From e3397fde9f5fc57405bd9494d7e003acc22d5d14 Mon Sep 17 00:00:00 2001 From: Quarto GHA Workflow Runner Date: Mon, 14 Oct 2024 21:48:32 +0000 Subject: [PATCH] Built site for gh-pages --- .nojekyll | 2 +- schedule/slides/15-LDA-and-QDA.html | 27 +- .../figure-revealjs/3class-comparison-1.png | Bin 76191 -> 47877 bytes .../figure-revealjs/3class-plot-1.png | Bin 93017 -> 59556 bytes .../figure-revealjs/plot-d1-1.png | Bin 43730 -> 24956 bytes .../figure-revealjs/qda-vs-lda-2class-1.png | Bin 68311 -> 39415 bytes .../figure-revealjs/unnamed-chunk-2-1.svg | 779 +++++++++--------- search.json | 8 +- sitemap.xml | 96 +-- 9 files changed, 456 insertions(+), 456 deletions(-) diff --git a/.nojekyll b/.nojekyll index 6026aa1..b6b86c4 100644 --- a/.nojekyll +++ b/.nojekyll @@ -1 +1 @@ -a9c76953 \ No newline at end of file +a6b74f38 \ No newline at end of file diff --git a/schedule/slides/15-LDA-and-QDA.html b/schedule/slides/15-LDA-and-QDA.html index 399d0fc..1dc0b37 100644 --- a/schedule/slides/15-LDA-and-QDA.html +++ b/schedule/slides/15-LDA-and-QDA.html @@ -398,7 +398,7 @@

15 LDA and QDA

Stat 406

Geoff Pleiss, Trevor Campbell

-

Last modified – 09 October 2023

+

Last modified – 14 October 2024

\[ \DeclareMathOperator*{\argmin}{argmin} \DeclareMathOperator*{\argmax}{argmax} @@ -424,33 +424,34 @@

15 LDA and QDA

\newcommand{\bls}{\widehat{\beta}_{ols}} \newcommand{\blt}{\widehat{\beta}^L_{s}} \newcommand{\bll}{\widehat{\beta}^L_{\lambda}} +\newcommand{\U}{\mathbf{U}} +\newcommand{\D}{\mathbf{D}} +\newcommand{\V}{\mathbf{V}} \]

Last time

We showed that with two classes, the Bayes’ classifier is

-

\[g_*(X) = \begin{cases} -1 & \textrm{ if } \frac{p_1(X)}{p_0(X)} > \frac{1-\pi}{\pi} \\ +

\[g_*(x) = \begin{cases} +1 & \textrm{ if } \frac{p_1(x)}{p_0(x)} > \frac{1-\pi}{\pi} \\ 0 & \textrm{ otherwise} \end{cases}\]

-

where \(p_1(X) = Pr(X \given Y=1)\), \(p_0(X) = Pr(X \given Y=0)\) and \(\pi = Pr(Y=1)\)

+

where \(p_1(x) = \Pr(X=x \given Y=1)\), \(p_0(x) = \Pr(X=x \given Y=0)\) and \(\pi = \Pr(Y=1)\)

-

For more than two classes.

-

\[g_*(X) = -\argmax_k \frac{\pi_k p_k(X)}{\sum_k \pi_k p_k(X)}\]

-

where \(p_k(X) = Pr(X \given Y=k)\) and \(\pi_k = P(Y=k)\)

+

For more than two classes:

+

\[g_*(x) = +\argmax_k \frac{\pi_k p_k(x)}{\sum_k \pi_k p_k(x)}\]

+

where \(p_k(x) = \Pr(X=x \given Y=k)\) and \(\pi_k = P(Y=k)\)

Estimating these

Let’s make some assumptions:

    -
  1. \(Pr(X\given Y=k) = \mbox{N}(\mu_k,\Sigma_k)\)
  2. +
  3. \(\Pr(X=x\given Y=k) = \mbox{N}(x; \mu_k,\Sigma_k)\)
  4. \(\Sigma_k = \Sigma_{k'} = \Sigma\)
-

This leads to Linear Discriminant Analysis (LDA), one of the oldest classifiers

-

LDA

@@ -606,10 +607,10 @@

Naïve Bayes

-

Assume that \(Pr(X | Y = k) = Pr(X_1 | Y = k)\cdots Pr(X_p | Y = k)\).

+

Assume that \(\Pr(X=x | Y = k) = \Pr(X_1=x_1 | Y = k)\cdots \Pr(X_p=x_p | Y = k)\).

That is, conditional on the class, the feature distribution is independent.

-

If we further assume that \(Pr(X_j | Y = k)\) is Gaussian,

+

If we further assume that \(\Pr(X_j=x_j | Y = k)\) is Gaussian,

This is the same as QDA but with \(\Sigma_k\) Diagonal.

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2023\n\\[\n\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\minimize}{minimize}\n\\DeclareMathOperator*{\\maximize}{maximize}\n\\DeclareMathOperator*{\\find}{find}\n\\DeclareMathOperator{\\st}{subject\\,\\,to}\n\\newcommand{\\E}{E}\n\\newcommand{\\Expect}[1]{\\E\\left[ #1 \\right]}\n\\newcommand{\\Var}[1]{\\mathrm{Var}\\left[ #1 \\right]}\n\\newcommand{\\Cov}[2]{\\mathrm{Cov}\\left[#1,\\ #2\\right]}\n\\newcommand{\\given}{\\ \\vert\\ }\n\\newcommand{\\X}{\\mathbf{X}}\n\\newcommand{\\x}{\\mathbf{x}}\n\\newcommand{\\y}{\\mathbf{y}}\n\\newcommand{\\P}{\\mathcal{P}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\norm}[1]{\\left\\lVert #1 \\right\\rVert}\n\\newcommand{\\snorm}[1]{\\lVert #1 \\rVert}\n\\newcommand{\\tr}[1]{\\mbox{tr}(#1)}\n\\newcommand{\\brt}{\\widehat{\\beta}^R_{s}}\n\\newcommand{\\brl}{\\widehat{\\beta}^R_{\\lambda}}\n\\newcommand{\\bls}{\\widehat{\\beta}_{ols}}\n\\newcommand{\\blt}{\\widehat{\\beta}^L_{s}}\n\\newcommand{\\bll}{\\widehat{\\beta}^L_{\\lambda}}\n\\]" + "text": "15 LDA and QDA\nStat 406\nGeoff Pleiss, Trevor Campbell\nLast modified – 14 October 2024\n\\[\n\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\minimize}{minimize}\n\\DeclareMathOperator*{\\maximize}{maximize}\n\\DeclareMathOperator*{\\find}{find}\n\\DeclareMathOperator{\\st}{subject\\,\\,to}\n\\newcommand{\\E}{E}\n\\newcommand{\\Expect}[1]{\\E\\left[ #1 \\right]}\n\\newcommand{\\Var}[1]{\\mathrm{Var}\\left[ #1 \\right]}\n\\newcommand{\\Cov}[2]{\\mathrm{Cov}\\left[#1,\\ #2\\right]}\n\\newcommand{\\given}{\\ \\vert\\ }\n\\newcommand{\\X}{\\mathbf{X}}\n\\newcommand{\\x}{\\mathbf{x}}\n\\newcommand{\\y}{\\mathbf{y}}\n\\newcommand{\\P}{\\mathcal{P}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\norm}[1]{\\left\\lVert #1 \\right\\rVert}\n\\newcommand{\\snorm}[1]{\\lVert #1 \\rVert}\n\\newcommand{\\tr}[1]{\\mbox{tr}(#1)}\n\\newcommand{\\brt}{\\widehat{\\beta}^R_{s}}\n\\newcommand{\\brl}{\\widehat{\\beta}^R_{\\lambda}}\n\\newcommand{\\bls}{\\widehat{\\beta}_{ols}}\n\\newcommand{\\blt}{\\widehat{\\beta}^L_{s}}\n\\newcommand{\\bll}{\\widehat{\\beta}^L_{\\lambda}}\n\\newcommand{\\U}{\\mathbf{U}}\n\\newcommand{\\D}{\\mathbf{D}}\n\\newcommand{\\V}{\\mathbf{V}}\n\\]" }, { "objectID": "schedule/slides/15-LDA-and-QDA.html#last-time", "href": "schedule/slides/15-LDA-and-QDA.html#last-time", "title": "UBC Stat406 2024W", "section": "Last time", - "text": "Last time\nWe showed that with two classes, the Bayes’ classifier is\n\\[g_*(X) = \\begin{cases}\n1 & \\textrm{ if } \\frac{p_1(X)}{p_0(X)} > \\frac{1-\\pi}{\\pi} \\\\\n0 & \\textrm{ otherwise}\n\\end{cases}\\]\nwhere \\(p_1(X) = Pr(X \\given Y=1)\\), \\(p_0(X) = Pr(X \\given Y=0)\\) and \\(\\pi = Pr(Y=1)\\)\n\nFor more than two classes.\n\\[g_*(X) =\n\\argmax_k \\frac{\\pi_k p_k(X)}{\\sum_k \\pi_k p_k(X)}\\]\nwhere \\(p_k(X) = Pr(X \\given Y=k)\\) and \\(\\pi_k = P(Y=k)\\)" + "text": "Last time\nWe showed that with two classes, the Bayes’ classifier is\n\\[g_*(x) = \\begin{cases}\n1 & \\textrm{ if } \\frac{p_1(x)}{p_0(x)} > \\frac{1-\\pi}{\\pi} \\\\\n0 & \\textrm{ otherwise}\n\\end{cases}\\]\nwhere \\(p_1(x) = \\Pr(X=x \\given Y=1)\\), \\(p_0(x) = \\Pr(X=x \\given Y=0)\\) and \\(\\pi = \\Pr(Y=1)\\)\n\nFor more than two classes:\n\\[g_*(x) =\n\\argmax_k \\frac{\\pi_k p_k(x)}{\\sum_k \\pi_k p_k(x)}\\]\nwhere \\(p_k(x) = \\Pr(X=x \\given Y=k)\\) and \\(\\pi_k = P(Y=k)\\)" }, { "objectID": "schedule/slides/15-LDA-and-QDA.html#estimating-these", "href": "schedule/slides/15-LDA-and-QDA.html#estimating-these", "title": "UBC Stat406 2024W", "section": "Estimating these", - "text": "Estimating these\nLet’s make some assumptions:\n\n\\(Pr(X\\given Y=k) = \\mbox{N}(\\mu_k,\\Sigma_k)\\)\n\\(\\Sigma_k = \\Sigma_{k'} = \\Sigma\\)\n\n\nThis leads to Linear Discriminant Analysis (LDA), one of the oldest classifiers" + "text": "Estimating these\nLet’s make some assumptions:\n\n\\(\\Pr(X=x\\given Y=k) = \\mbox{N}(x; \\mu_k,\\Sigma_k)\\)\n\\(\\Sigma_k = \\Sigma_{k'} = \\Sigma\\)\n\nThis leads to Linear Discriminant Analysis (LDA), one of the oldest classifiers" }, { "objectID": "schedule/slides/15-LDA-and-QDA.html#lda", @@ -3833,7 +3833,7 @@ "href": "schedule/slides/15-LDA-and-QDA.html#naïve-bayes", "title": "UBC Stat406 2024W", "section": "Naïve Bayes", - "text": "Naïve Bayes\nAssume that \\(Pr(X | Y = k) = Pr(X_1 | Y = k)\\cdots Pr(X_p | Y = k)\\).\nThat is, conditional on the class, the feature distribution is independent.\n\nIf we further assume that \\(Pr(X_j | Y = k)\\) is Gaussian,\nThis is the same as QDA but with \\(\\Sigma_k\\) Diagonal.\n\n\nDon’t have to assume Gaussian. Could do lots of stuff." + "text": "Naïve Bayes\nAssume that \\(\\Pr(X=x | Y = k) = \\Pr(X_1=x_1 | Y = k)\\cdots \\Pr(X_p=x_p | Y = k)\\).\nThat is, conditional on the class, the feature distribution is independent.\n\nIf we further assume that \\(\\Pr(X_j=x_j | Y = k)\\) is Gaussian,\nThis is the same as QDA but with \\(\\Sigma_k\\) Diagonal.\n\n\nDon’t have to assume Gaussian. 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