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April 10, 2022 19:06
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{ | |
"cells": [ | |
{ | |
"cell_type": "markdown", | |
"source": [ | |
"Michel Kiffel<br />\n", | |
"[email protected]<br />\n", | |
"This notebook supports [this blog article](https://mkffl.github.io/2022/03/02/Decisions-Part-3.html)" | |
], | |
"metadata": { | |
"id": "KPhuGzHOTnPv" | |
} | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": { | |
"id": "k0iuDR861Vsi" | |
}, | |
"source": [ | |
"## Objective\n", | |
"- Visual validation of ML model score calibration\n", | |
" - Calibration of log-likelihood ratios using logistic regression\n", | |
" - Validation using known score distributions to calculate the true ratios\n", | |
" - Visual inspection of calibration fit on a separate dataset of scores\n", | |
"\n", | |
"## Logistic regression calibration\n", | |
"- Estimate \n", | |
"$$\\text{llr}(x ; \\omega) = \\log \\frac{ p(x \\vert \\omega_1)}{p(x \\vert \\omega_0)}$$\n", | |
"- Using logistic regression, which by default estimates target class log-odds i.e.\n", | |
"$$\\text{log-odds}(\\omega; x) = \\log \\frac{ p(\\omega_1 \\vert x)}{p(\\omega_0 \\vert x)}$$\n", | |
"\n", | |
"- Conversion from log-odds to llr uses the relation log-odds = llr effective-prior (ep) hence llr = log-odds - ep\n", | |
"\n", | |
"## Score distributions\n", | |
"- I test two assumptions for the score class-conditional density\n", | |
" - Gaussian: logistic regression calibration works well as expected\n", | |
" - Skew-normal: lack of fit\n", | |
"\n", | |
"Main source is *Tutorial on logistic-regression calibration and fusion: Converting a score to a likelihood ratio* by GS Morrison\n" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": { | |
"id": "do-mtxdB1Vss" | |
}, | |
"outputs": [], | |
"source": [ | |
"# Stats\n", | |
"import numpy as np\n", | |
"import seaborn as sns\n", | |
"from scipy.special import logit, expit\n", | |
"from scipy.stats import norm as f_norm, skewnorm\n", | |
"\n", | |
"# Off the shelf stats models\n", | |
"from sklearn.linear_model import LogisticRegression as lr\n", | |
"\n", | |
"import pandas as pd\n", | |
"import matplotlib.pyplot as plt\n", | |
"from IPython.display import Image\n", | |
"from IPython.core.display import HTML" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": { | |
"id": "Zm67U0vf1Vs9" | |
}, | |
"outputs": [], | |
"source": [ | |
"# Utils\n", | |
"def reshape_to_1d(array):\n", | |
" \"\"\" Reshape [] to [[]], a requirement from sklearn. \"\"\"\n", | |
" return np.reshape(array,(-1, 1))\n", | |
"\n", | |
"def get_logistic_estimates(clf):\n", | |
" \"\"\" Find the logistic reg parameter estimates \"\"\"\n", | |
" return clf.intercept_[0], clf.coef_[0][0]\n", | |
"\n", | |
"def get_effective_prior(tarN, nontarN):\n", | |
" return logit(tarN/(tarN nontarN))\n", | |
"\n", | |
"def logistic_logodds(x, β_0, β_1):\n", | |
" return β_0 β_1*x\n", | |
"\n", | |
"def log_likelihood_ratio_density(tar_density, nontar_density):\n", | |
" def func(x):\n", | |
" return np.log(tar_density(x)/nontar_density(x))\n", | |
" \n", | |
" return func\n", | |
"\n", | |
"# Data\n", | |
"def generate_data(nontar_rv, tar_rv, nontarN, tarN):\n", | |
" \"\"\" Simulate the scores from two classes\n", | |
" \n", | |
" Args:\n", | |
" nontar_rv (scipy.continuous_dist): rv for w0 class\n", | |
" tar_rv (scipy.continuous_dist): rv for w1 class\n", | |
" \"\"\"\n", | |
" w0_sample = nontar_rv.rvs(nontarN)\n", | |
"\n", | |
" w1_sample = tar_rv.rvs(tarN)\n", | |
"\n", | |
" X = np.concatenate([w0_sample, w1_sample])\n", | |
"\n", | |
" y = [0 for d in w0_sample] [1 for d in w1_sample]\n", | |
" \n", | |
" return X, y\n", | |
"\n", | |
"# Validation\n", | |
"def logistic_calibration_validation(x, y, test_x, nontarN, tarN, nontar_rv, tar_rv):\n", | |
" \"\"\" Validate llr from logistic regression using the formula for two gaussians.\n", | |
"\n", | |
" Args:\n", | |
" x (np.array): raw scores\n", | |
" y (np.array): labels w0 or w1\n", | |
" test_x (np.array): validation dataset to visually inspect the calibrated scores\n", | |
" nontar_rv (scipy.continuous_dist): rv for w0\n", | |
" tar_rv (scipy.continuous_dist): rv for w1\n", | |
" \"\"\" \n", | |
" # True llr\n", | |
" llr_density = log_likelihood_ratio_density(tar_rv.pdf, nontar_rv.pdf)\n", | |
" \n", | |
" llr_true = [llr_density(x) for x in test_x]\n", | |
" \n", | |
" # Estimated llr\n", | |
" x_1d = reshape_to_1d(x)\n", | |
" \n", | |
" clf = lr(random_state=0).fit(x_1d, y)\n", | |
" \n", | |
" β_0, β_1 = get_logistic_estimates(clf)\n", | |
" \n", | |
" lo_preds = [logistic_logodds(x, β_0, β_1) for x in test_x]\n", | |
" \n", | |
" effective_prior = get_effective_prior(tarN, nontarN)\n", | |
" \n", | |
" llr_preds = [(lo - effective_prior) for lo in lo_preds]\n", | |
" \n", | |
" return llr_true, llr_preds\n", | |
"\n", | |
"\n", | |
"def plot_true_and_predicted_llr(test_x, llr_true, llr_preds, title):\n", | |
" (\n", | |
" sns.lineplot(\n", | |
" 'x', \n", | |
" 'value', \n", | |
" hue='variable', \n", | |
" data=(\n", | |
" pd.DataFrame({\n", | |
" \"x\": test_x, \n", | |
" \"llr_true\": llr_true, \n", | |
" \"llr_preds\": llr_preds})\n", | |
" .melt(\"x\")\n", | |
" )\n", | |
" )\n", | |
" .set_title(title)\n", | |
" )\n", | |
"\n", | |
"# main\n", | |
"def run(nontar_rv, tar_rv, nontarN, tarN, title):\n", | |
" X, y = generate_data(nontar_rv, tar_rv, nontarN, tarN)\n", | |
" \n", | |
" sns.distplot(X[:nontarN])\n", | |
" sns.distplot(X[nontarN:])\n", | |
" plt.title(\"Class-conditional score distributions\")\n", | |
" plt.show()\n", | |
"\n", | |
" # Validation sample scores\n", | |
" test_x = np.linspace(-1, 1, 15) #np.linspace(-3, 3, 40)\n", | |
"\n", | |
" llr_true, llr_preds = logistic_calibration_validation(X, y, test_x, nontarN, tarN, nontar_rv, tar_rv)\n", | |
" \n", | |
" plot_true_and_predicted_llr(test_x, llr_true, llr_preds, title)" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": { | |
"id": "6fB-NYZj1VtF", | |
"outputId": "fb3c2a39-24ad-494f-8802-aee06ef3b26b" | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
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"text/plain": [ | |
"<Figure size 432x288 with 1 Axes>" | |
] | |
}, | |
"metadata": { | |
"needs_background": "light" | |
}, | |
"output_type": "display_data" | |
}, | |
{ | |
"data": { | |
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"text/plain": [ | |
"<Figure size 432x288 with 1 Axes>" | |
] | |
}, | |
"metadata": { | |
"needs_background": "light" | |
}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"nontarN = 1_000\n", | |
"tarN = 1_000\n", | |
"nontar_rv = f_norm(-3, 0.5)\n", | |
"tar_rv = f_norm(-2, 0.5)\n", | |
"\n", | |
"run(nontar_rv, tar_rv, nontarN, tarN, \"Balanced dataset with normally distributed scores\")" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": { | |
"id": "s0JIJ_XW1VtG", | |
"outputId": "1e728a9f-1186-4950-cc8b-680431b25e56" | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
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"text/plain": [ | |
"<Figure size 432x288 with 1 Axes>" | |
] | |
}, | |
"metadata": { | |
"needs_background": "light" | |
}, | |
"output_type": "display_data" | |
}, | |
{ | |
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"text/plain": [ | |
"<Figure size 432x288 with 1 Axes>" | |
] | |
}, | |
"metadata": { | |
"needs_background": "light" | |
}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"tarN = 1_000\n", | |
"nontarN = int(7*tarN)\n", | |
"nontar_rv = f_norm(-3, 0.5)\n", | |
"tar_rv = f_norm(-2, 0.5)\n", | |
"\n", | |
"run(nontar_rv, tar_rv, nontarN, tarN, \"Imbalanced dataset with normally distributed scores\")" | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"execution_count": null, | |
"metadata": { | |
"id": "8i61UylH1VtI", | |
"outputId": "4dab1a14-2257-4f00-981a-497a840d8a1b" | |
}, | |
"outputs": [ | |
{ | |
"data": { | |
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"text/plain": [ | |
"<Figure size 432x288 with 1 Axes>" | |
] | |
}, | |
"metadata": { | |
"needs_background": "light" | |
}, | |
"output_type": "display_data" | |
}, | |
{ | |
"data": { | |
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"text/plain": [ | |
"<Figure size 432x288 with 1 Axes>" | |
] | |
}, | |
"metadata": { | |
"needs_background": "light" | |
}, | |
"output_type": "display_data" | |
} | |
], | |
"source": [ | |
"tarN = 1_000\n", | |
"nontarN = 1_000\n", | |
"nontar_rv = skewnorm(a=4, loc=-0.5, scale=0.5)\n", | |
"tar_rv = skewnorm(a=-4, loc=0.5, scale=0.5)\n", | |
"\n", | |
"run(nontar_rv, tar_rv, nontarN, tarN, \"Balanced dataset with skew-normally distributed scores\")" | |
] | |
} | |
], | |
"metadata": { | |
"kernelspec": { | |
"display_name": "Python 3", | |
"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": "3.8.5" | |
}, | |
"colab": { | |
"name": "Logistic Calibration.ipynb", | |
"provenance": [], | |
"collapsed_sections": [] | |
} | |
}, | |
"nbformat": 4, | |
"nbformat_minor": 0 | |
} |
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