update of week 43
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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 exercisesweek43.do.txt -->\n",
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"<!-- dom:TITLE: Exercises week 43 -->"
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"source": [
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"# Exercises week 43 \n",
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"**October 20-24, 2025**\n",
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"\n",
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"Date: **Deadline Friday October 24 at midnight**"
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]
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"source": [
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"# Overarching aims of the exercises weeks 43 and 44\n",
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"\n",
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"The aim of the exercises this week is to gain some confidence with\n",
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"ways to visualize the results of a classification problem. We will\n",
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"target three ways of setting up the analysis. The first and simplest\n",
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"one is the\n",
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"1. so-called confusion matrix, and the next is the\n",
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"\n",
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"2. ROC curve and finally the\n",
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"\n",
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"3. Cumulative gain curve.\n",
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"\n",
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"We will use Logistic Regression as method for the classification in\n",
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"this exercise. You can compare these results with those obtained with\n",
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"your neural network code from project 2 without a hidden layer.\n",
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"\n",
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"In these exercises we will use binary and multi-class data sets\n",
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"(the Iris data set from week 41).\n",
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"\n",
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"The underlying mathematics is described here."
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]
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"metadata": {
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"source": [
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"### Confusion Matrix\n",
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"\n",
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"A **confusion matrix** summarizes a classifier’s performance by\n",
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"tabulating predictions versus true labels. For binary classification,\n",
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"it is a $2\\times2$ table whose entries are counts of outcomes:"
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]
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"source": [
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"$$\n",
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"\\begin{array}{l|cc} & \\text{Predicted Positive} & \\text{Predicted Negative} \\\\ \\hline \\text{Actual Positive} & TP & FN \\\\ \\text{Actual Negative} & FP & TN \\end{array}.\n",
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"$$"
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]
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"source": [
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"Here TP (true positives) is the number of cases correctly predicted as\n",
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"positive, FP (false positives) is the number incorrectly predicted as\n",
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"positive, TN (true negatives) is correctly predicted negative, and FN\n",
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"(false negatives) is incorrectly predicted negative . In other words,\n",
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"“positive” means class 1 and “negative” means class 0; for example, TP\n",
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"occurs when the prediction and actual are both positive. Formally:"
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]
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"$$\n",
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"\\text{TPR} = \\frac{\\text{TP}}{\\text{TP} + \\text{FN}}, \\quad \\text{FPR} = \\frac{\\text{FP}}{\\text{FP} + \\text{TN}},\n",
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"$$"
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"source": [
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"where TPR and FPR are the true and false positive rates defined below.\n",
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"\n",
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"In multiclass classification with $K$ classes, the confusion matrix\n",
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"generalizes to a $K\\times K$ table. Entry $N_{ij}$ in the table is\n",
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"the count of instances whose true class is $i$ and whose predicted\n",
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"class is $j$. For example, a three-class confusion matrix can be written\n",
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"as:"
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]
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"source": [
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"$$\n",
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"\\begin{array}{c|ccc} & \\text{Pred Class 1} & \\text{Pred Class 2} & \\text{Pred Class 3} \\\\ \\hline \\text{Act Class 1} & N_{11} & N_{12} & N_{13} \\\\ \\text{Act Class 2} & N_{21} & N_{22} & N_{23} \\\\ \\text{Act Class 3} & N_{31} & N_{32} & N_{33} \\end{array}.\n",
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"$$"
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]
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"source": [
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"Here the diagonal entries $N_{ii}$ are the true positives for each\n",
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"class, and off-diagonal entries are misclassifications. This matrix\n",
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"allows computation of per-class metrics: e.g. for class $i$,\n",
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"$\\mathrm{TP}_i=N_{ii}$, $\\mathrm{FN}_i=\\sum_{j\\neq i}N_{ij}$,\n",
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"$\\mathrm{FP}_i=\\sum_{j\\neq i}N_{ji}$, and $\\mathrm{TN}_i$ is the sum of\n",
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"all remaining entries.\n",
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"\n",
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"As defined above, TPR and FPR come from the binary case. In binary\n",
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"terms with $P$ actual positives and $N$ actual negatives, one has"
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"source": [
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"$$\n",
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"\\text{TPR} = \\frac{TP}{P} = \\frac{TP}{TP+FN}, \\quad \\text{FPR} =\n",
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"\\frac{FP}{N} = \\frac{FP}{FP+TN},\n",
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"$$"
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"source": [
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"as used in standard confusion-matrix\n",
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"formulations. These rates will be used in constructing ROC curves."
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]
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"metadata": {
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"editable": true
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"source": [
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"### ROC Curve\n",
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"\n",
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"The Receiver Operating Characteristic (ROC) curve plots the trade-off\n",
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"between true positives and false positives as a discrimination\n",
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"threshold varies. Specifically, for a binary classifier that outputs\n",
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"a score or probability, one varies the threshold $t$ for declaring\n",
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"**positive**, and computes at each $t$ the true positive rate\n",
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"$\\mathrm{TPR}(t)$ and false positive rate $\\mathrm{FPR}(t)$ using the\n",
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"confusion matrix at that threshold. The ROC curve is then the graph\n",
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"of TPR versus FPR. By definition,"
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]
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"source": [
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"$$\n",
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"\\mathrm{TPR} = \\frac{TP}{TP+FN}, \\qquad \\mathrm{FPR} = \\frac{FP}{FP+TN},\n",
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"$$"
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]
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},
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"cell_type": "markdown",
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"source": [
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"where $TP,FP,TN,FN$ are counts determined by threshold $t$. A perfect\n",
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"classifier would reach the point (FPR=0, TPR=1) at some threshold.\n",
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"\n",
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"Formally, the ROC curve is obtained by plotting\n",
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"$(\\mathrm{FPR}(t),\\mathrm{TPR}(t))$ for all $t\\in[0,1]$ (or as $t$\n",
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"sweeps through the sorted scores). The Area Under the ROC Curve (AUC)\n",
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"quantifies the average performance over all thresholds. It can be\n",
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"interpreted probabilistically: $\\mathrm{AUC} =\n",
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"\\Pr\\bigl(s(X^+)>s(X^-)\\bigr)$, the probability that a random positive\n",
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"instance $X^+$ receives a higher score $s$ than a random negative\n",
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"instance $X^-$ . Equivalently, the AUC is the integral under the ROC\n",
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"curve:"
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]
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"source": [
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"$$\n",
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"\\mathrm{AUC} \\;=\\; \\int_{0}^{1} \\mathrm{TPR}(f)\\,df,\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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},
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"source": [
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"where $f$ ranges over FPR (or fraction of negatives). A model that guesses at random yields a diagonal ROC (AUC=0.5), whereas a perfect model yields AUC=1.0."
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]
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},
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"metadata": {
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"source": [
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"### Cumulative Gain\n",
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"\n",
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"The cumulative gain curve (or gains chart) evaluates how many\n",
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"positives are captured as one targets an increasing fraction of the\n",
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"population, sorted by model confidence. To construct it, sort all\n",
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"instances by decreasing predicted probability of the positive class.\n",
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"Then, for the top $\\alpha$ fraction of instances, compute the fraction\n",
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"of all actual positives that fall in this subset. In formula form, if\n",
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"$P$ is the total number of positive instances and $P(\\alpha)$ is the\n",
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"number of positives among the top $\\alpha$ of the data, the cumulative\n",
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"gain at level $\\alpha$ is"
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]
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"source": [
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"$$\n",
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"\\mathrm{Gain}(\\alpha) \\;=\\; \\frac{P(\\alpha)}{P}.\n",
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"$$"
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]
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"source": [
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"For example, cutting off at the top 10% of predictions yields a gain\n",
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"equal to (positives in top 10%) divided by (total positives) .\n",
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"Plotting $\\mathrm{Gain}(\\alpha)$ versus $\\alpha$ (often in percent)\n",
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"gives the gain curve. The baseline (random) curve is the diagonal\n",
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"$\\mathrm{Gain}(\\alpha)=\\alpha$, while an ideal model has a steep climb\n",
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"toward 1.\n",
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"\n",
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"A related measure is the {\\em lift}, often called the gain ratio. It is the ratio of the model’s capture rate to that of random selection. Equivalently,"
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]
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"source": [
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"$$\n",
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"\\mathrm{Lift}(\\alpha) \\;=\\; \\frac{\\mathrm{Gain}(\\alpha)}{\\alpha}.\n",
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"$$"
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]
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},
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"cell_type": "markdown",
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"source": [
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"A lift $>1$ indicates better-than-random targeting. In practice, gain\n",
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"and lift charts (used e.g.\\ in marketing or imbalanced classification)\n",
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"show how many positives can be “gained” by focusing on a fraction of\n",
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"the population ."
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"source": [
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"### Other measures: Precision, Recall, and the F$_1$ Measure\n",
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"\n",
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"Precision and recall (sensitivity) quantify binary classification\n",
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"accuracy in terms of positive predictions. They are defined from the\n",
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"confusion matrix as:"
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]
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"source": [
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"$$\n",
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"\\text{Precision} = \\frac{TP}{TP + FP}, \\qquad \\text{Recall} = \\frac{TP}{TP + FN}.\n",
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"$$"
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]
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},
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"cell_type": "markdown",
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"source": [
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"Precision is the fraction of predicted positives that are correct, and\n",
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"recall is the fraction of actual positives that are correctly\n",
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"identified . A high-precision classifier makes few false-positive\n",
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"errors, while a high-recall classifier makes few false-negative\n",
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"errors.\n",
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"\n",
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"The F$_1$ score (balanced F-measure) combines precision and recall into a single metric via their harmonic mean. The usual formula 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": "621a2e8b",
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"metadata": {
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"source": [
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"$$\n",
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"F_1 =2\\frac{\\text{Precision}\\times\\text{Recall}}{\\text{Precision} + \\text{Recall}}.\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "62eee54a",
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"metadata": {
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"editable": true
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"source": [
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"This can be shown to equal"
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]
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},
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{
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"cell_type": "markdown",
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"id": "7a6a2e7a",
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"metadata": {
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"editable": true
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"source": [
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"$$\n",
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"\\frac{2\\,TP}{2\\,TP + FP + FN}.\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "b96c9ff4",
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"metadata": {
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"editable": true
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"source": [
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"The F$_1$ score ranges from 0 (worst) to 1 (best), and balances the\n",
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"trade-off between precision and recall.\n",
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"\n",
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"For multi-class classification, one computes per-class\n",
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"precision/recall/F$_1$ (treating each class as “positive” in a\n",
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"one-vs-rest manner) and then averages. Common averaging methods are:\n",
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"\n",
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"Micro-averaging: Sum all true positives, false positives, and false negatives across classes, then compute precision/recall/F$_1$ from these totals.\n",
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"Macro-averaging: Compute the F$1$ score $F{1,i}$ for each class $i$ separately, then take the unweighted mean: $F_{1,\\mathrm{macro}} = \\frac{1}{K}\\sum_{i=1}^K F_{1,i}$ . This treats all classes equally regardless of size.\n",
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"Weighted-averaging: Like macro-average, but weight each class’s $F_{1,i}$ by its support $n_i$ (true count): $F_{1,\\mathrm{weighted}} = \\frac{1}{N}\\sum_{i=1}^K n_i F_{1,i}$, where $N=\\sum_i n_i$. This accounts for class imbalance by giving more weight to larger classes .\n",
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"\n",
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"Each of these averages has different use-cases. Micro-average is\n",
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"dominated by common classes, macro-average highlights performance on\n",
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"rare classes, and weighted-average is a compromise. These formulas\n",
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"and concepts allow rigorous evaluation of classifier performance in\n",
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"both binary and multi-class settings."
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]
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||||
},
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{
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"cell_type": "markdown",
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"id": "9274bf3f",
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"metadata": {
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"editable": true
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||||
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||||
"source": [
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"## Exercises\n",
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"\n",
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||||
"Here is a simple code example which uses the Logistic regression machinery from **scikit-learn**.\n",
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"At the end it sets up the confusion matrix and the ROC and cumulative gain curves.\n",
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"Feel free to use these functionalities (we don't expect you to write your own code for say the confusion matrix)."
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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": 1,
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"id": "be9ff0b9",
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"metadata": {
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||||
"collapsed": false,
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"editable": true
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"outputs": [],
|
||||
"source": [
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"%matplotlib inline\n",
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"\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"from sklearn.model_selection import train_test_split \n",
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||||
"# from sklearn.datasets import fill in the data set\n",
|
||||
"from sklearn.linear_model import LogisticRegression\n",
|
||||
"\n",
|
||||
"# Load the data, fill inn\n",
|
||||
"mydata.data = ?\n",
|
||||
"\n",
|
||||
"X_train, X_test, y_train, y_test = train_test_split(mydata.data,cancer.target,random_state=0)\n",
|
||||
"print(X_train.shape)\n",
|
||||
"print(X_test.shape)\n",
|
||||
"# Logistic Regression\n",
|
||||
"# define which type of problem, binary or multiclass\n",
|
||||
"logreg = LogisticRegression(solver='lbfgs')\n",
|
||||
"logreg.fit(X_train, y_train)\n",
|
||||
"\n",
|
||||
"from sklearn.preprocessing import LabelEncoder\n",
|
||||
"from sklearn.model_selection import cross_validate\n",
|
||||
"#Cross validation\n",
|
||||
"accuracy = cross_validate(logreg,X_test,y_test,cv=10)['test_score']\n",
|
||||
"print(accuracy)\n",
|
||||
"print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n",
|
||||
"\n",
|
||||
"import scikitplot as skplt\n",
|
||||
"y_pred = logreg.predict(X_test)\n",
|
||||
"skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n",
|
||||
"plt.show()\n",
|
||||
"y_probas = logreg.predict_proba(X_test)\n",
|
||||
"skplt.metrics.plot_roc(y_test, y_probas)\n",
|
||||
"plt.show()\n",
|
||||
"skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "51760b3e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### Exercise a)\n",
|
||||
"\n",
|
||||
"Convince yourself about the mathematics for the confusion matrix, the ROC and the cumlative gain curves for both a binary and a multiclass classification problem."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c1d42f5f",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### Exercise b)\n",
|
||||
"\n",
|
||||
"Use a binary classification data available from **scikit-learn**. As an example you can use\n",
|
||||
"the MNIST data set and just specialize to two numbers. To do so you can use the following code lines"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"id": "d20bb8be",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import load_digits\n",
|
||||
"digits = load_digits(n_class=2) # Load only two classes, e.g., 0 and 1\n",
|
||||
"X, y = digits.data, digits.target"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "828ea1cd",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Alternatively, you can use the _make$\\_$classification_\n",
|
||||
"functionality. This function generates a random $n$-class classification\n",
|
||||
"dataset, which can be configured for binary classification by setting\n",
|
||||
"n_classes=2. You can also control the number of samples, features,\n",
|
||||
"informative features, redundant features, and more."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"id": "d271f0ba",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import make_classification\n",
|
||||
"X, y = make_classification(n_samples=1000, n_features=20, n_informative=10, n_redundant=5, n_classes=2, random_state=42)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "0068b032",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"You can use this option for the multiclass case as well, see the next exercise.\n",
|
||||
"If you prefer to study other binary classification datasets, feel free\n",
|
||||
"to replace the above suggestions with your own dataset.\n",
|
||||
"\n",
|
||||
"Make plots of the confusion matrix, the ROC curve and the cumulative gain curve."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c45f5b41",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### Exercise c) week 43\n",
|
||||
"\n",
|
||||
"As a multiclass problem, we will use the Iris data set discussed in\n",
|
||||
"the exercises from weeks 41 and 42. This is a three-class data set and\n",
|
||||
"you can set it up using **scikit-learn**,"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"id": "3b045d56",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import load_iris\n",
|
||||
"iris = load_iris()\n",
|
||||
"X = iris.data # Features\n",
|
||||
"y = iris.target # Target labels"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "14cc859c",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Make plots of the confusion matrix, the ROC curve and the cumulative\n",
|
||||
"gain curve for this (or other) multiclass data set."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
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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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|
||||
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||||
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|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek43.html">Exercises week 43</a></li>
|
||||
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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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||||
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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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||||
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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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|
||||
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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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||||
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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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||||
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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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||||
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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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||||
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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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|
||||
|
||||
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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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||||
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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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|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek43.html">Exercises week 43</a></li>
|
||||
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||||
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||||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Projects</span></p>
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||||
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||||
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||||
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||||
<link rel="next" title="Project 1 on Machine Learning, deadline October 6 (midnight), 2025" href="project1.html" />
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||||
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Applied Data Analysis and Machine Learning
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Review of Statistics with Resampling Techniques and Linear Algebra</span></p>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek34.html">Exercises week 34</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week43.html">Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations</a></li>
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<h1>Exercises week 43</h1>
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<!-- Table of contents -->
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<h2> Contents </h2>
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<nav aria-label="Page">
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<ul class="visible nav section-nav flex-column">
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<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Exercises week 43</a></li>
|
||||
<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#overarching-aims-of-the-exercises-weeks-43-and-44">Overarching aims of the exercises weeks 43 and 44</a><ul class="visible nav section-nav flex-column">
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#confusion-matrix">Confusion Matrix</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#roc-curve">ROC Curve</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cumulative-gain">Cumulative Gain</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-measures-precision-recall-and-the-f-1-measure">Other measures: Precision, Recall, and the F<span class="math notranslate nohighlight">\(_1\)</span> Measure</a></li>
|
||||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercises">Exercises</a><ul class="nav section-nav flex-column">
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-a">Exercise a)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-b">Exercise b)</a></li>
|
||||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-c-week-43">Exercise c) week 43</a></li>
|
||||
</ul>
|
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<article class="bd-article">
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<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
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doconce format html exercisesweek43.do.txt -->
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<!-- dom:TITLE: Exercises week 43 --><section class="tex2jax_ignore mathjax_ignore" id="exercises-week-43">
|
||||
<h1>Exercises week 43<a class="headerlink" href="#exercises-week-43" title="Link to this heading">#</a></h1>
|
||||
<p><strong>October 20-24, 2025</strong></p>
|
||||
<p>Date: <strong>Deadline Friday October 24 at midnight</strong></p>
|
||||
</section>
|
||||
<section class="tex2jax_ignore mathjax_ignore" id="overarching-aims-of-the-exercises-weeks-43-and-44">
|
||||
<h1>Overarching aims of the exercises weeks 43 and 44<a class="headerlink" href="#overarching-aims-of-the-exercises-weeks-43-and-44" title="Link to this heading">#</a></h1>
|
||||
<p>The aim of the exercises this week is to gain some confidence with
|
||||
ways to visualize the results of a classification problem. We will
|
||||
target three ways of setting up the analysis. The first and simplest
|
||||
one is the</p>
|
||||
<ol class="arabic simple">
|
||||
<li><p>so-called confusion matrix, and the next is the</p></li>
|
||||
<li><p>ROC curve and finally the</p></li>
|
||||
<li><p>Cumulative gain curve.</p></li>
|
||||
</ol>
|
||||
<p>We will use Logistic Regression as method for the classification in
|
||||
this exercise. You can compare these results with those obtained with
|
||||
your neural network code from project 2 without a hidden layer.</p>
|
||||
<p>In these exercises we will use binary and multi-class data sets
|
||||
(the Iris data set from week 41).</p>
|
||||
<p>The underlying mathematics is described here.</p>
|
||||
<section id="confusion-matrix">
|
||||
<h2>Confusion Matrix<a class="headerlink" href="#confusion-matrix" title="Link to this heading">#</a></h2>
|
||||
<p>A <strong>confusion matrix</strong> summarizes a classifier’s performance by
|
||||
tabulating predictions versus true labels. For binary classification,
|
||||
it is a <span class="math notranslate nohighlight">\(2\times2\)</span> table whose entries are counts of outcomes:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}
|
||||
\begin{array}{l|cc} & \text{Predicted Positive} & \text{Predicted Negative} \\ \hline \text{Actual Positive} & TP & FN \\ \text{Actual Negative} & FP & TN \end{array}.
|
||||
\end{split}\]</div>
|
||||
<p>Here TP (true positives) is the number of cases correctly predicted as
|
||||
positive, FP (false positives) is the number incorrectly predicted as
|
||||
positive, TN (true negatives) is correctly predicted negative, and FN
|
||||
(false negatives) is incorrectly predicted negative . In other words,
|
||||
“positive” means class 1 and “negative” means class 0; for example, TP
|
||||
occurs when the prediction and actual are both positive. Formally:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\text{TPR} = \frac{\text{TP}}{\text{TP} + \text{FN}}, \quad \text{FPR} = \frac{\text{FP}}{\text{FP} + \text{TN}},
|
||||
\]</div>
|
||||
<p>where TPR and FPR are the true and false positive rates defined below.</p>
|
||||
<p>In multiclass classification with <span class="math notranslate nohighlight">\(K\)</span> classes, the confusion matrix
|
||||
generalizes to a <span class="math notranslate nohighlight">\(K\times K\)</span> table. Entry <span class="math notranslate nohighlight">\(N_{ij}\)</span> in the table is
|
||||
the count of instances whose true class is <span class="math notranslate nohighlight">\(i\)</span> and whose predicted
|
||||
class is <span class="math notranslate nohighlight">\(j\)</span>. For example, a three-class confusion matrix can be written
|
||||
as:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[\begin{split}
|
||||
\begin{array}{c|ccc} & \text{Pred Class 1} & \text{Pred Class 2} & \text{Pred Class 3} \\ \hline \text{Act Class 1} & N_{11} & N_{12} & N_{13} \\ \text{Act Class 2} & N_{21} & N_{22} & N_{23} \\ \text{Act Class 3} & N_{31} & N_{32} & N_{33} \end{array}.
|
||||
\end{split}\]</div>
|
||||
<p>Here the diagonal entries <span class="math notranslate nohighlight">\(N_{ii}\)</span> are the true positives for each
|
||||
class, and off-diagonal entries are misclassifications. This matrix
|
||||
allows computation of per-class metrics: e.g. for class <span class="math notranslate nohighlight">\(i\)</span>,
|
||||
<span class="math notranslate nohighlight">\(\mathrm{TP}_i=N_{ii}\)</span>, <span class="math notranslate nohighlight">\(\mathrm{FN}_i=\sum_{j\neq i}N_{ij}\)</span>,
|
||||
<span class="math notranslate nohighlight">\(\mathrm{FP}_i=\sum_{j\neq i}N_{ji}\)</span>, and <span class="math notranslate nohighlight">\(\mathrm{TN}_i\)</span> is the sum of
|
||||
all remaining entries.</p>
|
||||
<p>As defined above, TPR and FPR come from the binary case. In binary
|
||||
terms with <span class="math notranslate nohighlight">\(P\)</span> actual positives and <span class="math notranslate nohighlight">\(N\)</span> actual negatives, one has</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\text{TPR} = \frac{TP}{P} = \frac{TP}{TP+FN}, \quad \text{FPR} =
|
||||
\frac{FP}{N} = \frac{FP}{FP+TN},
|
||||
\]</div>
|
||||
<p>as used in standard confusion-matrix
|
||||
formulations. These rates will be used in constructing ROC curves.</p>
|
||||
</section>
|
||||
<section id="roc-curve">
|
||||
<h2>ROC Curve<a class="headerlink" href="#roc-curve" title="Link to this heading">#</a></h2>
|
||||
<p>The Receiver Operating Characteristic (ROC) curve plots the trade-off
|
||||
between true positives and false positives as a discrimination
|
||||
threshold varies. Specifically, for a binary classifier that outputs
|
||||
a score or probability, one varies the threshold <span class="math notranslate nohighlight">\(t\)</span> for declaring
|
||||
<strong>positive</strong>, and computes at each <span class="math notranslate nohighlight">\(t\)</span> the true positive rate
|
||||
<span class="math notranslate nohighlight">\(\mathrm{TPR}(t)\)</span> and false positive rate <span class="math notranslate nohighlight">\(\mathrm{FPR}(t)\)</span> using the
|
||||
confusion matrix at that threshold. The ROC curve is then the graph
|
||||
of TPR versus FPR. By definition,</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathrm{TPR} = \frac{TP}{TP+FN}, \qquad \mathrm{FPR} = \frac{FP}{FP+TN},
|
||||
\]</div>
|
||||
<p>where <span class="math notranslate nohighlight">\(TP,FP,TN,FN\)</span> are counts determined by threshold <span class="math notranslate nohighlight">\(t\)</span>. A perfect
|
||||
classifier would reach the point (FPR=0, TPR=1) at some threshold.</p>
|
||||
<p>Formally, the ROC curve is obtained by plotting
|
||||
<span class="math notranslate nohighlight">\((\mathrm{FPR}(t),\mathrm{TPR}(t))\)</span> for all <span class="math notranslate nohighlight">\(t\in[0,1]\)</span> (or as <span class="math notranslate nohighlight">\(t\)</span>
|
||||
sweeps through the sorted scores). The Area Under the ROC Curve (AUC)
|
||||
quantifies the average performance over all thresholds. It can be
|
||||
interpreted probabilistically: <span class="math notranslate nohighlight">\(\mathrm{AUC} =
|
||||
\Pr\bigl(s(X^+)>s(X^-)\bigr)\)</span>, the probability that a random positive
|
||||
instance <span class="math notranslate nohighlight">\(X^+\)</span> receives a higher score <span class="math notranslate nohighlight">\(s\)</span> than a random negative
|
||||
instance <span class="math notranslate nohighlight">\(X^-\)</span> . Equivalently, the AUC is the integral under the ROC
|
||||
curve:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathrm{AUC} \;=\; \int_{0}^{1} \mathrm{TPR}(f)\,df,
|
||||
\]</div>
|
||||
<p>where <span class="math notranslate nohighlight">\(f\)</span> ranges over FPR (or fraction of negatives). A model that guesses at random yields a diagonal ROC (AUC=0.5), whereas a perfect model yields AUC=1.0.</p>
|
||||
</section>
|
||||
<section id="cumulative-gain">
|
||||
<h2>Cumulative Gain<a class="headerlink" href="#cumulative-gain" title="Link to this heading">#</a></h2>
|
||||
<p>The cumulative gain curve (or gains chart) evaluates how many
|
||||
positives are captured as one targets an increasing fraction of the
|
||||
population, sorted by model confidence. To construct it, sort all
|
||||
instances by decreasing predicted probability of the positive class.
|
||||
Then, for the top <span class="math notranslate nohighlight">\(\alpha\)</span> fraction of instances, compute the fraction
|
||||
of all actual positives that fall in this subset. In formula form, if
|
||||
<span class="math notranslate nohighlight">\(P\)</span> is the total number of positive instances and <span class="math notranslate nohighlight">\(P(\alpha)\)</span> is the
|
||||
number of positives among the top <span class="math notranslate nohighlight">\(\alpha\)</span> of the data, the cumulative
|
||||
gain at level <span class="math notranslate nohighlight">\(\alpha\)</span> is</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathrm{Gain}(\alpha) \;=\; \frac{P(\alpha)}{P}.
|
||||
\]</div>
|
||||
<p>For example, cutting off at the top 10% of predictions yields a gain
|
||||
equal to (positives in top 10%) divided by (total positives) .
|
||||
Plotting <span class="math notranslate nohighlight">\(\mathrm{Gain}(\alpha)\)</span> versus <span class="math notranslate nohighlight">\(\alpha\)</span> (often in percent)
|
||||
gives the gain curve. The baseline (random) curve is the diagonal
|
||||
<span class="math notranslate nohighlight">\(\mathrm{Gain}(\alpha)=\alpha\)</span>, while an ideal model has a steep climb
|
||||
toward 1.</p>
|
||||
<p>A related measure is the {\em lift}, often called the gain ratio. It is the ratio of the model’s capture rate to that of random selection. Equivalently,</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\mathrm{Lift}(\alpha) \;=\; \frac{\mathrm{Gain}(\alpha)}{\alpha}.
|
||||
\]</div>
|
||||
<p>A lift <span class="math notranslate nohighlight">\(>1\)</span> indicates better-than-random targeting. In practice, gain
|
||||
and lift charts (used e.g.\ in marketing or imbalanced classification)
|
||||
show how many positives can be “gained” by focusing on a fraction of
|
||||
the population .</p>
|
||||
</section>
|
||||
<section id="other-measures-precision-recall-and-the-f-1-measure">
|
||||
<h2>Other measures: Precision, Recall, and the F<span class="math notranslate nohighlight">\(_1\)</span> Measure<a class="headerlink" href="#other-measures-precision-recall-and-the-f-1-measure" title="Link to this heading">#</a></h2>
|
||||
<p>Precision and recall (sensitivity) quantify binary classification
|
||||
accuracy in terms of positive predictions. They are defined from the
|
||||
confusion matrix as:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\text{Precision} = \frac{TP}{TP + FP}, \qquad \text{Recall} = \frac{TP}{TP + FN}.
|
||||
\]</div>
|
||||
<p>Precision is the fraction of predicted positives that are correct, and
|
||||
recall is the fraction of actual positives that are correctly
|
||||
identified . A high-precision classifier makes few false-positive
|
||||
errors, while a high-recall classifier makes few false-negative
|
||||
errors.</p>
|
||||
<p>The F<span class="math notranslate nohighlight">\(_1\)</span> score (balanced F-measure) combines precision and recall into a single metric via their harmonic mean. The usual formula is:</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
F_1 =2\frac{\text{Precision}\times\text{Recall}}{\text{Precision} + \text{Recall}}.
|
||||
\]</div>
|
||||
<p>This can be shown to equal</p>
|
||||
<div class="math notranslate nohighlight">
|
||||
\[
|
||||
\frac{2\,TP}{2\,TP + FP + FN}.
|
||||
\]</div>
|
||||
<p>The F<span class="math notranslate nohighlight">\(_1\)</span> score ranges from 0 (worst) to 1 (best), and balances the
|
||||
trade-off between precision and recall.</p>
|
||||
<p>For multi-class classification, one computes per-class
|
||||
precision/recall/F<span class="math notranslate nohighlight">\(_1\)</span> (treating each class as “positive” in a
|
||||
one-vs-rest manner) and then averages. Common averaging methods are:</p>
|
||||
<p>Micro-averaging: Sum all true positives, false positives, and false negatives across classes, then compute precision/recall/F<span class="math notranslate nohighlight">\(_1\)</span> from these totals.
|
||||
Macro-averaging: Compute the F<span class="math notranslate nohighlight">\(1\)</span> score <span class="math notranslate nohighlight">\(F{1,i}\)</span> for each class <span class="math notranslate nohighlight">\(i\)</span> separately, then take the unweighted mean: <span class="math notranslate nohighlight">\(F_{1,\mathrm{macro}} = \frac{1}{K}\sum_{i=1}^K F_{1,i}\)</span> . This treats all classes equally regardless of size.
|
||||
Weighted-averaging: Like macro-average, but weight each class’s <span class="math notranslate nohighlight">\(F_{1,i}\)</span> by its support <span class="math notranslate nohighlight">\(n_i\)</span> (true count): <span class="math notranslate nohighlight">\(F_{1,\mathrm{weighted}} = \frac{1}{N}\sum_{i=1}^K n_i F_{1,i}\)</span>, where <span class="math notranslate nohighlight">\(N=\sum_i n_i\)</span>. This accounts for class imbalance by giving more weight to larger classes .</p>
|
||||
<p>Each of these averages has different use-cases. Micro-average is
|
||||
dominated by common classes, macro-average highlights performance on
|
||||
rare classes, and weighted-average is a compromise. These formulas
|
||||
and concepts allow rigorous evaluation of classifier performance in
|
||||
both binary and multi-class settings.</p>
|
||||
</section>
|
||||
<section id="exercises">
|
||||
<h2>Exercises<a class="headerlink" href="#exercises" title="Link to this heading">#</a></h2>
|
||||
<p>Here is a simple code example which uses the Logistic regression machinery from <strong>scikit-learn</strong>.
|
||||
At the end it sets up the confusion matrix and the ROC and cumulative gain curves.
|
||||
Feel free to use these functionalities (we don’t expect you to write your own code for say the confusion matrix).</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>%matplotlib inline
|
||||
|
||||
import matplotlib.pyplot as plt
|
||||
import numpy as np
|
||||
from sklearn.model_selection import train_test_split
|
||||
# from sklearn.datasets import fill in the data set
|
||||
from sklearn.linear_model import LogisticRegression
|
||||
|
||||
# Load the data, fill inn
|
||||
mydata.data = ?
|
||||
|
||||
X_train, X_test, y_train, y_test = train_test_split(mydata.data,cancer.target,random_state=0)
|
||||
print(X_train.shape)
|
||||
print(X_test.shape)
|
||||
# Logistic Regression
|
||||
# define which type of problem, binary or multiclass
|
||||
logreg = LogisticRegression(solver='lbfgs')
|
||||
logreg.fit(X_train, y_train)
|
||||
|
||||
from sklearn.preprocessing import LabelEncoder
|
||||
from sklearn.model_selection import cross_validate
|
||||
#Cross validation
|
||||
accuracy = cross_validate(logreg,X_test,y_test,cv=10)['test_score']
|
||||
print(accuracy)
|
||||
print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
|
||||
|
||||
import scikitplot as skplt
|
||||
y_pred = logreg.predict(X_test)
|
||||
skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
|
||||
plt.show()
|
||||
y_probas = logreg.predict_proba(X_test)
|
||||
skplt.metrics.plot_roc(y_test, y_probas)
|
||||
plt.show()
|
||||
skplt.metrics.plot_cumulative_gain(y_test, y_probas)
|
||||
plt.show()
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<section id="exercise-a">
|
||||
<h3>Exercise a)<a class="headerlink" href="#exercise-a" title="Link to this heading">#</a></h3>
|
||||
<p>Convince yourself about the mathematics for the confusion matrix, the ROC and the cumlative gain curves for both a binary and a multiclass classification problem.</p>
|
||||
</section>
|
||||
<section id="exercise-b">
|
||||
<h3>Exercise b)<a class="headerlink" href="#exercise-b" title="Link to this heading">#</a></h3>
|
||||
<p>Use a binary classification data available from <strong>scikit-learn</strong>. As an example you can use
|
||||
the MNIST data set and just specialize to two numbers. To do so you can use the following code lines</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.datasets import load_digits
|
||||
digits = load_digits(n_class=2) # Load only two classes, e.g., 0 and 1
|
||||
X, y = digits.data, digits.target
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>Alternatively, you can use the <em>make<span class="math notranslate nohighlight">\(\_\)</span>classification</em>
|
||||
functionality. This function generates a random <span class="math notranslate nohighlight">\(n\)</span>-class classification
|
||||
dataset, which can be configured for binary classification by setting
|
||||
n_classes=2. You can also control the number of samples, features,
|
||||
informative features, redundant features, and more.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.datasets import make_classification
|
||||
X, y = make_classification(n_samples=1000, n_features=20, n_informative=10, n_redundant=5, n_classes=2, random_state=42)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>You can use this option for the multiclass case as well, see the next exercise.
|
||||
If you prefer to study other binary classification datasets, feel free
|
||||
to replace the above suggestions with your own dataset.</p>
|
||||
<p>Make plots of the confusion matrix, the ROC curve and the cumulative gain curve.</p>
|
||||
</section>
|
||||
<section id="exercise-c-week-43">
|
||||
<h3>Exercise c) week 43<a class="headerlink" href="#exercise-c-week-43" title="Link to this heading">#</a></h3>
|
||||
<p>As a multiclass problem, we will use the Iris data set discussed in
|
||||
the exercises from weeks 41 and 42. This is a three-class data set and
|
||||
you can set it up using <strong>scikit-learn</strong>,</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.datasets import load_iris
|
||||
iris = load_iris()
|
||||
X = iris.data # Features
|
||||
y = iris.target # Target labels
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
</div>
|
||||
<p>Make plots of the confusion matrix, the ROC curve and the cumulative
|
||||
gain curve for this (or other) multiclass data set.</p>
|
||||
</section>
|
||||
</section>
|
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#confusion-matrix">Confusion Matrix</a></li>
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|
||||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek43.html">Exercises week 43</a></li>
|
||||
|
||||
</ul>
|
||||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Projects</span></p>
|
||||
<ul class="nav bd-sidenav">
|
||||
|
||||
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|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "860d70d8",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)\n",
|
||||
"doconce format html exercisesweek43.do.txt -->\n",
|
||||
"<!-- dom:TITLE: Exercises week 43 -->"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "119c0988",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"# Exercises week 43 \n",
|
||||
"**October 20-24, 2025**\n",
|
||||
"\n",
|
||||
"Date: **Deadline Friday October 24 at midnight**"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "909887eb",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"# Overarching aims of the exercises weeks 43 and 44\n",
|
||||
"\n",
|
||||
"The aim of the exercises this week is to gain some confidence with\n",
|
||||
"ways to visualize the results of a classification problem. We will\n",
|
||||
"target three ways of setting up the analysis. The first and simplest\n",
|
||||
"one is the\n",
|
||||
"1. so-called confusion matrix, and the next is the\n",
|
||||
"\n",
|
||||
"2. ROC curve and finally the\n",
|
||||
"\n",
|
||||
"3. Cumulative gain curve.\n",
|
||||
"\n",
|
||||
"We will use Logistic Regression as method for the classification in\n",
|
||||
"this exercise. You can compare these results with those obtained with\n",
|
||||
"your neural network code from project 2 without a hidden layer.\n",
|
||||
"\n",
|
||||
"In these exercises we will use binary and multi-class data sets\n",
|
||||
"(the Iris data set from week 41).\n",
|
||||
"\n",
|
||||
"The underlying mathematics is described here."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "1e1cb4fb",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### Confusion Matrix\n",
|
||||
"\n",
|
||||
"A **confusion matrix** summarizes a classifier’s performance by\n",
|
||||
"tabulating predictions versus true labels. For binary classification,\n",
|
||||
"it is a $2\\times2$ table whose entries are counts of outcomes:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "7b090385",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\begin{array}{l|cc} & \\text{Predicted Positive} & \\text{Predicted Negative} \\\\ \\hline \\text{Actual Positive} & TP & FN \\\\ \\text{Actual Negative} & FP & TN \\end{array}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "1e14904b",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Here TP (true positives) is the number of cases correctly predicted as\n",
|
||||
"positive, FP (false positives) is the number incorrectly predicted as\n",
|
||||
"positive, TN (true negatives) is correctly predicted negative, and FN\n",
|
||||
"(false negatives) is incorrectly predicted negative . In other words,\n",
|
||||
"“positive” means class 1 and “negative” means class 0; for example, TP\n",
|
||||
"occurs when the prediction and actual are both positive. Formally:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "e93ea290",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\text{TPR} = \\frac{\\text{TP}}{\\text{TP} + \\text{FN}}, \\quad \\text{FPR} = \\frac{\\text{FP}}{\\text{FP} + \\text{TN}},\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c80bea5b",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"where TPR and FPR are the true and false positive rates defined below.\n",
|
||||
"\n",
|
||||
"In multiclass classification with $K$ classes, the confusion matrix\n",
|
||||
"generalizes to a $K\\times K$ table. Entry $N_{ij}$ in the table is\n",
|
||||
"the count of instances whose true class is $i$ and whose predicted\n",
|
||||
"class is $j$. For example, a three-class confusion matrix can be written\n",
|
||||
"as:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "a0f68f5f",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\begin{array}{c|ccc} & \\text{Pred Class 1} & \\text{Pred Class 2} & \\text{Pred Class 3} \\\\ \\hline \\text{Act Class 1} & N_{11} & N_{12} & N_{13} \\\\ \\text{Act Class 2} & N_{21} & N_{22} & N_{23} \\\\ \\text{Act Class 3} & N_{31} & N_{32} & N_{33} \\end{array}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "869669b2",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Here the diagonal entries $N_{ii}$ are the true positives for each\n",
|
||||
"class, and off-diagonal entries are misclassifications. This matrix\n",
|
||||
"allows computation of per-class metrics: e.g. for class $i$,\n",
|
||||
"$\\mathrm{TP}_i=N_{ii}$, $\\mathrm{FN}_i=\\sum_{j\\neq i}N_{ij}$,\n",
|
||||
"$\\mathrm{FP}_i=\\sum_{j\\neq i}N_{ji}$, and $\\mathrm{TN}_i$ is the sum of\n",
|
||||
"all remaining entries.\n",
|
||||
"\n",
|
||||
"As defined above, TPR and FPR come from the binary case. In binary\n",
|
||||
"terms with $P$ actual positives and $N$ actual negatives, one has"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "2abd82a7",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\text{TPR} = \\frac{TP}{P} = \\frac{TP}{TP+FN}, \\quad \\text{FPR} =\n",
|
||||
"\\frac{FP}{N} = \\frac{FP}{FP+TN},\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "2f79325c",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"as used in standard confusion-matrix\n",
|
||||
"formulations. These rates will be used in constructing ROC curves."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "0ce65a47",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### ROC Curve\n",
|
||||
"\n",
|
||||
"The Receiver Operating Characteristic (ROC) curve plots the trade-off\n",
|
||||
"between true positives and false positives as a discrimination\n",
|
||||
"threshold varies. Specifically, for a binary classifier that outputs\n",
|
||||
"a score or probability, one varies the threshold $t$ for declaring\n",
|
||||
"**positive**, and computes at each $t$ the true positive rate\n",
|
||||
"$\\mathrm{TPR}(t)$ and false positive rate $\\mathrm{FPR}(t)$ using the\n",
|
||||
"confusion matrix at that threshold. The ROC curve is then the graph\n",
|
||||
"of TPR versus FPR. By definition,"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "d750fdff",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{TPR} = \\frac{TP}{TP+FN}, \\qquad \\mathrm{FPR} = \\frac{FP}{FP+TN},\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "561bfb2c",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"where $TP,FP,TN,FN$ are counts determined by threshold $t$. A perfect\n",
|
||||
"classifier would reach the point (FPR=0, TPR=1) at some threshold.\n",
|
||||
"\n",
|
||||
"Formally, the ROC curve is obtained by plotting\n",
|
||||
"$(\\mathrm{FPR}(t),\\mathrm{TPR}(t))$ for all $t\\in[0,1]$ (or as $t$\n",
|
||||
"sweeps through the sorted scores). The Area Under the ROC Curve (AUC)\n",
|
||||
"quantifies the average performance over all thresholds. It can be\n",
|
||||
"interpreted probabilistically: $\\mathrm{AUC} =\n",
|
||||
"\\Pr\\bigl(s(X^+)>s(X^-)\\bigr)$, the probability that a random positive\n",
|
||||
"instance $X^+$ receives a higher score $s$ than a random negative\n",
|
||||
"instance $X^-$ . Equivalently, the AUC is the integral under the ROC\n",
|
||||
"curve:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "5ca722fe",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{AUC} \\;=\\; \\int_{0}^{1} \\mathrm{TPR}(f)\\,df,\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "30080a86",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"where $f$ ranges over FPR (or fraction of negatives). A model that guesses at random yields a diagonal ROC (AUC=0.5), whereas a perfect model yields AUC=1.0."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "9e627156",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### Cumulative Gain\n",
|
||||
"\n",
|
||||
"The cumulative gain curve (or gains chart) evaluates how many\n",
|
||||
"positives are captured as one targets an increasing fraction of the\n",
|
||||
"population, sorted by model confidence. To construct it, sort all\n",
|
||||
"instances by decreasing predicted probability of the positive class.\n",
|
||||
"Then, for the top $\\alpha$ fraction of instances, compute the fraction\n",
|
||||
"of all actual positives that fall in this subset. In formula form, if\n",
|
||||
"$P$ is the total number of positive instances and $P(\\alpha)$ is the\n",
|
||||
"number of positives among the top $\\alpha$ of the data, the cumulative\n",
|
||||
"gain at level $\\alpha$ is"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "3e9132ef",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{Gain}(\\alpha) \\;=\\; \\frac{P(\\alpha)}{P}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "75be6f5c",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"For example, cutting off at the top 10% of predictions yields a gain\n",
|
||||
"equal to (positives in top 10%) divided by (total positives) .\n",
|
||||
"Plotting $\\mathrm{Gain}(\\alpha)$ versus $\\alpha$ (often in percent)\n",
|
||||
"gives the gain curve. The baseline (random) curve is the diagonal\n",
|
||||
"$\\mathrm{Gain}(\\alpha)=\\alpha$, while an ideal model has a steep climb\n",
|
||||
"toward 1.\n",
|
||||
"\n",
|
||||
"A related measure is the {\\em lift}, often called the gain ratio. It is the ratio of the model’s capture rate to that of random selection. Equivalently,"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "e5525570",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\mathrm{Lift}(\\alpha) \\;=\\; \\frac{\\mathrm{Gain}(\\alpha)}{\\alpha}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "18ff8dc2",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"A lift $>1$ indicates better-than-random targeting. In practice, gain\n",
|
||||
"and lift charts (used e.g.\\ in marketing or imbalanced classification)\n",
|
||||
"show how many positives can be “gained” by focusing on a fraction of\n",
|
||||
"the population ."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c3d3fde8",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### Other measures: Precision, Recall, and the F$_1$ Measure\n",
|
||||
"\n",
|
||||
"Precision and recall (sensitivity) quantify binary classification\n",
|
||||
"accuracy in terms of positive predictions. They are defined from the\n",
|
||||
"confusion matrix as:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "f1f14c8e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\text{Precision} = \\frac{TP}{TP + FP}, \\qquad \\text{Recall} = \\frac{TP}{TP + FN}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "422cc743",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Precision is the fraction of predicted positives that are correct, and\n",
|
||||
"recall is the fraction of actual positives that are correctly\n",
|
||||
"identified . A high-precision classifier makes few false-positive\n",
|
||||
"errors, while a high-recall classifier makes few false-negative\n",
|
||||
"errors.\n",
|
||||
"\n",
|
||||
"The F$_1$ score (balanced F-measure) combines precision and recall into a single metric via their harmonic mean. The usual formula is:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "621a2e8b",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"F_1 =2\\frac{\\text{Precision}\\times\\text{Recall}}{\\text{Precision} + \\text{Recall}}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "62eee54a",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"This can be shown to equal"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "7a6a2e7a",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"$$\n",
|
||||
"\\frac{2\\,TP}{2\\,TP + FP + FN}.\n",
|
||||
"$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "b96c9ff4",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"The F$_1$ score ranges from 0 (worst) to 1 (best), and balances the\n",
|
||||
"trade-off between precision and recall.\n",
|
||||
"\n",
|
||||
"For multi-class classification, one computes per-class\n",
|
||||
"precision/recall/F$_1$ (treating each class as “positive” in a\n",
|
||||
"one-vs-rest manner) and then averages. Common averaging methods are:\n",
|
||||
"\n",
|
||||
"Micro-averaging: Sum all true positives, false positives, and false negatives across classes, then compute precision/recall/F$_1$ from these totals.\n",
|
||||
"Macro-averaging: Compute the F$1$ score $F{1,i}$ for each class $i$ separately, then take the unweighted mean: $F_{1,\\mathrm{macro}} = \\frac{1}{K}\\sum_{i=1}^K F_{1,i}$ . This treats all classes equally regardless of size.\n",
|
||||
"Weighted-averaging: Like macro-average, but weight each class’s $F_{1,i}$ by its support $n_i$ (true count): $F_{1,\\mathrm{weighted}} = \\frac{1}{N}\\sum_{i=1}^K n_i F_{1,i}$, where $N=\\sum_i n_i$. This accounts for class imbalance by giving more weight to larger classes .\n",
|
||||
"\n",
|
||||
"Each of these averages has different use-cases. Micro-average is\n",
|
||||
"dominated by common classes, macro-average highlights performance on\n",
|
||||
"rare classes, and weighted-average is a compromise. These formulas\n",
|
||||
"and concepts allow rigorous evaluation of classifier performance in\n",
|
||||
"both binary and multi-class settings."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "9274bf3f",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"## Exercises\n",
|
||||
"\n",
|
||||
"Here is a simple code example which uses the Logistic regression machinery from **scikit-learn**.\n",
|
||||
"At the end it sets up the confusion matrix and the ROC and cumulative gain curves.\n",
|
||||
"Feel free to use these functionalities (we don't expect you to write your own code for say the confusion matrix)."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"id": "be9ff0b9",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"%matplotlib inline\n",
|
||||
"\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"import numpy as np\n",
|
||||
"from sklearn.model_selection import train_test_split \n",
|
||||
"# from sklearn.datasets import fill in the data set\n",
|
||||
"from sklearn.linear_model import LogisticRegression\n",
|
||||
"\n",
|
||||
"# Load the data, fill inn\n",
|
||||
"mydata.data = ?\n",
|
||||
"\n",
|
||||
"X_train, X_test, y_train, y_test = train_test_split(mydata.data,cancer.target,random_state=0)\n",
|
||||
"print(X_train.shape)\n",
|
||||
"print(X_test.shape)\n",
|
||||
"# Logistic Regression\n",
|
||||
"# define which type of problem, binary or multiclass\n",
|
||||
"logreg = LogisticRegression(solver='lbfgs')\n",
|
||||
"logreg.fit(X_train, y_train)\n",
|
||||
"\n",
|
||||
"from sklearn.preprocessing import LabelEncoder\n",
|
||||
"from sklearn.model_selection import cross_validate\n",
|
||||
"#Cross validation\n",
|
||||
"accuracy = cross_validate(logreg,X_test,y_test,cv=10)['test_score']\n",
|
||||
"print(accuracy)\n",
|
||||
"print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n",
|
||||
"\n",
|
||||
"import scikitplot as skplt\n",
|
||||
"y_pred = logreg.predict(X_test)\n",
|
||||
"skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n",
|
||||
"plt.show()\n",
|
||||
"y_probas = logreg.predict_proba(X_test)\n",
|
||||
"skplt.metrics.plot_roc(y_test, y_probas)\n",
|
||||
"plt.show()\n",
|
||||
"skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "51760b3e",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### Exercise a)\n",
|
||||
"\n",
|
||||
"Convince yourself about the mathematics for the confusion matrix, the ROC and the cumlative gain curves for both a binary and a multiclass classification problem."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c1d42f5f",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### Exercise b)\n",
|
||||
"\n",
|
||||
"Use a binary classification data available from **scikit-learn**. As an example you can use\n",
|
||||
"the MNIST data set and just specialize to two numbers. To do so you can use the following code lines"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"id": "d20bb8be",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import load_digits\n",
|
||||
"digits = load_digits(n_class=2) # Load only two classes, e.g., 0 and 1\n",
|
||||
"X, y = digits.data, digits.target"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "828ea1cd",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Alternatively, you can use the _make$\\_$classification_\n",
|
||||
"functionality. This function generates a random $n$-class classification\n",
|
||||
"dataset, which can be configured for binary classification by setting\n",
|
||||
"n_classes=2. You can also control the number of samples, features,\n",
|
||||
"informative features, redundant features, and more."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"id": "d271f0ba",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import make_classification\n",
|
||||
"X, y = make_classification(n_samples=1000, n_features=20, n_informative=10, n_redundant=5, n_classes=2, random_state=42)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "0068b032",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"You can use this option for the multiclass case as well, see the next exercise.\n",
|
||||
"If you prefer to study other binary classification datasets, feel free\n",
|
||||
"to replace the above suggestions with your own dataset.\n",
|
||||
"\n",
|
||||
"Make plots of the confusion matrix, the ROC curve and the cumulative gain curve."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "c45f5b41",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"### Exercise c) week 43\n",
|
||||
"\n",
|
||||
"As a multiclass problem, we will use the Iris data set discussed in\n",
|
||||
"the exercises from weeks 41 and 42. This is a three-class data set and\n",
|
||||
"you can set it up using **scikit-learn**,"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"id": "3b045d56",
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"editable": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sklearn.datasets import load_iris\n",
|
||||
"iris = load_iris()\n",
|
||||
"X = iris.data # Features\n",
|
||||
"y = iris.target # Target labels"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "14cc859c",
|
||||
"metadata": {
|
||||
"editable": true
|
||||
},
|
||||
"source": [
|
||||
"Make plots of the confusion matrix, the ROC curve and the cumulative\n",
|
||||
"gain curve for this (or other) multiclass data set."
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 5
|
||||
}
|
||||
File diff suppressed because it is too large
Load Diff
Reference in New Issue
Block a user