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Morten Hjorth-Jensen
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"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
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"doconce format html exercisesweek38.do.txt -->\n",
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"source": [
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"# Exercises week 38\n",
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||||
"**September 18-22, 2023**\n",
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||||
"\n",
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||||
"Date: **Deadline is Sunday September 24 at midnight**"
|
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]
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},
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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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||||
"The aim of the exercises this week is to derive the equations for the bias-variance tradeoff to be used in project 1 as well as testing this for a simpler function using the bootstrap method. The exercises here can be reused in project 1 as well.\n",
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"\n",
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||||
"Consider a\n",
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||||
"dataset $\\mathcal{L}$ consisting of the data\n",
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||||
"$\\mathbf{X}_\\mathcal{L}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$.\n",
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||||
"\n",
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||||
"We assume that the true data is generated from a noisy model"
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]
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},
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"id": "d08c4671",
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"metadata": {
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"editable": true
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"source": [
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||||
"$$\n",
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||||
"\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}.\n",
|
||||
"$$"
|
||||
]
|
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},
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{
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"cell_type": "markdown",
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"id": "56f9ca3e",
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"metadata": {
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},
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"source": [
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||||
"Here $\\epsilon$ is normally distributed with mean zero and standard\n",
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||||
"deviation $\\sigma^2$.\n",
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||||
"\n",
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||||
"In our derivation of the ordinary least squares method we defined \n",
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||||
"an approximation to the function $f$ in terms of the parameters\n",
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||||
"$\\boldsymbol{\\beta}$ and the design matrix $\\boldsymbol{X}$ which embody our model,\n",
|
||||
"that is $\\boldsymbol{\\tilde{y}}=\\boldsymbol{X}\\boldsymbol{\\beta}$.\n",
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||||
"\n",
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||||
"The parameters $\\boldsymbol{\\beta}$ are in turn found by optimizing the mean\n",
|
||||
"squared error via the so-called cost function"
|
||||
]
|
||||
},
|
||||
{
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"cell_type": "markdown",
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"id": "ae36b494",
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"editable": true
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},
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"source": [
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||||
"$$\n",
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||||
"C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n",
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||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
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||||
"id": "be3eadf1",
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"metadata": {
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||||
"editable": true
|
||||
},
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||||
"source": [
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||||
"Here the expected value $\\mathbb{E}$ is the sample value. \n",
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"\n",
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||||
"Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a\n",
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"term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.\n",
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"That is, show that"
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||||
]
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||||
},
|
||||
{
|
||||
"cell_type": "markdown",
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"id": "ae9ebea0",
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"metadata": {
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"editable": true
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},
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||||
"source": [
|
||||
"$$\n",
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||||
"\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=(\\mathrm{Bias}[\\tilde{y}])^2+\\mathrm{var}[\\tilde{f}]+\\sigma^2,\n",
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||||
"$$"
|
||||
]
|
||||
},
|
||||
{
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||||
"cell_type": "markdown",
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"id": "d1cbae1b",
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"metadata": {
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},
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"source": [
|
||||
"with"
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||||
]
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||||
},
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{
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"cell_type": "markdown",
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"id": "d2e1f899",
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"metadata": {
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"editable": true
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},
|
||||
"source": [
|
||||
"$$\n",
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||||
"(\\mathrm{Bias}[\\tilde{y}])^2=\\left(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2,\n",
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||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
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||||
"id": "0486221c",
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||||
"metadata": {
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"editable": true
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},
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"source": [
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"and"
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]
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},
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{
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"cell_type": "markdown",
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"id": "86746df2",
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"metadata": {
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"editable": true
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||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{var}[\\tilde{f}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
|
||||
"$$"
|
||||
]
|
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},
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{
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"cell_type": "markdown",
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"id": "9aa6d3dc",
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"metadata": {
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||||
},
|
||||
"source": [
|
||||
"Explain what the terms mean and discuss their interpretations.\n",
|
||||
"\n",
|
||||
"Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by\n",
|
||||
"studying the MSE value as function of the complexity of your model. Use ordinary least squares only.\n",
|
||||
"\n",
|
||||
"Discuss the bias and variance trade-off as function\n",
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||||
"of your model complexity (the degree of the polynomial) and the number\n",
|
||||
"of data points, and possibly also your training and test data using the **bootstrap** resampling method.\n",
|
||||
"You can follow the code example in the jupyter-book at <https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff>.\n",
|
||||
"\n",
|
||||
"See also the whiteboard notes from week 37 at <https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf>"
|
||||
]
|
||||
}
|
||||
],
|
||||
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"nbformat": 4,
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"nbformat_minor": 5
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}
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Applied Data Analysis and Machine Learning
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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From Regression to Support Vector Machines
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3. Linear Regression
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4. Ridge and Lasso Regression
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5. Resampling Methods
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6. Logistic Regression
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7. Optimization, the central part of any Machine Learning algortithm
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8. Support Vector Machines, overarching aims
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</a>
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<p aria-level="2" class="caption" role="heading">
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Decision Trees, Ensemble Methods and Boosting
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9. Decision trees, overarching aims
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<a class="reference internal" href="chapter7.html">
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
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||||
</a>
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||||
</ul>
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<p aria-level="2" class="caption" role="heading">
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Dimensionality Reduction
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11. Basic ideas of the Principal Component Analysis (PCA)
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12. Clustering and Unsupervised Learning
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Deep Learning Methods
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13. Neural networks
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14. Building a Feed Forward Neural Network
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15. Solving Differential Equations with Deep Learning
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16. Convolutional Neural Networks
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||||
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||||
<a class="reference internal" href="chapter13.html">
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||||
17. Recurrent neural networks: Overarching view
|
||||
</a>
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||||
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|
||||
</ul>
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Week 34: Introduction to the course, Logistics and Practicalities
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Week 36: Statistical interpretation of Linear Regression and Resampling techniques
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<h1>Exercises week 38</h1>
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<!-- dom:TITLE: Exercises week 38 --><div class="tex2jax_ignore mathjax_ignore section" id="exercises-week-38">
|
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<h1>Exercises week 38<a class="headerlink" href="#exercises-week-38" title="Permalink to this headline">¶</a></h1>
|
||||
<p><strong>September 18-22, 2023</strong></p>
|
||||
<p>Date: <strong>Deadline is Sunday September 24 at midnight</strong></p>
|
||||
<div class="section" id="overarching-aims-of-the-exercises-this-week">
|
||||
<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>
|
||||
<p>The aim of the exercises this week is to derive the equations for the bias-variance tradeoff to be used in project 1 as well as testing this for a simpler function using the bootstrap method. The exercises here can be reused in project 1 as well.</p>
|
||||
<p>Consider a
|
||||
dataset <span class="math notranslate nohighlight">\(\mathcal{L}\)</span> consisting of the data
|
||||
<span class="math notranslate nohighlight">\(\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}\)</span>.</p>
|
||||
<p>We assume that the true data is generated from a noisy model</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}.
|
||||
\]</div>
|
||||
<p>Here <span class="math notranslate nohighlight">\(\epsilon\)</span> is normally distributed with mean zero and standard
|
||||
deviation <span class="math notranslate nohighlight">\(\sigma^2\)</span>.</p>
|
||||
<p>In our derivation of the ordinary least squares method we defined
|
||||
an approximation to the function <span class="math notranslate nohighlight">\(f\)</span> in terms of the parameters
|
||||
<span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> and the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> which embody our model,
|
||||
that is <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta}\)</span>.</p>
|
||||
<p>The parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> are in turn found by optimizing the mean
|
||||
squared error via the so-called cost function</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
|
||||
\]</div>
|
||||
<p>Here the expected value <span class="math notranslate nohighlight">\(\mathbb{E}\)</span> is the sample value.</p>
|
||||
<p>Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a
|
||||
term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.
|
||||
That is, show that</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=(\mathrm{Bias}[\tilde{y}])^2+\mathrm{var}[\tilde{f}]+\sigma^2,
|
||||
\]</div>
|
||||
<p>with</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
(\mathrm{Bias}[\tilde{y}])^2=\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2,
|
||||
\]</div>
|
||||
<p>and</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathrm{var}[\tilde{f}]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
|
||||
\]</div>
|
||||
<p>Explain what the terms mean and discuss their interpretations.</p>
|
||||
<p>Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by
|
||||
studying the MSE value as function of the complexity of your model. Use ordinary least squares only.</p>
|
||||
<p>Discuss the bias and variance trade-off as function
|
||||
of your model complexity (the degree of the polynomial) and the number
|
||||
of data points, and possibly also your training and test data using the <strong>bootstrap</strong> resampling method.
|
||||
You can follow the code example in the jupyter-book at <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff">https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff</a>.</p>
|
||||
<p>See also the whiteboard notes from week 37 at <a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf">https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf</a></p>
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"source": [
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"# Exercises week 38\n",
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||||
"**September 18-22, 2023**\n",
|
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"\n",
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"Date: **Deadline is Sunday September 24 at midnight**"
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"metadata": {
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},
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"source": [
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"## Overarching aims of the exercises this week\n",
|
||||
"\n",
|
||||
"The aim of the exercises this week is to derive the equations for the bias-variance tradeoff to be used in project 1 as well as testing this for a simpler function using the bootstrap method. The exercises here can be reused in project 1 as well.\n",
|
||||
"\n",
|
||||
"Consider a\n",
|
||||
"dataset $\\mathcal{L}$ consisting of the data\n",
|
||||
"$\\mathbf{X}_\\mathcal{L}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$.\n",
|
||||
"\n",
|
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"We assume that the true data is generated from a noisy model"
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]
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},
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{
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"cell_type": "markdown",
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"id": "d08c4671",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
|
||||
"\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}.\n",
|
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "56f9ca3e",
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"metadata": {
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"editable": true
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},
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"source": [
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"Here $\\epsilon$ is normally distributed with mean zero and standard\n",
|
||||
"deviation $\\sigma^2$.\n",
|
||||
"\n",
|
||||
"In our derivation of the ordinary least squares method we defined \n",
|
||||
"an approximation to the function $f$ in terms of the parameters\n",
|
||||
"$\\boldsymbol{\\beta}$ and the design matrix $\\boldsymbol{X}$ which embody our model,\n",
|
||||
"that is $\\boldsymbol{\\tilde{y}}=\\boldsymbol{X}\\boldsymbol{\\beta}$.\n",
|
||||
"\n",
|
||||
"The parameters $\\boldsymbol{\\beta}$ are in turn found by optimizing the mean\n",
|
||||
"squared error via the so-called cost function"
|
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]
|
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},
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{
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"cell_type": "markdown",
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"id": "ae36b494",
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"metadata": {
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"editable": true
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"source": [
|
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"$$\n",
|
||||
"C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "be3eadf1",
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"metadata": {
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"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Here the expected value $\\mathbb{E}$ is the sample value. \n",
|
||||
"\n",
|
||||
"Show that you can rewrite this in terms of a term which contains the variance of the model itself (the so-called variance term), a\n",
|
||||
"term which measures the deviation from the true data and the mean value of the model (the bias term) and finally the variance of the noise.\n",
|
||||
"That is, show that"
|
||||
]
|
||||
},
|
||||
{
|
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"cell_type": "markdown",
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"id": "ae9ebea0",
|
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"metadata": {
|
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"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=(\\mathrm{Bias}[\\tilde{y}])^2+\\mathrm{var}[\\tilde{f}]+\\sigma^2,\n",
|
||||
"$$"
|
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]
|
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},
|
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{
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"cell_type": "markdown",
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"id": "d1cbae1b",
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"metadata": {
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"editable": true
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"source": [
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"with"
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]
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{
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"cell_type": "markdown",
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"id": "d2e1f899",
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"metadata": {
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"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"(\\mathrm{Bias}[\\tilde{y}])^2=\\left(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2,\n",
|
||||
"$$"
|
||||
]
|
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},
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{
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"cell_type": "markdown",
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"id": "0486221c",
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"metadata": {
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"source": [
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"and"
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]
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"id": "86746df2",
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"metadata": {
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"editable": true
|
||||
},
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||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{var}[\\tilde{f}]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
|
||||
"$$"
|
||||
]
|
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},
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{
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"cell_type": "markdown",
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|
||||
"Explain what the terms mean and discuss their interpretations.\n",
|
||||
"\n",
|
||||
"Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by\n",
|
||||
"studying the MSE value as function of the complexity of your model. Use ordinary least squares only.\n",
|
||||
"\n",
|
||||
"Discuss the bias and variance trade-off as function\n",
|
||||
"of your model complexity (the degree of the polynomial) and the number\n",
|
||||
"of data points, and possibly also your training and test data using the **bootstrap** resampling method.\n",
|
||||
"You can follow the code example in the jupyter-book at <https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff>.\n",
|
||||
"\n",
|
||||
"See also the whiteboard notes from week 37 at <https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf>"
|
||||
]
|
||||
}
|
||||
],
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
- file: chapter4.ipynb
|
||||
- part: Advanced Topics
|
||||
numbered: true
|
||||
chapters:
|
||||
- file: chapter5.ipynb
|
||||
- file: chapter6.ipynb
|
||||
- file: chapter7.ipynb
|
||||
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|
||||
format: jb-book
|
||||
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|
||||
parts:
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
- file: textbooks
|
||||
- caption: Review of Statistics with Resampling Techniques and Linear Algebra
|
||||
numbered: true
|
||||
chapters:
|
||||
- file: statistics.ipynb
|
||||
- file: linalg.ipynb
|
||||
- caption: From Regression to Support Vector Machines
|
||||
numbered: true
|
||||
chapters:
|
||||
- file: chapter1.ipynb
|
||||
- file: chapter2.ipynb
|
||||
- file: chapter3.ipynb
|
||||
- file: chapter4.ipynb
|
||||
- file: chapteroptimization.ipynb
|
||||
- file: chapter5.ipynb
|
||||
- caption: Decision Trees, Ensemble Methods and Boosting
|
||||
numbered: true
|
||||
chapters:
|
||||
- file: chapter6.ipynb
|
||||
- file: chapter7.ipynb
|
||||
- caption: Dimensionality Reduction
|
||||
numbered: true
|
||||
chapters:
|
||||
- file: chapter8.ipynb
|
||||
- file: clustering.ipynb
|
||||
- caption: Deep Learning Methods
|
||||
numbered: true
|
||||
chapters:
|
||||
- file: chapter9.ipynb
|
||||
- file: chapter10.ipynb
|
||||
- file: chapter11.ipynb
|
||||
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|
||||
- file: chapter13.ipynb
|
||||
- caption: Weekly material, notes and exercises
|
||||
numbered: false
|
||||
chapters:
|
||||
- file: exercisesweek34.ipynb
|
||||
- file: week34.ipynb
|
||||
- file: exercisesweek35.ipynb
|
||||
- file: week35.ipynb
|
||||
- file: exercisesweek36.ipynb
|
||||
- file: week36.ipynb
|
||||
- file: exercisesweek37.ipynb
|
||||
- file: week37.ipynb
|
||||
- caption: Projects
|
||||
numbered: false
|
||||
chapters:
|
||||
- file: project1.ipynb
|
||||
|
||||
Reference in New Issue
Block a user