232 lines
38 KiB
Plaintext
232 lines
38 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"import pandas as pd\n",
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"\n",
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"%matplotlib inline\n",
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"%config InlineBackend.figure_format = \"svg\"\n",
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"\n",
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"from IPython.display import display, Math, Latex\n",
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"\n",
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"data = pd.read_csv(\"fires_thefts.csv\", names=[\"x\", \"y\"])\n",
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"\n",
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"x = data[\"x\"].to_numpy()\n",
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"y = data[\"y\"].to_numpy()\n",
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"\n",
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"# Hipoteza: funkcja liniowa jednej zmiennej\n",
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"def h(theta, x):\n",
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" return theta[0] + theta[1] * x\n",
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"\n",
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"# Funkcja kosztu\n",
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"def J(h, theta, x, y):\n",
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" m = len(y)\n",
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" return 1.0 / (2 * m) * sum((h(theta, x[i]) - y[i]) ** 2 for i in range(m))\n",
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"\n",
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"# Wyświetlanie macierzy w LaTeX-u\n",
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"def LatexMatrix(matrix):\n",
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" ltx = r\"\\left[\\begin{array}\"\n",
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" m, n = matrix.shape\n",
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" ltx += \"{\" + (\"r\" * n) + \"}\"\n",
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" for i in range(m):\n",
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" ltx += r\" & \".join([(\"%.4f\" % j.item()) for j in matrix[i]]) + r\" \\\\ \"\n",
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" ltx += r\"\\end{array}\\right]\"\n",
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" return ltx\n",
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"\n",
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"def gradient_descent(h, cost_fun, theta, x, y, alpha, eps):\n",
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" current_cost = cost_fun(h, theta, x, y)\n",
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" history = [\n",
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" [current_cost, theta]\n",
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" ] # zapiszmy wartości kosztu i parametrów, by potem zrobić wykres\n",
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" m = len(y)\n",
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" while True:\n",
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" new_theta = [\n",
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" theta[0] - alpha / float(m) * sum(h(theta, x[i]) - y[i] for i in range(m)),\n",
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" theta[1]\n",
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" - alpha / float(m) * sum((h(theta, x[i]) - y[i]) * x[i] for i in range(m)),\n",
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" ]\n",
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" theta = new_theta # jednoczesna aktualizacja - używamy zmiennej tymczasowej\n",
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" try:\n",
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" prev_cost = current_cost\n",
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" current_cost = cost_fun(h, theta, x, y)\n",
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" except OverflowError:\n",
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" break\n",
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" if abs(prev_cost - current_cost) <= eps:\n",
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" break\n",
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" history.append([current_cost, theta])\n",
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" return theta, history\n",
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"\n"
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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": 28,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/latex": [
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"$\\displaystyle \\large\\textrm{Wynik:}\\quad \\theta = \\left[\\begin{array}{r}16.9446 \\\\ 1.3160 \\\\ \\end{array}\\right] \\quad J(\\theta) = 180.4105 \\quad \\textrm{po 5369 iteracjach}$"
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],
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"text/plain": [
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"<IPython.core.display.Math object>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"best_theta, history = gradient_descent(h, J, [0.0, 0.0], x, y, alpha=0.003, eps=0.000001)\n",
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"\n",
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"display(\n",
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" Math(\n",
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" r\"\\large\\textrm{Wynik:}\\quad \\theta = \"\n",
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" + LatexMatrix(np.matrix(best_theta).reshape(2, 1))\n",
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" + (r\" \\quad J(\\theta) = %.4f\" % history[-1][0])\n",
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" + r\" \\quad \\textrm{po %d iteracjach}\" % len(history)\n",
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" )\n",
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")\n"
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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": 19,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"eps= 1.0, cost= 231.741, steps= 4\n",
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"eps= 0.1, cost= 226.569, steps= 52\n",
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"eps= 0.01, cost= 185.031, steps= 1115\n",
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"eps= 0.001, cost= 180.872, steps= 2179\n",
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"eps= 0.0001, cost= 180.456, steps= 3242\n",
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"eps= 1e-05, cost= 180.415, steps= 4306\n",
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"eps= 1e-06, cost= 180.411, steps= 5369\n",
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"eps= 1e-07, cost= 180.410, steps= 6433\n",
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"eps= 1e-08, cost= 180.410, steps= 7496\n",
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"eps= 1e-09, cost= 180.410, steps= 8560\n",
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"eps= 1e-10, cost= 180.410, steps= 9623\n",
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"eps= 1e-11, cost= 180.410, steps= 10687\n"
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]
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}
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],
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"source": [
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"\n",
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"epss = [10.0**(-n) for n in range(0, 12)]\n",
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"alpha=0.003\n",
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"costs = []\n",
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"lengths = []\n",
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"for eps in epss:\n",
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" theta_best, history = gradient_descent(\n",
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" h, J, [0.0, 0.0], x, y, alpha, eps)\n",
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" cost = history[-1][0]\n",
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" steps = len(history)\n",
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" print(f\"{eps=:7}, {cost=:15.3f}, {steps=:6}\")\n",
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" costs.append(cost)\n",
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" lengths.append(steps)\n",
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"\n"
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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": 20,
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"metadata": {},
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"outputs": [],
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"source": [
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"def eps_cost_steps_plot(eps, costs, steps):\n",
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" \"\"\"Wykres kosztu i liczby kroków w zależności od eps\"\"\"\n",
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" fig, ax1 = plt.subplots()\n",
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" ax2 = ax1.twinx()\n",
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" ax1.plot(eps, steps, \"--s\", color=\"green\")\n",
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" ax2.plot(eps, costs, \":o\", color=\"orange\")\n",
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" ax1.set_xscale(\"log\")\n",
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" ax1.set_xlabel(\"eps\")\n",
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" ax1.set_ylabel(\"liczba kroków\", color=\"green\")\n",
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" ax2.set_ylabel(\"koszt\", color=\"orange\")\n",
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" plt.show()\n"
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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": 21,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/svg+xml": "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n<svg xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"463.77625pt\" height=\"310.86825pt\" viewBox=\"0 0 463.77625 310.86825\" xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\">\n <metadata>\n <rdf:RDF xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n <cc:Work>\n <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n <dc:date>2022-10-31T17:08:50.563047</dc:date>\n <dc:format>image/svg+xml</dc:format>\n <dc:creator>\n <cc:Agent>\n <dc:title>Matplotlib v3.6.1, https://matplotlib.org/</dc:title>\n </cc:Agent>\n </dc:creator>\n </cc:Work>\n </rdf:RDF>\n </metadata>\n <defs>\n <style type=\"text/css\">*{stroke-linejoin: round; stroke-linecap: butt}</style>\n </defs>\n <g id=\"figure_1\">\n <g id=\"patch_1\">\n <path d=\"M 0 310.86825 \nL 463.77625 310.86825 \nL 463.77625 0 \nL 0 0 \nz\n\" style=\"fill: #ffffff\"/>\n </g>\n <g id=\"axes_1\">\n <g id=\"patch_2\">\n <path d=\"M 59.690625 273.312 \nL 416.810625 273.312 \nL 416.810625 7.2 \nL 59.690625 7.2 \nz\n\" style=\"fill: #ffffff\"/>\n </g>\n <g id=\"matplotlib.axis_1\">\n <g id=\"xtick_1\">\n <g id=\"line2d_1\">\n <defs>\n <path id=\"m958b897915\" d=\"M 0 0 \nL 0 3.5 \n\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n </defs>\n <g>\n <use xlink:href=\"#m958b897915\" x=\"105.437402\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n </g>\n </g>\n <g id=\"text_1\">\n <!-- $\\mathdefault{10^{-10}}$ -->\n <g transform=\"translate(91.487402 287.910437) scale(0.1 -0.1)\">\n <defs>\n <path id=\"DejaVuSans-31\" d=\"M 794 531 \nL 1825 531 \nL 1825 4091 \nL 703 3866 \nL 703 4441 \nL 1819 4666 \nL 2450 4666 \nL 2450 531 \nL 3481 531 \nL 3481 0 \nL 794 0 \nL 794 531 \nz\n\" transform=\"scale(0.015625)\"/>\n <path id=\"DejaVuSans-30\" d=\"M 2034 4250 \nQ 1547 4250 1301 3770 \nQ 1056 3291 1056 2328 \nQ 1056 1369 1301 889 \nQ 1547 409 2034 409 \nQ 2525 409 2770 889 \nQ 3016 1369 3016 2328 \nQ 3016 3291 2770 3770 \nQ 2525 4250 2034 4250 \nz\nM 2034 4750 \nQ 2819 4750 3233 4129 \nQ 3647 3509 3647 2328 \nQ 3647 1150 3233 529 \nQ 2819 -91 2034 -91 \nQ 1250 -91 836 529 \nQ 422 1150 422 2328 \nQ 422 3509 836 4129 \nQ 1250 4750 2034 4750 \nz\n\" transform=\"scale(0.015625)\"/>\n <path id=\"DejaVuSans-2212\" d=\"M 678 2272 \nL 4684 2272 \nL 4684 1741 \nL 678 1741 \nL 678 2272 \nz\n\" transform=\"scale(0.015625)\"/>\n </defs>\n <use xlink:href=\"#DejaVuSans-31\" transform=\"translate(0 0.765625)\"/>\n <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(63.623047 0.765625)\"/>\n <use xlink:href=\"#DejaVuSans-2212\" transform=\"translate(128.203125 39.046875) scale(0.7)\"/>\n <use xlink:href=\"#DejaVuSans-31\" transform=\"translate(186.855469 39.046875) scale(0.7)\"/>\n <use xlink:href=\"#DejaVuSans-30\" transform=\"translate(231.391602 39.046875) scale(0.7)\"/>\n </g>\n </g>\n </g>\n <g id=\"xtick_2\">\n <g id=\"line2d_2\">\n <g>\n <use xlink:href=\"#m958b897915\" x=\"164.465501\" y=\"273.312\" style=\"stroke: #000000; stroke-width: 0.8\"/>\n </g>\n </g>\n <g id=\"text_2\">\n <!-- $\\mathdefault{10^{-8}}$ -->\n <g transform=\"translate(152.715501 287.910437) scale(0.1 -0.1)\">\n <defs>\n <path id=\"DejaVuSans-38\" d=\"M 2034 2216 \nQ 1584 2216 1326 1975 \nQ 1069 1734 1069 1313 \nQ 1069 891 1326 650 \nQ 1584 409 2034 409 \nQ 2484 409 2743 651 \nQ 3003 894 3003 1313 \nQ 3003 1734 2745 1975 \nQ 2488 2216 2034 2216 \nz\nM 1403 2484 \nQ 997 2584 770 2862 \nQ 544 3141 544 3541 \nQ 544 4100 942 4425 \nQ 1341 4750 2034 4750 \nQ 2731 4750 3128 4425 \nQ 3525 4100 3525 3541 \nQ 3525 3141 3298 2862 \nQ 3072 2584 2669 2484 \nQ 3125 2378 3379 2068 \nQ
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"text/plain": [
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"<Figure size 640x480 with 2 Axes>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"eps_cost_steps_plot(epss, costs, lengths)\n"
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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": 22,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[16.835521154474677, 1.3214970549417684]\n",
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"Liczba pozarów - 50 Przewidywalna liczba włamań - 82.91037390156309\n",
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"Liczba pozarów - 100 Przewidywalna liczba włamań - 148.98522664865152\n",
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"Liczba pozarów - 200 Przewidywalna liczba włamań - 281.13493214282835\n"
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]
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}
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],
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"source": [
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"example_x = [50, 100, 200]\n",
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"print(best_theta)\n",
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"example_y = [h(best_theta, ex) for ex in example_x]\n",
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"for i in range(3):\n",
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" print(f\"Liczba pozarów - {example_x[i]} \"\n",
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" f\"Przewidywalna liczba włamań - {example_y[i]}\")"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3.10.8 ('venv': venv)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.10.8"
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},
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"orig_nbformat": 4,
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"vscode": {
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||
|
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"interpreter": {
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||
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"hash": "f37486876ad4b243625dcab03485f0edb2a22cb7fa9db711ceb1161e85adf5f1"
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||
|
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}
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||
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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