6388 lines
269 KiB
Plaintext
6388 lines
269 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"### Uczenie maszynowe\n",
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"# 4. Regresja logistyczna"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"**Uwaga**: Wbrew nazwie, *regresja* logistyczna jest algorytmem służącym do rozwiązywania problemów *klasyfikacji* (wcale nie problemów *regresji*!)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "notes"
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}
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},
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"source": [
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"Do demonstracji metody regresji ligistycznej wykorzystamy klasyczny zbiór danych <a href=\"https://en.wikipedia.org/wiki/Iris_flower_data_set\">*Iris flower data set*</a>, składający się ze 150 przykładów wartości 4 cech dla 3 gatunków irysów (kosaćców)."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"### *Iris flower data set*\n",
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"\n",
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"* 150 przykładów\n",
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"* 4 cechy\n",
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"* 3 kategorie"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"| <img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/5/56/Kosaciec_szczecinkowaty_Iris_setosa.jpg/450px-Kosaciec_szczecinkowaty_Iris_setosa.jpg\"> | <img style=\"float: right;\" src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Iris_virginica.jpg/736px-Iris_virginica.jpg\"> | <img style=\"float: right;\" src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Blue_Flag%2C_Ottawa.jpg/600px-Blue_Flag%2C_Ottawa.jpg\"> |\n",
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"| :--- | :--- | :--- |\n",
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"| *Iris setosa* | *Iris virginica* | *Iris versicolor* |\n",
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"| kosaciec szczecinkowy | kosaciec amerykański | kosaciec różnobarwny |\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"4 cechy:\n",
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" * długość działek kielicha (*sepal length*, `sl`)\n",
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" * szerokość działek kielicha (*sepal width*, `sw`)\n",
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" * długość płatka (*petal length*, `pl`)\n",
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" * szerokość płatka (*petal width*, `pw`)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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||
"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## 4.1. Dwuklasowa regresja logistyczna"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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||
"slide_type": "subslide"
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||
}
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||
},
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"source": [
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"Zacznijmy od najprostszego przypadku:\n",
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" * ograniczmy się do **2** klas\n",
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" * ograniczmy się do **1** zmiennej"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"→ dwuklasowa regresja logistyczna jednej zmiennej"
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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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"slideshow": {
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"slide_type": "notes"
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}
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},
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"outputs": [],
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"source": [
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"# Przydatne importy\n",
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"\n",
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"import numpy as np\n",
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"import matplotlib\n",
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"import matplotlib.pyplot as plt\n",
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"import pandas\n",
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"import ipywidgets as widgets\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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"# Przydatne funkcje\n",
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"\n",
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"def LatexMatrix(matrix):\n",
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" \"\"\"Wyświetlanie macierzy w LaTeX-u\"\"\"\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 h(theta, X):\n",
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" \"\"\"Hipoteza (wersja macierzowa)\"\"\"\n",
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" return X * theta\n",
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"\n",
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"def regdots(X, y, xlabel, ylabel):\n",
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" \"\"\"Wykres danych (wersja macierzowa)\"\"\"\n",
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" fig = plt.figure(figsize=(16*.6, 9*.6))\n",
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" ax = fig.add_subplot(111)\n",
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" fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
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" ax.scatter([X[:, 1]], [y], c='r', s=50, label='Dane')\n",
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" \n",
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" ax.set_xlabel(xlabel)\n",
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" ax.set_ylabel(ylabel)\n",
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" ax.margins(.05, .05)\n",
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" plt.ylim(y.min() - 1, y.max() + 1)\n",
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" plt.xlim(np.min(X[:, 1]) - 1, np.max(X[:, 1]) + 1)\n",
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" return fig\n",
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"\n",
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"def regline(fig, fun, theta, X):\n",
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" \"\"\"Wykres krzywej regresji (wersja macierzowa)\"\"\"\n",
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" ax = fig.axes[0]\n",
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" x0 = np.min(X[:, 1]) - 1.0\n",
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" x1 = np.max(X[:, 1]) + 1.0\n",
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" L = [x0, x1]\n",
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" LX = np.matrix([1, x0, 1, x1]).reshape(2, 2)\n",
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" ax.plot(L, fun(theta, LX), linewidth='2',\n",
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" label=(r'$y={theta0:.2}{op}{theta1:.2}x$'.format(\n",
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" theta0=float(theta[0][0]),\n",
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" theta1=(float(theta[1][0]) if theta[1][0] >= 0 else float(-theta[1][0])),\n",
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" op='+' if theta[1][0] >= 0 else '-')))\n",
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"\n",
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||
"def legend(fig):\n",
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||
" \"\"\"Legenda wykresu\"\"\"\n",
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||
" ax = fig.axes[0]\n",
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" handles, labels = ax.get_legend_handles_labels()\n",
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||
" # try-except block is a fix for a bug in Poly3DCollection\n",
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||
" try:\n",
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" fig.legend(handles, labels, fontsize='15', loc='lower right')\n",
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||
" except AttributeError:\n",
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" pass\n",
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"\n",
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"def J(theta,X,y):\n",
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" \"\"\"Wersja macierzowa funkcji kosztu\"\"\"\n",
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" m = len(y)\n",
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" J = 1.0 / (2.0 * m) * ((X * theta - y).T * ( X * theta - y))\n",
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" return J.item()\n",
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"\n",
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"def dJ(theta,X,y):\n",
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" \"\"\"Wersja macierzowa gradientu funkcji kosztu\"\"\"\n",
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" return 1.0 / len(y) * (X.T * (X * theta - y)) \n",
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"\n",
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||
"def GD(fJ, fdJ, theta, X, y, alpha=0.1, eps=10**-3):\n",
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" \"\"\"Implementacja algorytmu gradientu prostego za pomocą numpy i macierzy\"\"\"\n",
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||
" current_cost = fJ(theta, X, y)\n",
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||
" while True:\n",
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" theta = theta - alpha * fdJ(theta, X, y) # implementacja wzoru\n",
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" current_cost, prev_cost = fJ(theta, X, y), current_cost\n",
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||
" if current_cost > 10000:\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",
|
||
" return theta\n",
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||
"\n",
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"theta_start = np.matrix([0, 0]).reshape(2, 1)\n",
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"\n",
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||
"def threshold(fig, theta):\n",
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" \"\"\"Funkcja, która rysuje próg\"\"\"\n",
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||
" x_thr = (0.5 - theta.item(0)) / theta.item(1)\n",
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" ax = fig.axes[0]\n",
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" ax.plot([x_thr, x_thr], [-1, 2],\n",
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||
" color='orange', linestyle='dashed',\n",
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" label=u'próg: $x={:.2F}$'.format(x_thr))"
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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": 2,
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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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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" sl sw pl pw Gatunek\n",
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||
"0 5.2 3.4 1.4 0.2 Iris-setosa\n",
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||
"1 5.1 3.7 1.5 0.4 Iris-setosa\n",
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||
"2 6.7 3.1 5.6 2.4 Iris-virginica\n",
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||
"3 6.5 3.2 5.1 2.0 Iris-virginica\n",
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||
"4 4.9 2.5 4.5 1.7 Iris-virginica\n",
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||
"5 6.0 2.7 5.1 1.6 Iris-versicolor\n"
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||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"# Wczytanie pełnych (oryginalnych) danych\n",
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||
"\n",
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||
"data_iris = pandas.read_csv(\"iris.csv\")\n",
|
||
"print(data_iris[:6])\n"
|
||
]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {
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"scrolled": true,
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
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||
"text": [
|
||
" dł. płatka Iris setosa?\n",
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||
"0 1.4 1\n",
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||
"1 1.5 1\n",
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||
"2 5.6 0\n",
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||
"3 5.1 0\n",
|
||
"4 4.5 0\n",
|
||
"5 5.1 0\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"# Ograniczenie danych do 2 klas i 1 cechy\n",
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"\n",
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"data_iris_setosa = pandas.DataFrame()\n",
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"data_iris_setosa[\"dł. płatka\"] = data_iris[\"pl\"] # \"pl\" oznacza \"petal length\"\n",
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"data_iris_setosa[\"Iris setosa?\"] = data_iris[\"Gatunek\"].apply(\n",
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||
" lambda x: 1 if x == \"Iris-setosa\" else 0\n",
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")\n",
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||
"print(data_iris_setosa[:6])\n"
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]
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||
},
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||
{
|
||
"cell_type": "code",
|
||
"execution_count": 4,
|
||
"metadata": {
|
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"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"import numpy as np\n",
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||
"\n",
|
||
"# Przygotowanie danych\n",
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||
"m, n_plus_1 = data_iris_setosa.values.shape\n",
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||
"n = n_plus_1 - 1\n",
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||
"Xn = data_iris_setosa.values[:, 0:n].reshape(m, n)\n",
|
||
"\n",
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||
"X = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
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||
"y = np.matrix(data_iris_setosa.values[:, 1]).reshape(m, 1)\n",
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||
"\n",
|
||
"# Regresja liniowa\n",
|
||
"theta_lin = GD(J, dJ, theta_start, X, y, alpha=0.03, eps=0.000001)\n"
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||
]
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},
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{
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"cell_type": "code",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"#### Próba zastosowania regresji liniowej do problemu klasyfikacji\n",
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"\n",
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||
"Najpierw z ciekawości sprawdźmy, co otrzymalibyśmy, gdybyśmy zastosowali regresję liniową do problemu klasyfikacji."
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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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"\n",
|
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"\n",
|
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"def plot_logistic():\n",
|
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" \"\"\"Wykres funkcji logistycznej\"\"\"\n",
|
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|
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|
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|
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|
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|
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"source": [
|
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|
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]
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"Funkcja regresji logistycznej dla pojedynczego przykładu o cechach wyrażonych wektorem $x$:\n",
|
||
"\n",
|
||
"$$h_\\theta(x) = g(\\theta^T \\, x) = \\dfrac{1}{1 + e^{-\\theta^T x}}$$"
|
||
]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
|
||
"Dla całej macierzy cech $X$:\n",
|
||
"\n",
|
||
"$$h_\\theta(X) = g(X \\, \\theta) = \\dfrac{1}{1 + e^{-X \\theta}}$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
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"execution_count": 11,
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"metadata": {
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"slideshow": {
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"slide_type": "skip"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def h(theta, X):\n",
|
||
" \"\"\"Funkcja regresji logistcznej\"\"\"\n",
|
||
" return 1.0 / (1.0 + np.exp(-X * theta))\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
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"metadata": {
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"slideshow": {
|
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"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Funkcja kosztu dla regresji logistycznej:\n",
|
||
"\n",
|
||
"$$J(\\theta) = -\\dfrac{1}{m} \\left( \\sum_{i=1}^{m} y^{(i)} \\log h_\\theta( x^{(i)} ) + \\left( 1 - y^{(i)} \\right) \\log \\left( 1 - h_\\theta (x^{(i)}) \\right) \\right)$$"
|
||
]
|
||
},
|
||
{
|
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"cell_type": "markdown",
|
||
"metadata": {
|
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"slideshow": {
|
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"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Gradient dla regresji logistycznej (wersja macierzowa):\n",
|
||
"\n",
|
||
"$$\\nabla J(\\theta) = \\frac{1}{|\\vec y|} X^T \\left( h_\\theta(X) - \\vec y \\right)$$\n",
|
||
"\n",
|
||
"(Jedyna różnica między gradientem dla regresji logistycznej a gradientem dla regresji liniowej to postać $h_\\theta$)."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 12,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def J(h, theta, X, y):\n",
|
||
" \"\"\"Funkcja kosztu dla regresji logistycznej\"\"\"\n",
|
||
" m = len(y)\n",
|
||
" h_val = h(theta, X)\n",
|
||
" s1 = np.multiply(y, np.log(h_val))\n",
|
||
" s2 = np.multiply((1 - y), np.log(1 - h_val))\n",
|
||
" return -np.sum(s1 + s2, axis=0) / m\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 13,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def dJ(h, theta, X, y):\n",
|
||
" \"\"\"Gradient dla regresji logistycznej\"\"\"\n",
|
||
" return 1.0 / len(y) * (X.T * (h(theta, X) - y))\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 14,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def GD(h, fJ, fdJ, theta, X, y, alpha=0.01, eps=10**-3, max_steps=10000):\n",
|
||
" \"\"\"Metoda gradientu prostego dla regresji logistycznej\"\"\"\n",
|
||
" curr_cost = fJ(h, theta, X, y)\n",
|
||
" history = [[curr_cost, theta]]\n",
|
||
" while True:\n",
|
||
" # oblicz nowe theta\n",
|
||
" theta = theta - alpha * fdJ(h, theta, X, y)\n",
|
||
" # raportuj poziom błędu\n",
|
||
" prev_cost = curr_cost\n",
|
||
" curr_cost = fJ(h, theta, X, y)\n",
|
||
" # kryteria stopu\n",
|
||
" if abs(prev_cost - curr_cost) <= eps:\n",
|
||
" break\n",
|
||
" if len(history) > max_steps:\n",
|
||
" break\n",
|
||
" history.append([curr_cost, theta])\n",
|
||
" return theta, history\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 15,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"Koszt: [[0.05755617]]\n",
|
||
"theta = [[ 5.02530461]\n",
|
||
" [-1.99174803]]\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"# Uruchomienie metody gradientu prostego dla regresji logistycznej\n",
|
||
"theta_best, history = GD(\n",
|
||
" h, J, dJ, theta_start, X, y, alpha=0.1, eps=10**-7, max_steps=1000\n",
|
||
")\n",
|
||
"print(f\"Koszt: {history[-1][0]}\")\n",
|
||
"print(f\"theta = {theta_best}\")\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 16,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def scalar_logistic_regression_function(theta, x):\n",
|
||
" \"\"\"Funkcja regresji logistycznej (wersja skalarna)\"\"\"\n",
|
||
" return 1.0 / (1.0 + np.exp(-(theta.item(0) + theta.item(1) * x)))\n",
|
||
"\n",
|
||
"\n",
|
||
"def threshold_val(fig, x_thr):\n",
|
||
" \"\"\"Rysowanie progu\"\"\"\n",
|
||
" ax = fig.axes[0]\n",
|
||
" ax.plot(\n",
|
||
" [x_thr, x_thr],\n",
|
||
" [-1, 2],\n",
|
||
" color=\"orange\",\n",
|
||
" linestyle=\"dashed\",\n",
|
||
" label=\"próg: $x={:.2F}$\".format(x_thr),\n",
|
||
" )\n",
|
||
"\n",
|
||
"\n",
|
||
"def logistic_regline(fig, theta, X):\n",
|
||
" \"\"\"Wykres krzywej regresji logistycznej\"\"\"\n",
|
||
" ax = fig.axes[0]\n",
|
||
" x0 = np.min(X[:, 1]) - 1.0\n",
|
||
" x1 = np.max(X[:, 1]) + 1.0\n",
|
||
" Arg = np.arange(x0, x1, 0.1)\n",
|
||
" Val = scalar_logistic_regression_function(theta, Arg)\n",
|
||
" ax.plot(Arg, Val, linewidth=\"2\")\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 17,
|
||
"metadata": {
|
||
"scrolled": true,
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
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"outputs": [
|
||
{
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|
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|
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|
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|
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"fig = regdots(X, y, xlabel=\"x\", ylabel=\"Iris setosa?\")\n",
|
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|
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}
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},
|
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"source": [
|
||
"Traktujemy wartość $h_\\theta(x)$ jako prawdopodobieństwo zdefiniowane w następujący sposób:\n",
|
||
"\n",
|
||
"$$ h_\\theta(x) = P(y = 1 \\, | \\, x; \\theta) $$"
|
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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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"source": [
|
||
"Jeżeli $h_\\theta(x) > 0.5$, to dla takiego $x$ będziemy przewidywać wartość $y = 1$.\n",
|
||
"W przeciwnym wypadku uprzewidzimy $y = 0$."
|
||
]
|
||
},
|
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{
|
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|
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|
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"slideshow": {
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"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Dlaczego możemy traktować wartość funkcji regresji logistycznej jako prawdopodobieństwo?\n",
|
||
"\n",
|
||
"Można o tym poczytać w zewnętrznych źródłach, np. https://towardsdatascience.com/logit-of-logistic-regression-understanding-the-fundamentals-f384152a33d1"
|
||
]
|
||
},
|
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{
|
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|
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|
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"slideshow": {
|
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"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Dwuklasowa regresja logistyczna: więcej cech"
|
||
]
|
||
},
|
||
{
|
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"cell_type": "markdown",
|
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"metadata": {
|
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"slideshow": {
|
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"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Jak postąpić, jeżeli będziemy mieli więcej niż jedną cechę $x$?\n",
|
||
"\n",
|
||
"Weźmy teraz wszystkie cechy występujące w zbiorze *Iris*:\n",
|
||
"* długość płatków (`pl`, *petal length*)\n",
|
||
"* szerokość płatków (`pw`, *petal width*)\n",
|
||
"* długość działek kielicha (`sl`, *sepal length*)\n",
|
||
"* szerokość działek kielicha (`sw`, *sepal width*)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 18,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
" pl pw sl sw Iris setosa?\n",
|
||
"0 1.4 0.2 5.2 3.4 1\n",
|
||
"1 1.5 0.4 5.1 3.7 1\n",
|
||
"2 5.6 2.4 6.7 3.1 0\n",
|
||
"3 5.1 2.0 6.5 3.2 0\n",
|
||
"4 4.5 1.7 4.9 2.5 0\n",
|
||
"5 5.1 1.6 6.0 2.7 0\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"data_iris_setosa_multi = pandas.DataFrame()\n",
|
||
"for feature in [\"pl\", \"pw\", \"sl\", \"sw\"]:\n",
|
||
" data_iris_setosa_multi[feature] = data_iris[feature]\n",
|
||
"data_iris_setosa_multi[\"Iris setosa?\"] = data_iris[\"Gatunek\"].apply(\n",
|
||
" lambda x: 1 if x == \"Iris-setosa\" else 0\n",
|
||
")\n",
|
||
"print(data_iris_setosa_multi[:6])\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 19,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"[[1. 1.4 0.2 5.2 3.4]\n",
|
||
" [1. 1.5 0.4 5.1 3.7]\n",
|
||
" [1. 5.6 2.4 6.7 3.1]\n",
|
||
" [1. 5.1 2. 6.5 3.2]\n",
|
||
" [1. 4.5 1.7 4.9 2.5]\n",
|
||
" [1. 5.1 1.6 6. 2.7]]\n",
|
||
"[[1.]\n",
|
||
" [1.]\n",
|
||
" [0.]\n",
|
||
" [0.]\n",
|
||
" [0.]\n",
|
||
" [0.]]\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"# Przygotowanie danych\n",
|
||
"m, n_plus_1 = data_iris_setosa_multi.values.shape\n",
|
||
"n = n_plus_1 - 1\n",
|
||
"Xn = data_iris_setosa_multi.values[:, 0:n].reshape(m, n)\n",
|
||
"\n",
|
||
"X = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
|
||
"y = np.matrix(data_iris_setosa_multi.values[:, n]).reshape(m, 1)\n",
|
||
"\n",
|
||
"print(X[:6])\n",
|
||
"print(y[:6])\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 20,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Podział danych na zbiór trenujący i testowy\n",
|
||
"XTrain, XTest = X[:100], X[100:]\n",
|
||
"yTrain, yTest = y[:100], y[100:]\n",
|
||
"\n",
|
||
"# Macierz parametrów początkowych\n",
|
||
"theta_start = np.ones(5).reshape(5, 1)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 21,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"Koszt: [[0.006797]]\n",
|
||
"theta = [[ 1.11414027]\n",
|
||
" [-2.89324615]\n",
|
||
" [-0.66543637]\n",
|
||
" [ 0.14887292]\n",
|
||
" [ 2.13284493]]\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"theta_best, history = GD(\n",
|
||
" h, J, dJ, theta_start, XTrain, yTrain, alpha=0.1, eps=10**-7, max_steps=1000\n",
|
||
")\n",
|
||
"print(f\"Koszt: {history[-1][0]}\")\n",
|
||
"print(f\"theta = {theta_best}\")\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Funkcja decyzyjna regresji logistycznej\n",
|
||
"\n",
|
||
"Funkcja decyzyjna mówi o tym, kiedy nasz algorytm będzie przewidywał $y = 1$, a kiedy $y = 0$:\n",
|
||
"\n",
|
||
"$$ c(x) := \\left\\{ \n",
|
||
"\\begin{array}{ll}\n",
|
||
"1, & \\mbox{gdy } P(y=1 \\, | \\, x; \\theta) > 0.5 \\\\\n",
|
||
"0 & \\mbox{w przeciwnym przypadku}\n",
|
||
"\\end{array}\\right.\n",
|
||
"$$\n",
|
||
"\n",
|
||
"$$ P(y=1 \\,| \\, x; \\theta) = h_\\theta(x) $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 22,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"theta = [[ 1.11414027]\n",
|
||
" [-2.89324615]\n",
|
||
" [-0.66543637]\n",
|
||
" [ 0.14887292]\n",
|
||
" [ 2.13284493]]\n",
|
||
"x0 = [[1. 6.3 1.8 7.3 2.9]]\n",
|
||
"h(x0) = 1.606143695982487e-05\n",
|
||
"c(x0) = (0, 1.606143695982487e-05)\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"def classifyBi(theta, X):\n",
|
||
" \"\"\"Funkcja decyzyjna regresji logistycznej\"\"\"\n",
|
||
" prob = h(theta, X).item()\n",
|
||
" return (1, prob) if prob > 0.5 else (0, prob)\n",
|
||
"\n",
|
||
"\n",
|
||
"print(f\"theta = {theta_best}\")\n",
|
||
"print(f\"x0 = {XTest[0]}\")\n",
|
||
"print(f\"h(x0) = {h(theta_best, XTest[0]).item()}\")\n",
|
||
"print(f\"c(x0) = {classifyBi(theta_best, XTest[0])}\")\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Obliczmy teraz skuteczność modelu (więcej na ten temat na następnym wykładzie, poświęconym metodom ewaluacji)."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 23,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"0 <=> 0 -- prob: 0.0000\n",
|
||
"1 <=> 1 -- prob: 0.9816\n",
|
||
"0 <=> 0 -- prob: 0.0001\n",
|
||
"0 <=> 0 -- prob: 0.0005\n",
|
||
"0 <=> 0 -- prob: 0.0001\n",
|
||
"1 <=> 1 -- prob: 0.9936\n",
|
||
"0 <=> 0 -- prob: 0.0059\n",
|
||
"0 <=> 0 -- prob: 0.0992\n",
|
||
"0 <=> 0 -- prob: 0.0001\n",
|
||
"0 <=> 0 -- prob: 0.0001\n",
|
||
"\n",
|
||
"Accuracy: 1.0\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"correct = 0\n",
|
||
"for i, rest in enumerate(yTest):\n",
|
||
" cls, prob = classifyBi(theta_best, XTest[i])\n",
|
||
" if i < 10:\n",
|
||
" print(f\"{yTest[i].item():1.0f} <=> {cls} -- prob: {prob:6.4f}\")\n",
|
||
" correct += cls == yTest[i].item()\n",
|
||
"accuracy = correct / len(XTest)\n",
|
||
"\n",
|
||
"print(f\"\\nAccuracy: {accuracy}\")\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"## 4.2. Wieloklasowa regresja logistyczna"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Przykład: wszystkie cechy ze zbioru *Iris*, wszystkie 3 klasy ze zbioru *Iris*."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 24,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"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>sl</th>\n",
|
||
" <th>sw</th>\n",
|
||
" <th>pl</th>\n",
|
||
" <th>pw</th>\n",
|
||
" <th>Gatunek</th>\n",
|
||
" </tr>\n",
|
||
" </thead>\n",
|
||
" <tbody>\n",
|
||
" <tr>\n",
|
||
" <th>0</th>\n",
|
||
" <td>5.2</td>\n",
|
||
" <td>3.4</td>\n",
|
||
" <td>1.4</td>\n",
|
||
" <td>0.2</td>\n",
|
||
" <td>Iris-setosa</td>\n",
|
||
" </tr>\n",
|
||
" <tr>\n",
|
||
" <th>1</th>\n",
|
||
" <td>5.1</td>\n",
|
||
" <td>3.7</td>\n",
|
||
" <td>1.5</td>\n",
|
||
" <td>0.4</td>\n",
|
||
" <td>Iris-setosa</td>\n",
|
||
" </tr>\n",
|
||
" <tr>\n",
|
||
" <th>2</th>\n",
|
||
" <td>6.7</td>\n",
|
||
" <td>3.1</td>\n",
|
||
" <td>5.6</td>\n",
|
||
" <td>2.4</td>\n",
|
||
" <td>Iris-virginica</td>\n",
|
||
" </tr>\n",
|
||
" <tr>\n",
|
||
" <th>3</th>\n",
|
||
" <td>6.5</td>\n",
|
||
" <td>3.2</td>\n",
|
||
" <td>5.1</td>\n",
|
||
" <td>2.0</td>\n",
|
||
" <td>Iris-virginica</td>\n",
|
||
" </tr>\n",
|
||
" <tr>\n",
|
||
" <th>4</th>\n",
|
||
" <td>4.9</td>\n",
|
||
" <td>2.5</td>\n",
|
||
" <td>4.5</td>\n",
|
||
" <td>1.7</td>\n",
|
||
" <td>Iris-virginica</td>\n",
|
||
" </tr>\n",
|
||
" <tr>\n",
|
||
" <th>5</th>\n",
|
||
" <td>6.0</td>\n",
|
||
" <td>2.7</td>\n",
|
||
" <td>5.1</td>\n",
|
||
" <td>1.6</td>\n",
|
||
" <td>Iris-versicolor</td>\n",
|
||
" </tr>\n",
|
||
" </tbody>\n",
|
||
"</table>\n",
|
||
"</div>"
|
||
],
|
||
"text/plain": [
|
||
" sl sw pl pw Gatunek\n",
|
||
"0 5.2 3.4 1.4 0.2 Iris-setosa\n",
|
||
"1 5.1 3.7 1.5 0.4 Iris-setosa\n",
|
||
"2 6.7 3.1 5.6 2.4 Iris-virginica\n",
|
||
"3 6.5 3.2 5.1 2.0 Iris-virginica\n",
|
||
"4 4.9 2.5 4.5 1.7 Iris-virginica\n",
|
||
"5 6.0 2.7 5.1 1.6 Iris-versicolor"
|
||
]
|
||
},
|
||
"execution_count": 24,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"import pandas\n",
|
||
"\n",
|
||
"data_iris = pandas.read_csv(\"iris.csv\")\n",
|
||
"data_iris[:6]\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 25,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"X = [[1. 5.2 3.4 1.4 0.2]\n",
|
||
" [1. 5.1 3.7 1.5 0.4]\n",
|
||
" [1. 6.7 3.1 5.6 2.4]\n",
|
||
" [1. 6.5 3.2 5.1 2. ]]\n",
|
||
"y = [['Iris-setosa']\n",
|
||
" ['Iris-setosa']\n",
|
||
" ['Iris-virginica']\n",
|
||
" ['Iris-virginica']]\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"# Przygotowanie danych\n",
|
||
"\n",
|
||
"import numpy as np\n",
|
||
"\n",
|
||
"features = [\"sl\", \"sw\", \"pl\", \"pw\"]\n",
|
||
"m = len(data_iris)\n",
|
||
"X = np.matrix(data_iris[features])\n",
|
||
"X0 = np.ones(m).reshape(m, 1)\n",
|
||
"X = np.hstack((X0, X))\n",
|
||
"y = np.matrix(data_iris[[\"Gatunek\"]]).reshape(m, 1)\n",
|
||
"\n",
|
||
"print(\"X = \", X[:4])\n",
|
||
"print(\"y = \", y[:4])\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Zamieńmy etykiety tekstowe w tablicy $y$ na wektory jednostkowe (*one-hot vectors*):\n",
|
||
"\n",
|
||
"$$\n",
|
||
"\\begin{array}{ccc}\n",
|
||
"\\mbox{\"Iris-setosa\"} & \\mapsto & \\left[ \\begin{array}{ccc} 1 & 0 & 0 \\\\ \\end{array} \\right] \\\\\n",
|
||
"\\mbox{\"Iris-virginica\"} & \\mapsto & \\left[ \\begin{array}{ccc} 0 & 1 & 0 \\\\ \\end{array} \\right] \\\\\n",
|
||
"\\mbox{\"Iris-versicolor\"} & \\mapsto & \\left[ \\begin{array}{ccc} 0 & 0 & 1 \\\\ \\end{array} \\right] \\\\\n",
|
||
"\\end{array}\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Wówczas zamiast wektora $y$ otrzymamy macierz $Y$:\n",
|
||
"\n",
|
||
"$$\n",
|
||
"y \\; = \\;\n",
|
||
"\\left[\n",
|
||
"\\begin{array}{c}\n",
|
||
"y^{(1)} \\\\\n",
|
||
"y^{(2)} \\\\\n",
|
||
"y^{(3)} \\\\\n",
|
||
"y^{(4)} \\\\\n",
|
||
"y^{(5)} \\\\\n",
|
||
"\\vdots \\\\\n",
|
||
"\\end{array}\n",
|
||
"\\right]\n",
|
||
"\\; = \\;\n",
|
||
"\\left[\n",
|
||
"\\begin{array}{c}\n",
|
||
"\\mbox{\"Iris-setosa\"} \\\\\n",
|
||
"\\mbox{\"Iris-setosa\"} \\\\\n",
|
||
"\\mbox{\"Iris-virginica\"} \\\\\n",
|
||
"\\mbox{\"Iris-versicolor\"} \\\\\n",
|
||
"\\mbox{\"Iris-virginica\"} \\\\\n",
|
||
"\\vdots \\\\\n",
|
||
"\\end{array}\n",
|
||
"\\right]\n",
|
||
"\\quad \\mapsto \\quad\n",
|
||
"Y \\; = \\;\n",
|
||
"\\left[\n",
|
||
"\\begin{array}{ccc}\n",
|
||
"1 & 0 & 0 \\\\\n",
|
||
"1 & 0 & 0 \\\\\n",
|
||
"0 & 1 & 0 \\\\\n",
|
||
"0 & 0 & 1 \\\\\n",
|
||
"0 & 1 & 0 \\\\\n",
|
||
"\\vdots & \\vdots & \\vdots \\\\\n",
|
||
"\\end{array}\n",
|
||
"\\right]\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 26,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def mapY(y, cls):\n",
|
||
" m = len(y)\n",
|
||
" yBi = np.matrix(np.zeros(m)).reshape(m, 1)\n",
|
||
" yBi[y == cls] = 1.0\n",
|
||
" return yBi\n",
|
||
"\n",
|
||
"\n",
|
||
"def indicatorMatrix(y):\n",
|
||
" classes = np.unique(y.tolist())\n",
|
||
" m = len(y)\n",
|
||
" k = len(classes)\n",
|
||
" Y = np.matrix(np.zeros((m, k)))\n",
|
||
" for i, cls in enumerate(classes):\n",
|
||
" Y[:, i] = mapY(y, cls)\n",
|
||
" return Y\n",
|
||
"\n",
|
||
"\n",
|
||
"# Macierz jednostkowa\n",
|
||
"Y = indicatorMatrix(y)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 27,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Podział danych na zbiór trenujący i testowy\n",
|
||
"XTrain, XTest = X[:100], X[100:]\n",
|
||
"YTrain, YTest = Y[:100], Y[100:]\n",
|
||
"\n",
|
||
"# Macierz parametrów początkowych - niech skłąda się z samych jedynek\n",
|
||
"theta_start = np.ones(5).reshape(5, 1)\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Regresja logistyczna jest metodą rozwiązywania problemów klasyfikacji **dwuklasowej**."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
"Aby znaleźć rozwiązanie problemu klasyfikacji **wieloklasowej** metodą regresji logistycznej, trzeba przekształcić problem na zbiór problemów klasyfikacji dwuklasowej."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"source": [
|
||
"Alternatywnie, można użyć **wielomianowej regresji logistycznej** (zob. https://machinelearningmastery.com/multinomial-logistic-regression-with-python)."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Od regresji logistycznej dwuklasowej do wieloklasowej\n",
|
||
"\n",
|
||
"* Irysy są przydzielone do trzech klas: _Iris-setosa_ (0), _Iris-versicolor_ (1), _Iris-virginica_ (2).\n",
|
||
"* Wiemy, jak stworzyć klasyfikatory dwuklasowe typu _Iris-setosa_ vs. _Nie-Iris-setosa_ (tzw. *one-vs-all*).\n",
|
||
"* Możemy stworzyć trzy klasyfikatory $h_{\\theta_1}, h_{\\theta_2}, h_{\\theta_3}$ (otrzymując trzy zestawy parametrów $\\theta$) i wybrać klasę o najwyższym prawdopodobieństwie."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "notes"
|
||
}
|
||
},
|
||
"source": [
|
||
"Pomoże nam w tym funkcja *softmax*, która jest uogólnieniem funkcji logistycznej na większą liczbę wymiarów."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Funkcja _softmax_\n",
|
||
"\n",
|
||
"Odpowiednikiem funkcji logistycznej dla wieloklasowej regresji logistycznej jest funkcja $\\mathrm{softmax}$:\n",
|
||
"\n",
|
||
"$$ \\textrm{softmax} \\colon \\mathbb{R}^k \\to [0,1]^k $$\n",
|
||
"\n",
|
||
"$$ \\textrm{softmax}(z_1,z_2,\\dots,z_k) = \\left( \\dfrac{e^{z_1}}{\\sum_{i=1}^{k}e^{z_i}}, \\dfrac{e^{z_2}}{\\sum_{i=1}^{k}e^{z_i}}, \\ldots, \\dfrac{e^{z_k}}{\\sum_{i=1}^{k}e^{z_i}} \\right) $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"$$ \\textrm{softmax}( \\left[ \\begin{array}{c} \\theta_1^T x \\\\ \\theta_2^T x \\\\ \\vdots \\\\ \\theta_k^T x \\end{array} \\right] ) = \\left[ \\begin{array}{c} P(y=1 \\, | \\, x;\\theta_1,\\ldots,\\theta_k) \\\\ P(y=2 \\, | \\, x;\\theta_1,\\ldots,\\theta_k) \\\\ \\vdots \\\\ P(y=k \\, | \\, x;\\theta_1,\\ldots,\\theta_k) \\end{array} \\right] $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 28,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def softmax(X):\n",
|
||
" \"\"\"Funkcja softmax (wersja macierzowa)\"\"\"\n",
|
||
" return np.exp(X) / np.sum(np.exp(X))\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Wartości funkcji $\\mathrm{softmax}$ sumują się do 1:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 29,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"0.9999999999999999\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"Z = np.matrix([[2.1, 0.5, 0.8, 0.9, 3.2]])\n",
|
||
"P = softmax(Z)\n",
|
||
"print(np.sum(P))\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 30,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def multiple_binary_classifiers(X, Y):\n",
|
||
" n = X.shape[1]\n",
|
||
" thetas = []\n",
|
||
" # Dla każdej klasy wytrenujmy osobny klasyfikator dwuklasowy.\n",
|
||
" for c in range(Y.shape[1]):\n",
|
||
" YBi = Y[:, c]\n",
|
||
" theta = np.matrix(np.random.random(n)).reshape(n, 1)\n",
|
||
" # Macierz parametrów theta obliczona dla każdej klasy osobno.\n",
|
||
" theta_best, history = GD(h, J, dJ, theta, X, YBi, alpha=0.1, eps=10**-4)\n",
|
||
" thetas.append(theta_best)\n",
|
||
" return thetas\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 31,
|
||
"metadata": {
|
||
"scrolled": true,
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"Otrzymana macierz parametrów theta dla klasy 0:\n",
|
||
" [[ 0.44809895]\n",
|
||
" [-0.02470655]\n",
|
||
" [ 1.73058264]\n",
|
||
" [-1.97727025]\n",
|
||
" [-0.3624556 ]] \n",
|
||
"\n",
|
||
"Otrzymana macierz parametrów theta dla klasy 1:\n",
|
||
" [[ 0.5642055 ]\n",
|
||
" [ 0.11085065]\n",
|
||
" [-1.01510744]\n",
|
||
" [ 0.55275035]\n",
|
||
" [-0.70237456]] \n",
|
||
"\n",
|
||
"Otrzymana macierz parametrów theta dla klasy 2:\n",
|
||
" [[-0.44893077]\n",
|
||
" [-1.60497843]\n",
|
||
" [-1.86196767]\n",
|
||
" [ 2.36049057]\n",
|
||
" [ 2.5959696 ]] \n",
|
||
"\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"# Macierze theta dla każdej klasy\n",
|
||
"thetas = multiple_binary_classifiers(XTrain, YTrain)\n",
|
||
"for c, theta in enumerate(thetas):\n",
|
||
" print(f\"Otrzymana macierz parametrów theta dla klasy {c}:\\n\", theta, \"\\n\")\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Funkcja decyzyjna wieloklasowej regresji logistycznej\n",
|
||
"\n",
|
||
"$$ c = \\mathop{\\textrm{arg}\\,\\textrm{max}}_{i \\in \\{1, \\ldots ,k\\}} P(y=i|x;\\theta_1,\\ldots,\\theta_k) $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 32,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def classify(thetas, X, debug=False):\n",
|
||
" regs = np.array([(X * theta).item() for theta in thetas])\n",
|
||
" if debug:\n",
|
||
" print(\"Po zastosowaniu regresji: \", regs)\n",
|
||
" probs = softmax(regs)\n",
|
||
" if debug:\n",
|
||
" print(\"Otrzymane prawdopodobieństwa: \", np.around(probs, decimals=3))\n",
|
||
" result = np.argmax(probs)\n",
|
||
" if debug:\n",
|
||
" print(\"Wybrana klasa: \", result)\n",
|
||
" return result\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 33,
|
||
"metadata": {
|
||
"scrolled": true,
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"Dla x = [[1. 7.3 2.9 6.3 1.8]]:\n",
|
||
"Po zastosowaniu regresji: [-7.82279188 0.64765672 1.97885627]\n",
|
||
"Otrzymane prawdopodobieństwa: [0. 0.209 0.791]\n",
|
||
"Wybrana klasa: 2\n",
|
||
"Obliczone y = 2\n",
|
||
"Oczekiwane y = 2\n",
|
||
"\n",
|
||
"Dla x = [[1. 4.8 3. 1.4 0.3]]:\n",
|
||
"Po zastosowaniu regresji: [ 2.6443404 -1.38589555 -9.65525259]\n",
|
||
"Otrzymane prawdopodobieństwa: [0.983 0.017 0. ]\n",
|
||
"Wybrana klasa: 0\n",
|
||
"Obliczone y = 0\n",
|
||
"Oczekiwane y = 0\n",
|
||
"\n",
|
||
"Dla x = [[1. 7.1 3. 5.9 2.1]]:\n",
|
||
"Po zastosowaniu regresji: [-6.96262088 0.09216334 1.94824984]\n",
|
||
"Otrzymane prawdopodobieństwa: [0. 0.135 0.865]\n",
|
||
"Wybrana klasa: 2\n",
|
||
"Obliczone y = 2\n",
|
||
"Oczekiwane y = 2\n",
|
||
"\n",
|
||
"Dla x = [[1. 5.9 3. 5.1 1.8]]:\n",
|
||
"Po zastosowaniu regresji: [-5.24242014 -0.27234536 1.20704063]\n",
|
||
"Otrzymane prawdopodobieństwa: [0.001 0.185 0.813]\n",
|
||
"Wybrana klasa: 2\n",
|
||
"Obliczone y = 2\n",
|
||
"Oczekiwane y = 2\n",
|
||
"\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"for i in range(4):\n",
|
||
" print(f\"Dla x = {XTest[i]}:\")\n",
|
||
" YPredicted = classify(thetas, XTest[i], debug=True)\n",
|
||
" print(f\"Obliczone y = {YPredicted}\")\n",
|
||
" print(f\"Oczekiwane y = {np.argmax(YTest[i])}\")\n",
|
||
" print() \n"
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"celltoolbar": "Slideshow",
|
||
"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": "3.10.6"
|
||
},
|
||
"livereveal": {
|
||
"start_slideshow_at": "selected",
|
||
"theme": "white"
|
||
},
|
||
"vscode": {
|
||
"interpreter": {
|
||
"hash": "916dbcbb3f70747c44a77c7bcd40155683ae19c65e1c03b4aa3499c5328201f1"
|
||
}
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 4
|
||
}
|