From 5fddfdc72e955908db3f0e0f7fd4eaeb8b4d4f02 Mon Sep 17 00:00:00 2001 From: Carlos Camara Date: Thu, 19 Oct 2023 16:27:08 +0100 Subject: [PATCH] Built site for gh-pages --- .nojekyll | 2 +- .../IM939_Lab_3_2_Linear_Regression.html | 256 ++++++++++-------- .../figure-html/cell-12-output-1.png | Bin 106107 -> 106211 bytes .../figure-html/cell-14-output-2.png | Bin 129991 -> 130108 bytes .../figure-html/cell-16-output-1.png | Bin 160180 -> 159923 bytes index.html | 2 +- search.json | 6 +- sitemap.xml | 88 +++--- 8 files changed, 194 insertions(+), 160 deletions(-) diff --git a/.nojekyll b/.nojekyll index b37f113..3de82c3 100644 --- a/.nojekyll +++ b/.nojekyll @@ -1 +1 @@ -51169f94 \ No newline at end of file +3914b05c \ No newline at end of file diff --git a/content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression.html b/content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression.html index 930fe71..e676083 100644 --- a/content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression.html +++ b/content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression.html @@ -653,7 +653,7 @@

ice.head() 
-
+
@@ -724,7 +724,7 @@

And we can get some summary statistics from the numerical attributes:

ice.describe()
-
+
@@ -829,7 +829,7 @@

plt.scatter(ice.mo, ice.extent, color = 'red') plt.xlabel('Year') plt.ylabel('Extent')

-
+
Text(0, 0.5, 'Extent')
@@ -858,12 +858,46 @@

-
print (ice[(ice.data_type != 'Goddard')
-          & (ice.data_type != 'NRTSI-G')])
-
-
     year  mo data_type region  extent    area
-9    1988   1     -9999      N -9999.0 -9999.0
-397  1987  12     -9999      N -9999.0 -9999.0
+
ice[(ice.data_type != 'Goddard') & (ice.data_type != 'NRTSI-G')]
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
yearmodata_typeregionextentarea
919881-9999N-9999.0-9999.0
397198712-9999N-9999.0-9999.0
+ +
@@ -886,20 +920,20 @@

-
# We can easily clean the data now:
-ice2 = ice[ice.data_type != '-9999'].copy()
-
-print ('shape:', ice2.shape)
+
# We can easily clean the data now:
+ice2 = ice[ice.data_type != '-9999'].copy()
+
+print ('shape:', ice2.shape)
shape: (422, 6)
-
# And repeat the plot, without the outliers
-plt.scatter(ice2.mo, ice2.extent, color = 'red')
-plt.xlabel('Month')
-plt.ylabel('Extent')
-
+
# And repeat the plot, without the outliers
+plt.scatter(ice2.mo, ice2.extent, color = 'red')
+plt.xlabel('Month')
+plt.ylabel('Extent')
+
Text(0, 0.5, 'Extent')
@@ -907,7 +941,7 @@

-
sns.relplot(ice2, x = "mo", y = "extent", aspect = 2)
+
sns.relplot(ice2, x = "mo", y = "extent", aspect = 2)
/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead
   if pd.api.types.is_categorical_dtype(vector):
@@ -933,10 +967,10 @@ 

Now that we have a clean dataset, we can use Seaborn’s lmplot() function comparing month vs extent.

The lmplot() function from the Seaborn module is intended for exploring linear relationships of different forms in multidimensional datesets. Input data must be in a Pandas DataFrame. To plot them, we provide the predictor (mo) and response (extent) variable names along with the dataset (ice2).

-
sns.lmplot(ice2, x = "mo", y = "extent", aspect=2);
-
-# Uncomment below to save the resulting plot.
-#plt.savefig("figs/CleanedByMonth.png", dpi = 300, bbox_inches = 'tight')
+
sns.lmplot(ice2, x = "mo", y = "extent", aspect=2);
+
+# Uncomment below to save the resulting plot.
+#plt.savefig("figs/CleanedByMonth.png", dpi = 300, bbox_inches = 'tight')

@@ -945,16 +979,16 @@

12.3.1 Normalization

-
# Compute the mean for each month.
-month_means = ice2.groupby('mo').extent.mean()
-
-# Compute the variance for each month.
-month_variances = ice2.groupby('mo').extent.var()
-
-# Show the values:
-print('Means:', month_means)
-print('\n') # Add a new line between the two prints, so they are easily distinguishible.
-print ('Variances:',month_variances)
+
# Compute the mean for each month.
+month_means = ice2.groupby('mo').extent.mean()
+
+# Compute the variance for each month.
+month_variances = ice2.groupby('mo').extent.var()
+
+# Show the values:
+print('Means:', month_means)
+print('\n') # Add a new line between the two prints, so they are easily distinguishible.
+print ('Variances:',month_variances)
Means: mo
 1     14.479429
@@ -990,69 +1024,69 @@ 

To capture variation per month, we can compute mean for the i-th interval of time (using 1979-2014) and subtract it from the set of extent values for that month . This can be converted to a relative pecentage difference by dividing it by the total avareage (1979-2014) and multiplying by 100.

-
# Data normalization
-for i in range(12):
-    ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-    
-sns.lmplot(ice2 , x = "mo", y = "extent", height = 5.2, aspect = 2);
-#plt.savefig("figs/IceExtentNormalizedByMonth.png", dpi = 300, bbox_inches = 'tight')
+
# Data normalization
+for i in range(12):
+    ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
+    
+sns.lmplot(ice2 , x = "mo", y = "extent", height = 5.2, aspect = 2);
+#plt.savefig("figs/IceExtentNormalizedByMonth.png", dpi = 300, bbox_inches = 'tight')
-
/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+
/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
   ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()
-/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: 
+/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: 
 A value is trying to be set on a copy of a slice from a DataFrame
 
 See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
@@ -1063,8 +1097,8 @@ 

-
print ('mean:', ice2.extent.mean())
-print ('var:', ice2.extent.var())
+
print ('mean:', ice2.extent.mean())
+print ('var:', ice2.extent.var())
mean: -7.745252569896827e-16
 var: 31.983239774968798
@@ -1072,8 +1106,8 @@

Let us consider the entire year

-
sns.lmplot(ice2, x = "year", y = "extent", height = 5.2, aspect = 2);
-#plt.savefig("figs/IceExtentAllMonthsByYearlmplot.png", dpi = 300, bbox_inches = 'tight')
+
sns.lmplot(ice2, x = "year", y = "extent", height = 5.2, aspect = 2);
+#plt.savefig("figs/IceExtentAllMonthsByYearlmplot.png", dpi = 300, bbox_inches = 'tight')

@@ -1083,9 +1117,9 @@

12.3.2 Pearson’s correlation

Let’s calculate Pearson’s correlation coefficient and the p-value for testing non-correlation.

-
import scipy.stats
-scipy.stats.pearsonr(ice2.year.values, ice2.extent.values)
-
+
import scipy.stats
+scipy.stats.pearsonr(ice2.year.values, ice2.extent.values)
+
PearsonRResult(statistic=-0.8183500709897178, pvalue=4.4492318168687107e-103)
@@ -1096,17 +1130,17 @@

For that we use using Scikit-learn, library that provides a variety of both supervised and unsupervised machine learning techniques. Scikit-learn provides an object-oriented interface centered around the concept of an Estimator. The Estimator.fit method sets the state of the estimator based on the training data. Usually, the data is comprised of a two-dimensional numpy array \(X\) of shape (n_samples, n_predictors) that holds the so-called feature matrix and a one-dimensional numpy array \(\textbf{y}\) that holds the responses. Some estimators allow the user to control the fitting behavior. For example, the sklearn.linear_model.LinearRegression estimator allows the user to specify whether or not to fit an intercept term. This is done by setting the corresponding constructor arguments of the estimator object. During the fitting process, the state of the estimator is stored in instance attributes that have a trailing underscore (_). For example, the coefficients of a LinearRegression estimator are stored in the attribute coef_.

Estimators that can generate predictions provide a Estimator.predict method. In the case of regression, Estimator.predict will return the predicted regression values, \(\hat{\textbf{y}}\).

-
from sklearn.linear_model import LinearRegression
-
-est = LinearRegression(fit_intercept = True)
-
-x = ice2[['year']]
-y = ice2[['extent']]
-
-est.fit(x, y)
-
-print("Coefficients:", est.coef_)
-print ("Intercept:", est.intercept_)
+
from sklearn.linear_model import LinearRegression
+
+est = LinearRegression(fit_intercept = True)
+
+x = ice2[['year']]
+y = ice2[['extent']]
+
+est.fit(x, y)
+
+print("Coefficients:", est.coef_)
+print ("Intercept:", est.intercept_)
Coefficients: [[-0.45275459]]
 Intercept: [903.71640207]
@@ -1114,22 +1148,22 @@

We can evaluate the model fitting by computing the mean squared error (\(MSE\)) and the coefficient of determination (\(R^2\)) of the model. The coefficient \(R^2\) is defined as \((1 - \textbf{u}/\textbf{v})\), where \(\textbf{u}\) is the residual sum of squares \(\sum (\textbf{y} - \hat{\textbf{y}})^2\) and \(\textbf{v}\) is the regression sum of squares \(\sum (\textbf{y} - \bar{\textbf{y}})^2\), where \(\bar{\textbf{y}}\) is the mean. The best possible score for \(R^2\) is 1.0: lower values are worse. These measures can provide a quantitative answer to the question we are facing: Is there a negative trend in the evolution of sea ice extent over recent years?

-
from sklearn import metrics
-
-# Analysis for all months together.
-x = ice2[['year']]
-y = ice2[['extent']]
-model = LinearRegression()
-model.fit(x, y)
-y_hat = model.predict(x)
-plt.plot(x, y,'o', alpha = 0.5)
-plt.plot(x, y_hat, 'r', alpha = 0.5)
-plt.xlabel('year')
-plt.ylabel('extent (All months)')
-print ("MSE:", metrics.mean_squared_error(y_hat, y))
-print ("R^2:", metrics.r2_score(y_hat, y))
-print ("var:", y.var())
-#plt.savefig("figs/IceExtentLinearRegressionAllMonthsByYearPrediction.png", dpi = 300, bbox_inches = 'tight')
+
from sklearn import metrics
+
+# Analysis for all months together.
+x = ice2[['year']]
+y = ice2[['extent']]
+model = LinearRegression()
+model.fit(x, y)
+y_hat = model.predict(x)
+plt.plot(x, y,'o', alpha = 0.5)
+plt.plot(x, y_hat, 'r', alpha = 0.5)
+plt.xlabel('year')
+plt.ylabel('extent (All months)')
+print ("MSE:", metrics.mean_squared_error(y_hat, y))
+print ("R^2:", metrics.r2_score(y_hat, y))
+print ("var:", y.var())
+#plt.savefig("figs/IceExtentLinearRegressionAllMonthsByYearPrediction.png", dpi = 300, bbox_inches = 'tight')
MSE: 10.539131639803488
 R^2: 0.5067870382100226
@@ -1142,26 +1176,26 @@ 

We can conclude that the data show a long-term negative trend in recent years.

-
# Analysis for a particular month.
-#For January
-jan = ice2[ice2.mo == 1];
-
-x = jan[['year']]
-y = jan[['extent']]
-
-model = LinearRegression()
-model.fit(x, y)
-
-y_hat = model.predict(x)
-
-plt.figure()
-plt.plot(x, y,'-o', alpha = 0.5)
-plt.plot(x, y_hat, 'r', alpha = 0.5)
-plt.xlabel('year')
-plt.ylabel('extent (January)')
-
-print ("MSE:", metrics.mean_squared_error(y_hat, y))
-print ("R^2:", metrics.r2_score(y_hat, y))
+
# Analysis for a particular month.
+#For January
+jan = ice2[ice2.mo == 1];
+
+x = jan[['year']]
+y = jan[['extent']]
+
+model = LinearRegression()
+model.fit(x, y)
+
+y_hat = model.predict(x)
+
+plt.figure()
+plt.plot(x, y,'-o', alpha = 0.5)
+plt.plot(x, y_hat, 'r', alpha = 0.5)
+plt.xlabel('year')
+plt.ylabel('extent (January)')
+
+print ("MSE:", metrics.mean_squared_error(y_hat, y))
+print ("R^2:", metrics.r2_score(y_hat, y))
MSE: 3.8395160752867565
 R^2: 0.7810636041396216
@@ -1172,12 +1206,12 @@

We can also estimate the extent value for 2025. For that we use the function predict of the model.

-
X = np.array(2025) 
-y_hat = model.predict(X.reshape(-1, 1))
-j = 1 # January
-# Original value (before normalization)
-y_hat = (y_hat*month_means.mean()/100) + month_means[j]
-print ("Prediction of extent for January 2025 (in millions of square km):", y_hat)
+
X = np.array(2025) 
+y_hat = model.predict(X.reshape(-1, 1))
+j = 1 # January
+# Original value (before normalization)
+y_hat = (y_hat*month_means.mean()/100) + month_means[j]
+print ("Prediction of extent for January 2025 (in millions of square km):", y_hat)
Prediction of extent for January 2025 (in millions of square km): [[13.14449923]]
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z^FwvmEos(7-Jp?0K{t`oZ*>1%&@On+;W}~FeuJcsNo0!#HG?($Osem}-B647;pjCj zVck3p|2Dff6}vz;R!84E{qj{`4=Y5Qf3nOz{?sId_`} z?o$|aFp~RqOw!fYVyvLyM~4Q_hp$bM<&~Ty+ThdXrVpxA$c7jzT9nC!4RUXRZsPq_ z>eq0=))U7(&~4bEHV=9lkfTysyoq Qk1K)EH~Kkar_H(l0ls|bPyhe` diff --git a/index.html b/index.html index 12bf6a7..83e72dc 100644 --- a/index.html +++ b/index.html @@ -631,7 +631,7 @@

Acknowledgements

-

This handbook has been created by Carlos Cámara-Menoyo and Cagatay Turkay, based on the materials from previous years created by Cagatay Turkay and James Tripp.

+

This handbook has been created by Carlos Cámara-Menoyo and Cagatay Turkay, based on the materials from previous years created by Cagatay Turkay, James Tripp and Zofia Bednarowska-Michaiel.

Licence

diff --git a/search.json b/search.json index 1d746a7..756ffc9 100644 --- a/search.json +++ b/search.json @@ -25,7 +25,7 @@ "href": "index.html#acknowledgements", "title": "Data Science Across Disciplines", "section": "Acknowledgements", - "text": "Acknowledgements\nThis handbook has been created by Carlos Cámara-Menoyo and Cagatay Turkay, based on the materials from previous years created by Cagatay Turkay and James Tripp." + "text": "Acknowledgements\nThis handbook has been created by Carlos Cámara-Menoyo and Cagatay Turkay, based on the materials from previous years created by Cagatay Turkay, James Tripp and Zofia Bednarowska-Michaiel." }, { "objectID": "index.html#licence", @@ -466,14 +466,14 @@ "href": "content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression.html#case-study-climate-change-and-sea-ice-extent", "title": "12  Lab: Regressions", "section": "12.2 Case study: Climate Change and Sea Ice Extent", - "text": "12.2 Case study: Climate Change and Sea Ice Extent\nRemember, we are trying to answer the following research question: Has there been a decrease in the amount of ice in the last years?\n\n12.2.1 Data assessment\nFirst, let’s load the SeaIce.txt dataset that is already in the data folder1. It is a text file, but unlike csv files, where columns are separated by commas (,), it is a Tab separated file, where each Tab delimites the following columns:1 The original dataset was downloaded from https://nsidc.org/data/g02135/versions/3, which provides useful metadata information, as well as API services.\n\nYear: 4-digit year\nmo: 1- or 2-digit month\ndata_type: Input data set (Goddard/NRTSI-G)\nregion: Hemisphere that this data covers (N: Northern; S: Southern)\nextent: Sea ice extent in millions of square km\narea: Sea ice area in millions of square km\n\nOnce we upload the data, we can create a DataFrame using Pandas using the well known read_csv() function, but in this case, because columns are not separated by commas as expected, but Tabs, we will need to use the delim_whitespace=True argument.\n\nimport pandas as pd\nice = pd.read_csv('data/raw/SeaIce.txt', delim_whitespace=True)\n\nice.info()\n\n<class 'pandas.core.frame.DataFrame'>\nRangeIndex: 424 entries, 0 to 423\nData columns (total 6 columns):\n # Column Non-Null Count Dtype \n--- ------ -------------- ----- \n 0 year 424 non-null int64 \n 1 mo 424 non-null int64 \n 2 data_type 424 non-null object \n 3 region 424 non-null object \n 4 extent 424 non-null float64\n 5 area 424 non-null float64\ndtypes: float64(2), int64(2), object(2)\nmemory usage: 20.0+ KB\n\n\n\nice.head() \n\n\n\n\n\n\n\n\nyear\nmo\ndata_type\nregion\nextent\narea\n\n\n\n\n0\n1979\n1\nGoddard\nN\n15.54\n12.33\n\n\n1\n1980\n1\nGoddard\nN\n14.96\n11.85\n\n\n2\n1981\n1\nGoddard\nN\n15.03\n11.82\n\n\n3\n1982\n1\nGoddard\nN\n15.26\n12.11\n\n\n4\n1983\n1\nGoddard\nN\n15.10\n11.92\n\n\n\n\n\n\n\nAnd we can get some summary statistics from the numerical attributes:\n\nice.describe()\n\n\n\n\n\n\n\n\nyear\nmo\nextent\narea\n\n\n\n\ncount\n424.000000\n424.000000\n424.000000\n424.000000\n\n\nmean\n1996.000000\n6.500000\n-35.443066\n-37.921108\n\n\nstd\n10.214716\n3.474323\n686.736905\n686.566381\n\n\nmin\n1978.000000\n1.000000\n-9999.000000\n-9999.000000\n\n\n25%\n1987.000000\n3.000000\n9.272500\n6.347500\n\n\n50%\n1996.000000\n6.500000\n12.385000\n9.895000\n\n\n75%\n2005.000000\n10.000000\n14.540000\n12.222500\n\n\nmax\n2014.000000\n12.000000\n16.450000\n13.840000\n\n\n\n\n\n\n\n\n\n\n\n\n\nWarning\n\n\n\nDid we receive a negative mean for extent and area? What this could possibly mean? Probably, inspecting those attributes visually could give us a clue.\n\n\n\n\n12.2.2 Data visualisation to explore data\nWe will use Seaborn’s lmplot() function to explore linear relationship of different forms, e.g. relationship between the month of the year (variable) and extent (responses):\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pandas as pd\nimport seaborn as sns\n\n\n# Visualize the data\nplt.scatter(ice.mo, ice.extent, color = 'red')\nplt.xlabel('Year')\nplt.ylabel('Extent')\n\nText(0, 0.5, 'Extent')\n\n\n\n\n\n\n\n\n\n\n\nWarning\n\n\n\nWe detect some outlier or missing data. This might have to do with those negative mean values that we detected previously.\n\n\nWe can use numpy’s function np.unique() to find the unique elements of an array.\n\nprint ('Different values in data_type field:', np.unique(ice.data_type.values)) # there is a -9999 value!\n\nDifferent values in data_type field: ['-9999' 'Goddard' 'NRTSI-G']\n\n\nLet’s see what type of data we have, other than the ones printed above\n\nprint (ice[(ice.data_type != 'Goddard')\n & (ice.data_type != 'NRTSI-G')])\n\n year mo data_type region extent area\n9 1988 1 -9999 N -9999.0 -9999.0\n397 1987 12 -9999 N -9999.0 -9999.0\n\n\n\n\n12.2.3 Data cleaning\nWe checked all the values and notice -9999 values in data_type field which should contain Goddard or NRTSI-G (some type of input dataset).\nIn this case, we will clean them by creating a copy of the original dataframe that does not include these instances.\n\n\n\n\n\n\nDataframe copies vs instances\n\n\n\nUnless we do not explicitly create a copy of a dataframe, when subsetting a dataframes we are actually creating instances. Whereas copies are totally independent objects from the original one, instances are reduced “views” from the original, meaning that if we change a value on the instance, we are also changing the value on the original data frame, which may not be what we wanted to do.\nAn explanation of how (and why) do we need to create copies can be found here: https://www.statology.org/pandas-copy-dataframe/\n\n\n\n# We can easily clean the data now:\nice2 = ice[ice.data_type != '-9999'].copy()\n\nprint ('shape:', ice2.shape)\n\nshape: (422, 6)\n\n\n\n# And repeat the plot, without the outliers\nplt.scatter(ice2.mo, ice2.extent, color = 'red')\nplt.xlabel('Month')\nplt.ylabel('Extent')\n\nText(0, 0.5, 'Extent')\n\n\n\n\n\n\nsns.relplot(ice2, x = \"mo\", y = \"extent\", aspect = 2)\n\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):" + "text": "12.2 Case study: Climate Change and Sea Ice Extent\nRemember, we are trying to answer the following research question: Has there been a decrease in the amount of ice in the last years?\n\n12.2.1 Data assessment\nFirst, let’s load the SeaIce.txt dataset that is already in the data folder1. It is a text file, but unlike csv files, where columns are separated by commas (,), it is a Tab separated file, where each Tab delimites the following columns:1 The original dataset was downloaded from https://nsidc.org/data/g02135/versions/3, which provides useful metadata information, as well as API services.\n\nYear: 4-digit year\nmo: 1- or 2-digit month\ndata_type: Input data set (Goddard/NRTSI-G)\nregion: Hemisphere that this data covers (N: Northern; S: Southern)\nextent: Sea ice extent in millions of square km\narea: Sea ice area in millions of square km\n\nOnce we upload the data, we can create a DataFrame using Pandas using the well known read_csv() function, but in this case, because columns are not separated by commas as expected, but Tabs, we will need to use the delim_whitespace=True argument.\n\nimport pandas as pd\nice = pd.read_csv('data/raw/SeaIce.txt', delim_whitespace=True)\n\nice.info()\n\n<class 'pandas.core.frame.DataFrame'>\nRangeIndex: 424 entries, 0 to 423\nData columns (total 6 columns):\n # Column Non-Null Count Dtype \n--- ------ -------------- ----- \n 0 year 424 non-null int64 \n 1 mo 424 non-null int64 \n 2 data_type 424 non-null object \n 3 region 424 non-null object \n 4 extent 424 non-null float64\n 5 area 424 non-null float64\ndtypes: float64(2), int64(2), object(2)\nmemory usage: 20.0+ KB\n\n\n\nice.head() \n\n\n\n\n\n\n\n\nyear\nmo\ndata_type\nregion\nextent\narea\n\n\n\n\n0\n1979\n1\nGoddard\nN\n15.54\n12.33\n\n\n1\n1980\n1\nGoddard\nN\n14.96\n11.85\n\n\n2\n1981\n1\nGoddard\nN\n15.03\n11.82\n\n\n3\n1982\n1\nGoddard\nN\n15.26\n12.11\n\n\n4\n1983\n1\nGoddard\nN\n15.10\n11.92\n\n\n\n\n\n\n\nAnd we can get some summary statistics from the numerical attributes:\n\nice.describe()\n\n\n\n\n\n\n\n\nyear\nmo\nextent\narea\n\n\n\n\ncount\n424.000000\n424.000000\n424.000000\n424.000000\n\n\nmean\n1996.000000\n6.500000\n-35.443066\n-37.921108\n\n\nstd\n10.214716\n3.474323\n686.736905\n686.566381\n\n\nmin\n1978.000000\n1.000000\n-9999.000000\n-9999.000000\n\n\n25%\n1987.000000\n3.000000\n9.272500\n6.347500\n\n\n50%\n1996.000000\n6.500000\n12.385000\n9.895000\n\n\n75%\n2005.000000\n10.000000\n14.540000\n12.222500\n\n\nmax\n2014.000000\n12.000000\n16.450000\n13.840000\n\n\n\n\n\n\n\n\n\n\n\n\n\nWarning\n\n\n\nDid we receive a negative mean for extent and area? What this could possibly mean? Probably, inspecting those attributes visually could give us a clue.\n\n\n\n\n12.2.2 Data visualisation to explore data\nWe will use Seaborn’s lmplot() function to explore linear relationship of different forms, e.g. relationship between the month of the year (variable) and extent (responses):\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport pandas as pd\nimport seaborn as sns\n\n\n# Visualize the data\nplt.scatter(ice.mo, ice.extent, color = 'red')\nplt.xlabel('Year')\nplt.ylabel('Extent')\n\nText(0, 0.5, 'Extent')\n\n\n\n\n\n\n\n\n\n\n\nWarning\n\n\n\nWe detect some outlier or missing data. This might have to do with those negative mean values that we detected previously.\n\n\nWe can use numpy’s function np.unique() to find the unique elements of an array.\n\nprint ('Different values in data_type field:', np.unique(ice.data_type.values)) # there is a -9999 value!\n\nDifferent values in data_type field: ['-9999' 'Goddard' 'NRTSI-G']\n\n\nLet’s see what type of data we have, other than the ones printed above\n\nice[(ice.data_type != 'Goddard') & (ice.data_type != 'NRTSI-G')]\n\n\n\n\n\n\n\n\nyear\nmo\ndata_type\nregion\nextent\narea\n\n\n\n\n9\n1988\n1\n-9999\nN\n-9999.0\n-9999.0\n\n\n397\n1987\n12\n-9999\nN\n-9999.0\n-9999.0\n\n\n\n\n\n\n\n\n\n12.2.3 Data cleaning\nWe checked all the values and notice -9999 values in data_type field which should contain Goddard or NRTSI-G (some type of input dataset).\nIn this case, we will clean them by creating a copy of the original dataframe that does not include these instances.\n\n\n\n\n\n\nDataframe copies vs instances\n\n\n\nUnless we do not explicitly create a copy of a dataframe, when subsetting a dataframes we are actually creating instances. Whereas copies are totally independent objects from the original one, instances are reduced “views” from the original, meaning that if we change a value on the instance, we are also changing the value on the original data frame, which may not be what we wanted to do.\nAn explanation of how (and why) do we need to create copies can be found here: https://www.statology.org/pandas-copy-dataframe/\n\n\n\n# We can easily clean the data now:\nice2 = ice[ice.data_type != '-9999'].copy()\n\nprint ('shape:', ice2.shape)\n\nshape: (422, 6)\n\n\n\n# And repeat the plot, without the outliers\nplt.scatter(ice2.mo, ice2.extent, color = 'red')\nplt.xlabel('Month')\nplt.ylabel('Extent')\n\nText(0, 0.5, 'Extent')\n\n\n\n\n\n\nsns.relplot(ice2, x = \"mo\", y = \"extent\", aspect = 2)\n\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/seaborn/_oldcore.py:1498: FutureWarning: is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead\n if pd.api.types.is_categorical_dtype(vector):" }, { "objectID": "content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression.html#regression-model-fit", "href": "content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression.html#regression-model-fit", "title": "12  Lab: Regressions", "section": "12.3 Regression model fit", - "text": "12.3 Regression model fit\nNow that we have a clean dataset, we can use Seaborn’s lmplot() function comparing month vs extent.\nThe lmplot() function from the Seaborn module is intended for exploring linear relationships of different forms in multidimensional datesets. Input data must be in a Pandas DataFrame. To plot them, we provide the predictor (mo) and response (extent) variable names along with the dataset (ice2).\n\nsns.lmplot(ice2, x = \"mo\", y = \"extent\", aspect=2);\n\n# Uncomment below to save the resulting plot.\n#plt.savefig(\"figs/CleanedByMonth.png\", dpi = 300, bbox_inches = 'tight')\n\n\n\n\nAbove you can see ice extent data by month. You can see a monthly fluctuation of the sea ice extent, as would be expected for the different seasons of the year. In order to run regression, and avoid this fluctuation we can normalize data. This will let us see the evolution of the extent over the years.\n\n12.3.1 Normalization\n\n# Compute the mean for each month.\nmonth_means = ice2.groupby('mo').extent.mean()\n\n# Compute the variance for each month.\nmonth_variances = ice2.groupby('mo').extent.var()\n\n# Show the values:\nprint('Means:', month_means)\nprint('\\n') # Add a new line between the two prints, so they are easily distinguishible.\nprint ('Variances:',month_variances)\n\nMeans: mo\n1 14.479429\n2 15.298889\n3 15.491714\n4 14.766000\n5 13.396000\n6 11.860000\n7 9.601143\n8 7.122286\n9 6.404857\n10 8.809143\n11 10.964722\n12 13.059429\nName: extent, dtype: float64\n\n\nVariances: mo\n1 0.304906\n2 0.295804\n3 0.237209\n4 0.215378\n5 0.189901\n6 0.247918\n7 0.679175\n8 0.824577\n9 1.143902\n10 0.630361\n11 0.412511\n12 0.284870\nName: extent, dtype: float64\n\n\nTo capture variation per month, we can compute mean for the i-th interval of time (using 1979-2014) and subtract it from the set of extent values for that month . This can be converted to a relative pecentage difference by dividing it by the total avareage (1979-2014) and multiplying by 100.\n\n# Data normalization\nfor i in range(12):\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n \nsns.lmplot(ice2 , x = \"mo\", y = \"extent\", height = 5.2, aspect = 2);\n#plt.savefig(\"figs/IceExtentNormalizedByMonth.png\", dpi = 300, bbox_inches = 'tight')\n\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_73230/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n\n\n\n\n\n\nprint ('mean:', ice2.extent.mean())\nprint ('var:', ice2.extent.var())\n\nmean: -7.745252569896827e-16\nvar: 31.983239774968798\n\n\nLet us consider the entire year\n\nsns.lmplot(ice2, x = \"year\", y = \"extent\", height = 5.2, aspect = 2);\n#plt.savefig(\"figs/IceExtentAllMonthsByYearlmplot.png\", dpi = 300, bbox_inches = 'tight')\n\n\n\n\n\n\n12.3.2 Pearson’s correlation\nLet’s calculate Pearson’s correlation coefficient and the p-value for testing non-correlation.\n\nimport scipy.stats\nscipy.stats.pearsonr(ice2.year.values, ice2.extent.values)\n\nPearsonRResult(statistic=-0.8183500709897178, pvalue=4.4492318168687107e-103)\n\n\n\n\n12.3.3 Simple OLS\nWe can also compute the trend as a simple linear regression (OLS) and quantitatively evaluate it.\nFor that we use using Scikit-learn, library that provides a variety of both supervised and unsupervised machine learning techniques. Scikit-learn provides an object-oriented interface centered around the concept of an Estimator. The Estimator.fit method sets the state of the estimator based on the training data. Usually, the data is comprised of a two-dimensional numpy array \\(X\\) of shape (n_samples, n_predictors) that holds the so-called feature matrix and a one-dimensional numpy array \\(\\textbf{y}\\) that holds the responses. Some estimators allow the user to control the fitting behavior. For example, the sklearn.linear_model.LinearRegression estimator allows the user to specify whether or not to fit an intercept term. This is done by setting the corresponding constructor arguments of the estimator object. During the fitting process, the state of the estimator is stored in instance attributes that have a trailing underscore (_). For example, the coefficients of a LinearRegression estimator are stored in the attribute coef_.\nEstimators that can generate predictions provide a Estimator.predict method. In the case of regression, Estimator.predict will return the predicted regression values, \\(\\hat{\\textbf{y}}\\).\n\nfrom sklearn.linear_model import LinearRegression\n\nest = LinearRegression(fit_intercept = True)\n\nx = ice2[['year']]\ny = ice2[['extent']]\n\nest.fit(x, y)\n\nprint(\"Coefficients:\", est.coef_)\nprint (\"Intercept:\", est.intercept_)\n\nCoefficients: [[-0.45275459]]\nIntercept: [903.71640207]\n\n\nWe can evaluate the model fitting by computing the mean squared error (\\(MSE\\)) and the coefficient of determination (\\(R^2\\)) of the model. The coefficient \\(R^2\\) is defined as \\((1 - \\textbf{u}/\\textbf{v})\\), where \\(\\textbf{u}\\) is the residual sum of squares \\(\\sum (\\textbf{y} - \\hat{\\textbf{y}})^2\\) and \\(\\textbf{v}\\) is the regression sum of squares \\(\\sum (\\textbf{y} - \\bar{\\textbf{y}})^2\\), where \\(\\bar{\\textbf{y}}\\) is the mean. The best possible score for \\(R^2\\) is 1.0: lower values are worse. These measures can provide a quantitative answer to the question we are facing: Is there a negative trend in the evolution of sea ice extent over recent years?\n\nfrom sklearn import metrics\n\n# Analysis for all months together.\nx = ice2[['year']]\ny = ice2[['extent']]\nmodel = LinearRegression()\nmodel.fit(x, y)\ny_hat = model.predict(x)\nplt.plot(x, y,'o', alpha = 0.5)\nplt.plot(x, y_hat, 'r', alpha = 0.5)\nplt.xlabel('year')\nplt.ylabel('extent (All months)')\nprint (\"MSE:\", metrics.mean_squared_error(y_hat, y))\nprint (\"R^2:\", metrics.r2_score(y_hat, y))\nprint (\"var:\", y.var())\n#plt.savefig(\"figs/IceExtentLinearRegressionAllMonthsByYearPrediction.png\", dpi = 300, bbox_inches = 'tight')\n\nMSE: 10.539131639803488\nR^2: 0.5067870382100226\nvar: extent 31.98324\ndtype: float64\n\n\n\n\n\nWe can conclude that the data show a long-term negative trend in recent years.\n\n# Analysis for a particular month.\n#For January\njan = ice2[ice2.mo == 1];\n\nx = jan[['year']]\ny = jan[['extent']]\n\nmodel = LinearRegression()\nmodel.fit(x, y)\n\ny_hat = model.predict(x)\n\nplt.figure()\nplt.plot(x, y,'-o', alpha = 0.5)\nplt.plot(x, y_hat, 'r', alpha = 0.5)\nplt.xlabel('year')\nplt.ylabel('extent (January)')\n\nprint (\"MSE:\", metrics.mean_squared_error(y_hat, y))\nprint (\"R^2:\", metrics.r2_score(y_hat, y))\n\nMSE: 3.8395160752867565\nR^2: 0.7810636041396216\n\n\n\n\n\nWe can also estimate the extent value for 2025. For that we use the function predict of the model.\n\nX = np.array(2025) \ny_hat = model.predict(X.reshape(-1, 1))\nj = 1 # January\n# Original value (before normalization)\ny_hat = (y_hat*month_means.mean()/100) + month_means[j]\nprint (\"Prediction of extent for January 2025 (in millions of square km):\", y_hat)\n\nPrediction of extent for January 2025 (in millions of square km): [[13.14449923]]\n\n\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/sklearn/base.py:465: UserWarning: X does not have valid feature names, but LinearRegression was fitted with feature names\n warnings.warn(\n\n\nPrediction of extent for January 2025 (in millions of square km): 13.14449923" + "text": "12.3 Regression model fit\nNow that we have a clean dataset, we can use Seaborn’s lmplot() function comparing month vs extent.\nThe lmplot() function from the Seaborn module is intended for exploring linear relationships of different forms in multidimensional datesets. Input data must be in a Pandas DataFrame. To plot them, we provide the predictor (mo) and response (extent) variable names along with the dataset (ice2).\n\nsns.lmplot(ice2, x = \"mo\", y = \"extent\", aspect=2);\n\n# Uncomment below to save the resulting plot.\n#plt.savefig(\"figs/CleanedByMonth.png\", dpi = 300, bbox_inches = 'tight')\n\n\n\n\nAbove you can see ice extent data by month. You can see a monthly fluctuation of the sea ice extent, as would be expected for the different seasons of the year. In order to run regression, and avoid this fluctuation we can normalize data. This will let us see the evolution of the extent over the years.\n\n12.3.1 Normalization\n\n# Compute the mean for each month.\nmonth_means = ice2.groupby('mo').extent.mean()\n\n# Compute the variance for each month.\nmonth_variances = ice2.groupby('mo').extent.var()\n\n# Show the values:\nprint('Means:', month_means)\nprint('\\n') # Add a new line between the two prints, so they are easily distinguishible.\nprint ('Variances:',month_variances)\n\nMeans: mo\n1 14.479429\n2 15.298889\n3 15.491714\n4 14.766000\n5 13.396000\n6 11.860000\n7 9.601143\n8 7.122286\n9 6.404857\n10 8.809143\n11 10.964722\n12 13.059429\nName: extent, dtype: float64\n\n\nVariances: mo\n1 0.304906\n2 0.295804\n3 0.237209\n4 0.215378\n5 0.189901\n6 0.247918\n7 0.679175\n8 0.824577\n9 1.143902\n10 0.630361\n11 0.412511\n12 0.284870\nName: extent, dtype: float64\n\n\nTo capture variation per month, we can compute mean for the i-th interval of time (using 1979-2014) and subtract it from the set of extent values for that month . This can be converted to a relative pecentage difference by dividing it by the total avareage (1979-2014) and multiplying by 100.\n\n# Data normalization\nfor i in range(12):\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n \nsns.lmplot(ice2 , x = \"mo\", y = \"extent\", height = 5.2, aspect = 2);\n#plt.savefig(\"figs/IceExtentNormalizedByMonth.png\", dpi = 300, bbox_inches = 'tight')\n\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n/var/folders/7v/zl9mv52s3ls94kntlt_l9ryh0000gq/T/ipykernel_75501/2141240604.py:3: SettingWithCopyWarning: \nA value is trying to be set on a copy of a slice from a DataFrame\n\nSee the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n ice2.extent[ice2.mo == i+1] = 100*(ice2.extent[ice2.mo == i+1] - month_means[i+1])/month_means.mean()\n\n\n\n\n\n\nprint ('mean:', ice2.extent.mean())\nprint ('var:', ice2.extent.var())\n\nmean: -7.745252569896827e-16\nvar: 31.983239774968798\n\n\nLet us consider the entire year\n\nsns.lmplot(ice2, x = \"year\", y = \"extent\", height = 5.2, aspect = 2);\n#plt.savefig(\"figs/IceExtentAllMonthsByYearlmplot.png\", dpi = 300, bbox_inches = 'tight')\n\n\n\n\n\n\n12.3.2 Pearson’s correlation\nLet’s calculate Pearson’s correlation coefficient and the p-value for testing non-correlation.\n\nimport scipy.stats\nscipy.stats.pearsonr(ice2.year.values, ice2.extent.values)\n\nPearsonRResult(statistic=-0.8183500709897178, pvalue=4.4492318168687107e-103)\n\n\n\n\n12.3.3 Simple OLS\nWe can also compute the trend as a simple linear regression (OLS) and quantitatively evaluate it.\nFor that we use using Scikit-learn, library that provides a variety of both supervised and unsupervised machine learning techniques. Scikit-learn provides an object-oriented interface centered around the concept of an Estimator. The Estimator.fit method sets the state of the estimator based on the training data. Usually, the data is comprised of a two-dimensional numpy array \\(X\\) of shape (n_samples, n_predictors) that holds the so-called feature matrix and a one-dimensional numpy array \\(\\textbf{y}\\) that holds the responses. Some estimators allow the user to control the fitting behavior. For example, the sklearn.linear_model.LinearRegression estimator allows the user to specify whether or not to fit an intercept term. This is done by setting the corresponding constructor arguments of the estimator object. During the fitting process, the state of the estimator is stored in instance attributes that have a trailing underscore (_). For example, the coefficients of a LinearRegression estimator are stored in the attribute coef_.\nEstimators that can generate predictions provide a Estimator.predict method. In the case of regression, Estimator.predict will return the predicted regression values, \\(\\hat{\\textbf{y}}\\).\n\nfrom sklearn.linear_model import LinearRegression\n\nest = LinearRegression(fit_intercept = True)\n\nx = ice2[['year']]\ny = ice2[['extent']]\n\nest.fit(x, y)\n\nprint(\"Coefficients:\", est.coef_)\nprint (\"Intercept:\", est.intercept_)\n\nCoefficients: [[-0.45275459]]\nIntercept: [903.71640207]\n\n\nWe can evaluate the model fitting by computing the mean squared error (\\(MSE\\)) and the coefficient of determination (\\(R^2\\)) of the model. The coefficient \\(R^2\\) is defined as \\((1 - \\textbf{u}/\\textbf{v})\\), where \\(\\textbf{u}\\) is the residual sum of squares \\(\\sum (\\textbf{y} - \\hat{\\textbf{y}})^2\\) and \\(\\textbf{v}\\) is the regression sum of squares \\(\\sum (\\textbf{y} - \\bar{\\textbf{y}})^2\\), where \\(\\bar{\\textbf{y}}\\) is the mean. The best possible score for \\(R^2\\) is 1.0: lower values are worse. These measures can provide a quantitative answer to the question we are facing: Is there a negative trend in the evolution of sea ice extent over recent years?\n\nfrom sklearn import metrics\n\n# Analysis for all months together.\nx = ice2[['year']]\ny = ice2[['extent']]\nmodel = LinearRegression()\nmodel.fit(x, y)\ny_hat = model.predict(x)\nplt.plot(x, y,'o', alpha = 0.5)\nplt.plot(x, y_hat, 'r', alpha = 0.5)\nplt.xlabel('year')\nplt.ylabel('extent (All months)')\nprint (\"MSE:\", metrics.mean_squared_error(y_hat, y))\nprint (\"R^2:\", metrics.r2_score(y_hat, y))\nprint (\"var:\", y.var())\n#plt.savefig(\"figs/IceExtentLinearRegressionAllMonthsByYearPrediction.png\", dpi = 300, bbox_inches = 'tight')\n\nMSE: 10.539131639803488\nR^2: 0.5067870382100226\nvar: extent 31.98324\ndtype: float64\n\n\n\n\n\nWe can conclude that the data show a long-term negative trend in recent years.\n\n# Analysis for a particular month.\n#For January\njan = ice2[ice2.mo == 1];\n\nx = jan[['year']]\ny = jan[['extent']]\n\nmodel = LinearRegression()\nmodel.fit(x, y)\n\ny_hat = model.predict(x)\n\nplt.figure()\nplt.plot(x, y,'-o', alpha = 0.5)\nplt.plot(x, y_hat, 'r', alpha = 0.5)\nplt.xlabel('year')\nplt.ylabel('extent (January)')\n\nprint (\"MSE:\", metrics.mean_squared_error(y_hat, y))\nprint (\"R^2:\", metrics.r2_score(y_hat, y))\n\nMSE: 3.8395160752867565\nR^2: 0.7810636041396216\n\n\n\n\n\nWe can also estimate the extent value for 2025. For that we use the function predict of the model.\n\nX = np.array(2025) \ny_hat = model.predict(X.reshape(-1, 1))\nj = 1 # January\n# Original value (before normalization)\ny_hat = (y_hat*month_means.mean()/100) + month_means[j]\nprint (\"Prediction of extent for January 2025 (in millions of square km):\", y_hat)\n\nPrediction of extent for January 2025 (in millions of square km): [[13.14449923]]\n\n\n/Users/u2071219/anaconda3/envs/IM939/lib/python3.11/site-packages/sklearn/base.py:465: UserWarning: X does not have valid feature names, but LinearRegression was fitted with feature names\n warnings.warn(\n\n\nPrediction of extent for January 2025 (in millions of square km): 13.14449923" }, { "objectID": "content/labs/Lab_3/IM939_Lab_3_2_Linear_Regression.html#more-information", diff --git a/sitemap.xml b/sitemap.xml index 5f802b4..8b3c805 100644 --- a/sitemap.xml +++ b/sitemap.xml @@ -2,178 +2,178 @@ https://warwickcim.github.io/IM939_handbook/index.html - 2023-10-19T14:24:42.858Z + 2023-10-19T15:27:04.455Z https://warwickcim.github.io/IM939_handbook/content/about/teaching_staff.html - 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