updating typos
This commit is contained in:
Morten Hjorth-Jensen
@@ -2,7 +2,7 @@
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@@ -14,7 +14,7 @@
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@@ -27,25 +27,34 @@
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"metadata": {
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"source": [
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"## Overarching aims of the exercises this week\n",
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"\n",
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"This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of [Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer](https://www.springer.com/gp/book/9780387848570)). The exercise is also a part of project 1 and can be reused in the theory part of the project.\n",
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"This exercise deals with various mean values and variances in linear\n",
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"regression method (here it may be useful to look up chapter 3,\n",
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"equation (3.8) of [Trevor Hastie, Robert Tibshirani, Jerome\n",
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"H. Friedman, The Elements of Statistical Learning,\n",
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"Springer](https://www.springer.com/gp/book/9780387848570)). The\n",
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"exercise is also a part of project 1 and can be reused in the theory\n",
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"part of the project.\n",
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"\n",
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"For more discussions on Ridge regression and calculation of expectation values, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n",
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"For more discussions on Ridge regression and calculation of\n",
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"expectation values, [Wessel van\n",
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"Wieringen's](https://arxiv.org/abs/1509.09169) article is highly\n",
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"recommended.\n",
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"\n",
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"The assumption we have made is \n",
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"that there exists a continuous function $f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim N(0, \\sigma^2)$\n",
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"which describes our data"
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"The assumption we have made is that there exists a continuous function\n",
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"$f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim N(0,\n",
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"\\sigma^2)$ which describes our data"
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]
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},
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{
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"cell_type": "markdown",
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@@ -57,7 +66,7 @@
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@@ -68,7 +77,7 @@
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@@ -80,7 +89,7 @@
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@@ -90,7 +99,7 @@
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"cell_type": "markdown",
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"id": "4b1eb811",
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"id": "3b491679",
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@@ -102,7 +111,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "036a5440",
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"id": "93a49739",
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@@ -114,7 +123,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "f7fb8094",
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"id": "a3fa80db",
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@@ -125,7 +134,7 @@
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"id": "d7d92995",
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"id": "e6e2defd",
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"metadata": {
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"editable": true
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@@ -137,7 +146,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "29eb9701",
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"id": "9aa19afa",
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@@ -150,7 +159,7 @@
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{
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"cell_type": "markdown",
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"id": "aa106511",
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"id": "5fa82a47",
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"metadata": {
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"editable": true
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@@ -162,7 +171,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "58be1091",
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"id": "33099abb",
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"metadata": {
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@@ -172,7 +181,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "4cb58e41",
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"id": "5d03f001",
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@@ -184,7 +193,7 @@
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},
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{
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"cell_type": "markdown",
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"id": "a04df28d",
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@@ -195,7 +204,7 @@
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{
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"cell_type": "markdown",
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"id": "f0b84941",
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@@ -207,20 +216,20 @@
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},
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{
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"cell_type": "markdown",
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"id": "5d425a79",
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"id": "60370a63",
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"metadata": {
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},
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"source": [
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"$$\n",
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"\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\\n",
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"\\mathbb{E} \\big[ \\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\\n",
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"\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "fb89cfab",
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"id": "af251217",
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"metadata": {
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"editable": true
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@@ -233,19 +242,19 @@
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},
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{
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"cell_type": "markdown",
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"id": "aea8801c",
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"id": "02e978da",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T}\\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
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"\\mbox{Var}[\\hat{\\boldsymbol{\\beta}}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T}\\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "0e885693",
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File diff suppressed because it is too large
Load Diff
@@ -438,11 +438,20 @@ doconce format html exercisesweek37.do.txt -->
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<p>Date: <strong>Deadline is Sunday September 17 at midnight</strong></p>
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<div class="section" id="overarching-aims-of-the-exercises-this-week">
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<h2>Overarching aims of the exercises this week<a class="headerlink" href="#overarching-aims-of-the-exercises-this-week" title="Permalink to this headline">¶</a></h2>
|
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<p>This exercise deals with various mean values and variances in linear regression method (here it may be useful to look up chapter 3, equation (3.8) of <a class="reference external" href="https://www.springer.com/gp/book/9780387848570">Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer</a>). The exercise is also a part of project 1 and can be reused in the theory part of the project.</p>
|
||||
<p>For more discussions on Ridge regression and calculation of expectation values, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van Wieringen’s</a> article is highly recommended.</p>
|
||||
<p>The assumption we have made is
|
||||
that there exists a continuous function <span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> and a normal distributed error <span class="math notranslate nohighlight">\(\boldsymbol{\varepsilon}\sim N(0, \sigma^2)\)</span>
|
||||
which describes our data</p>
|
||||
<p>This exercise deals with various mean values and variances in linear
|
||||
regression method (here it may be useful to look up chapter 3,
|
||||
equation (3.8) of <a class="reference external" href="https://www.springer.com/gp/book/9780387848570">Trevor Hastie, Robert Tibshirani, Jerome
|
||||
H. Friedman, The Elements of Statistical Learning,
|
||||
Springer</a>). The
|
||||
exercise is also a part of project 1 and can be reused in the theory
|
||||
part of the project.</p>
|
||||
<p>For more discussions on Ridge regression and calculation of
|
||||
expectation values, <a class="reference external" href="https://arxiv.org/abs/1509.09169">Wessel van
|
||||
Wieringen’s</a> article is highly
|
||||
recommended.</p>
|
||||
<p>The assumption we have made is that there exists a continuous function
|
||||
<span class="math notranslate nohighlight">\(f(\boldsymbol{x})\)</span> and a normal distributed error <span class="math notranslate nohighlight">\(\boldsymbol{\varepsilon}\sim N(0,
|
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\sigma^2)\)</span> which describes our data</p>
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||||
<div class="math notranslate nohighlight">
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||||
\[
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\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon}
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@@ -488,7 +497,7 @@ A given parameter <span class="math notranslate nohighlight">\(\beta_j\)</span>
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<p>Show that</p>
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||||
<div class="math notranslate nohighlight">
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||||
\[
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||||
\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\
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||||
\mathbb{E} \big[ \hat{\boldsymbol{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\
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\boldsymbol{\beta}^{\mathrm{OLS}}.
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||||
\]</div>
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<p>We see clearly that
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@@ -496,7 +505,7 @@ A given parameter <span class="math notranslate nohighlight">\(\beta_j\)</span>
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<p>Show also that the variance is</p>
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||||
<div class="math notranslate nohighlight">
|
||||
\[
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||||
\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T}\mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
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\mbox{Var}[\hat{\boldsymbol{\beta}}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T}\mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
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||||
\]</div>
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<p>and it is easy to see that if the parameter <span class="math notranslate nohighlight">\(\lambda\)</span> goes to infinity then the variance of Ridge parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> goes to zero.</p>
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</div>
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File diff suppressed because one or more lines are too long
@@ -1646,7 +1646,7 @@ Since we are not using <strong>Scikit-Learn</strong> here we can define our own
|
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</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
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||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.996535469511469
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9952714635635723
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
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||||
@@ -1663,7 +1663,7 @@ Since we are not using <strong>Scikit-Learn</strong> here we can define our own
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||||
</div>
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||||
</div>
|
||||
<div class="cell_output docutils container">
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||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.0077750600806588575
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.009039197557276654
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1678,23 +1678,31 @@ Since we are not using <strong>Scikit-Learn</strong> here we can define our own
|
||||
</div>
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||||
</div>
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||||
<div class="cell_output docutils container">
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||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.05751737 0.02644748 0.02533184 0.02193035 0.02521356 0.00712899
|
||||
0.00857688 0.0101339 0.01097227 0.01132947 0.01644452 0.01655455
|
||||
0.00771046 0.05177021 0.01374437 0.04208274 0.01667873 0.00228986
|
||||
0.00749579 0.00852965 0.01628823 0.0341899 0.01178253 0.00626339
|
||||
0.00514913 0.00791019 0.00363986 0.00471966 0.00390805 0.00910241
|
||||
0.00232255 0.02670606 0.03224523 0.01637147 0.00918914 0.02705154
|
||||
0.00321694 0.01904583 0.0177181 0.00538287 0.02291121 0.01028255
|
||||
0.04566103 0.00954257 0.00011551 0.0283589 0.00953728 0.03399521
|
||||
0.01205286 0.02251524 0.00353655 0.02344605 0.05288681 0.01950574
|
||||
0.00330907 0.01492855 0.01472766 0.03952093 0.01491299 0.00187154
|
||||
0.02897168 0.00072037 0.00895315 0.02775293 0.01547992 0.04117184
|
||||
0.0116726 0.03330335 0.01515966 0.0070381 0.01317074 0.00654702
|
||||
0.00329264 0.02502101 0.00226569 0.05029894 0.01929643 0.03028379
|
||||
0.053994 0.03413302 0.01854824 0.00393744 0.0600658 0.01855624
|
||||
0.0590702 0.01826743 0.01039549 0.02151219 0.00928016 0.03536062
|
||||
0.00503218 0.08526352 0.00506765 0.02609885 0.04771105 0.0010059
|
||||
0.00659545 0.00143188 0.01489692 0.08237141]
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[6.11430166e-02 7.59428035e-03 7.64295978e-03 5.34442654e-02
|
||||
2.53861506e-02 7.46543658e-02 2.79339933e-02 1.32946226e-02
|
||||
3.80320028e-03 2.40474117e-02 4.36028477e-02 3.24382528e-02
|
||||
1.26022420e-02 5.06848116e-03 1.96793569e-02 1.04826413e-04
|
||||
9.25580178e-03 1.80413812e-02 1.22519792e-02 1.01505087e-02
|
||||
2.67180442e-02 6.71316703e-02 7.68415876e-03 3.33174939e-02
|
||||
1.43570786e-02 8.43459933e-03 4.11046407e-03 9.10453697e-04
|
||||
4.33142087e-03 7.64507671e-02 2.19696497e-02 1.18460525e-02
|
||||
4.35665906e-02 3.62818572e-02 1.61018347e-03 3.78738205e-02
|
||||
4.57031517e-03 3.11279984e-02 1.07745809e-02 7.09848429e-03
|
||||
2.30006341e-03 4.62535124e-02 6.70493970e-02 6.26624267e-03
|
||||
4.23399553e-02 1.88690267e-02 7.46184816e-03 1.75934097e-02
|
||||
3.60820168e-02 1.09322328e-02 3.16749353e-02 2.30626515e-02
|
||||
5.76196895e-04 1.51000420e-02 1.95456434e-03 1.22138768e-02
|
||||
2.52945186e-03 2.12804129e-03 3.32695433e-03 1.90886410e-02
|
||||
1.84291049e-02 1.78104004e-02 6.34498455e-02 1.50032818e-02
|
||||
3.43525003e-02 1.52033394e-03 3.95921014e-02 7.67148008e-02
|
||||
2.40893740e-02 6.75638425e-02 1.22105472e-02 7.72706640e-03
|
||||
3.00721135e-02 8.77081499e-02 1.40684484e-02 4.57081718e-02
|
||||
3.51190686e-02 7.42092776e-02 4.03105744e-02 1.08994018e-02
|
||||
1.37071826e-02 3.69106251e-02 1.94966662e-03 1.71112140e-02
|
||||
3.12669602e-03 3.01610677e-02 1.72139847e-02 2.75703556e-02
|
||||
2.23333773e-02 2.76892410e-03 8.05886306e-02 1.15545946e-01
|
||||
4.88029792e-02 4.59648068e-02 1.98750794e-02 2.69178261e-03
|
||||
1.07925220e-02 1.13151043e-03 5.00620642e-03 3.69871367e-03]
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1763,15 +1771,15 @@ but now splitting the data into a training set and a test set.</p>
|
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</div>
|
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</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 1.95976303 0.48974963 2.82202765 3.79224555 -2.14248744]
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 2.04038959 -0.81343612 7.94304103 -3.83770135 1.70073443]
|
||||
Training R2
|
||||
0.9947613223847728
|
||||
0.9954441212986554
|
||||
Training MSE
|
||||
0.010596520060222926
|
||||
0.008653750612762521
|
||||
Test R2
|
||||
0.989645675273948
|
||||
0.99255249100346
|
||||
Test MSE
|
||||
0.01889099091713886
|
||||
0.016970019460234076
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
@@ -519,6 +519,16 @@ const thebe_selector_output = ".output, .cell_output"
|
||||
Material for lecture Thursday September 7
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#important-technicalities-more-on-rescaling-data">
|
||||
Important technicalities: More on Rescaling data
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#test-function-for-what-happens-with-ols-ridge-and-lasso">
|
||||
Test Function for what happens with OLS, Ridge and Lasso
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">
|
||||
Linking the regression analysis with a statistical interpretation
|
||||
@@ -602,16 +612,6 @@ const thebe_selector_output = ".output, .cell_output"
|
||||
Bayes’ Theorem and Ridge and Lasso Regression
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#test-function-for-what-happens-with-ols-ridge-and-lasso">
|
||||
Test Function for what happens with OLS, Ridge and Lasso
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#invoking-bayes-theorem">
|
||||
Invoking Bayes’ theorem
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#ridge-and-bayes">
|
||||
Ridge and Bayes
|
||||
@@ -790,6 +790,16 @@ const thebe_selector_output = ".output, .cell_output"
|
||||
Material for lecture Thursday September 7
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#important-technicalities-more-on-rescaling-data">
|
||||
Important technicalities: More on Rescaling data
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#test-function-for-what-happens-with-ols-ridge-and-lasso">
|
||||
Test Function for what happens with OLS, Ridge and Lasso
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#linking-the-regression-analysis-with-a-statistical-interpretation">
|
||||
Linking the regression analysis with a statistical interpretation
|
||||
@@ -873,16 +883,6 @@ const thebe_selector_output = ".output, .cell_output"
|
||||
Bayes’ Theorem and Ridge and Lasso Regression
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#test-function-for-what-happens-with-ols-ridge-and-lasso">
|
||||
Test Function for what happens with OLS, Ridge and Lasso
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#invoking-bayes-theorem">
|
||||
Invoking Bayes’ theorem
|
||||
</a>
|
||||
</li>
|
||||
<li class="toc-h2 nav-item toc-entry">
|
||||
<a class="reference internal nav-link" href="#ridge-and-bayes">
|
||||
Ridge and Bayes
|
||||
@@ -920,16 +920,19 @@ doconce format html week36.do.txt --no_mako -->
|
||||
</li>
|
||||
<li><p>Material for the lecture on Thursday September 7</p>
|
||||
<ul>
|
||||
<li><p>Linear Regression and links with Statistics, Resampling methods</p></li>
|
||||
<li><p>Recommended Reading: Goodfellow et al chapter 3 on probability theory, see URL:””</p></li>
|
||||
<li><p>Technicalities related to scaling and other issues with data handling</p></li>
|
||||
<li><p>Linear Regression and links with Statistics</p></li>
|
||||
<li><p><a class="reference external" href="https://www.deeplearningbook.org/">Recommended Reading: Goodfellow et al chapter 3 on probability theory</a></p></li>
|
||||
<li><p>See also Murphy, sections 2.4 (Gaussian distributions) and 3.2 (Bayesian Statistics, basis)</p></li>
|
||||
<li><p><a class="reference external" href="https://youtu.be/Kc20CFK0z7Y">Video of lecture</a></p></li>
|
||||
<li><p><a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep7.pdf">Whiteboard notes</a></p></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
</div>
|
||||
<div class="section" id="material-for-the-active-learning-sessions-tuesday-and-wednesday">
|
||||
<h2>Material for the active learning sessions Tuesday and Wednesday<a class="headerlink" href="#material-for-the-active-learning-sessions-tuesday-and-wednesday" title="Permalink to this headline">¶</a></h2>
|
||||
<p>The material here contains a summary from last Week and discussion of SVD, Ridge and Lasso regression with examples</p>
|
||||
<p>The material here contains a summary from last week and discussion of SVD, Ridge and Lasso regression with examples</p>
|
||||
</div>
|
||||
<div class="section" id="linear-regression-and-the-svd">
|
||||
<h2>Linear Regression and the SVD<a class="headerlink" href="#linear-regression-and-the-svd" title="Permalink to this headline">¶</a></h2>
|
||||
@@ -1700,6 +1703,531 @@ C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\ve
|
||||
<div class="section" id="material-for-lecture-thursday-september-7">
|
||||
<h2>Material for lecture Thursday September 7<a class="headerlink" href="#material-for-lecture-thursday-september-7" title="Permalink to this headline">¶</a></h2>
|
||||
</div>
|
||||
<div class="section" id="important-technicalities-more-on-rescaling-data">
|
||||
<h2>Important technicalities: More on Rescaling data<a class="headerlink" href="#important-technicalities-more-on-rescaling-data" title="Permalink to this headline">¶</a></h2>
|
||||
<p>When you are comparing your own code with for example <strong>Scikit-Learn</strong>’s
|
||||
library, there are some technicalities to keep in mind. The examples
|
||||
here demonstrate some of these aspects with potential pitfalls.</p>
|
||||
<p>The discussion here focuses on the role of the intercept, how we can
|
||||
set up the design matrix, what scaling we should use and other topics
|
||||
which tend confuse us.</p>
|
||||
<p>The intercept can be interpreted as the expected value of our
|
||||
target/output variables when all other predictors are set to zero.
|
||||
Thus, if we cannot assume that the expected outputs/targets are zero
|
||||
when all predictors are zero (the columns in the design matrix), it
|
||||
may be a bad idea to implement a model which penalizes the intercept.
|
||||
Furthermore, in for example Ridge and Lasso regression, the default solutions
|
||||
from the library <strong>Scikit-Learn</strong> (when not shrinking <span class="math notranslate nohighlight">\(\beta_0\)</span>) for the unknown parameters
|
||||
<span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>, are derived under the assumption that both <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> and
|
||||
<span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> are zero centered, that is we subtract the mean values.</p>
|
||||
<p>If our predictors represent different scales, then it is important to
|
||||
standardize the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> by subtracting the mean of each
|
||||
column from the corresponding column and dividing the column with its
|
||||
standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,
|
||||
the results may differ.</p>
|
||||
<p>The
|
||||
<a class="reference external" href="https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html">Standardscaler</a>
|
||||
function in <strong>Scikit-Learn</strong> does this for us. For the data sets we
|
||||
have been studying in our various examples, the data are in many cases
|
||||
already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a
|
||||
survey of your data, with a critical assessment of them in case you need to scale the data.</p>
|
||||
<p>If you need to scale the data, not doing so will give an <em>unfair</em>
|
||||
penalization of the parameters since their magnitude depends on the
|
||||
scale of their corresponding predictor.</p>
|
||||
<p>Suppose as an example that you
|
||||
you have an input variable given by the heights of different persons.
|
||||
Human height might be measured in inches or meters or
|
||||
kilometers. If measured in kilometers, a standard linear regression
|
||||
model with this predictor would probably give a much bigger
|
||||
coefficient term, than if measured in millimeters.
|
||||
This can clearly lead to problems in evaluating the cost/loss functions.</p>
|
||||
<p>Keep in mind that when you transform your data set before training a model, the same transformation needs to be done
|
||||
on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="sd">"""</span>
|
||||
<span class="sd">#Model training, we compute the mean value of y and X</span>
|
||||
<span class="sd">y_train_mean = np.mean(y_train)</span>
|
||||
<span class="sd">X_train_mean = np.mean(X_train,axis=0)</span>
|
||||
<span class="sd">X_train = X_train - X_train_mean</span>
|
||||
<span class="sd">y_train = y_train - y_train_mean</span>
|
||||
|
||||
<span class="sd"># The we fit our model with the training data</span>
|
||||
<span class="sd">trained_model = some_model.fit(X_train,y_train)</span>
|
||||
|
||||
|
||||
<span class="sd">#Model prediction, we need also to transform our data set used for the prediction.</span>
|
||||
<span class="sd">X_test = X_test - X_train_mean #Use mean from training data</span>
|
||||
<span class="sd">y_pred = trained_model(X_test)</span>
|
||||
<span class="sd">y_pred = y_pred + y_train_mean</span>
|
||||
<span class="sd">"""</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>Let us try to understand what this may imply mathematically when we
|
||||
subtract the mean values, also known as <em>zero centering</em>. For
|
||||
simplicity, we will focus on ordinary regression, as done in the above example.</p>
|
||||
<p>The cost/loss function for regression is</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,.
|
||||
\]</div>
|
||||
<p>Recall also that we use the squared value. This expression can lead to an
|
||||
increased penalty for higher differences between predicted and
|
||||
output/target values.</p>
|
||||
<p>What we have done is to single out the <span class="math notranslate nohighlight">\(\beta_0\)</span> term in the
|
||||
definition of the mean squared error (MSE). The design matrix <span class="math notranslate nohighlight">\(X\)</span>
|
||||
does in this case not contain any intercept column. When we take the
|
||||
derivative with respect to <span class="math notranslate nohighlight">\(\beta_0\)</span>, we want the derivative to obey</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\frac{\partial C}{\partial \beta_j} = 0,
|
||||
\]</div>
|
||||
<p>for all <span class="math notranslate nohighlight">\(j\)</span>. For <span class="math notranslate nohighlight">\(\beta_0\)</span> we have</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right).
|
||||
\]</div>
|
||||
<p>Multiplying away the constant <span class="math notranslate nohighlight">\(2/n\)</span>, we obtain</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j.
|
||||
\]</div>
|
||||
<p>Let us specialize first to the case where we have only two parameters <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>.
|
||||
Our result for <span class="math notranslate nohighlight">\(\beta_0\)</span> simplifies then to</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1.
|
||||
\]</div>
|
||||
<p>We obtain then</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}.
|
||||
\]</div>
|
||||
<p>If we define</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mu_{\boldsymbol{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1},
|
||||
\]</div>
|
||||
<p>and the mean value of the outputs as</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i,
|
||||
\]</div>
|
||||
<p>we have</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\beta_0 = \mu_y - \beta_1\mu_{\boldsymbol{x}_1}.
|
||||
\]</div>
|
||||
<p>In the general case with more parameters than <span class="math notranslate nohighlight">\(\beta_0\)</span> and <span class="math notranslate nohighlight">\(\beta_1\)</span>, we have</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j.
|
||||
\]</div>
|
||||
<p>We can rewrite the latter equation as</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\boldsymbol{x}_j}\beta_j,
|
||||
\]</div>
|
||||
<p>where we have defined</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mu_{\boldsymbol{x}_j}=\frac{1}{n}\sum_{i=0}^{n-1} X_{ij},
|
||||
\]</div>
|
||||
<p>the mean value for all elements of the column vector <span class="math notranslate nohighlight">\(\boldsymbol{x}_j\)</span>.</p>
|
||||
<p>Replacing <span class="math notranslate nohighlight">\(y_i\)</span> with <span class="math notranslate nohighlight">\(y_i - y_i - \overline{\boldsymbol{y}}\)</span> and centering also our design matrix results in a cost function (in vector-matrix disguise)</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}).
|
||||
\]</div>
|
||||
<p>If we minimize with respect to <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> we have then</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}},
|
||||
\]</div>
|
||||
<p>where <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}}\)</span>
|
||||
and <span class="math notranslate nohighlight">\(\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj}\)</span>.</p>
|
||||
<p>For Ridge regression we need to add <span class="math notranslate nohighlight">\(\lambda \boldsymbol{\beta}^T\boldsymbol{\beta}\)</span> to the cost function and get then</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}.
|
||||
\]</div>
|
||||
<p>What does this mean? And why do we insist on all this? Let us look at some examples.</p>
|
||||
<p>This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (<em>code example thanks to Øyvind Sigmundson Schøyen</em>). Here our scaling of the data is done by subtracting the mean values only.
|
||||
Note also that we do not split the data into training and test.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
|
||||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
|
||||
|
||||
|
||||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2021</span><span class="p">)</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">MSE</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">y_model</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
|
||||
|
||||
|
||||
<span class="k">def</span> <span class="nf">fit_beta</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y</span>
|
||||
|
||||
|
||||
<span class="n">true_beta</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">3.7</span><span class="p">]</span>
|
||||
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">)</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span>
|
||||
<span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="n">x</span> <span class="o">**</span> <span class="n">p</span> <span class="o">*</span> <span class="n">b</span> <span class="k">for</span> <span class="n">p</span><span class="p">,</span> <span class="n">b</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">true_beta</span><span class="p">)]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span>
|
||||
<span class="p">)</span> <span class="o">+</span> <span class="mf">0.1</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
|
||||
|
||||
<span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">degree</span><span class="p">))</span>
|
||||
|
||||
<span class="c1"># Include the intercept in the design matrix</span>
|
||||
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">degree</span><span class="p">):</span>
|
||||
<span class="n">X</span><span class="p">[:,</span> <span class="n">p</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span> <span class="o">**</span> <span class="n">p</span>
|
||||
|
||||
<span class="n">beta</span> <span class="o">=</span> <span class="n">fit_beta</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||||
|
||||
<span class="c1"># Intercept is included in the design matrix</span>
|
||||
<span class="n">skl</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">(</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"True beta: </span><span class="si">{</span><span class="n">true_beta</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Fitted beta: </span><span class="si">{</span><span class="n">beta</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Sklearn fitted beta: </span><span class="si">{</span><span class="n">skl</span><span class="o">.</span><span class="n">coef_</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||||
<span class="n">ypredictOwn</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta</span>
|
||||
<span class="n">ypredictSKL</span> <span class="o">=</span> <span class="n">skl</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"MSE with intercept column"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">ypredictOwn</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"MSE with intercept column from SKL"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">ypredictSKL</span><span class="p">))</span>
|
||||
|
||||
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Data"</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Fit"</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">skl</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Sklearn (fit_intercept=False)"</span><span class="p">)</span>
|
||||
|
||||
|
||||
<span class="c1"># Do not include the intercept in the design matrix</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span>
|
||||
|
||||
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
|
||||
<span class="n">X</span><span class="p">[:,</span> <span class="n">p</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span> <span class="o">**</span> <span class="p">(</span><span class="n">p</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
|
||||
|
||||
<span class="c1"># Intercept is not included in the design matrix</span>
|
||||
<span class="n">skl</span> <span class="o">=</span> <span class="n">LinearRegression</span><span class="p">(</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||||
|
||||
<span class="c1"># Use centered values for X and y when computing coefficients</span>
|
||||
<span class="n">y_offset</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">average</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||||
<span class="n">X_offset</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">average</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||||
|
||||
<span class="n">beta</span> <span class="o">=</span> <span class="n">fit_beta</span><span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="n">X_offset</span><span class="p">,</span> <span class="n">y</span> <span class="o">-</span> <span class="n">y_offset</span><span class="p">)</span>
|
||||
<span class="n">intercept</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_offset</span> <span class="o">-</span> <span class="n">X_offset</span> <span class="o">@</span> <span class="n">beta</span><span class="p">)</span>
|
||||
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Manual intercept: </span><span class="si">{</span><span class="n">intercept</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Fitted beta (wiothout intercept): </span><span class="si">{</span><span class="n">beta</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Sklearn intercept: </span><span class="si">{</span><span class="n">skl</span><span class="o">.</span><span class="n">intercept_</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Sklearn fitted beta (without intercept): </span><span class="si">{</span><span class="n">skl</span><span class="o">.</span><span class="n">coef_</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
|
||||
<span class="n">ypredictOwn</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta</span>
|
||||
<span class="n">ypredictSKL</span> <span class="o">=</span> <span class="n">skl</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"MSE with Manual intercept"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">ypredictOwn</span><span class="o">+</span><span class="n">intercept</span><span class="p">))</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"MSE with Sklearn intercept"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y</span><span class="p">,</span><span class="n">ypredictSKL</span><span class="p">))</span>
|
||||
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta</span> <span class="o">+</span> <span class="n">intercept</span><span class="p">,</span> <span class="s2">"--"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Fit (manual intercept)"</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">skl</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="s2">"--"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Sklearn (fit_intercept=True)"</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||||
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>The intercept is the value of our output/target variable
|
||||
when all our features are zero and our function crosses the <span class="math notranslate nohighlight">\(y\)</span>-axis (for a one-dimensional case).</p>
|
||||
<p>Printing the MSE, we see first that both methods give the same MSE, as
|
||||
they should. However, when we move to for example Ridge regression,
|
||||
the way we treat the intercept may give a larger or smaller MSE,
|
||||
meaning that the MSE can be penalized by the value of the
|
||||
intercept. Not including the intercept in the fit, means that the
|
||||
regularization term does not include <span class="math notranslate nohighlight">\(\beta_0\)</span>. For different values
|
||||
of <span class="math notranslate nohighlight">\(\lambda\)</span>, this may lead to different MSE values.</p>
|
||||
<p>To remind the reader, the regularization term, with the intercept in Ridge regression, is given by</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2,
|
||||
\]</div>
|
||||
<p>but when we take out the intercept, this equation becomes</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2.
|
||||
\]</div>
|
||||
<p>For Lasso regression we have</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\lambda \vert\vert \boldsymbol{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert.
|
||||
\]</div>
|
||||
<p>It means that, when scaling the design matrix and the outputs/targets,
|
||||
by subtracting the mean values, we have an optimization problem which
|
||||
is not penalized by the intercept. The MSE value can then be smaller
|
||||
since it focuses only on the remaining quantities. If we however bring
|
||||
back the intercept, we will get a MSE which then contains the
|
||||
intercept.</p>
|
||||
<p>Armed with this wisdom, we attempt first to simply set the intercept equal to <strong>False</strong> in our implementation of Ridge regression for our well-known vanilla data set.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">MSE</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">y_model</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
|
||||
|
||||
|
||||
<span class="c1"># A seed just to ensure that the random numbers are the same for every run.</span>
|
||||
<span class="c1"># Useful for eventual debugging.</span>
|
||||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">3155</span><span class="p">)</span>
|
||||
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
|
||||
|
||||
<span class="n">Maxpolydegree</span> <span class="o">=</span> <span class="mi">20</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="n">Maxpolydegree</span><span class="p">))</span>
|
||||
<span class="c1">#We include explicitely the intercept column</span>
|
||||
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Maxpolydegree</span><span class="p">):</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="n">degree</span>
|
||||
<span class="c1"># We split the data in test and training data</span>
|
||||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
|
||||
|
||||
<span class="n">p</span> <span class="o">=</span> <span class="n">Maxpolydegree</span>
|
||||
<span class="n">I</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
|
||||
<span class="c1"># Decide which values of lambda to use</span>
|
||||
<span class="n">nlambdas</span> <span class="o">=</span> <span class="mi">6</span>
|
||||
<span class="n">MSEOwnRidgePredict</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="n">MSERidgePredict</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="n">lambdas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">):</span>
|
||||
<span class="n">lmb</span> <span class="o">=</span> <span class="n">lambdas</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
<span class="n">OwnRidgeBeta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X_train</span><span class="o">+</span><span class="n">lmb</span><span class="o">*</span><span class="n">I</span><span class="p">)</span> <span class="o">@</span> <span class="n">X_train</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y_train</span>
|
||||
<span class="c1"># Note: we include the intercept column and no scaling</span>
|
||||
<span class="n">RegRidge</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">lmb</span><span class="p">,</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||||
<span class="n">RegRidge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="c1"># and then make the prediction</span>
|
||||
<span class="n">ytildeOwnRidge</span> <span class="o">=</span> <span class="n">X_train</span> <span class="o">@</span> <span class="n">OwnRidgeBeta</span>
|
||||
<span class="n">ypredictOwnRidge</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">@</span> <span class="n">OwnRidgeBeta</span>
|
||||
<span class="n">ytildeRidge</span> <span class="o">=</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||||
<span class="n">ypredictRidge</span> <span class="o">=</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||||
<span class="n">MSEOwnRidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictOwnRidge</span><span class="p">)</span>
|
||||
<span class="n">MSERidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictRidge</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Beta values for own Ridge implementation"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">OwnRidgeBeta</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Beta values for Scikit-Learn Ridge implementation"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">RegRidge</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"MSE values for own Ridge implementation"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSEOwnRidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"MSE values for Scikit-Learn Ridge implementation"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSERidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
||||
|
||||
<span class="c1"># Now plot the results</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">MSEOwnRidgePredict</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'MSE own Ridge Test'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">MSERidgePredict</span><span class="p">,</span> <span class="s1">'g'</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'MSE Ridge Test'</span><span class="p">)</span>
|
||||
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'log10(lambda)'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'MSE'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>The results here agree when we force <strong>Scikit-Learn</strong>’s Ridge function to include the first column in our design matrix.
|
||||
We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.
|
||||
What happens if we do not include the intercept in our fit?
|
||||
Let us see how we can change this code by zero centering.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">MSE</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">y_model</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
|
||||
<span class="c1"># A seed just to ensure that the random numbers are the same for every run.</span>
|
||||
<span class="c1"># Useful for eventual debugging.</span>
|
||||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">315</span><span class="p">)</span>
|
||||
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
|
||||
|
||||
<span class="n">Maxpolydegree</span> <span class="o">=</span> <span class="mi">20</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="n">Maxpolydegree</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
|
||||
|
||||
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">Maxpolydegree</span><span class="p">):</span> <span class="c1">#No intercept column</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="n">degree</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">degree</span><span class="p">)</span>
|
||||
|
||||
<span class="c1"># We split the data in test and training data</span>
|
||||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
|
||||
|
||||
<span class="c1">#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable</span>
|
||||
<span class="n">X_train_mean</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||||
<span class="c1">#Center by removing mean from each feature</span>
|
||||
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">X_train</span> <span class="o">-</span> <span class="n">X_train_mean</span>
|
||||
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">-</span> <span class="n">X_train_mean</span>
|
||||
<span class="c1">#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)</span>
|
||||
<span class="c1">#Remove the intercept from the training data.</span>
|
||||
<span class="n">y_scaler</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="n">y_train_scaled</span> <span class="o">=</span> <span class="n">y_train</span> <span class="o">-</span> <span class="n">y_scaler</span>
|
||||
|
||||
<span class="n">p</span> <span class="o">=</span> <span class="n">Maxpolydegree</span><span class="o">-</span><span class="mi">1</span>
|
||||
<span class="n">I</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="n">p</span><span class="p">)</span>
|
||||
<span class="c1"># Decide which values of lambda to use</span>
|
||||
<span class="n">nlambdas</span> <span class="o">=</span> <span class="mi">6</span>
|
||||
<span class="n">MSEOwnRidgePredict</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="n">MSERidgePredict</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">)</span>
|
||||
|
||||
<span class="n">lambdas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">):</span>
|
||||
<span class="n">lmb</span> <span class="o">=</span> <span class="n">lambdas</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
<span class="n">OwnRidgeBeta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X_train_scaled</span><span class="o">+</span><span class="n">lmb</span><span class="o">*</span><span class="n">I</span><span class="p">)</span> <span class="o">@</span> <span class="n">X_train_scaled</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="p">(</span><span class="n">y_train_scaled</span><span class="p">)</span>
|
||||
<span class="n">intercept_</span> <span class="o">=</span> <span class="n">y_scaler</span> <span class="o">-</span> <span class="n">X_train_mean</span><span class="nd">@OwnRidgeBeta</span> <span class="c1">#The intercept can be shifted so the model can predict on uncentered data</span>
|
||||
<span class="c1">#Add intercept to prediction</span>
|
||||
<span class="n">ypredictOwnRidge</span> <span class="o">=</span> <span class="n">X_test_scaled</span> <span class="o">@</span> <span class="n">OwnRidgeBeta</span> <span class="o">+</span> <span class="n">y_scaler</span>
|
||||
<span class="n">RegRidge</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">lmb</span><span class="p">)</span>
|
||||
<span class="n">RegRidge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="n">ypredictRidge</span> <span class="o">=</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||||
<span class="n">MSEOwnRidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictOwnRidge</span><span class="p">)</span>
|
||||
<span class="n">MSERidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictRidge</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Beta values for own Ridge implementation"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">OwnRidgeBeta</span><span class="p">)</span> <span class="c1">#Intercept is given by mean of target variable</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Beta values for Scikit-Learn Ridge implementation"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">RegRidge</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Intercept from own implementation:'</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">intercept_</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Intercept from Scikit-Learn Ridge implementation'</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">RegRidge</span><span class="o">.</span><span class="n">intercept_</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"MSE values for own Ridge implementation"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSEOwnRidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"MSE values for Scikit-Learn Ridge implementation"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSERidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
||||
|
||||
|
||||
<span class="c1"># Now plot the results</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">MSEOwnRidgePredict</span><span class="p">,</span> <span class="s1">'b--'</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'MSE own Ridge Test'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">MSERidgePredict</span><span class="p">,</span> <span class="s1">'g--'</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'MSE SL Ridge Test'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'log10(lambda)'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'MSE'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>We see here, when compared to the code which includes explicitely the
|
||||
intercept column, that our MSE value is actually smaller. This is
|
||||
because the regularization term does not include the intercept value
|
||||
<span class="math notranslate nohighlight">\(\beta_0\)</span> in the fitting. This applies to Lasso regularization as
|
||||
well. It means that our optimization is now done only with the
|
||||
centered matrix and/or vector that enter the fitting procedure.</p>
|
||||
</div>
|
||||
<div class="section" id="test-function-for-what-happens-with-ols-ridge-and-lasso">
|
||||
<h2>Test Function for what happens with OLS, Ridge and Lasso<a class="headerlink" href="#test-function-for-what-happens-with-ols-ridge-and-lasso" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Hitherto we have discussed Ridge and Lasso regression in terms of a
|
||||
linear analysis. This may to many of you feel rather technical and
|
||||
perhaps not that intuitive. The question is whether we can develop a
|
||||
more intuitive way of understanding what Ridge and Lasso express.</p>
|
||||
<p>Before we proceed let us perform a Ridge, Lasso and OLS analysis of a polynomial fit.</p>
|
||||
<p>We will play around with a study of the values for the optimal
|
||||
parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> using OLS, Ridge and Lasso regression. For
|
||||
OLS, you will notice as function of the noise and polynomial degree,
|
||||
that the parameters <span class="math notranslate nohighlight">\(\beta\)</span> will fluctuate from order to order in the
|
||||
polynomial fit and that for larger and larger polynomial degrees of freedom, the parameters will tend to increase in value for OLS.</p>
|
||||
<p>For Ridge and Lasso regression, the higher order parameters will typically be reduced, providing thereby less fluctuations from one order to another one.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">R2</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span> <span class="n">y_model</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">y_model</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_data</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">def</span> <span class="nf">MSE</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">y_model</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
|
||||
|
||||
<span class="c1"># Make data set.</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="mi">10000</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
|
||||
<span class="n">Maxpolydegree</span> <span class="o">=</span> <span class="mi">5</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">Maxpolydegree</span><span class="p">))</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||||
|
||||
|
||||
<span class="k">for</span> <span class="n">polydegree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">Maxpolydegree</span><span class="p">):</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="n">polydegree</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">polydegree</span><span class="p">)</span>
|
||||
|
||||
<span class="c1"># We split the data in test and training data</span>
|
||||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
|
||||
|
||||
<span class="c1"># matrix inversion to find beta</span>
|
||||
<span class="n">OLSbeta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X_train</span><span class="p">)</span> <span class="o">@</span> <span class="n">X_train</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y_train</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">OLSbeta</span><span class="p">)</span>
|
||||
<span class="n">ypredictOLS</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">@</span> <span class="n">OLSbeta</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test MSE OLS"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictOLS</span><span class="p">))</span>
|
||||
<span class="c1"># Repeat now for Lasso and Ridge regression and various values of the regularization parameter using Scikit-Learn</span>
|
||||
<span class="c1"># Decide which values of lambda to use</span>
|
||||
<span class="n">nlambdas</span> <span class="o">=</span> <span class="mi">4</span>
|
||||
<span class="n">MSERidgePredict</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="n">MSELassoPredict</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="n">lambdas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">):</span>
|
||||
<span class="n">lmb</span> <span class="o">=</span> <span class="n">lambdas</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
<span class="c1"># Make the fit using Ridge and Lasso</span>
|
||||
<span class="n">RegRidge</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">lmb</span><span class="p">,</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||||
<span class="n">RegRidge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="n">RegLasso</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Lasso</span><span class="p">(</span><span class="n">lmb</span><span class="p">,</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||||
<span class="n">RegLasso</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="c1"># and then make the prediction</span>
|
||||
<span class="n">ypredictRidge</span> <span class="o">=</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||||
<span class="n">ypredictLasso</span> <span class="o">=</span> <span class="n">RegLasso</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||||
<span class="c1"># Compute the MSE and print it</span>
|
||||
<span class="n">MSERidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictRidge</span><span class="p">)</span>
|
||||
<span class="n">MSELassoPredict</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictLasso</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">lmb</span><span class="p">,</span><span class="n">RegRidge</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">lmb</span><span class="p">,</span><span class="n">RegLasso</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
|
||||
<span class="c1"># Now plot the results</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">MSERidgePredict</span><span class="p">,</span> <span class="s1">'b'</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'MSE Ridge Test'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">MSELassoPredict</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'MSE Lasso Test'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'log10(lambda)'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'MSE'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>How can we understand this?</p>
|
||||
</div>
|
||||
<div class="section" id="linking-the-regression-analysis-with-a-statistical-interpretation">
|
||||
<h2>Linking the regression analysis with a statistical interpretation<a class="headerlink" href="#linking-the-regression-analysis-with-a-statistical-interpretation" title="Permalink to this headline">¶</a></h2>
|
||||
<p>We will now couple the discussions of ordinary least squares, Ridge
|
||||
@@ -2036,93 +2564,6 @@ p(Y=1\vert X=1)=\frac{p(X=1\vert Y=1)p(Y=1)}{p(X=1\vert Y=1)p(Y=1)+p(X=1\vert Y=
|
||||
</div>
|
||||
<div class="section" id="bayes-theorem-and-ridge-and-lasso-regression">
|
||||
<h2>Bayes’ Theorem and Ridge and Lasso Regression<a class="headerlink" href="#bayes-theorem-and-ridge-and-lasso-regression" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Hitherto we have discussed Ridge and Lasso regression in terms of a
|
||||
linear analysis. This may to many of you feel rather technical and
|
||||
perhaps not that intuitive. The question is whether we can develop a
|
||||
more intuitive way of understanding what Ridge and Lasso express.</p>
|
||||
<p>Before we proceed let us perform a Ridge, Lasso and OLS analysis of a polynomial fit.</p>
|
||||
</div>
|
||||
<div class="section" id="test-function-for-what-happens-with-ols-ridge-and-lasso">
|
||||
<h2>Test Function for what happens with OLS, Ridge and Lasso<a class="headerlink" href="#test-function-for-what-happens-with-ols-ridge-and-lasso" title="Permalink to this headline">¶</a></h2>
|
||||
<p>We will play around with a study of the values for the optimal
|
||||
parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> using OLS, Ridge and Lasso regression. For
|
||||
OLS, you will notice as function of the noise and polynomial degree,
|
||||
that the parameters <span class="math notranslate nohighlight">\(\beta\)</span> will fluctuate from order to order in the
|
||||
polynomial fit and that for larger and larger polynomial degrees of freedom, the parameters will tend to increase in value for OLS.</p>
|
||||
<p>For Ridge and Lasso regression, the higher order parameters will typically be reduced, providing thereby less fluctuations from one order to another one.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
|
||||
|
||||
<span class="k">def</span> <span class="nf">R2</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span> <span class="n">y_model</span><span class="p">):</span>
|
||||
<span class="k">return</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">y_model</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_data</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">def</span> <span class="nf">MSE</span><span class="p">(</span><span class="n">y_data</span><span class="p">,</span><span class="n">y_model</span><span class="p">):</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="n">y_model</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">((</span><span class="n">y_data</span><span class="o">-</span><span class="n">y_model</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
|
||||
|
||||
<span class="c1"># Make data set.</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="mi">10000</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
|
||||
|
||||
<span class="n">Maxpolydegree</span> <span class="o">=</span> <span class="mi">5</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="n">Maxpolydegree</span><span class="p">))</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
|
||||
|
||||
<span class="k">for</span> <span class="n">polydegree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">Maxpolydegree</span><span class="p">):</span>
|
||||
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">polydegree</span><span class="p">):</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">degree</span><span class="p">)</span>
|
||||
|
||||
|
||||
<span class="c1"># We split the data in test and training data</span>
|
||||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
|
||||
|
||||
<span class="c1"># matrix inversion to find beta</span>
|
||||
<span class="n">OLSbeta</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X_train</span><span class="p">)</span> <span class="o">@</span> <span class="n">X_train</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">y_train</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">OLSbeta</span><span class="p">)</span>
|
||||
<span class="n">ypredictOLS</span> <span class="o">=</span> <span class="n">X_test</span> <span class="o">@</span> <span class="n">OLSbeta</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test MSE OLS"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictOLS</span><span class="p">))</span>
|
||||
<span class="c1"># Repeat now for Lasso and Ridge regression and various values of the regularization parameter using Scikit-Learn</span>
|
||||
<span class="c1"># Decide which values of lambda to use</span>
|
||||
<span class="n">nlambdas</span> <span class="o">=</span> <span class="mi">4</span>
|
||||
<span class="n">MSERidgePredict</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="n">MSELassoPredict</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="n">lambdas</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">nlambdas</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">nlambdas</span><span class="p">):</span>
|
||||
<span class="n">lmb</span> <span class="o">=</span> <span class="n">lambdas</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
|
||||
<span class="c1"># Make the fit using Ridge and Lasso</span>
|
||||
<span class="n">RegRidge</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">lmb</span><span class="p">,</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||||
<span class="n">RegRidge</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="n">RegLasso</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Lasso</span><span class="p">(</span><span class="n">lmb</span><span class="p">,</span><span class="n">fit_intercept</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||||
<span class="n">RegLasso</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="c1"># and then make the prediction</span>
|
||||
<span class="n">ypredictRidge</span> <span class="o">=</span> <span class="n">RegRidge</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||||
<span class="n">ypredictLasso</span> <span class="o">=</span> <span class="n">RegLasso</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||||
<span class="c1"># Compute the MSE and print it</span>
|
||||
<span class="n">MSERidgePredict</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictRidge</span><span class="p">)</span>
|
||||
<span class="n">MSELassoPredict</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">MSE</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span><span class="n">ypredictLasso</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">lmb</span><span class="p">,</span><span class="n">RegRidge</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">lmb</span><span class="p">,</span><span class="n">RegLasso</span><span class="o">.</span><span class="n">coef_</span><span class="p">)</span>
|
||||
<span class="c1"># Now plot the results</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">MSERidgePredict</span><span class="p">,</span> <span class="s1">'b'</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'MSE Ridge Test'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">lambdas</span><span class="p">),</span> <span class="n">MSELassoPredict</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">,</span> <span class="n">label</span> <span class="o">=</span> <span class="s1">'MSE Lasso Test'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">'log10(lambda)'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">'MSE'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>How can we understand this?</p>
|
||||
</div>
|
||||
<div class="section" id="invoking-bayes-theorem">
|
||||
<h2>Invoking Bayes’ theorem<a class="headerlink" href="#invoking-bayes-theorem" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Using Bayes’ theorem we can gain a better intuition about Ridge and Lasso regression.</p>
|
||||
<p>For ordinary least squares we postulated that the maximum likelihood for the doamin of events <span class="math notranslate nohighlight">\(\boldsymbol{D}\)</span> (one-dimensional case)</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
|
||||
Reference in New Issue
Block a user