update
This commit is contained in:
Morten Hjorth-Jensen
@@ -346,7 +346,7 @@ $$
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<p>In this case the standard OLS results in </p>
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$$
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\boldsymbol{\theta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
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\boldsymbol{\theta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y},
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$$
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<p>and</p>
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@@ -360,7 +360,7 @@ the Ridge estimator converges to zero when the hyperparameter goes to
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infinity.
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</p>
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<p>We will come back to more interpreations after we have gone through some of the statistical analysis part. </p>
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<p>We will come back to more interpretions after we have gone through some of the statistical analysis part. </p>
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<p>For more discussions of Ridge and Lasso regression, <a href="https://arxiv.org/abs/1509.09169" target="_self">Wessel van Wieringen's</a> article is highly recommended.
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Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_self">Mehta et al's article</a> is also recommended.
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@@ -1226,7 +1226,7 @@ $$
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<p>In this case the standard OLS results in </p>
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<p> <br>
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$$
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\boldsymbol{\theta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
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\boldsymbol{\theta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y},
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$$
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<p> <br>
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@@ -1243,7 +1243,7 @@ the Ridge estimator converges to zero when the hyperparameter goes to
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infinity.
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</p>
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<p>We will come back to more interpreations after we have gone through some of the statistical analysis part. </p>
|
||||
<p>We will come back to more interpretions after we have gone through some of the statistical analysis part. </p>
|
||||
|
||||
<p>For more discussions of Ridge and Lasso regression, <a href="https://arxiv.org/abs/1509.09169" target="_blank">Wessel van Wieringen's</a> article is highly recommended.
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Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et al's article</a> is also recommended.
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@@ -1177,7 +1177,7 @@ $$
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<p>In this case the standard OLS results in </p>
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$$
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\boldsymbol{\theta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
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\boldsymbol{\theta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y},
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$$
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<p>and</p>
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@@ -1191,7 +1191,7 @@ the Ridge estimator converges to zero when the hyperparameter goes to
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infinity.
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</p>
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<p>We will come back to more interpreations after we have gone through some of the statistical analysis part. </p>
|
||||
<p>We will come back to more interpretions after we have gone through some of the statistical analysis part. </p>
|
||||
|
||||
<p>For more discussions of Ridge and Lasso regression, <a href="https://arxiv.org/abs/1509.09169" target="_blank">Wessel van Wieringen's</a> article is highly recommended.
|
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Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et al's article</a> is also recommended.
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@@ -1254,7 +1254,7 @@ $$
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<p>In this case the standard OLS results in </p>
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$$
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\boldsymbol{\theta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}_i^T\boldsymbol{y},
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\boldsymbol{\theta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y},
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$$
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<p>and</p>
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@@ -1268,7 +1268,7 @@ the Ridge estimator converges to zero when the hyperparameter goes to
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infinity.
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</p>
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<p>We will come back to more interpreations after we have gone through some of the statistical analysis part. </p>
|
||||
<p>We will come back to more interpretions after we have gone through some of the statistical analysis part. </p>
|
||||
|
||||
<p>For more discussions of Ridge and Lasso regression, <a href="https://arxiv.org/abs/1509.09169" target="_blank">Wessel van Wieringen's</a> article is highly recommended.
|
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Similarly, <a href="https://arxiv.org/abs/1803.08823" target="_blank">Mehta et al's article</a> is also recommended.
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-360
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@@ -1,357 +0,0 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "2c329a04",
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"metadata": {
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"editable": true
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},
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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 exercisesweek37.do.txt -->\n",
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"<!-- dom:TITLE: Week 37: Linear Regression and Gradient descent -->"
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]
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},
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{
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"cell_type": "markdown",
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"id": "432a2e5c",
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"metadata": {
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"editable": true
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},
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"source": [
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"# Week 37: Linear Regression and Gradient descent\n",
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"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo, Norway\n",
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"\n",
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"Date: **September 1-5, 2025**"
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]
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},
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{
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"cell_type": "markdown",
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"id": "8352513e",
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"metadata": {
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"editable": true
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},
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"source": [
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"## Ridge regression and a new Synthetic Dataset\n",
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"\n",
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"We create a synthetic linear regression dataset with a sparse\n",
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"underlying relationship. This means we have many features but only a\n",
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"few of them actually contribute to the target. In our example, we’ll\n",
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"use 10 features with only 3 non-zero weights in the true model. This\n",
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"way, the target is generated as a linear combination of a few features\n",
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"(with known coefficients) plus some random noise. The steps we include are:\n",
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"\n",
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"Decide on the number of samples and features (e.g. 100 samples, 10 features).\n",
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"Define the **true** coefficient vector with mostly zeros (for sparsity). For example, we set $\\hat{\\boldsymbol{\\theta}} = [5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0]$, meaning only features 0, 1, and 6 have a real effect on y.\n",
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"\n",
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"Then we sample feature values for $\\boldsymbol{X}$ randomly (e.g. from a normal distribution). We use a normal distribution so features are roughly centered around 0.\n",
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"Then we compute the target values $y$ using the linear combination $\\boldsymbol{X}\\hat{\\boldsymbol{\\theta}}$ and add some noise (to simulate measurement error or unexplained variance).\n",
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"\n",
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"Below is the code to generate the dataset:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "d7eebc41",
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"metadata": {
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"collapsed": false,
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"editable": true,
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"jupyter": {
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"outputs_hidden": false
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}
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},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"\n",
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"# Set random seed for reproducibility\n",
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"np.random.seed(0)\n",
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"\n",
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"# Define dataset size\n",
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"n_samples = 100\n",
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"n_features = 10\n",
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"\n",
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"# Define true coefficients (sparse linear relationship)\n",
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"theta_true = np.array([5.0, -3.0, 0.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0])\n",
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"\n",
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"# Generate feature matrix X (n_samples x n_features) with random values\n",
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"X = np.random.randn(n_samples, n_features) # standard normal distribution\n",
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"\n",
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"# Generate target values y with a linear combination of X and theta_true, plus noise\n",
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"noise = 0.5 * np.random.randn(n_samples) # Gaussian noise\n",
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"y = X @ theta_true + noise"
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]
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},
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{
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"cell_type": "markdown",
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"id": "1640310e",
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"metadata": {
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"editable": true
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},
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"source": [
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"This code produces a dataset where only features 0, 1, and 6\n",
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"significantly influence $\\boldsymbol{y}$. The rest of the features have zero true\n",
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"coefficient, so they only contribute noise. For example, feature 0 has\n",
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"a true weight of 5.0, feature 1 has -3.0, and feature 6 has 2.0, so\n",
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"the expected relationship is:"
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]
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},
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{
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"cell_type": "markdown",
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"id": "60bce352",
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"metadata": {
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"editable": true
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},
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"source": [
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"$$\n",
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"y \\approx 5 \\times X_0 \\;-\\; 3 \\times X_1 \\;+\\; 2 \\times X_6 \\;+\\; \\text{noise}.\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "75322f31",
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"metadata": {
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"editable": true
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||||
},
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"source": [
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"Before fitting a regression model, it is good practice to normalize or\n",
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"standardize the features. This ensures all features are on a\n",
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"comparable scale, which is especially important when using\n",
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"regularization. Here we will perform standardization, scaling each\n",
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"feature to have mean 0 and standard deviation 1:\n",
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"\n",
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"Compute the mean and standard deviation of each column (feature) in $bm{X}$.\n",
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"Subtract the mean and divide by the standard deviation for each feature.\n",
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"\n",
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"We also center the target $\\boldsymbol{y}$ to mean $0$. Centering $\\boldsymbol{y}$\n",
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"(and each feature) means the model won’t require a separate intercept\n",
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"term – the data is shifted such that the intercept is effectively 0\n",
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". (In practice, one could include an intercept in the model and not\n",
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"penalize it, but here we simplify by centering.)"
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]
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||||
},
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||||
{
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||||
"cell_type": "code",
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||||
"execution_count": 3,
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||||
"id": "4b3f945a",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true,
|
||||
"jupyter": {
|
||||
"outputs_hidden": false
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||||
}
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||||
},
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||||
"outputs": [],
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"source": [
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"# Standardize features (zero mean, unit variance for each feature)\n",
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"X_mean = X.mean(axis=0)\n",
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"X_std = X.std(axis=0)\n",
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"X_std[X_std == 0] = 1 # safeguard to avoid division by zero for constant features\n",
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"X_norm = (X - X_mean) / X_std\n",
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"\n",
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||||
"# Center the target to zero mean (optional, to simplify intercept handling)\n",
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"y_mean = y.mean()\n",
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"y_centered = y - y_mean"
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||||
]
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||||
},
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||||
{
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||||
"cell_type": "markdown",
|
||||
"id": "12aec8b3",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"After this preprocessing, each column of $\\boldsymbol{X}_norm$ has mean zero and standard deviation $1$\n",
|
||||
"and $\\boldsymbol{y}_centered$ has mean 0. This makes the optimization landscape\n",
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"nicer and ensures the regularization penalty $\\lambda \\sum_j\n",
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"\\beta_j^2$ treats each coefficient fairly (since features are on the\n",
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||||
"same scale)."
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]
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||||
},
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||||
{
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||||
"cell_type": "code",
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||||
"execution_count": 7,
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||||
"id": "ffdecd03",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true,
|
||||
"jupyter": {
|
||||
"outputs_hidden": false
|
||||
}
|
||||
},
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||||
"outputs": [
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||||
{
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"name": "stdout",
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||||
"output_type": "stream",
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"text": [
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||||
"Closed-form Ridge coefficients: [ 4.98041921 -2.85683951 -0.01549982 0.15658961 -0.07810233 -0.05094777\n",
|
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" 1.75377987 0.00679113 0.0343071 -0.04883458]\n"
|
||||
]
|
||||
}
|
||||
],
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||||
"source": [
|
||||
"# Set regularization parameter\n",
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"lam = 1.0\n",
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"\n",
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"# Closed-form Ridge solution: w = (X^T X + lam * I)^{-1} X^T y\n",
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"I = np.eye(n_features)\n",
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"theta_closed_form = np.linalg.inv(X_norm.T.dot(X_norm) + lam * I).dot(X_norm.T).dot(y_centered)\n",
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"\n",
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||||
"print(\"Closed-form Ridge coefficients:\", theta_closed_form)"
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]
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||||
},
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||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "33ff39a5",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"This computes the ridge regression coefficients directly. The identity\n",
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"matrix $I$ has the same size as $X^T X$ (which is n_features x\n",
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"n_features), and lam * I adds $\\lambda$ to the diagonal of $X^T X. We\n",
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||||
"then invert this matrix and multiply by $X^T y. The result\n",
|
||||
"for $\\boldsymbol{\\theta}$ is a NumPy array of shape (n_features,) containing the\n",
|
||||
"fitted weights.\n",
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"\n",
|
||||
"Alternatively, we can fit the ridge regression model using gradient\n",
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||||
"descent. This is useful to visualize the iterative convergence and is\n",
|
||||
"necessary if $n$ and $p$ are so large that the closed-form might be\n",
|
||||
"too slow or memory-intensive. We derive the gradients from the cost\n",
|
||||
"function defined above. The gradient of the ridge cost with respect to\n",
|
||||
"the weight vector $w$ is:\n",
|
||||
"\n",
|
||||
"Below is the code for gradient descent implementation of ridge:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 8,
|
||||
"id": "5b72dc82",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true,
|
||||
"jupyter": {
|
||||
"outputs_hidden": false
|
||||
}
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
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||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Gradient Descent Ridge coefficients: [ 4.98041921 -2.85683951 -0.01549982 0.15658961 -0.07810233 -0.05094777\n",
|
||||
" 1.75377987 0.00679113 0.0343071 -0.04883458]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Gradient descent parameters\n",
|
||||
"alpha = 0.1\n",
|
||||
"num_iters = 1000\n",
|
||||
"\n",
|
||||
"# Initialize weights for gradient descent\n",
|
||||
"theta = np.zeros(n_features)\n",
|
||||
"\n",
|
||||
"# Arrays to store history for plotting\n",
|
||||
"cost_history = np.zeros(num_iters)\n",
|
||||
"\n",
|
||||
"# Gradient descent loop\n",
|
||||
"m = n_samples # number of examples\n",
|
||||
"for t in range(num_iters):\n",
|
||||
" # Compute prediction error\n",
|
||||
" error = X_norm.dot(theta) - y_centered # shape (m,)\n",
|
||||
" # Compute cost (MSE + regularization) for monitoring\n",
|
||||
" cost = (1/(2*m)) * np.dot(error, error) + (lam/(2*m)) * np.dot(theta, theta)\n",
|
||||
" cost_history[t] = cost\n",
|
||||
" # Compute gradient\n",
|
||||
" grad = (1/m) * (X_norm.T.dot(error) + lam * theta)\n",
|
||||
" # Update weights\n",
|
||||
" theta = theta - alpha * grad\n",
|
||||
"\n",
|
||||
"# After the loop, theta contains the fitted coefficients\n",
|
||||
"theta_gd = theta\n",
|
||||
"print(\"Gradient Descent Ridge coefficients:\", theta_gd)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "1fc5b8d0",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Let us confirm that the two approaches (closed-form and gradient\n",
|
||||
"descent) give similar results, and then evaluate the model. First,\n",
|
||||
"compare the learned coefficients to the true coefficients:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
|
||||
"id": "a510e4e6",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true,
|
||||
"jupyter": {
|
||||
"outputs_hidden": false
|
||||
}
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"True coefficients: [ 5. -3. 0. 0. 0. 0. 2. 0. 0. 0.]\n",
|
||||
"Closed-form learned coefficients: [ 4.98041921 -2.85683951 -0.01549982 0.15658961 -0.07810233 -0.05094777\n",
|
||||
" 1.75377987 0.00679113 0.0343071 -0.04883458]\n",
|
||||
"Gradient descent learned coefficients: [ 4.98041921 -2.85683951 -0.01549982 0.15658961 -0.07810233 -0.05094777\n",
|
||||
" 1.75377987 0.00679113 0.0343071 -0.04883458]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"print(\"True coefficients:\", theta_true)\n",
|
||||
"print(\"Closed-form learned coefficients:\", theta_closed_form)\n",
|
||||
"print(\"Gradient descent learned coefficients:\", theta_gd)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "07081487",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"If everything worked correctly, the learned coefficients should be\n",
|
||||
"close to the true values [5.0, -3.0, 0.0, …, 2.0, …] that we used to\n",
|
||||
"generate the data. Keep in mind that due to regularization and noise,\n",
|
||||
"the learned values will not exactly equal the true ones, but they\n",
|
||||
"should be in the same ballpark."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3 (ipykernel)",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.9.15"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
@@ -928,7 +928,7 @@ For the sake of simplicity, let us assume that the design matrix is orthonormal,
|
||||
In this case the standard OLS results in
|
||||
!bt
|
||||
\[
|
||||
\bm{\theta}^{\mathrm{OLS}} = \bm{X}^T\bm{y}=\sum_{i=0}^{n-1}\bm{u}_i\bm{u}_i^T\bm{y},
|
||||
\bm{\theta}^{\mathrm{OLS}} = \bm{X}^T\bm{y},
|
||||
\]
|
||||
!et
|
||||
|
||||
@@ -944,7 +944,7 @@ that is the Ridge estimator scales the OLS estimator by the inverse of a factor
|
||||
the Ridge estimator converges to zero when the hyperparameter goes to
|
||||
infinity.
|
||||
|
||||
We will come back to more interpreations after we have gone through some of the statistical analysis part.
|
||||
We will come back to more interpretions after we have gone through some of the statistical analysis part.
|
||||
|
||||
For more discussions of Ridge and Lasso regression, "Wessel van Wieringen's":"https://arxiv.org/abs/1509.09169" article is highly recommended.
|
||||
Similarly, "Mehta et al's article":"https://arxiv.org/abs/1803.08823" is also recommended.
|
||||
|
||||
Reference in New Issue
Block a user