,0.8172906757328554,0.9914863945944594,0.7463045643893275,0.6781476655269286,0.8004168093975221,0.774229380616265,0.16910766800065336,0.9864564022139799,0.6910730316727558,0.32886708741313564,0.7133898997076908,0.49594779002486655,0.396563015154336,0.20642770206675431,0.5225096452066463,0.6140769885924622,0.7866186068480033,0.796592565106617,0.3048171261265402,0.1576407183768771,0.6270498517704332,0.9911963596850787,0.22684467556289112,0.961528351314202,0.910718057599455,0.7983379080630263,0.8679564692974379,0.12277498980308077,0.12598208990571658,0.02153992649143721,0.13784776677467625,0.47207601460427084,0.19818568020248295,0.31420960427393496,0.255896116843374,0.01892841411139623,0.10181002110291204,0.04258779220364117,0.03465987991939612,0.0041283425710421096,0.18763208651235252,0.060667973657022145,0.45923567832443857,0.015143845565283809,0.14397832489629325,0.03723759610216404,0.05457274444805482],"type":"scatter"}], {"template":{"data":{"histogram2dcontour":[{"type":"histogram2dcontour","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"choropleth":[{"type":"choropleth","colorbar":{"outlinewidth":0,"ticks":""}}],"histogram2d":[{"type":"histogram2d","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"heatmap":[{"type":"heatmap","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"heatmapgl":[{"type":"heatmapgl","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"contourcarpet":[{"type":"contourcarpet","colorbar":{"outlinewidth":0,"ticks":""}}],"contour":[{"type":"contour","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"surface":[{"type":"surface","colorbar":{"outlinewidth":0,"ticks":""},"colorscale":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]]}],"mesh3d":[{"type":"mesh3d","colorbar":{"outlinewidth":0,"ticks":""}}],"scatter":[{"fillpattern":{"fillmode":"overlay","size":10,"solidity":0.2},"type":"scatter"}],"parcoords":[{"type":"parcoords","line":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterpolargl":[{"type":"scatterpolargl","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"bar":[{"error_x":{"color":"#2a3f5f"},"error_y":{"color":"#2a3f5f"},"marker":{"line":{"color":"#E5ECF6","width":0.5},"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"bar"}],"scattergeo":[{"type":"scattergeo","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterpolar":[{"type":"scatterpolar","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"histogram":[{"marker":{"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"histogram"}],"scattergl":[{"type":"scattergl","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatter3d":[{"type":"scatter3d","line":{"colorbar":{"outlinewidth":0,"ticks":""}},"marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scattermapbox":[{"type":"scattermapbox","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scatterternary":[{"type":"scatterternary","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"scattercarpet":[{"type":"scattercarpet","marker":{"colorbar":{"outlinewidth":0,"ticks":""}}}],"carpet":[{"aaxis":{"endlinecolor":"#2a3f5f","gridcolor":"white","linecolor":"white","minorgridcolor":"white","startlinecolor":"#2a3f5f"},"baxis":{"endlinecolor":"#2a3f5f","gridcolor":"white","linecolor":"white","minorgridcolor":"white","startlinecolor":"#2a3f5f"},"type":"carpet"}],"table":[{"cells":{"fill":{"color":"#EBF0F8"},"line":{"color":"white"}},"header":{"fill":{"color":"#C8D4E3"},"line":{"color":"white"}},"type":"table"}],"barpolar":[{"marker":{"line":{"color":"#E5ECF6","width":0.5},"pattern":{"fillmode":"overlay","size":10,"solidity":0.2}},"type":"barpolar"}],"pie":[{"automargin":true,"type":"pie"}]},"layout":{"autotypenumbers":"strict","colorway":["#636efa","#EF553B","#00cc96","#ab63fa","#FFA15A","#19d3f3","#FF6692","#B6E880","#FF97FF","#FECB52"],"font":{"color":"#2a3f5f"},"hovermode":"closest","hoverlabel":{"align":"left"},"paper_bgcolor":"white","plot_bgcolor":"#E5ECF6","polar":{"bgcolor":"#E5ECF6","angularaxis":{"gridcolor":"white","linecolor":"white","ticks":""},"radialaxis":{"gridcolor":"white","linecolor":"white","ticks":""}},"ternary":{"bgcolor":"#E5ECF6","aaxis":{"gridcolor":"white","linecolor":"white","ticks":""},"baxis":{"gridcolor":"white","linecolor":"white","ticks":""},"caxis":{"gridcolor":"white","linecolor":"white","ticks":""}},"coloraxis":{"colorbar":{"outlinewidth":0,"ticks":""}},"colorscale":{"sequential":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]],"sequentialminus":[[0.0,"#0d0887"],[0.1111111111111111,"#46039f"],[0.2222222222222222,"#7201a8"],[0.3333333333333333,"#9c179e"],[0.4444444444444444,"#bd3786"],[0.5555555555555556,"#d8576b"],[0.6666666666666666,"#ed7953"],[0.7777777777777778,"#fb9f3a"],[0.8888888888888888,"#fdca26"],[1.0,"#f0f921"]],"diverging":[[0,"#8e0152"],[0.1,"#c51b7d"],[0.2,"#de77ae"],[0.3,"#f1b6da"],[0.4,"#fde0ef"],[0.5,"#f7f7f7"],[0.6,"#e6f5d0"],[0.7,"#b8e186"],[0.8,"#7fbc41"],[0.9,"#4d9221"],[1,"#276419"]]},"xaxis":{"gridcolor":"white","linecolor":"white","ticks":"","title":{"standoff":15},"zerolinecolor":"white","automargin":true,"zerolinewidth":2},"yaxis":{"gridcolor":"white","linecolor":"white","ticks":"","title":{"standoff":15},"zerolinecolor":"white","automargin":true,"zerolinewidth":2},"scene":{"xaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2},"yaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2},"zaxis":{"backgroundcolor":"#E5ECF6","gridcolor":"white","linecolor":"white","showbackground":true,"ticks":"","zerolinecolor":"white","gridwidth":2}},"shapedefaults":{"line":{"color":"#2a3f5f"}},"annotationdefaults":{"arrowcolor":"#2a3f5f","arrowhead":0,"arrowwidth":1},"geo":{"bgcolor":"white","landcolor":"#E5ECF6","subunitcolor":"white","showland":true,"showlakes":true,"lakecolor":"white"},"title":{"x":0.05},"mapbox":{"style":"light"}}},"title":{"text":"\u1794\u179a\u1798\u17b6\u1798\u17c9\u17bc\u178a\u17c2\u179b\u1793\u17b7\u1784 \u1791\u17b7\u1793\u17d2\u1793\u1793\u17d0\u1799\u179f\u17d2\u179a\u1784\u17cb\u1796\u17b8\u179a\u1794\u17b6\u1799\u1794\u17d2\u179a\u17bc\u1794\u17b6\u1794\u1787\u17b6\u17a2\u1793\u17bb\u1782\u1793\u17cd\u1793\u17c3\u1780\u17b6\u179a\u17c9\u17c1\u179a\u1794\u179f\u17cb\u178e\u1798 x"},"width":650,"height":400,"xaxis":{"title":{"text":"||x||^2"}},"yaxis":{"title":{"text":"\u178f\u1798\u17d2\u179b\u17c3 ouput"}},"annotations":[{"arrowhead":1,"showarrow":false,"text":"MSE = 0.244","x":59.087001917590854,"y":0.4947832766965156}]}, {"responsive": true} ).then(function(){
var gd = document.getElementById('876335fa-c159-40c7-a860-9a8e35cdd39e');
var x = new MutationObserver(function (mutations, observer) {{
var display = window.getComputedStyle(gd).display;
if (!display || display === 'none') {{
console.log([gd, 'removed!']);
Plotly.purge(gd);
observer.disconnect();
}}
}});
// Listen for the removal of the full notebook cells
var notebookContainer = gd.closest('#notebook-container');
if (notebookContainer) {{
x.observe(notebookContainer, {childList: true});
}}
// Listen for the clearing of the current output cell
var outputEl = gd.closest('.output');
if (outputEl) {{
x.observe(outputEl, {childList: true});
}}
}) }; }); </script> </div>
<p>រូបទី៥៖ ទិន្នន័យតាមរបាយប្រូបាប (ដោយប្រើវិធីសាស្រ្ត <a href="https://en.wikipedia.org/wiki/Rejection_sampling">Rejection sampling</a>) និង ការព្យាករណ៍ដោយបរមាម៉ូដែល ។ យើងឃើញថាការព្យារកណ៍កាន់តែសុក្រិតនៅពេលណម <span class="math inline">\(X\)</span> យកតម្លៃកាន់តែធំ នេះក៏ព្រោះតែកាលណាណម <span class="math inline">\(X\)</span> មានតម្លៃធំ ដង់សុីតេរួមមានតម្លៃសឹងតែសូន្យគ្រប់ទីកន្លែងទៅហើយ លើកលែងតែចំពោះ <span class="math inline">\(Y\)</span> នៅក្បែរៗនឹង <span class="math inline">\(0\)</span> ប៉ុណ្ណោះ។ ក្នុងករណីនេះ ទាំងទិន្នន័យដែលបង្កើតតាមរបាយប្រូបាបរួមនិង ម៉ូដែលសុទ្ធប្រើប្រាស់តម្លៃ <span class="math inline">\(Y\)</span> ប្រហាក់ប្រហែលគ្នា ដែលធ្វើឲ្យម៉ូដែលព្យាករណ៍បានល្អ។</p>
</div>
</div>
<hr>
<p>ផ្នែកលក្ខខណ្ឌនៃសង្ឃឹមគណិតមានលក្ខខណ្ឌអាចជាវត្ថុគណិតវិទ្យាផ្សេងក្រៅពី វុិចទ័រចៃដន្យ។ ខាងក្រោមនេះជានិយមនៃសង្ឃឹមគណិតមានលក្ខខណ្ឌធៀបនឹង <a href="https://en.wikipedia.org/wiki/%CE%A3-algebra"><span class="math inline">\(\sigma\)</span> ពីជគណិតមួយ</a> មួយ។</p>
<blockquote class="blockquote">
<p><strong>និយមន័យ.៦.</strong> បើ <span class="math inline">\(\cal F\)</span> ជា <span class="math inline">\(\sigma\)</span> ពីជគណិតរងនៃ <span class="math inline">\(\sigma\)</span> ពីជគណិតបង្ករដោយ <span class="math inline">\({Y}\)</span> តាងដោយ <span class="math inline">\(\sigma(Y)\)</span> នោះគេមានសង្ឃឹមគណិតមានលក្ខខណ្ឌធៀបនឹង (ឬ ដោយដឹងពត៌មានកំណត់ដោយ <span class="math inline">\(\sigma\)</span> ពីជគណិត) <span class="math inline">\(\cal F\)</span> តាងដោយ <span class="math inline">\(\mathbb{E}(Y|{\cal F})\)</span> តែមួយគត់ដែលជាអថេរចៃដន្យ<a href="https://en.wikipedia.org/wiki/Measurable_function">វាស់បាន</a>ធៀបនឹង <span class="math inline">\({\cal F}\)</span> ផ្ទៀតផ្ទាត់៖ <span class="math display">\[\mathbb{E}[\mathbb{1}_{\{A\}}\mathbb{E}(Y|{\cal F})]=\mathbb{E}[\mathbb{1}_{\{A\}}Y]\]</span> ចំពោះគ្រប់ព្រឹត្តិការណ៍ <span class="math inline">\(A\in{\cal F}\)</span> ។ ជាសម្គាល់៖<br> ១. <span class="math inline">\(\mathbb{1}_{\{A\}}\)</span> ជាអនុគមន៍សម្គាល់នៃ <span class="math inline">\(A\)</span> ដែល <span class="math inline">\(\mathbb{1}_{\{A\}}(\omega)=1\)</span> បើ <span class="math inline">\(\omega\in A\)</span> និង <span class="math inline">\(\mathbb{1}_{\{A\}}(\omega)=0\)</span> បើ <span class="math inline">\(\omega\notin A\)</span>។<br> ២. <span class="math inline">\(\mathbb{E}(Y|X)=\mathbb{E}(Y|\sigma(X))\)</span> ។</p>
</blockquote>
<p>នេះមានន័យថា <span class="math inline">\(\mathbb{E}(Y|{\cal F})\)</span> ជាអថេរចៃដន្យវាស់បានធៀបនឹង <span class="math inline">\({\cal F}\)</span> ដែលសង្ឃឹមគណិតនៃបង្រួមរបស់វាលើគ្រប់ព្រឹត្តិការណ៍ <span class="math inline">\(A\)</span> ត្រួតសុីគ្នានឹងសង្ឃឹមគណិតនៃបង្រួមរបស់ <span class="math inline">\(X\)</span> ទៅលើព្រឹត្តិការណ៍ដូចគ្នា ចំពោះគ្រប់ <span class="math inline">\(A\in{\cal F}\)</span> ទាំងអស់។</p>
<p>សង្ឃឹមគណិតមានលក្ខខណ្ឌក៏ដូចជាសង្ឃឹមគណិតគ្មានលក្ខខណ្ឌដែរ ហេតុនេះវាផ្ទៀងផ្ទាត់លក្ខណៈទាំងឡាយនៃសង្ឃឹមគណិតទូទៅដូចជាលក្ខណៈលីនេអ៊ែរនិង លក្ខណៈម៉ូណូតូនជាដើម។ ខាងក្រោមជាលក្ខណៈពិសេសមួយសម្រាប់ឲ្យយើងស្រាយពីលក្ខណៈបរមានៃ បរមាម៉ូដែល <span class="math inline">\(\eta\)</span> នៅក្នុងកិច្ចការបែបតម្រែតម្រង់តម្លៃមាន MSE ជារង្វាស់នៃកំហុស។</p>
<blockquote class="blockquote">
<p><strong>លក្ខណៈ Tower នៃសង្ឃឹមគណិតមានលក្ខខណ្ឌ</strong>៖ បើ <span class="math inline">\({\cal F}\subset{\cal G}\subset \sigma(X)\)</span> ជា <span class="math inline">\(\sigma\)</span> ពីជគណិតរងពីរនៃ <span class="math inline">\(\sigma(X)\)</span> នោះយើងបាន៖ <span class="math display">\[\mathbb{E}[\mathbb{E}(X|{\cal G})|{\cal F}]=\mathbb{E}(X|{\cal F})\ \text{។}\]</span></p>
</blockquote>
<p>លក្ខណៈ Tower មានន័យថាការគណនាសង្ឃឹមគណិតធៀបនឹងលំហរធំសិន រួចហើយបន្តគណនាធៀបនឹងលំហតូចមានតម្លៃស្មើនឹងសង្ឃឹមគណិតមានលក្ខខណ្ឌធៀបនឹងលំហតូចតែម្តង។ នេះប្រៀបដូចជាការធ្វើចំណោលកែងចំណុចពីលើលំហធំទៅលើលំហតូចដែរ។</p>
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="./figures/con_exp.png" class="img-fluid figure-img"></p>
<figcaption class="figure-caption">រូបទី៦. លក្ខណៈ Tower នៃសង្ឃឹមគណិតមានលក្ខខណ្ឌអាចប្រដូចនឹងចំណោលកែងក្នុងរូបខាងលើ។ ចំណុចពណ៌ខៀវតាងឲ្យ <span class="math inline">\(X\)</span> ហើយចំណុចក្រហមជាសង្ឃឹមគណិតមានលក្ខខណ្ឌធៀបនឹង <span class="math inline">\(\cal G\)</span> (ប្លង់) គឺ <span class="math inline">\(\mathbb{E}(X|{\cal G})\)</span> និង ចំណុចពណ៌ខ្មៅជាសង្ឃឹមគណិតមានលក្ខខណ្ឌធៀបនឹង <span class="math inline">\(\cal F\)</span> (បន្ទាត់) គឺ <span class="math inline">\(\mathbb{E}(X|{\cal F})\)</span> ។ ការធ្វើចំំណោលកែងពីចំណុចខៀវគឺ <span class="math inline">\(X\)</span> ទៅលើ <span class="math inline">\(\cal G\)</span> ដោយឲ្យជាចំណុចក្រហមគឺ <span class="math inline">\(\mathbb{E}(X|{\cal G})\)</span> រួចបន្តធ្វើចំណោលទៅលើ <span class="math inline">\(\cal F\)</span> ដោយឲ្យជា <span class="math inline">\(\mathbb{E}[\mathbb{E}(X|{\cal G})|{\cal F}]\)</span> គឺដូចនឹងការធ្វើចំណោលពី <span class="math inline">\(X\)</span> ទៅលើ <span class="math inline">\(\cal F\)</span> ដោយឲ្យជា <span class="math inline">\(\mathbb{E}(X|{\cal F})\)</span> តែម្តង។</figcaption>
</figure>
</div>
<hr>
<p><strong>ឧទាហរណ៍.៥.</strong> តាមលក្ខណៈខាងលើគេបាន៖</p>
<p>១. <span class="math inline">\(\mathbb{E}(Y|{\cal F})=Y\)</span> បើ <span class="math inline">\(Y\)</span> វាស់បានធៀបនឹង <span class="math inline">\({\cal F}\)</span> ។ ជាពិសេស <span class="math inline">\(\mathbb{E}(f(X)|X)=f(X)\)</span> ចំពោះគ្រប់អនុគមន៍ <span class="math inline">\(f\)</span> ដែលសង្ឃឹមគណិតរបស់វាអាចកំណត់បាន (ព្រោះ <span class="math inline">\(X\)</span> វាស់បានធៀបនឹង <span class="math inline">\(\sigma(X)\)</span>) ។</p>
<p>២. <span class="math inline">\(\mathbb{E}(Y|{\cal F})=\mathbb{E}(Y)\)</span> លុះត្រាតែ <span class="math inline">\(Y\)</span> មិនអាស្រ័យនឹង <span class="math inline">\({\cal F}\)</span> ដែលកំណត់តាងដោយ <span class="math inline">\(Y{\perp\!\!\!\!\perp} {\cal F}\)</span> ។</p>
<p>៣. <span class="math inline">\(\mathbb{E}[\mathbb{E}(Y|{\cal F})]=\mathbb{E}(Y)\)</span> ព្រោះបើយើងតាងលំហសំណាក់ដោយ <span class="math inline">\(\Omega\)</span> នោះតាមនិយមនន័យនៃសង្ឃឹមគណិតមានលក្ខខណ្ឌគេបាន៖ <span class="math display">\[
\begin{align}
\mathbb{E}(Y)&=\mathbb{E}[\mathbb{1}_{\{\Omega\}}Y]\\
&=\mathbb{E}[\mathbb{1}_{\{\Omega\}}\mathbb{E}(Y|{\cal F})],\qquad(\Omega\in{\cal F})\\
&=\mathbb{E}[\mathbb{E}(Y|{\cal F})],\qquad(\mathbb{1}_{\{\Omega\}}=1)
\end{align}
\]</span></p>
<hr>
<blockquote class="blockquote">
<p><strong>ចុងក្រោយបំផុតនៃផ្នែកនេះ យើងនឹងស្រាយបរមាភាពនៃ <span class="math inline">\(\eta\)</span> នៃសមីការ <a href="#eq-minimal-loss" class="quarto-xref">Equation 3</a> ។ ចំពោះ <span class="math inline">\(f\)</span> ទូទៅធាតុនៃ <span class="math inline">\(\cal M\)</span> គេបាន៖ <span class="math display">\[
\begin{align}
\text{MSE}(f)&=\mathbb{E}[(Y-f(X))^2]\\
&=\mathbb{E}[(Y-\eta(X)+\eta(X)-f(X))^2]\\
&=\mathbb{E}[(Y-\eta(X))^2]+2\mathbb{E}[(Y-\eta(X))(\eta(X)-f(X))]+\mathbb{E}[(\eta(X)-f(X))^2]
\end{align}
\]</span> ដោយពិនិត្យតួរកណ្តាលនៃកន្សោមខាងលើនិងប្រើលក្ខណៈសង្ឃឹមគណិតមានលក្ខខណ្ឌ យើងបាន៖ <span class="math display">\[
\begin{align}
\mathbb{E}[(Y-\eta(X))(\eta(X)-f(X))]&=\mathbb{E}[\mathbb{E}[(Y-\eta(X))(\eta(X)-f(X))|X]]\\
&=\mathbb{E}[(\eta(X)-f(X))\mathbb{E}[(Y-\eta(X))|X]]\\
&=\mathbb{E}[(\eta(X)-f(X))(\underbrace{\mathbb{E}(Y|X)-\eta(X)}_{0})]\\
&=0
\end{align}
\]</span> ដែលបន្ទាត់ទីពីរពិតយោងទៅតាមលក្ខណៈទី១ នៃ <em>ឧទាហរណ៍.៥.</em> ខាងលើ។ ហេតុនេះយើងបាន៖ <span class="math display">\[
\begin{align}
\text{MSE}(f)&=\mathbb{E}[(Y-\eta(X))^2]+\mathbb{E}[(\eta(X)-f(X))^2]\\
&\geq\mathbb{E}[(Y-\eta(X))^2],\qquad (\mathbb{E}[(\eta(X)-f(X))^2]\geq 0)\\
&=\text{MSE}(\eta)
\end{align}
\]</span> បញ្ហាត្រូវបានស្រាយបញ្ជាក់ដោយសារ <span class="math inline">\(\eta\in{\cal M}\)</span> ។</strong> <span class="math inline">\(\blacksquare\)</span></p>
</blockquote>
<hr>
<blockquote class="blockquote">
<p>🗝️ <strong>អត្ថន័យសំខាន់នៃទ្រឹស្តីខាងលើគឺមានន័យថា <span class="math inline">\(\eta(X)=\mathbb{E}(Y|X)\)</span> ជាចំណោលកែងនៃ <span class="math inline">\(Y\)</span> ទៅលើលំហនៃពត៌មានដែលបង្ករដោយ <span class="math inline">\(X\)</span> ព្រោះថា <span class="math inline">\(\eta(X)\)</span> ជាអនុគមន៍នៃ <span class="math inline">\(X\)</span> ដែលមានចម្ងាយទៅ <span class="math inline">\(Y\)</span> ខ្លីបំផុត (ចំពោះចម្ងាយ <span class="math inline">\(d(X,Y)=(\mathbb{E}[(X-Y)^2])^{(1/2)}, X,Y\in\mathbb{R}\)</span>) ។</strong></p>
</blockquote>
<hr>
</section>
</section>
</main>
<!-- /main column -->
<script id="quarto-html-after-body" type="application/javascript">
window.document.addEventListener("DOMContentLoaded", function (event) {
const toggleBodyColorMode = (bsSheetEl) => {
const mode = bsSheetEl.getAttribute("data-mode");
const bodyEl = window.document.querySelector("body");
if (mode === "dark") {
bodyEl.classList.add("quarto-dark");
bodyEl.classList.remove("quarto-light");
} else {
bodyEl.classList.add("quarto-light");
bodyEl.classList.remove("quarto-dark");
}
}
const toggleBodyColorPrimary = () => {
const bsSheetEl = window.document.querySelector("link#quarto-bootstrap");
if (bsSheetEl) {
toggleBodyColorMode(bsSheetEl);
}
}
toggleBodyColorPrimary();
const icon = "";
const anchorJS = new window.AnchorJS();
anchorJS.options = {
placement: 'right',
icon: icon
};
anchorJS.add('.anchored');
const isCodeAnnotation = (el) => {
for (const clz of el.classList) {
if (clz.startsWith('code-annotation-')) {
return true;
}
}
return false;
}
const clipboard = new window.ClipboardJS('.code-copy-button', {
text: function(trigger) {
const codeEl = trigger.previousElementSibling.cloneNode(true);
for (const childEl of codeEl.children) {
if (isCodeAnnotation(childEl)) {
childEl.remove();
}
}
return codeEl.innerText;
}
});
clipboard.on('success', function(e) {
// button target
const button = e.trigger;
// don't keep focus
button.blur();
// flash "checked"
button.classList.add('code-copy-button-checked');
var currentTitle = button.getAttribute("title");
button.setAttribute("title", "Copied!");
let tooltip;
if (window.bootstrap) {
button.setAttribute("data-bs-toggle", "tooltip");
button.setAttribute("data-bs-placement", "left");
button.setAttribute("data-bs-title", "Copied!");
tooltip = new bootstrap.Tooltip(button,
{ trigger: "manual",
customClass: "code-copy-button-tooltip",
offset: [0, -8]});
tooltip.show();
}
setTimeout(function() {
if (tooltip) {
tooltip.hide();
button.removeAttribute("data-bs-title");
button.removeAttribute("data-bs-toggle");
button.removeAttribute("data-bs-placement");
}
button.setAttribute("title", currentTitle);
button.classList.remove('code-copy-button-checked');
}, 1000);
// clear code selection
e.clearSelection();
});
function tippyHover(el, contentFn, onTriggerFn, onUntriggerFn) {
const config = {
allowHTML: true,
maxWidth: 500,
delay: 100,
arrow: false,
appendTo: function(el) {
return el.parentElement;
},
interactive: true,
interactiveBorder: 10,
theme: 'quarto',
placement: 'bottom-start',
};
if (contentFn) {
config.content = contentFn;
}
if (onTriggerFn) {
config.onTrigger = onTriggerFn;
}
if (onUntriggerFn) {
config.onUntrigger = onUntriggerFn;
}
window.tippy(el, config);
}
const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
for (var i=0; i<noterefs.length; i++) {
const ref = noterefs[i];
tippyHover(ref, function() {
// use id or data attribute instead here
let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
try { href = new URL(href).hash; } catch {}
const id = href.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
return note.innerHTML;
});
}
const xrefs = window.document.querySelectorAll('a.quarto-xref');
const processXRef = (id, note) => {
// Strip column container classes
const stripColumnClz = (el) => {
el.classList.remove("page-full", "page-columns");
if (el.children) {
for (const child of el.children) {
stripColumnClz(child);
}
}
}
stripColumnClz(note)
if (id.startsWith('sec-')) {
// Special case sections, only their first couple elements
const container = document.createElement("div");
if (note.children && note.children.length > 2) {
for (let i = 0; i < 2; i++) {
container.appendChild(note.children[i].cloneNode(true));
}
return container.innerHTML
} else {
return note.innerHTML;
}
} else {
// Remove any anchor links if they are present
const anchorLink = note.querySelector('a.anchorjs-link');
if (anchorLink) {
anchorLink.remove();
}
return note.innerHTML;
}
}
for (var i=0; i<xrefs.length; i++) {
const xref = xrefs[i];
tippyHover(xref, undefined, function(instance) {
instance.disable();
let url = xref.getAttribute('href');
let hash = undefined;
if (url.startsWith('#')) {
hash = url;
} else {
try { hash = new URL(url).hash; } catch {}
}
if (hash) {
const id = hash.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
if (note !== null) {
try {
const html = processXRef(id, note.cloneNode(true));
instance.setContent(html);
} finally {
instance.enable();
instance.show();
}
} else {
// See if we can fetch this
fetch(url.split('#')[0])
.then(res => res.text())
.then(html => {
const parser = new DOMParser();
const htmlDoc = parser.parseFromString(html, "text/html");
const note = htmlDoc.getElementById(id);
if (note !== null) {
const html = processXRef(id, note);
instance.setContent(html);
}
}).finally(() => {
instance.enable();
instance.show();
});
}
}
}, function(instance) {
});
}
let selectedAnnoteEl;
const selectorForAnnotation = ( cell, annotation) => {
let cellAttr = 'data-code-cell="' + cell + '"';
let lineAttr = 'data-code-annotation="' + annotation + '"';
const selector = 'span[' + cellAttr + '][' + lineAttr + ']';
return selector;
}
const selectCodeLines = (annoteEl) => {
const doc = window.document;
const targetCell = annoteEl.getAttribute("data-target-cell");
const targetAnnotation = annoteEl.getAttribute("data-target-annotation");
const annoteSpan = window.document.querySelector(selectorForAnnotation(targetCell, targetAnnotation));
const lines = annoteSpan.getAttribute("data-code-lines").split(",");
const lineIds = lines.map((line) => {
return targetCell + "-" + line;
})
let top = null;
let height = null;
let parent = null;
if (lineIds.length > 0) {
//compute the position of the single el (top and bottom and make a div)
const el = window.document.getElementById(lineIds[0]);
top = el.offsetTop;
height = el.offsetHeight;
parent = el.parentElement.parentElement;
if (lineIds.length > 1) {
const lastEl = window.document.getElementById(lineIds[lineIds.length - 1]);
const bottom = lastEl.offsetTop + lastEl.offsetHeight;
height = bottom - top;
}
if (top !== null && height !== null && parent !== null) {
// cook up a div (if necessary) and position it
let div = window.document.getElementById("code-annotation-line-highlight");
if (div === null) {
div = window.document.createElement("div");
div.setAttribute("id", "code-annotation-line-highlight");
div.style.position = 'absolute';
parent.appendChild(div);
}
div.style.top = top - 2 + "px";
div.style.height = height + 4 + "px";
div.style.left = 0;
let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
if (gutterDiv === null) {
gutterDiv = window.document.createElement("div");
gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
gutterDiv.style.position = 'absolute';
const codeCell = window.document.getElementById(targetCell);
const gutter = codeCell.querySelector('.code-annotation-gutter');
gutter.appendChild(gutterDiv);
}
gutterDiv.style.top = top - 2 + "px";
gutterDiv.style.height = height + 4 + "px";
}
selectedAnnoteEl = annoteEl;
}
};
const unselectCodeLines = () => {
const elementsIds = ["code-annotation-line-highlight", "code-annotation-line-highlight-gutter"];
elementsIds.forEach((elId) => {
const div = window.document.getElementById(elId);
if (div) {
div.remove();
}
});
selectedAnnoteEl = undefined;
};
// Handle positioning of the toggle
window.addEventListener(
"resize",
throttle(() => {
console.log("RESIZE");
elRect = undefined;
if (selectedAnnoteEl) {
selectCodeLines(selectedAnnoteEl);
}
}, 10)
);
function throttle(fn, ms) {
let throttle = false;
let timer;
return (...args) => {
if(!throttle) { // first call gets through
fn.apply(this, args);
throttle = true;
} else { // all the others get throttled
if(timer) clearTimeout(timer); // cancel #2
timer = setTimeout(() => {
fn.apply(this, args);
timer = throttle = false;
}, ms);
}
};
}
// Attach click handler to the DT
const annoteDls = window.document.querySelectorAll('dt[data-target-cell]');
for (const annoteDlNode of annoteDls) {
annoteDlNode.addEventListener('click', (event) => {
const clickedEl = event.target;
if (clickedEl !== selectedAnnoteEl) {
unselectCodeLines();
const activeEl = window.document.querySelector('dt[data-target-cell].code-annotation-active');
if (activeEl) {
activeEl.classList.remove('code-annotation-active');
}
selectCodeLines(clickedEl);
clickedEl.classList.add('code-annotation-active');
} else {
// Unselect the line
unselectCodeLines();
clickedEl.classList.remove('code-annotation-active');
}