forked from pms/uczenie-maszynowe
6194 lines
288 KiB
Plaintext
6194 lines
288 KiB
Plaintext
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{
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"cells": [
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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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"### AITech — Uczenie maszynowe\n",
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"# 3. Regresja logistyczna"
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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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"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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"## 3.1. 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 pl\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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"# 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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"# Hipoteza (wersja macierzowa)\n",
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"def hMx(theta, X):\n",
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" return X * theta\n",
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"\n",
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"# Wykres danych (wersja macierzowa)\n",
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"def regdotsMx(X, y, xlabel, ylabel): \n",
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" fig = pl.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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" pl.ylim(y.min() - 1, y.max() + 1)\n",
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" pl.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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"# Wykres krzywej regresji (wersja macierzowa)\n",
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"def reglineMx(fig, fun, theta, X):\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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"# Legenda wykresu\n",
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"def legend(fig):\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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"# Wersja macierzowa funkcji kosztu\n",
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"def JMx(theta,X,y):\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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"# Wersja macierzowa gradientu funkcji kosztu\n",
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"def dJMx(theta,X,y):\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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"# Implementacja algorytmu gradientu prostego za pomocą numpy i macierzy\n",
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"def GDMx(fJ, fdJ, theta, X, y, alpha=0.1, eps=10**-3):\n",
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" current_cost = fJ(theta, X, y)\n",
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" logs = [[current_cost, theta]]\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",
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" logs.append([current_cost, theta]) \n",
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" return theta, logs\n",
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"\n",
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"thetaStartMx = np.matrix([0, 0]).reshape(2, 1)\n",
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"\n",
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"# Funkcja, która rysuje próg\n",
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"def threshold(fig, theta):\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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]
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}
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],
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"source": [
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"# Wczytanie pełnych (oryginalnych) danych\n",
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"\n",
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"data_iris = pandas.read_csv('iris.csv')\n",
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"print(data_iris[:6])"
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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": 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": [
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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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" 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",
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"4 4.5 0\n",
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"5 5.1 0\n"
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]
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}
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],
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"source": [
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"# 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(lambda x: 1 if x=='Iris-setosa' else 0)\n",
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"print(data_iris_setosa[:6])"
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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": 4,
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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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"import numpy as np\n",
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"\n",
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"# 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",
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"\n",
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"XMx3 = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
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"yMx3 = np.matrix(data_iris_setosa.values[:, 1]).reshape(m, 1)\n",
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"\n",
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"# Regresja liniowa\n",
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"theta_e3, logs3 = GDMx(JMx, dJMx, thetaStartMx, XMx3, yMx3, alpha=0.03, eps=0.000001)"
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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": 5,
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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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"fig = regdotsMx(XMx3, yMx3, 'x', 'Iris setosa?')\n",
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"A gdyby tak przyjąć, że klasyfikator zwraca $1$ dla $h(x) > 0.5$ i $0$ w przeciwnym przypadku?"
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]
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"source": [
|
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"fig = regdotsMx(XMx3, yMx3, 'x', 'Iris setosa?')\n",
|
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|
"theta_e3, logs3 = GDMx(JMx, dJMx, thetaStartMx, XMx3, yMx3, alpha=0.03, eps=0.000001)\n",
|
|||
|
|
"reglineMx(fig, hMx, theta_e3, XMx3)\n",
|
|||
|
|
"threshold(fig, theta_e3) # pomarańczowa linia oznacza granicę między klasą \"1\" a klasą \"0\" wyznaczoną przez próg \"h(x) = 0.5\"\n",
|
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|
"legend(fig)"
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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": [
|
|||
|
|
" * Krzywa regresji liniowej jest niezbyt dopasowana do danych klasyfikacyjnych.\n",
|
|||
|
|
" * Zastosowanie progu $y = 0.5$ nie zawsze pomaga uzyskać sensowny rezultat.\n",
|
|||
|
|
" * $h(x)$ może przyjmować wartości mniejsze od $0$ i większe od $1$ – jak interpretować takie wyniki?\n",
|
|||
|
|
"\n",
|
|||
|
|
"Wniosek: w przypadku problemów klasyfikacyjnych regresja liniowa nie wydaje się najlepszym rozwiązaniem."
|
|||
|
|
]
|
|||
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},
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{
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"cell_type": "markdown",
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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": [
|
|||
|
|
"Wprowadźmy zatem pewne modyfikacje do naszego modelu."
|
|||
|
|
]
|
|||
|
|
},
|
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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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|||
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|
"Zdefiniujmy następującą funkcję, którą będziemy nazywać funkcją *logistyczną* (albo *sigmoidalną*):"
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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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|||
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|
"**Funkcja logistyczna (sigmoidalna)**:\n",
|
|||
|
|
"\n",
|
|||
|
|
"$$g(x) = \\dfrac{1}{1+e^{-x}}$$"
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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": 8,
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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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"outputs": [],
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"source": [
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|||
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|
"# Funkjca logistycza\n",
|
|||
|
|
"\n",
|
|||
|
|
"def logistic(x):\n",
|
|||
|
|
" return 1.0 / (1.0 + np.exp(-x))"
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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": 9,
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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": [
|
|||
|
|
"import matplotlib.pyplot as plt\n",
|
|||
|
|
"def plot_logistic():\n",
|
|||
|
|
" x = np.linspace(-5,5,200)\n",
|
|||
|
|
" y = logistic(x)\n",
|
|||
|
|
"\n",
|
|||
|
|
" fig = plt.figure(figsize=(7,5))\n",
|
|||
|
|
" ax = fig.add_subplot(111)\n",
|
|||
|
|
" plt.ylim(-.1,1.1)\n",
|
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|
|
" ax.plot(x, y, linewidth='2')"
|
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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": [
|
|||
|
|
"Wykres funkcji logistycznej $g(x) = \\dfrac{1}{1+e^{-x}}$:"
|
|||
|
|
]
|
|||
|
|
},
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|
|
{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {
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"scrolled": true,
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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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"plot_logistic()"
|
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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": [
|
|||
|
|
"Funkcja logistyczna przekształca zbiór liczb rzeczywistych $\\mathbb{R}$ w przedział otwarty $(0, 1)$."
|
|||
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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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|||
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|
},
|
|||
|
|
"source": [
|
|||
|
|
"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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{
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|||
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"cell_type": "markdown",
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|||
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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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|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Dla całej macierzy cech $X$:\n",
|
|||
|
|
"\n",
|
|||
|
|
"$$h_\\theta(X) = g(X \\, \\theta) = \\dfrac{1}{1 + e^{-X \\theta}}$$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
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|
"cell_type": "code",
|
|||
|
|
"execution_count": 11,
|
|||
|
|
"metadata": {
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|
"slideshow": {
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|||
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|
"slide_type": "skip"
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|||
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|
}
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|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Funkcja regresji logistcznej\n",
|
|||
|
|
"def h(theta, X):\n",
|
|||
|
|
" return 1.0/(1.0 + np.exp(-X * theta))"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
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|
"metadata": {
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"slideshow": {
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|||
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"slide_type": "subslide"
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|||
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|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Funkcja kosztu dla regresji logistycznej:\n",
|
|||
|
|
"\n",
|
|||
|
|
"$$J(\\theta) = -\\dfrac{1}{m} \\left( \\sum_{i=1}^{m} y^{(i)} \\log h_\\theta( x^{(i)} ) + \\left( 1 - y^{(i)} \\right) \\log \\left( 1 - h_\\theta (x^{(i)}) \\right) \\right)$$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
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|
|
"slideshow": {
|
|||
|
|
"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": [
|
|||
|
|
"# Funkcja kosztu dla regresji logistycznej\n",
|
|||
|
|
"def J(h, theta, X, y):\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"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 13,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "fragment"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Gradient dla regresji logistycznej\n",
|
|||
|
|
"def dJ(h, theta, X, y):\n",
|
|||
|
|
" return 1.0 / len(y) * (X.T * (h(theta, X) - y))"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 14,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Metoda gradientu prostego dla regresji logistycznej\n",
|
|||
|
|
"def GD(h, fJ, fdJ, theta, X, y, alpha=0.01, eps=10**-3, maxSteps=10000):\n",
|
|||
|
|
" errorCurr = fJ(h, theta, X, y)\n",
|
|||
|
|
" errors = [[errorCurr, theta]]\n",
|
|||
|
|
" while True:\n",
|
|||
|
|
" # oblicz nowe theta\n",
|
|||
|
|
" theta = theta - alpha * fdJ(h, theta, X, y)\n",
|
|||
|
|
" # raportuj poziom błędu\n",
|
|||
|
|
" errorCurr, errorPrev = fJ(h, theta, X, y), errorCurr\n",
|
|||
|
|
" # kryteria stopu\n",
|
|||
|
|
" if abs(errorPrev - errorCurr) <= eps:\n",
|
|||
|
|
" break\n",
|
|||
|
|
" if len(errors) > maxSteps:\n",
|
|||
|
|
" break\n",
|
|||
|
|
" errors.append([errorCurr, theta]) \n",
|
|||
|
|
" return theta, errors"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 15,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"error = [[0.05755617]]\n",
|
|||
|
|
"theta = [[ 5.02530461]\n",
|
|||
|
|
" [-1.99174803]]\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"# Uruchomienie metody gradientu prostego dla regresji logistycznej\n",
|
|||
|
|
"thetaBest, errors = GD(h, J, dJ, thetaStartMx, XMx3, yMx3, \n",
|
|||
|
|
" alpha=0.1, eps=10**-7, maxSteps=1000)\n",
|
|||
|
|
"print(\"error =\", errors[-1][0])\n",
|
|||
|
|
"print(\"theta =\", thetaBest)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 16,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Funkcja regresji logistycznej (wersja skalarna)\n",
|
|||
|
|
"def scalar_logistic_regression_function(theta, x):\n",
|
|||
|
|
" return 1.0/(1.0 + np.exp(-(theta.item(0) + theta.item(1) * x)))\n",
|
|||
|
|
"\n",
|
|||
|
|
"# Rysowanie progu\n",
|
|||
|
|
"def threshold_val(fig, x_thr):\n",
|
|||
|
|
" ax = fig.axes[0]\n",
|
|||
|
|
" ax.plot([x_thr, x_thr], [-1, 2],\n",
|
|||
|
|
" color='orange', linestyle='dashed',\n",
|
|||
|
|
" label=u'próg: $x={:.2F}$'.format(x_thr))\n",
|
|||
|
|
"\n",
|
|||
|
|
"# Wykres krzywej regresji logistycznej\n",
|
|||
|
|
"def logistic_regline(fig, theta, X):\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')"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 17,
|
|||
|
|
"metadata": {
|
|||
|
|
"scrolled": true,
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
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|
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"<!-- Created with matplotlib (https://matplotlib.org/) -->\r\n",
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" <defs>\r\n",
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|
" <style type=\"text/css\">\r\n",
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"*{stroke-linecap:butt;stroke-linejoin:round;}\r\n",
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.153849\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"451.132077\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"395.136127\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"185.151317\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"409.135115\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.153849\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"458.13157\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"143.154355\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"143.154355\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"143.154355\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"521.127013\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.153849\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"388.136634\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"402.135621\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"479.130051\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"136.154862\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"360.138659\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"381.13714\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"346.139672\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"325.141191\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"381.13714\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"367.138153\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"304.14271\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"332.140684\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"311.142203\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"143.154355\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"360.138659\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"500.128532\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"339.140178\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"339.140178\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"374.137646\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"381.13714\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"185.151317\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"164.152836\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"360.138659\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"388.136634\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"395.136127\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"381.13714\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"164.152836\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
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|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"423.134102\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"423.134102\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"367.138153\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"332.140684\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"164.152836\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"171.15233\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"444.132583\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"381.13714\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"367.138153\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
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|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"332.140684\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"388.136634\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"353.139165\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"339.140178\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.153849\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"283.144229\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"402.135621\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
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" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"164.152836\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"346.139672\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"122.155874\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"164.152836\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"164.152836\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"367.138153\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"409.135115\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"325.141191\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"388.136634\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"479.130051\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"430.133596\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"353.139165\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"437.133089\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"409.135115\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"451.132077\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"444.132583\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"143.154355\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"318.141697\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"479.130051\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"458.13157\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"374.137646\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"367.138153\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"395.136127\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"409.135115\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"143.154355\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"416.134608\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"409.135115\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"437.133089\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"493.129039\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.153849\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"465.131064\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"409.135115\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"444.132583\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"171.15233\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"367.138153\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"262.145748\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"444.132583\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"472.130558\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"171.15233\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.153849\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"332.140684\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"521.127013\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"346.139672\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.153849\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"416.134608\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"360.138659\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"367.138153\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"171.15233\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"444.132583\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"283.144229\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"402.135621\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.153849\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"402.135621\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.153849\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"395.136127\" xlink:href=\"#me8c38f96c6\" y=\"218.359219\"/>\r\n",
|
|||
|
|
" <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"157.153343\" xlink:href=\"#me8c38f96c6\" y=\"114.679219\"/>\r\n",
|
|||
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"text/plain": [
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]
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"metadata": {
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}
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],
|
|||
|
|
"source": [
|
|||
|
|
"fig = regdotsMx(XMx3, yMx3, xlabel='x', ylabel='Iris setosa?')\n",
|
|||
|
|
"logistic_regline(fig, thetaBest, XMx3)\n",
|
|||
|
|
"threshold_val(fig, 2.5)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
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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": [
|
|||
|
|
"Traktujemy wartość $h_\\theta(x)$ jako prawdopodobieństwo, że cecha przyjmie wartość pozytywną:\n",
|
|||
|
|
"\n",
|
|||
|
|
"$$ h_\\theta(x) = P(y = 1 \\, | \\, x; \\theta) $$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "fragment"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Jeżeli $h_\\theta(x) > 0.5$, to dla takiego $x$ będziemy przewidywać wartość $y = 1$.\n",
|
|||
|
|
"W przeciwnym wypadku uprzewidzimy $y = 0$."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"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"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "slide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"### Dwuklasowa regresja logistyczna: więcej cech"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Jak postąpić, jeżeli będziemy mieli więcej niż jedną cechę $x$?\n",
|
|||
|
|
"\n",
|
|||
|
|
"Weźmy teraz wszystkie cechy występujące w zbiorze *Iris*."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 18,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
" dł. płatków szer. płatków dł. dz. k. szer. dz. k. 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",
|
|||
|
|
"data_iris_setosa_multi['dł. płatków'] = data_iris['pl'] # \"pl\" oznacza \"petal length\" (długość płatków)\n",
|
|||
|
|
"data_iris_setosa_multi['szer. płatków'] = data_iris['pw'] # \"pw\" oznacza \"petal width\" (szerokość płatków)\n",
|
|||
|
|
"data_iris_setosa_multi['dł. dz. k.'] = data_iris['sl'] # \"sl\" oznacza \"sepal length\" (długość działek kielicha)\n",
|
|||
|
|
"data_iris_setosa_multi['szer. dz. k.'] = data_iris['sw'] # \"sw\" oznacza \"sepal width\" (szerokość działek kielicha)\n",
|
|||
|
|
"data_iris_setosa_multi['Iris setosa?'] = data_iris['Gatunek'].apply(lambda x: 1 if x=='Iris-setosa' else 0)\n",
|
|||
|
|
"print(data_iris_setosa_multi[:6])"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 19,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"%matplotlib inline\n",
|
|||
|
|
"\n",
|
|||
|
|
"import matplotlib.pyplot as plt\n",
|
|||
|
|
"import seaborn"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 20,
|
|||
|
|
"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",
|
|||
|
|
"XMx4 = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
|
|||
|
|
"yMx4 = np.matrix(data_iris_setosa_multi.values[:, n]).reshape(m, 1)\n",
|
|||
|
|
"\n",
|
|||
|
|
"print(XMx4[:6])\n",
|
|||
|
|
"print(yMx4[:6])"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 21,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Podział danych na zbiór trenujący i testowy\n",
|
|||
|
|
"XTrain, XTest = XMx4[:100], XMx4[100:]\n",
|
|||
|
|
"yTrain, yTest = yMx4[:100], yMx4[100:]\n",
|
|||
|
|
"\n",
|
|||
|
|
"# Macierz parametrów początkowych\n",
|
|||
|
|
"thetaTemp = np.ones(5).reshape(5,1)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 22,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"error = [[0.006797]]\n",
|
|||
|
|
"theta = [[ 1.11414027]\n",
|
|||
|
|
" [-2.89324615]\n",
|
|||
|
|
" [-0.66543637]\n",
|
|||
|
|
" [ 0.14887292]\n",
|
|||
|
|
" [ 2.13284493]]\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"thetaBest, errors = GD(h, J, dJ, thetaTemp, XTrain, yTrain, \n",
|
|||
|
|
" alpha=0.1, eps=10**-7, maxSteps=1000)\n",
|
|||
|
|
"print(\"error =\", errors[-1][0])\n",
|
|||
|
|
"print(\"theta =\", thetaBest)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"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",
|
|||
|
|
"$$ c = \\left\\{ \n",
|
|||
|
|
"\\begin{array}{ll}\n",
|
|||
|
|
"1, & \\mbox{gdy } P(y=1 \\, | \\, x; \\theta) > 0.5 \\\\\n",
|
|||
|
|
"0 & \\mbox{w przeciwnym przypadku}\n",
|
|||
|
|
"\\end{array}\\right.\n",
|
|||
|
|
"$$\n",
|
|||
|
|
"\n",
|
|||
|
|
"$$ P(y=1 \\,| \\, x; \\theta) = h_\\theta(x) $$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 23,
|
|||
|
|
"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) = 1.6061436959824898e-05\n",
|
|||
|
|
"c(x0) = (0, 1.6061436959824898e-05) \n",
|
|||
|
|
"\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"def classifyBi(theta, X):\n",
|
|||
|
|
" prob = h(theta, X).item()\n",
|
|||
|
|
" return (1, prob) if prob > 0.5 else (0, prob)\n",
|
|||
|
|
"\n",
|
|||
|
|
"print(\"theta =\", thetaBest)\n",
|
|||
|
|
"print(\"x0 =\", XTest[0])\n",
|
|||
|
|
"print(\"h(x0) =\", h(thetaBest, XTest[0]).item())\n",
|
|||
|
|
"print(\"c(x0) =\", classifyBi(thetaBest, 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": 24,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"0 <=> 0 -- prob: 0.0\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": [
|
|||
|
|
"acc = 0.0\n",
|
|||
|
|
"for i, rest in enumerate(yTest):\n",
|
|||
|
|
" cls, prob = classifyBi(thetaBest, XTest[i])\n",
|
|||
|
|
" if i < 10:\n",
|
|||
|
|
" print(int(yTest[i].item()), \"<=>\", cls, \"-- prob:\", round(prob, 4))\n",
|
|||
|
|
" acc += cls == yTest[i].item()\n",
|
|||
|
|
"\n",
|
|||
|
|
"print(\"\\nAccuracy:\", acc / len(XTest))"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "slide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"## 3.2. 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": 25,
|
|||
|
|
"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": 25,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"import pandas\n",
|
|||
|
|
"data_iris = pandas.read_csv('iris.csv')\n",
|
|||
|
|
"data_iris[:6]"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 26,
|
|||
|
|
"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])"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"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": 27,
|
|||
|
|
"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.\n",
|
|||
|
|
" return yBi\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",
|
|||
|
|
"# one-hot matrix\n",
|
|||
|
|
"Y = indicatorMatrix(y)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 28,
|
|||
|
|
"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",
|
|||
|
|
"thetaTemp = np.ones(5).reshape(5,1)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"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": 29,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Zapis macierzowy funkcji softmax\n",
|
|||
|
|
"def softmax(X):\n",
|
|||
|
|
" return np.exp(X) / np.sum(np.exp(X))"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Wartości funkcji $\\mathrm{softmax}$ sumują się do 1:"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 30,
|
|||
|
|
"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)) "
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 31,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"Otrzymana macierz parametrów theta dla klasy 0:\n",
|
|||
|
|
" [[ 0.68590262]\n",
|
|||
|
|
" [ 0.39948964]\n",
|
|||
|
|
" [ 1.13312933]\n",
|
|||
|
|
" [-2.17550597]\n",
|
|||
|
|
" [-0.53088875]] \n",
|
|||
|
|
"\n",
|
|||
|
|
"Otrzymana macierz parametrów theta dla klasy 1:\n",
|
|||
|
|
" [[ 0.95431453]\n",
|
|||
|
|
" [ 0.07249434]\n",
|
|||
|
|
" [-1.07233395]\n",
|
|||
|
|
" [ 0.53801787]\n",
|
|||
|
|
" [-0.65001214]] \n",
|
|||
|
|
"\n",
|
|||
|
|
"Otrzymana macierz parametrów theta dla klasy 2:\n",
|
|||
|
|
" [[-0.66101185]\n",
|
|||
|
|
" [-1.40133883]\n",
|
|||
|
|
" [-2.01776182]\n",
|
|||
|
|
" [ 2.18505283]\n",
|
|||
|
|
" [ 2.74690482]] \n",
|
|||
|
|
"\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"# Dla każdej klasy wytrenujmy osobny klasyfikator dwuklasowy.\n",
|
|||
|
|
"\n",
|
|||
|
|
"def trainMaxEnt(X, Y):\n",
|
|||
|
|
" n = X.shape[1]\n",
|
|||
|
|
" thetas = []\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",
|
|||
|
|
" thetaBest, errors = GD(h, J, dJ, theta, \n",
|
|||
|
|
" X, YBi, alpha=0.1, eps=10**-4)\n",
|
|||
|
|
" thetas.append(thetaBest)\n",
|
|||
|
|
" return thetas\n",
|
|||
|
|
"\n",
|
|||
|
|
"# Macierze theta dla każdej klasy\n",
|
|||
|
|
"thetas = trainMaxEnt(XTrain, YTrain);\n",
|
|||
|
|
"for c, theta in enumerate(thetas):\n",
|
|||
|
|
" print(f\"Otrzymana macierz parametrów theta dla klasy {c}:\\n\", theta, \"\\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": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"Dla x = [[1. 7.3 2.9 6.3 1.8]]:\n",
|
|||
|
|
"Po zastosowaniu regresji: [-7.77303536 0.59324542 1.96796697]\n",
|
|||
|
|
"Otrzymane prawdopodobieństwa: [0. 0.202 0.798]\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: [ 2.79786587 -1.35649314 -9.55757825]\n",
|
|||
|
|
"Otrzymane prawdopodobieństwa: [0.985 0.015 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: [-7.02868459 0.06130237 1.99650886]\n",
|
|||
|
|
"Otrzymane prawdopodobieństwa: [0. 0.126 0.874]\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: [-5.60840075 -0.26110148 1.10600174]\n",
|
|||
|
|
"Otrzymane prawdopodobieństwa: [0.001 0.203 0.796]\n",
|
|||
|
|
"Wybrana klasa: 2\n",
|
|||
|
|
"Obliczone y = 2\n",
|
|||
|
|
"Oczekiwane y = 2\n",
|
|||
|
|
"\n",
|
|||
|
|
"Dla x = [[1. 6.1 2.6 5.6 1.4]]:\n",
|
|||
|
|
"Po zastosowaniu regresji: [-6.85715204 0.71134476 1.62660319]\n",
|
|||
|
|
"Otrzymane prawdopodobieństwa: [0. 0.286 0.714]\n",
|
|||
|
|
"Wybrana klasa: 2\n",
|
|||
|
|
"Obliczone y = 2\n",
|
|||
|
|
"Oczekiwane y = 2\n",
|
|||
|
|
"\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"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",
|
|||
|
|
"\n",
|
|||
|
|
"for i in range(5):\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()"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"metadata": {
|
|||
|
|
"celltoolbar": "Slideshow",
|
|||
|
|
"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": "3.8.3"
|
|||
|
|
},
|
|||
|
|
"livereveal": {
|
|||
|
|
"start_slideshow_at": "selected",
|
|||
|
|
"theme": "white"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"nbformat": 4,
|
|||
|
|
"nbformat_minor": 4
|
|||
|
|
}
|