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{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Exercise: Classification of air showers measured with the MAGIC telescope\n", "\n", "The [MAGIC telescope](https://en.wikipedia.org/wiki/MAGIC_(telescope)) is a Cherenkov telescope situated on La Palma, one of the Canary Islands. The [MAGIC machine learning dataset](https://archive.ics.uci.edu/ml/datasets/magic+gamma+telescope) can be obtained from [UC Irvine Machine Learning Repository](https://archive.ics.uci.edu/ml/index.php).\n", "\n", "The task is to separate signal events (gamma showers) and background events (hadron showers) based on the features of a measured Cherenkov shower.\n", "\n", "The features of a shower are:\n", "\n", " 1. fLength: continuous # major axis of ellipse [mm]\n", " 2. fWidth: continuous # minor axis of ellipse [mm] \n", " 3. fSize: continuous # 10-log of sum of content of all pixels [in #phot]\n", " 4. fConc: continuous # ratio of sum of two highest pixels over fSize [ratio]\n", " 5. fConc1: continuous # ratio of highest pixel over fSize [ratio]\n", " 6. fAsym: continuous # distance from highest pixel to center, projected onto major axis [mm]\n", " 7. fM3Long: continuous # 3rd root of third moment along major axis [mm] \n", " 8. fM3Trans: continuous # 3rd root of third moment along minor axis [mm]\n", " 9. fAlpha: continuous # angle of major axis with vector to origin [deg]\n", " 10. fDist: continuous # distance from origin to center of ellipse [mm]\n", " 11. class: g,h # gamma (signal), hadron (background)\n", "\n", "g = gamma (signal): 12332\n", "h = hadron (background): 6688\n", "\n", "For technical reasons, the number of h events is underestimated.\n", "In the real data, the h class represents the majority of the events.\n", "\n", "You can find further information about the MAGIC telescope and the data discrimination studies in the following [paper](https://reader.elsevier.com/reader/sd/pii/S0168900203025051?token=8A02764E2448BDC5E4DD0ED53A301295162A6E9C8F223378E8CF80B187DBFD98BD3B642AB83886944002206EB1688FF4) (R. K. Bock et al., \"Methods for multidimensional event classification: a case studyusing images from a Cherenkov gamma-ray telescope\" NIM A 516 (2004) 511-528) (You need to be within the university network to get free access.) " ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "import pandas as pd\n", "import numpy as np\n", "from sklearn.model_selection import train_test_split" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "filename = \"https://www.physi.uni-heidelberg.de/~reygers/lectures/2021/ml/data/magic04_data.txt\"\n", "df = pd.read_csv(filename, engine='python')" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# use categories 1 and 0 insted of \"g\" and \"h\"\n", "df['class'] = df['class'].map({'g': 1, 'h': 0})" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "<div>\n", "<style scoped>\n", " .dataframe tbody tr th:only-of-type {\n", " vertical-align: middle;\n", " }\n", "\n", " .dataframe tbody tr th {\n", " vertical-align: top;\n", " }\n", "\n", " .dataframe thead th {\n", " text-align: right;\n", " }\n", "</style>\n", "<table border=\"1\" class=\"dataframe\">\n", " <thead>\n", " <tr style=\"text-align: right;\">\n", " <th></th>\n", " <th>fLength</th>\n", " <th>fWidth</th>\n", " <th>fSize</th>\n", " <th>fConc</th>\n", " <th>fConc1</th>\n", " <th>fAsym</th>\n", " <th>fM3Long</th>\n", " <th>fM3Trans</th>\n", " <th>fAlpha</th>\n", " <th>fDist</th>\n", " <th>class</th>\n", " </tr>\n", " </thead>\n", " <tbody>\n", " <tr>\n", " <th>0</th>\n", " <td>28.7967</td>\n", " <td>16.0021</td>\n", " <td>2.6449</td>\n", " <td>0.3918</td>\n", " <td>0.1982</td>\n", " <td>27.7004</td>\n", " <td>22.0110</td>\n", " <td>-8.2027</td>\n", " <td>40.0920</td>\n", " <td>81.8828</td>\n", " <td>1</td>\n", " </tr>\n", " <tr>\n", " <th>1</th>\n", " <td>31.6036</td>\n", " <td>11.7235</td>\n", " <td>2.5185</td>\n", " <td>0.5303</td>\n", " <td>0.3773</td>\n", " <td>26.2722</td>\n", " <td>23.8238</td>\n", " <td>-9.9574</td>\n", " <td>6.3609</td>\n", " <td>205.2610</td>\n", " <td>1</td>\n", " </tr>\n", " <tr>\n", " <th>2</th>\n", " <td>162.0520</td>\n", " <td>136.0310</td>\n", " <td>4.0612</td>\n", " <td>0.0374</td>\n", " <td>0.0187</td>\n", " <td>116.7410</td>\n", " <td>-64.8580</td>\n", " <td>-45.2160</td>\n", " <td>76.9600</td>\n", " <td>256.7880</td>\n", " <td>1</td>\n", " </tr>\n", " <tr>\n", " <th>3</th>\n", " <td>23.8172</td>\n", " <td>9.5728</td>\n", " <td>2.3385</td>\n", " <td>0.6147</td>\n", " <td>0.3922</td>\n", " <td>27.2107</td>\n", " <td>-6.4633</td>\n", " <td>-7.1513</td>\n", " <td>10.4490</td>\n", " <td>116.7370</td>\n", " <td>1</td>\n", " </tr>\n", " <tr>\n", " <th>4</th>\n", " <td>75.1362</td>\n", " <td>30.9205</td>\n", " <td>3.1611</td>\n", " <td>0.3168</td>\n", " <td>0.1832</td>\n", " <td>-5.5277</td>\n", " <td>28.5525</td>\n", " <td>21.8393</td>\n", " <td>4.6480</td>\n", " <td>356.4620</td>\n", " <td>1</td>\n", " </tr>\n", " </tbody>\n", "</table>\n", "</div>" ], "text/plain": [ " fLength fWidth fSize fConc fConc1 fAsym fM3Long fM3Trans \\\n", "0 28.7967 16.0021 2.6449 0.3918 0.1982 27.7004 22.0110 -8.2027 \n", "1 31.6036 11.7235 2.5185 0.5303 0.3773 26.2722 23.8238 -9.9574 \n", "2 162.0520 136.0310 4.0612 0.0374 0.0187 116.7410 -64.8580 -45.2160 \n", "3 23.8172 9.5728 2.3385 0.6147 0.3922 27.2107 -6.4633 -7.1513 \n", "4 75.1362 30.9205 3.1611 0.3168 0.1832 -5.5277 28.5525 21.8393 \n", "\n", " fAlpha fDist class \n", "0 40.0920 81.8828 1 \n", "1 6.3609 205.2610 1 \n", "2 76.9600 256.7880 1 \n", "3 10.4490 116.7370 1 \n", "4 4.6480 356.4620 1 " ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### a) Create for each variable a figure with a plot for gammas and hadrons overlayed." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [], "source": [ "df0 = df[df['class'] == 0] # hadron data set\n", "df1 = df[df['class'] == 1] # gamma data set" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "<matplotlib.legend.Legend at 0x122031130>" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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"text/plain": [ "<Figure size 1080x360 with 10 Axes>" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, axs = plt.subplots(nrows=2, ncols=5, figsize=(15,5))\n", "plt.subplots_adjust(hspace = 0.3, wspace=0.3)\n", "\n", "for i in range(10):\n", " kx = i // 5\n", " ky = i % 5\n", " axs[kx, ky].set_xlabel(df0.columns[i])\n", " df0.iloc[:,i].hist(ax = axs[kx, ky], bins = 50, alpha=0.5, density=True, label='h')\n", " df1.iloc[:,i].hist(ax = axs[kx, ky], bins = 50, alpha=0.5, density=True, label='g')\n", "\n", "axs[0, 0].legend()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### b) Create training and test data set. The tast data should amount to 50\\% of the total data set." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [], "source": [ "y = df['class'].values\n", "X = df[[col for col in df.columns if col!=\"class\"]]\n", "\n", "### YOUR CODE ### \n", "\n", "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.5, shuffle=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### c) Define the logistic regressor and fit the training data" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "from sklearn import linear_model\n", "\n", "# define logistic regressor\n", "\n", "### YOUR CODE ###\n", "\n", "logreg=linear_model.LogisticRegression(fit_intercept=True,\n", " penalty='none',\n", " max_iter=1000,\n", " tol=1E-5)\n", "\n", "# fit training data\n", "\n", "### YOUR CODE ###\n", "\n", "import time\n", "start_time = time.time()\n", "logreg.fit(X_train, y_train)\n", "run_time = time.time() - start_time" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [], "source": [ "y_pred = logreg.predict(X_test)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### d) Determine the Model Accuracy, the AUC score and the Run time" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Model Accuracy: 78.99%\n", "AUC score: 0.74\n", "Run time: 0.22 sec\n", "\n", "\n" ] } ], "source": [ "from sklearn.metrics import roc_auc_score\n", "\n", "### YOUR CODE ###\n", "\n", "print(\"Model Accuracy: {:.2f}%\".format(100*logreg.score(X_test, y_test)))\n", "print(\"AUC score: {:.2f}\".format(roc_auc_score(y_test,y_pred)))\n", "print(\"Run time: {:.2f} sec\\n\\n\".format(run_time))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### e) Plot the ROC curve (Backgropund Rejection vs signal efficiency)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "image/png": "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 "text/plain": [ "<Figure size 432x288 with 1 Axes>" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "from sklearn.metrics import roc_curve\n", "%matplotlib inline\n", "\n", "y_pred_prob = logreg.predict_proba(X_test) # predicted probabilities\n", "\n", "### YOUR CODE ###\n", "\n", "fpr, tpr, _ = roc_curve(y_test, y_pred_prob[:,1])\n", "plt.plot(tpr, 1-fpr)\n", "plt.xlabel('Signal efficiency')\n", "plt.ylabel('Background Rejection');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### f) Plot the Signal efficiency vs. the Background efficiency and compare it to the corresponding plot in the paper" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "image/png": "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 "text/plain": [ "<Figure size 432x288 with 1 Axes>" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "### YOUR CODE ###\n", "plt.plot(fpr, tpr)\n", "plt.xscale(\"log\")\n", "plt.xlabel('Background efficiency')\n", "plt.ylabel('Signal efficiency');" ] } ], "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": "3.8.16" } }, "nbformat": 4, "nbformat_minor": 4}
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