Morten Hjorth-Jensen
648 lines
19 KiB
Plaintext
648 lines
19 KiB
Plaintext
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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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"# 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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"source": [
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"# Overarching aims of the exercises for week 43\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. The next one is the so-called\n",
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"\n",
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"2. ROC curve. Finally we have 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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"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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"$$\n",
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"\\begin{array}{l|cc} & \\text{Predicted Positive} & \\text{Predicted Negative} \\\\ \\hline \\text{Actual Positive} & TP & FN \\\\ \\text{Actual Negative} & FP & TN \\end{array}.\n",
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"$$"
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"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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"$$\n",
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"\\begin{array}{c|ccc} & \\text{Pred Class 1} & \\text{Pred Class 2} & \\text{Pred Class 3} \\\\ \\hline \\text{Act Class 1} & N_{11} & N_{12} & N_{13} \\\\ \\text{Act Class 2} & N_{21} & N_{22} & N_{23} \\\\ \\text{Act Class 3} & N_{31} & N_{32} & N_{33} \\end{array}.\n",
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"$$"
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]
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"cell_type": "markdown",
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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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"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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"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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"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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"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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"$$\n",
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"\\mathrm{AUC} \\;=\\; \\int_{0}^{1} \\mathrm{TPR}(f)\\,df,\n",
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"$$"
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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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"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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"$$\n",
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"\\mathrm{Gain}(\\alpha) \\;=\\; \\frac{P(\\alpha)}{P}.\n",
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"$$"
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]
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},
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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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"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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"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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"$$\n",
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"\\text{Precision} = \\frac{TP}{TP + FP}, \\qquad \\text{Recall} = \\frac{TP}{TP + FN}.\n",
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"$$"
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]
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"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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"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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"metadata": {
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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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"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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"cell_type": "markdown",
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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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"cell_type": "markdown",
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"metadata": {
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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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"metadata": {
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"collapsed": false,
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"outputs": [],
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"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",
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"from sklearn.linear_model import LogisticRegression\n",
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"\n",
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"# Load the data, fill inn\n",
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"mydata.data = ?\n",
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"\n",
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"X_train, X_test, y_train, y_test = train_test_split(mydata.data,cancer.target,random_state=0)\n",
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"print(X_train.shape)\n",
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"print(X_test.shape)\n",
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"# Logistic Regression\n",
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"# define which type of problem, binary or multiclass\n",
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"logreg = LogisticRegression(solver='lbfgs')\n",
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"logreg.fit(X_train, y_train)\n",
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"\n",
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"from sklearn.preprocessing import LabelEncoder\n",
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"from sklearn.model_selection import cross_validate\n",
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"#Cross validation\n",
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"accuracy = cross_validate(logreg,X_test,y_test,cv=10)['test_score']\n",
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"print(accuracy)\n",
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"print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n",
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"\n",
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"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,
|
||
"jupyter": {
|
||
"outputs_hidden": false
|
||
}
|
||
},
|
||
"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,
|
||
"jupyter": {
|
||
"outputs_hidden": false
|
||
}
|
||
},
|
||
"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,
|
||
"jupyter": {
|
||
"outputs_hidden": false
|
||
}
|
||
},
|
||
"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": {
|
||
"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
|
||
}
|