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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 16-20, 2024**\n",
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"\n",
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"Date: **Deadline is Friday September 20 at midnight**"
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]
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},
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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",
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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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"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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||||
},
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{
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"cell_type": "markdown",
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"id": "36a8765e",
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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",
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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",
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"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",
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||||
"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": "68dd52df",
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"editable": true
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},
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||||
"source": [
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||||
"$$\n",
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||||
"C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n",
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||||
"$$"
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||||
]
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||||
},
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||||
{
|
||||
"cell_type": "markdown",
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||||
"id": "68a07606",
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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 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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||||
},
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||||
{
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||||
"cell_type": "markdown",
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"id": "fbbb3fd7",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathrm{Bias}[\\tilde{y}]+\\mathrm{var}[\\tilde{y}]+\\sigma^2,\n",
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"$$"
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]
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},
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "5ecc0f22",
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"metadata": {
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"editable": true
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},
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"source": [
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"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": "8c17bd0a",
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"metadata": {
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"editable": true
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},
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||||
"source": [
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||||
"$$\n",
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"\\mathrm{Bias}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right],\n",
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"$$"
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||||
]
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||||
},
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||||
{
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||||
"cell_type": "markdown",
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||||
"id": "55ab4f0c",
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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": "98f93c68",
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"metadata": {
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||||
"editable": true
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||||
},
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"source": [
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"$$\n",
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||||
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
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||||
"$$"
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||||
]
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},
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{
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||||
"cell_type": "markdown",
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"id": "d6d099d4",
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"metadata": {
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"editable": true
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||||
},
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||||
"source": [
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||||
"Explain what the terms mean and discuss their interpretations.\n",
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"\n",
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"Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by\n",
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||||
"studying the MSE value as function of the complexity of your model. Use ordinary least squares only.\n",
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||||
"\n",
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||||
"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",
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||||
"of data points, and possibly also your training and test data using the **bootstrap** resampling method.\n",
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||||
"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>."
|
||||
]
|
||||
}
|
||||
],
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||||
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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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Teachers and Grading
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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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2. Linear Algebra, Handling of Arrays and more Python Features
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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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</a>
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<a class="reference internal" href="chapter3.html">
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5. Resampling Methods
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</a>
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<li class="toctree-l1">
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<a class="reference internal" href="chapter4.html">
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6. Logistic Regression
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</a>
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<a class="reference internal" href="chapteroptimization.html">
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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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<span class="caption-text">
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Decision Trees, Ensemble Methods and Boosting
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9. Decision trees, overarching aims
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</li>
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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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</span>
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11. Basic ideas of the Principal Component Analysis (PCA)
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<li class="toctree-l1">
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<a class="reference internal" href="clustering.html">
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12. Clustering and Unsupervised Learning
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||||
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<p aria-level="2" class="caption" role="heading">
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Deep Learning Methods
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||||
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13. Neural networks
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14. Building a Feed Forward Neural Network
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||||
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15. Solving Differential Equations with Deep Learning
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||||
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<a class="reference internal" href="chapter12.html">
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16. Convolutional Neural Networks
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||||
</a>
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||||
<li class="toctree-l1">
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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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||||
<p aria-level="2" class="caption" role="heading">
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||||
<span class="caption-text">
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Weekly material, notes and exercises
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<a class="reference internal" href="exercisesweek34.html">
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Exercises week 34
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Week 34: Introduction to the course, Logistics and Practicalities
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Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression
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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">
|
||||
<h1>Exercises week 38<a class="headerlink" href="#exercises-week-38" title="Permalink to this headline">¶</a></h1>
|
||||
<p><strong>September 16-20, 2024</strong></p>
|
||||
<p>Date: <strong>Deadline is Friday September 20 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">
|
||||
\[
|
||||
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathrm{Bias}[\tilde{y}]+\mathrm{var}[\tilde{y}]+\sigma^2,
|
||||
\]</div>
|
||||
<p>with</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathrm{Bias}[\tilde{y}]=\mathbb{E}\left[\left(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right],
|
||||
\]</div>
|
||||
<p>and</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathrm{var}[\tilde{y}]=\mathbb{E}\left[\left(\tilde{\boldsymbol{y}}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\right)^2\right]=\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2.
|
||||
\]</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>
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@@ -288,6 +288,11 @@ const thebe_selector_output = ".output, .cell_output"
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Week 37: Statistical interpretations and Resampling Methods
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Exercises week 38
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@@ -961,9 +966,9 @@ doconce format html week37.do.txt --no_mako -->
|
||||
<!-- todo add link to videos and add link to Van Wieringens notes --><div class="section" id="plans-for-week-37-lecture-monday">
|
||||
<h2>Plans for week 37, lecture Monday<a class="headerlink" href="#plans-for-week-37-lecture-monday" title="Permalink to this headline">¶</a></h2>
|
||||
<p><strong>Material for the lecture on Monday September 9.</strong></p>
|
||||
<!-- * [Video of Lecture](https://youtu.be/YOBBr_toYxc) -->
|
||||
<!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf) -->
|
||||
<ul class="simple">
|
||||
<li><p><a class="reference external" href="https://youtu.be/omLmp_kkie0">Video of Lecture</a></p></li>
|
||||
<li><p><a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember9.pdf">Whiteboard notes</a></p></li>
|
||||
<li><p>Statistical interpretation of Ridge and Lasso regression, see also slides from last week</p></li>
|
||||
<li><p>Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff (this may partly be discussed during the exercise sessions as well.</p></li>
|
||||
<li><p>Readings and Videos:</p>
|
||||
@@ -1618,7 +1623,7 @@ theorem.</p>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Bootstrap Statistics :
|
||||
original bias std. error
|
||||
100.057 14.8058 100.061 0.148629
|
||||
100.106 15.0037 100.104 0.149019
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1843,7 +1848,9 @@ Error: 0.10398646080125035
|
||||
Bias^2: 0.1007711427354898
|
||||
Var: 0.0032153180657605116
|
||||
0.10398646080125035 >= 0.1007711427354898 + 0.0032153180657605116 = 0.10398646080125032
|
||||
Polynomial degree: 3
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 3
|
||||
Error: 0.06547790180152355
|
||||
Bias^2: 0.06208238634231949
|
||||
Var: 0.0033955154592040936
|
||||
@@ -1875,7 +1882,9 @@ Error: 0.017355848195593347
|
||||
Bias^2: 0.010331721306655127
|
||||
Var: 0.007024126888938232
|
||||
0.017355848195593347 >= 0.010331721306655127 + 0.007024126888938232 = 0.01735584819559336
|
||||
Polynomial degree: 9
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 9
|
||||
Error: 0.02660572763718093
|
||||
Bias^2: 0.010018312644137363
|
||||
Var: 0.016587414993043573
|
||||
@@ -1904,7 +1913,7 @@ Var: 0.20867052175034223
|
||||
0.22842468702219465 >= 0.01975416527185249 + 0.20867052175034223 = 0.2284246870221947
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/week37_139_3.png" src="_images/week37_139_3.png" />
|
||||
<img alt="_images/week37_139_5.png" src="_images/week37_139_5.png" />
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</div>
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||||
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||||
</div>
|
||||
@@ -2300,12 +2309,12 @@ Mean squared error on test data: 1.07641937
|
||||
Degree of polynomial: 12
|
||||
Mean squared error on training data: 0.00805074
|
||||
Mean squared error on test data: 0.04295757
|
||||
Degree of polynomial: 13
|
||||
Mean squared error on training data: 0.00781918
|
||||
Mean squared error on test data: 0.56965674
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 14
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 13
|
||||
Mean squared error on training data: 0.00781918
|
||||
Mean squared error on test data: 0.56965674
|
||||
Degree of polynomial: 14
|
||||
Mean squared error on training data: 0.00465099
|
||||
Mean squared error on test data: 0.28443039
|
||||
Degree of polynomial: 15
|
||||
@@ -2363,9 +2372,9 @@ Mean squared error on training data: 0.00063866
|
||||
Mean squared error on test data: 3099.60342978
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_88787/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_92606/626635268.py:73: RuntimeWarning: divide by zero encountered in log10
|
||||
plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
|
||||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_88787/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
|
||||
/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_92606/626635268.py:74: RuntimeWarning: divide by zero encountered in log10
|
||||
plt.plot(polynomial, np.log10(testerror), label='Test Error')
|
||||
</pre></div>
|
||||
</div>
|
||||
@@ -2450,7 +2459,7 @@ Mean squared error on test data: 3099.60342978
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_88787/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/var/folders/td/3yk470mj5p931p9dtkk0y6jw0000gn/T/ipykernel_92606/3817475779.py:63: RuntimeWarning: divide by zero encountered in log10
|
||||
plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
|
||||
</pre></div>
|
||||
</div>
|
||||
@@ -2644,10 +2653,10 @@ This means the variance we obtain with the standard OLS will always for <span cl
|
||||
<p class="prev-next-title">Exercises week 37</p>
|
||||
</div>
|
||||
</a>
|
||||
<a class='right-next' id="next-link" href="project1.html" title="next page">
|
||||
<a class='right-next' id="next-link" href="exercisesweek38.html" title="next page">
|
||||
<div class="prev-next-info">
|
||||
<p class="prev-next-subtitle">next</p>
|
||||
<p class="prev-next-title">Project 1 on Machine Learning, deadline October 7 (midnight), 2024</p>
|
||||
<p class="prev-next-title">Exercises week 38</p>
|
||||
</div>
|
||||
<i class="fas fa-angle-right"></i>
|
||||
</a>
|
||||
|
||||
@@ -0,0 +1,181 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "f9bcf943",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
|
||||
"doconce format html exercisesweek38.do.txt -->\n",
|
||||
"<!-- dom:TITLE: Exercises week 38 -->"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "ead6d6d6",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"# Exercises week 38\n",
|
||||
"**September 16-20, 2024**\n",
|
||||
"\n",
|
||||
"Date: **Deadline is Friday September 20 at midnight**"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "b6ce9344",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## 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",
|
||||
"We assume that the true data is generated from a noisy model"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "29371f21",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "36a8765e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"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"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "68dd52df",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "68a07606",
|
||||
"metadata": {
|
||||
"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"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "fbbb3fd7",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathrm{Bias}[\\tilde{y}]+\\mathrm{var}[\\tilde{y}]+\\sigma^2,\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "5ecc0f22",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"with"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "8c17bd0a",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{Bias}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right],\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "55ab4f0c",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"and"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "98f93c68",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "d6d099d4",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"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",
|
||||
"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>."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
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|
||||
"nbformat_minor": 5
|
||||
}
|
||||
File diff suppressed because one or more lines are too long
@@ -16,14 +16,15 @@
|
||||
#
|
||||
# **Material for the lecture on Monday September 9.**
|
||||
#
|
||||
# <!-- * [Video of Lecture](https://youtu.be/YOBBr_toYxc) -->
|
||||
# <!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf) -->
|
||||
# * [Video of Lecture](https://youtu.be/omLmp_kkie0)
|
||||
#
|
||||
# * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesSeptember9.pdf)
|
||||
#
|
||||
# * Statistical interpretation of Ridge and Lasso regression, see also slides from last week
|
||||
#
|
||||
# * Resampling techniques, Bootstrap and cross validation and bias-variance tradeoff (this may partly be discussed during the exercise sessions as well.
|
||||
#
|
||||
# * Readings and Videos:
|
||||
#
|
||||
# * Raschka et al, pages 175-192
|
||||
#
|
||||
# * Hastie et al Chapter 7, here we recommend 7.1-7.5 and 7.10 (cross-validation) and 7.11 (bootstrap). See <https://link.springer.com/book/10.1007/978-0-387-84858-7>.
|
||||
|
||||
Binary file not shown.
|
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|
After Width: | Height: | Size: 29 KiB |
@@ -49,6 +49,7 @@ parts:
|
||||
- file: week36.ipynb
|
||||
- file: exercisesweek37.ipynb
|
||||
- file: week37.ipynb
|
||||
- file: exercisesweek38.ipynb
|
||||
- caption: Projects
|
||||
numbered: false
|
||||
chapters:
|
||||
|
||||
@@ -0,0 +1,181 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
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"id": "f9bcf943",
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||||
"metadata": {
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||||
"editable": true
|
||||
},
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||||
"source": [
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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",
|
||||
"<!-- dom:TITLE: Exercises week 38 -->"
|
||||
]
|
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},
|
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{
|
||||
"cell_type": "markdown",
|
||||
"id": "ead6d6d6",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"# Exercises week 38\n",
|
||||
"**September 16-20, 2024**\n",
|
||||
"\n",
|
||||
"Date: **Deadline is Friday September 20 at midnight**"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "b6ce9344",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## 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",
|
||||
"We assume that the true data is generated from a noisy model"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "29371f21",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "36a8765e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"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"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "68dd52df",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "68a07606",
|
||||
"metadata": {
|
||||
"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"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "fbbb3fd7",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathrm{Bias}[\\tilde{y}]+\\mathrm{var}[\\tilde{y}]+\\sigma^2,\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "5ecc0f22",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"with"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "8c17bd0a",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{Bias}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right],\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "55ab4f0c",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"and"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "98f93c68",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{var}[\\tilde{y}]=\\mathbb{E}\\left[\\left(\\tilde{\\boldsymbol{y}}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]\\right)^2\\right]=\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "d6d099d4",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"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",
|
||||
"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>."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
+180
-179
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Load Diff
@@ -1,6 +1,6 @@
|
||||
TITLE: Exercises week 38
|
||||
AUTHOR: September 18-22, 2023
|
||||
DATE: Deadline is Sunday September 24 at midnight
|
||||
AUTHOR: September 16-20, 2024
|
||||
DATE: Deadline is Friday September 20 at midnight
|
||||
|
||||
|
||||
===== Overarching aims of the exercises this week =====
|
||||
@@ -59,7 +59,6 @@ and
|
||||
!et
|
||||
|
||||
|
||||
|
||||
Explain what the terms mean and discuss their interpretations.
|
||||
|
||||
Perform then a bias-variance analysis of a simple one-dimensional (or other models of your choice) function by
|
||||
@@ -71,7 +70,3 @@ of data points, and possibly also your training and test data using the _bootstr
|
||||
You can follow the code example in the jupyter-book at URL:"https://compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/chapter3.html#the-bias-variance-tradeoff".
|
||||
|
||||
|
||||
See also the whiteboard notes from week 37 at URL:"https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesSep14.pdf"
|
||||
|
||||
|
||||
|
||||
|
||||
Reference in New Issue
Block a user