18796 lines
1.1 MiB
18796 lines
1.1 MiB
{
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"cells": [
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{
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"cell_type": "markdown",
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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 UMZ 2019/2020\n",
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"### 24 marca 2020\n",
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"# 3. 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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"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": "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": "slide"
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}
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},
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"source": [
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"Zacznijmy od najprostszego przypadku: chcemy nasze dane przypisać do jednej z **dwóch** klas.\n",
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"W tym celu użyjemy regresji logistycznej **dwuklasowej**."
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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 style=\"float: right;\" src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/5/56/Kosaciec_szczecinkowaty_Iris_setosa.jpg/450px-Kosaciec_szczecinkowaty_Iris_setosa.jpg\">\n",
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"\n",
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"### Przykład: kosaciec szczecinkowy (_Iris setosa_)\n",
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"\n",
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"Mamy dane dotyczące długości płatków i chcielibyśmy na tej podstawie określić, czy jest to roślina z gatunku _Iris setosa_"
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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 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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"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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"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": "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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"Czy możemy tu zastosować regresję liniową?\n",
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"\n",
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"Spróbujmy:"
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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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"outputs": [
|
||
{
|
||
"name": "stdout",
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||
"output_type": "stream",
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||
"text": [
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"(150, 2)\n"
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]
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}
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||
],
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||
"source": [
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||
"print(data_iris_setosa.values.shape)"
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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": "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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||
"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)"
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||
]
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||
},
|
||
{
|
||
"cell_type": "code",
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"execution_count": 6,
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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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"source": [
|
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"fig = regdotsMx(XMx3, yMx3, 'x', 'Iris setosa?')\n",
|
||
"theta_e3, logs3 = GDMx(JMx, dJMx, thetaStartMx, XMx3, yMx3, alpha=0.03, eps=0.000001)\n",
|
||
"reglineMx(fig, hMx, theta_e3, XMx3)\n",
|
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"threshold(fig, theta_e3)\n",
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"legend(fig)"
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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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"source": [
|
||
"Zdefiniujmy sobie następującą funkcję, którą będziemy nazywać funkcją *logistyczną* (albo *sigmoidalną*):"
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]
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{
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"**Funkcja logistyczna (sigmoidalna)**:\n",
|
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"\n",
|
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"$$g(x) = \\dfrac{1}{1+e^{-x}}$$"
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]
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"metadata": {
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"slide_type": "fragment"
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}
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},
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"source": [
|
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"# Funkjca logistycza\n",
|
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"\n",
|
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"def logistic(x):\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": "notes"
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}
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},
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"outputs": [],
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"source": [
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"import matplotlib.pyplot as plt\n",
|
||
"def plot_logistic():\n",
|
||
" x = np.linspace(-5,5,200)\n",
|
||
" y = logistic(x)\n",
|
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"\n",
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" fig = plt.figure(figsize=(7,5))\n",
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|
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{
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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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{
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"Funkcja logistyczna przekształca zbiór liczb rzeczywistych $\\mathbb{R}$ w przedział otwarty $(0, 1)$."
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"Funkcja regresji logistycznej:\n",
|
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"\n",
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"\n",
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"outputs": [],
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"source": [
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"# Funkcja regresji logistcznej\n",
|
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"def h(theta, X):\n",
|
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]
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{
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"#print(XMx3)\n",
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]
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},
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{
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}
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"Funkcja kosztu dla regresji logistycznej:\n",
|
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"\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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}
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},
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"source": [
|
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"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 $h_\\theta$)."
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]
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},
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{
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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": [],
|
||
"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",
|
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" return -np.sum(s1 + s2, axis=0) / m"
|
||
]
|
||
},
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||
{
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"cell_type": "code",
|
||
"execution_count": 38,
|
||
"metadata": {
|
||
"slideshow": {
|
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"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))"
|
||
]
|
||
},
|
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{
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||
"cell_type": "code",
|
||
"execution_count": 39,
|
||
"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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"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",
|
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" return theta, errors"
|
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]
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},
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{
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"execution_count": 40,
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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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"error = [[0.05755617]]\n",
|
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"theta = [[ 5.02530461]\n",
|
||
" [-1.99174803]]\n"
|
||
]
|
||
}
|
||
],
|
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"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)"
|
||
]
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||
},
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{
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"cell_type": "code",
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"execution_count": 16,
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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": [
|
||
"# 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",
|
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" ax = fig.axes[0]\n",
|
||
" ax.plot([x_thr, x_thr], [-1, 2],\n",
|
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" 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')"
|
||
]
|
||
},
|
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{
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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": [
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||
"Traktujemy wartość $h_\\theta(x)$ jako prawdopodobieństwo, że cecha przyjmie wartość pozytywną:\n",
|
||
"\n",
|
||
"$$ h_\\theta(x) = P(y = 1 \\, | \\, x; \\theta) $$"
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]
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},
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"metadata": {},
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"source": [
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"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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{
|
||
"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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},
|
||
"source": [
|
||
"### Dwuklasowa regresja logistyczna: więcej cech"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
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"metadata": {
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"slideshow": {
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"slide_type": "notes"
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}
|
||
},
|
||
"source": [
|
||
"Jak postąpić, jeżeli będziemy mieli więcej niż jedną cechę $x$?"
|
||
]
|
||
},
|
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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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"source": [
|
||
"Weźmy pod uwagę następujące cechy:\n",
|
||
" * długość działek kielicha\n",
|
||
" * szerokość działek kielicha\n",
|
||
" * długość płatka\n",
|
||
" * szerokość płatka"
|
||
]
|
||
},
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{
|
||
"cell_type": "code",
|
||
"execution_count": 18,
|
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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": [
|
||
{
|
||
"name": "stdout",
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||
"output_type": "stream",
|
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"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": {
|
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"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"
|
||
}
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<seaborn.axisgrid.PairGrid at 0x1acd8f5fac8>"
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"58.032452\" xlink:href=\"#mf298374ff8\" y=\"351.447393\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"55.989686\" xlink:href=\"#mf298374ff8\" y=\"319.238622\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"58.032452\" xlink:href=\"#mf298374ff8\" y=\"364.330901\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"51.904155\" xlink:href=\"#mf298374ff8\" y=\"357.889147\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"55.989686\" xlink:href=\"#mf298374ff8\" y=\"351.447393\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"51.904155\" xlink:href=\"#mf298374ff8\" y=\"348.226516\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"55.989686\" xlink:href=\"#mf298374ff8\" y=\"361.110024\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"53.946921\" xlink:href=\"#mf298374ff8\" y=\"364.330901\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"60.075218\" xlink:href=\"#mf298374ff8\" y=\"335.343008\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"60.075218\" xlink:href=\"#mf298374ff8\" y=\"351.447393\"/>\r\n",
|
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"53.946921\" xlink:href=\"#mf298374ff8\" y=\"357.889147\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"53.946921\" xlink:href=\"#mf298374ff8\" y=\"348.226516\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"55.989686\" xlink:href=\"#mf298374ff8\" y=\"361.110024\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"55.989686\" xlink:href=\"#mf298374ff8\" y=\"361.110024\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"60.075218\" xlink:href=\"#mf298374ff8\" y=\"338.563885\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"53.946921\" xlink:href=\"#mf298374ff8\" y=\"325.680377\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"53.946921\" xlink:href=\"#mf298374ff8\" y=\"354.66827\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"55.989686\" xlink:href=\"#mf298374ff8\" y=\"341.784762\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"58.032452\" xlink:href=\"#mf298374ff8\" y=\"361.110024\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"51.904155\" xlink:href=\"#mf298374ff8\" y=\"386.87704\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"53.946921\" xlink:href=\"#mf298374ff8\" y=\"351.447393\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"55.989686\" xlink:href=\"#mf298374ff8\" y=\"328.901254\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"47.818624\" xlink:href=\"#mf298374ff8\" y=\"364.330901\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"49.861389\" xlink:href=\"#mf298374ff8\" y=\"332.122131\"/>\r\n",
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" <use style=\"fill:#ff7f0e;stroke:#ffffff;stroke-width:0.75;\" x=\"53.946921\" xlink:href=\"#mf298374ff8\" y=\"348.226516\"/>\r\n",
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" <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"66.203515\" xlink:href=\"#m2e52862573\" y=\"403.7\"/>\r\n",
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" </g>\r\n",
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" <g id=\"xtick_49\">\r\n",
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" <g id=\"line2d_85\">\r\n",
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" <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"188.769454\" xlink:href=\"#m2e52862573\" y=\"403.7\"/>\r\n",
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" </g>\r\n",
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" <!-- 4 -->\r\n",
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]
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},
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"metadata": {
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},
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}
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],
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"source": [
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"# Poniższy wykres przedstawia zależności między wszystkimi cechami\n",
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"\n",
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"seaborn.pairplot(\n",
|
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" data_iris_setosa_multi,\n",
|
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" vars=[c for c in data_iris_setosa_multi.columns if c != 'Iris setosa?'], \n",
|
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" hue='Iris setosa?', height=1.5, aspect=1.75)"
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]
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},
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{
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"execution_count": 21,
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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": [
|
||
"[[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",
|
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" [1. 5.1 1.6 6. 2.7]]\n",
|
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"[[1.]\n",
|
||
" [1.]\n",
|
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" [0.]\n",
|
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" [0.]\n",
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" [0.]\n",
|
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" [0.]]\n"
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]
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}
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],
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"source": [
|
||
"# Przygotowanie danych\n",
|
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"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",
|
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"\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",
|
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"\n",
|
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"print(XMx4[:6])\n",
|
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"print(yMx4[: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": 22,
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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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"source": [
|
||
"# Podział danych na zbiór trenujący i testowy\n",
|
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"XTrain, XTest = XMx4[:100], XMx4[100:]\n",
|
||
"yTrain, yTest = yMx4[:100], yMx4[100:]\n",
|
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"\n",
|
||
"# Macierz parametrów początkowych\n",
|
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"thetaTemp = np.ones(5).reshape(5,1)"
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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": 23,
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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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"error = [[0.006797]]\n",
|
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"theta = [[ 1.11414027]\n",
|
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" [-2.89324615]\n",
|
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" [-0.66543637]\n",
|
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" [ 0.14887292]\n",
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" [ 2.13284493]]\n"
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]
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}
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],
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"source": [
|
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"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)"
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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": [
|
||
"### 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",
|
||
"$$ P(y=1 \\,| \\, x; \\theta) = h_\\theta(x) $$"
|
||
]
|
||
},
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{
|
||
"cell_type": "code",
|
||
"execution_count": 24,
|
||
"metadata": {
|
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"slideshow": {
|
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"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"
|
||
]
|
||
}
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],
|
||
"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\")"
|
||
]
|
||
},
|
||
{
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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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},
|
||
"source": [
|
||
"### Skuteczność"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 25,
|
||
"metadata": {},
|
||
"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"
|
||
]
|
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}
|
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],
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"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": [
|
||
"## 3.4. Wieloklasowa regresja logistyczna"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Przykład: gatunki irysów (kosaćców)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"<img style=\"float: right;\" src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/5/56/Kosaciec_szczecinkowaty_Iris_setosa.jpg/450px-Kosaciec_szczecinkowaty_Iris_setosa.jpg\">\n",
|
||
"\n",
|
||
"Kosaciec szczecinkowy (_Iris setosa_)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"<img style=\"float: right;\" src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Iris_virginica.jpg/736px-Iris_virginica.jpg\">\n",
|
||
"\n",
|
||
"Kosaciec amerykański (_Iris virginica_)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"<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",
|
||
"\n",
|
||
"Kosaciec różnobarwny (_Iris versicolor_)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Cechy:\n",
|
||
" * długość działek kielicha\n",
|
||
" * szerokość działek kielicha\n",
|
||
" * długość płatka\n",
|
||
" * szerokość płatka"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Wczytanie danych"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 26,
|
||
"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": 26,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"import pandas\n",
|
||
"data_iris = pandas.read_csv('iris.csv')\n",
|
||
"data_iris[:6]"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"source": [
|
||
"Przygotowanie danych"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 27,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "fragment"
|
||
}
|
||
},
|
||
"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": [
|
||
"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": 28,
|
||
"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": 29,
|
||
"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\n",
|
||
"thetaTemp = np.ones(5).reshape(5,1)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 30,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stderr",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"C:\\Users\\pawel\\anaconda3\\lib\\site-packages\\seaborn\\axisgrid.py:2079: UserWarning: The `size` parameter has been renamed to `height`; please update your code.\n",
|
||
" warnings.warn(msg, UserWarning)\n"
|
||
]
|
||
},
|
||
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|
||
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" <use style=\"fill:#2ca02c;stroke:#ffffff;stroke-width:0.75;\" x=\"114.855344\" xlink:href=\"#m3b1275ea06\" y=\"413.511741\"/>\r\n",
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"source": [
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"g = seaborn.pairplot(data_iris, hue='Gatunek', size=1.75, aspect=1.5) "
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]
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},
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{
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"slide_type": "slide"
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}
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},
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"source": [
|
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"### Od regresji logistycznej dwuklasowej do wieloklasowej\n",
|
||
"\n",
|
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"* Irysy są przydzielone do trzech klas: _Iris-setosa_ (0), _Iris-versicolor_ (1), _Iris-virginica_ (2).\n",
|
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"* Wiemy, jak stworzyć klasyfikatory 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."
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]
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}
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},
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"source": [
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"### Funkcja _softmax_\n",
|
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"\n",
|
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"Odpowiednikiem funkcji logistycznej dla wieloklasowej regresji logistycznej jest funkcja $\\mathrm{softmax}$:\n",
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"\n",
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"$$ \\textrm{softmax}(k,x_1,\\dots,x_n) = \\dfrac{e^{x_k}}{\\sum_{i=i}^{n}e^{x_i}} $$\n",
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"\n",
|
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"$$P(y=c|x;\\theta_1,\\ldots,\\theta_k) = \\textrm{softmax}(c,\\theta_1^Tx,\\ldots,\\theta_k^Tx)$$"
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]
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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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"X5 = [[1. 5.2 3.4 1.4 0.2]\n",
|
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" [1. 5.1 3.7 1.5 0.4]\n",
|
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" [1. 6.7 3.1 5.6 2.4]]\n",
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"softmax = [[0.00175241 0.11686208 0.01931717 0.0026143 0.00078741]\n",
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" [0.00175241 0.10574119 0.02607546 0.00288924 0.00096175]\n",
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" [0.00175241 0.52373952 0.01431051 0.17433774 0.00710639]]\n"
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]
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}
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],
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"source": [
|
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"# Zapis macierzowy funkcji softmax\n",
|
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"def softmax(X):\n",
|
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" return np.exp(X) / np.sum(np.exp(X))\n",
|
||
"\n",
|
||
"X5 = X[:3]\n",
|
||
"print(\"X5 =\", X5)\n",
|
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"print(\"softmax =\", softmax(X5))"
|
||
]
|
||
},
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{
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"cell_type": "code",
|
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"execution_count": 32,
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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": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"[[2.1]\n",
|
||
" [0.5]\n",
|
||
" [0.8]\n",
|
||
" [0.9]\n",
|
||
" [3.2]]\n",
|
||
"Suma X = 7.500000000000001 \n",
|
||
"\n",
|
||
"[[0.20921428]\n",
|
||
" [0.04223963]\n",
|
||
" [0.05701754]\n",
|
||
" [0.06301413]\n",
|
||
" [0.62851442]]\n",
|
||
"Suma P = 0.9999999999999999\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"XN = np.matrix([2.1, 0.5, 0.8, 0.9, 3.2]).reshape(5,1)\n",
|
||
"P = softmax(XN)\n",
|
||
"print(XN)\n",
|
||
"print(\"Suma X =\", np.sum(XN), \"\\n\")\n",
|
||
"print(P)\n",
|
||
"print(\"Suma P =\", np.sum(P)) "
|
||
]
|
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},
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{
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"cell_type": "code",
|
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"execution_count": 33,
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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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},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"[[ 0.73973637]\n",
|
||
" [ 0.40248388]\n",
|
||
" [ 1.13832166]\n",
|
||
" [-2.27811895]\n",
|
||
" [-0.2706796 ]] \n",
|
||
"\n",
|
||
"[[ 0.79885563]\n",
|
||
" [-0.05169838]\n",
|
||
" [-0.89917914]\n",
|
||
" [ 0.70287215]\n",
|
||
" [-0.87160557]] \n",
|
||
"\n",
|
||
"[[-0.59026003]\n",
|
||
" [-1.76321652]\n",
|
||
" [-1.56383243]\n",
|
||
" [ 2.41036901]\n",
|
||
" [ 2.60626226]] \n",
|
||
"\n"
|
||
]
|
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}
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],
|
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"source": [
|
||
"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",
|
||
" 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",
|
||
"thetas = trainMaxEnt(XTrain, YTrain);\n",
|
||
"for theta in thetas:\n",
|
||
" print(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": 34,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"YTest = [[0. 0. 1.]\n",
|
||
" [1. 0. 0.]\n",
|
||
" [0. 0. 1.]\n",
|
||
" [0. 0. 1.]\n",
|
||
" [0. 0. 1.]\n",
|
||
" [1. 0. 0.]]\n",
|
||
"YTestCls = [[2.]\n",
|
||
" [0.]\n",
|
||
" [2.]\n",
|
||
" [2.]\n",
|
||
" [2.]\n",
|
||
" [0.]] \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(\"regs =\", regs)\n",
|
||
" probs = softmax(regs)\n",
|
||
" if debug:\n",
|
||
" print(\"probs =\", np.around(probs,decimals=3))\n",
|
||
" return np.argmax(probs), probs\n",
|
||
"\n",
|
||
"print(\"YTest =\", YTest[:6])\n",
|
||
"YTestCls = YTest * np.matrix((0,1,2)).T\n",
|
||
"print(\"YTestCls =\", YTestCls[:6], \"\\n\")"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "slide"
|
||
}
|
||
},
|
||
"source": [
|
||
"### Ewaluacja"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Dokonajmy teraz ewaluacji przewidywań naszego algorytmu. Użyjemy w tym celu metryki **skuteczności** (ang. *accuracy*).\n",
|
||
"\n",
|
||
"(Więcej o metrykach na jednym z następnych wykładów)."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 35,
|
||
"metadata": {
|
||
"slideshow": {
|
||
"slide_type": "subslide"
|
||
}
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"regs = [-7.86037111 0.67304249 1.87974212]\n",
|
||
"probs = [0. 0.23 0.77]\n",
|
||
"2 <=> 2 -- True [0.0, 0.2303, 0.7697]\n",
|
||
"regs = [ 2.81605359 -1.42429467 -9.58880135]\n",
|
||
"probs = [0.986 0.014 0. ]\n",
|
||
"0 <=> 0 -- True [0.9858, 0.0142, 0.0]\n",
|
||
"regs = [-6.99699202 0.05083372 1.89373326]\n",
|
||
"probs = [0. 0.137 0.863]\n",
|
||
"2 <=> 2 -- True [0.0001, 0.1367, 0.8632]\n",
|
||
"regs = [-5.57627364 -0.18794428 1.2994192 ]\n",
|
||
"probs = [0.001 0.184 0.815]\n",
|
||
"2 <=> 2 -- True [0.0008, 0.1842, 0.815]\n",
|
||
"regs = [-6.98189316 0.861466 1.73498847]\n",
|
||
"probs = [0. 0.294 0.705]\n",
|
||
"2 <=> 2 -- True [0.0001, 0.2945, 0.7054]\n",
|
||
"regs = [ 3.37152977 -2.14087384 -11.07044352]\n",
|
||
"probs = [0.996 0.004 0. ]\n",
|
||
"0 <=> 0 -- True [0.996, 0.004, 0.0]\n",
|
||
"\n",
|
||
"Accuracy = 1.0\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"acc = 0.0\n",
|
||
"for i in range(len(YTestCls)):\n",
|
||
" cls, probs = classify(thetas, XTest[i], i < 6)\n",
|
||
" correctCls = int(YTestCls[i].item())\n",
|
||
" if i < 6:\n",
|
||
" print(correctCls, \" <=>\", cls, \" -- \", cls == correctCls, np.round(probs, 4).tolist())\n",
|
||
" acc += correctCls == cls\n",
|
||
"print(\"\\nAccuracy =\", acc/len(XTest))"
|
||
]
|
||
}
|
||
],
|
||
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|
||
"celltoolbar": "Slideshow",
|
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|
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"display_name": "Python 3",
|
||
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|
||
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|
||
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|
||
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|
||
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|
||
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|
||
"version": 3
|
||
},
|
||
"file_extension": ".py",
|
||
"mimetype": "text/x-python",
|
||
"name": "python",
|
||
"nbconvert_exporter": "python",
|
||
"pygments_lexer": "ipython3",
|
||
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|
||
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|
||
"livereveal": {
|
||
"start_slideshow_at": "selected",
|
||
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||
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