uczenie-maszynowe/wyk/03_Regresja_liniowa_2.ipynb
2022-10-14 11:34:46 +02:00

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{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Uczenie maszynowe\n",
"# 3. Regresja liniowa część 2"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## 3.1. Regresja liniowa wielu zmiennych"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"source": [
"Do przewidywania wartości $y$ możemy użyć więcej niż jednej cechy $x$:"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Przykład ceny mieszkań"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"y : price x1: isNew x2: rooms x3: floor x4: location x5: sqrMetres\n",
"476118.0 False 3 1 Centrum 78 \n",
"459531.0 False 3 2 Sołacz 62 \n",
"411557.0 False 3 0 Sołacz 15 \n",
"496416.0 False 4 0 Sołacz 14 \n",
"406032.0 False 3 0 Sołacz 15 \n",
"450026.0 False 3 1 Naramowice 80 \n",
"571229.15 False 2 4 Wilda 39 \n",
"325000.0 False 3 1 Grunwald 54 \n",
"268229.0 False 2 1 Grunwald 90 \n"
]
}
],
"source": [
"import csv\n",
"\n",
"reader = csv.reader(open(\"data02_train.tsv\", encoding=\"utf-8\"), delimiter=\"\\t\")\n",
"for i, row in enumerate(list(reader)[:10]):\n",
" if i == 0:\n",
" print(\n",
" \" \".join(\n",
" [\n",
" \"{}: {:8}\".format(\"x\" + str(j) if j > 0 else \"y \", entry)\n",
" for j, entry in enumerate(row)\n",
" ]\n",
" )\n",
" )\n",
" else:\n",
" print(\" \".join([\"{:12}\".format(entry) for entry in row]))\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$ x^{(2)} = ({\\rm \"False\"}, 3, 2, {\\rm \"Sołacz\"}, 62), \\quad x_3^{(2)} = 2 $$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Hipoteza"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"W naszym przypadku (wybraliśmy 5 cech):\n",
"\n",
"$$ h_\\theta(x) = \\theta_0 + \\theta_1 x_1 + \\theta_2 x_2 + \\theta_3 x_3 + \\theta_4 x_4 + \\theta_5 x_5 $$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"W ogólności ($n$ cech):\n",
"\n",
"$$ h_\\theta(x) = \\theta_0 + \\theta_1 x_1 + \\theta_2 x_2 + \\ldots + \\theta_n x_n $$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Jeżeli zdefiniujemy $x_0 = 1$, będziemy mogli powyższy wzór zapisać w bardziej kompaktowy sposób:\n",
"\n",
"$$\n",
"\\begin{array}{rcl}\n",
"h_\\theta(x)\n",
" & = & \\theta_0 x_0 + \\theta_1 x_1 + \\theta_2 x_2 + \\ldots + \\theta_n x_n \\\\\n",
" & = & \\displaystyle\\sum_{i=0}^{n} \\theta_i x_i \\\\\n",
" & = & \\theta^T \\, x \\\\\n",
" & = & x^T \\, \\theta \\\\\n",
"\\end{array}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Metoda gradientu prostego notacja macierzowa"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"source": [
"Metoda gradientu prostego przyjmie bardzo elegancką formę, jeżeli do jej zapisu użyjemy wektorów i macierzy."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\n",
"X=\\left[\\begin{array}{cc}\n",
"1 & \\left( \\vec x^{(1)} \\right)^T \\\\\n",
"1 & \\left( \\vec x^{(2)} \\right)^T \\\\\n",
"\\vdots & \\vdots\\\\\n",
"1 & \\left( \\vec x^{(m)} \\right)^T \\\\\n",
"\\end{array}\\right] \n",
"= \\left[\\begin{array}{cccc}\n",
"1 & x_1^{(1)} & \\cdots & x_n^{(1)} \\\\\n",
"1 & x_1^{(2)} & \\cdots & x_n^{(2)} \\\\\n",
"\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
"1 & x_1^{(m)} & \\cdots & x_n^{(m)} \\\\\n",
"\\end{array}\\right]\n",
"\\quad\n",
"\\vec{y} = \n",
"\\left[\\begin{array}{c}\n",
"y^{(1)}\\\\\n",
"y^{(2)}\\\\\n",
"\\vdots\\\\\n",
"y^{(m)}\\\\\n",
"\\end{array}\\right]\n",
"\\quad\n",
"\\theta = \\left[\\begin{array}{c}\n",
"\\theta_0\\\\\n",
"\\theta_1\\\\\n",
"\\vdots\\\\\n",
"\\theta_n\\\\\n",
"\\end{array}\\right]\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {
"slideshow": {
"slide_type": "skip"
}
},
"outputs": [],
"source": [
"# Wersje macierzowe funkcji rysowania wykresów punktowych oraz krzywej regresyjnej\n",
"\n",
"\n",
"def hMx(theta, X):\n",
" return X * theta\n",
"\n",
"\n",
"def regdotsMx(X, y):\n",
" fig = plt.figure(figsize=(16 * 0.6, 9 * 0.6))\n",
" ax = fig.add_subplot(111)\n",
" fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
" ax.scatter([X[:, 1]], [y], c=\"r\", s=50, label=\"Dane\")\n",
"\n",
" ax.set_xlabel(\"Populacja\")\n",
" ax.set_ylabel(\"Zysk\")\n",
" ax.margins(0.05, 0.05)\n",
" plt.ylim(y.min() - 1, y.max() + 1)\n",
" plt.xlim(np.min(X[:, 1]) - 1, np.max(X[:, 1]) + 1)\n",
" return fig\n",
"\n",
"\n",
"def reglineMx(fig, fun, theta, X):\n",
" ax = fig.axes[0]\n",
" x0, x1 = np.min(X[:, 1]), np.max(X[:, 1])\n",
" L = [x0, x1]\n",
" LX = np.matrix([1, x0, 1, x1]).reshape(2, 2)\n",
" ax.plot(\n",
" L,\n",
" fun(theta, LX),\n",
" linewidth=\"2\",\n",
" label=(\n",
" r\"$y={theta0:.2}{op}{theta1:.2}x$\".format(\n",
" theta0=float(theta[0][0]),\n",
" theta1=(\n",
" float(theta[1][0]) if theta[1][0] >= 0 else float(-theta[1][0])\n",
" ),\n",
" op=\"+\" if theta[1][0] >= 0 else \"-\",\n",
" )\n",
" ),\n",
" )\n"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 1. 3. 1. 78.]\n",
" [ 1. 3. 2. 62.]\n",
" [ 1. 3. 0. 15.]\n",
" [ 1. 4. 0. 14.]\n",
" [ 1. 3. 0. 15.]]\n",
"(1339, 4)\n",
"\n",
"[[476118.]\n",
" [459531.]\n",
" [411557.]\n",
" [496416.]\n",
" [406032.]]\n",
"(1339, 1)\n"
]
}
],
"source": [
"# Wczytwanie danych z pliku za pomocą numpy regresja liniowa wielu zmiennych notacja macierzowa\n",
"\n",
"import pandas\n",
"\n",
"data = pandas.read_csv(\n",
" \"data02_train.tsv\", delimiter=\"\\t\", usecols=[\"price\", \"rooms\", \"floor\", \"sqrMetres\"]\n",
")\n",
"m, n_plus_1 = data.values.shape\n",
"n = n_plus_1 - 1\n",
"Xn = data.values[:, 1:].reshape(m, n)\n",
"\n",
"# Dodaj kolumnę jedynek do macierzy\n",
"XMx = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
"yMx = np.matrix(data.values[:, 0]).reshape(m, 1)\n",
"\n",
"print(XMx[:5])\n",
"print(XMx.shape)\n",
"\n",
"print()\n",
"\n",
"print(yMx[:5])\n",
"print(yMx.shape)\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Funkcja kosztu notacja macierzowa"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$J(\\theta)=\\dfrac{1}{2|\\vec y|}\\left(X\\theta-\\vec{y}\\right)^T\\left(X\\theta-\\vec{y}\\right)$$ \n"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\Large J(\\theta) = 85104141370.9717$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from IPython.display import display, Math, Latex\n",
"\n",
"\n",
"def JMx(theta, X, y):\n",
" \"\"\"Wersja macierzowa funkcji kosztu\"\"\"\n",
" m = len(y)\n",
" J = 1.0 / (2.0 * m) * ((X * theta - y).T * (X * theta - y))\n",
" return J.item()\n",
"\n",
"\n",
"thetaMx = np.matrix([10, 90, -1, 2.5]).reshape(4, 1)\n",
"\n",
"cost = JMx(thetaMx, XMx, yMx)\n",
"display(Math(r\"\\Large J(\\theta) = %.4f\" % cost))\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Gradient notacja macierzowa"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$\\nabla J(\\theta) = \\frac{1}{|\\vec y|} X^T\\left(X\\theta-\\vec y\\right)$$"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\large \\theta = \\left[\\begin{array}{r}10.0000 \\\\ 90.0000 \\\\ -1.0000 \\\\ 2.5000 \\\\ \\end{array}\\right]\\quad\\large \\nabla J(\\theta) = \\left[\\begin{array}{r}-373492.7442 \\\\ -1075656.5086 \\\\ -989554.4921 \\\\ -23806475.6561 \\\\ \\end{array}\\right]$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from IPython.display import display, Math, Latex\n",
"\n",
"\n",
"def dJMx(theta, X, y):\n",
" \"\"\"Wersja macierzowa gradientu funckji kosztu\"\"\"\n",
" return 1.0 / len(y) * (X.T * (X * theta - y))\n",
"\n",
"\n",
"thetaMx = np.matrix([10, 90, -1, 2.5]).reshape(4, 1)\n",
"\n",
"display(\n",
" Math(\n",
" r\"\\large \\theta = \"\n",
" + LatexMatrix(thetaMx)\n",
" + r\"\\quad\"\n",
" + r\"\\large \\nabla J(\\theta) = \"\n",
" + LatexMatrix(dJMx(thetaMx, XMx, yMx))\n",
" )\n",
")\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Algorytm gradientu prostego notacja macierzowa"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"$$ \\theta := \\theta - \\alpha \\, \\nabla J(\\theta) $$"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\large\\textrm{Wynik:}\\quad \\theta = \\left[\\begin{array}{r}17446.2135 \\\\ 86476.7960 \\\\ -1374.8950 \\\\ 2165.0689 \\\\ \\end{array}\\right] \\quad J(\\theta) = 10324864803.1591 \\quad \\textrm{po 374575 iteracjach}$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Implementacja algorytmu gradientu prostego za pomocą numpy i macierzy\n",
"\n",
"\n",
"def GDMx(fJ, fdJ, theta, X, y, alpha, eps):\n",
" current_cost = fJ(theta, X, y)\n",
" history = [[current_cost, theta]]\n",
" while True:\n",
" theta = theta - alpha * fdJ(theta, X, y) # implementacja wzoru\n",
" current_cost, prev_cost = fJ(theta, X, y), current_cost\n",
" if abs(prev_cost - current_cost) <= eps:\n",
" break\n",
" if current_cost > prev_cost:\n",
" print(\"Długość kroku (alpha) jest zbyt duża!\")\n",
" break\n",
" history.append([current_cost, theta])\n",
" return theta, history\n",
"\n",
"\n",
"thetaStartMx = np.zeros((n + 1, 1))\n",
"\n",
"# Zmieniamy wartości alpha (rozmiar kroku) oraz eps (kryterium stopu)\n",
"thetaBestMx, history = GDMx(JMx, dJMx, thetaStartMx, XMx, yMx, alpha=0.0001, eps=0.1)\n",
"\n",
"######################################################################\n",
"display(\n",
" Math(\n",
" r\"\\large\\textrm{Wynik:}\\quad \\theta = \"\n",
" + LatexMatrix(thetaBestMx)\n",
" + (r\" \\quad J(\\theta) = %.4f\" % history[-1][0])\n",
" + r\" \\quad \\textrm{po %d iteracjach}\" % len(history)\n",
" )\n",
")\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## 3.2. Metoda gradientu prostego w praktyce"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"### Kryterium stopu"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"source": [
"Algorytm gradientu prostego polega na wykonywaniu określonych kroków w pętli. Pytanie brzmi: kiedy należy zatrzymać wykonywanie tej pętli?\n",
"\n",
"W każdej kolejnej iteracji wartość funkcji kosztu maleje o coraz mniejszą wartość.\n",
"Parametr `eps` określa, jaka wartość graniczna tej różnicy jest dla nas wystarczająca:\n",
"\n",
" * Im mniejsza wartość `eps`, tym dokładniejszy wynik, ale dłuższy czas działania algorytmu.\n",
" * Im większa wartość `eps`, tym krótszy czas działania algorytmu, ale mniej dokładny wynik."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Na wykresie zobaczymy porównanie regresji dla różnych wartości `eps`"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
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"text/plain": [
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"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Wczytwanie danych z pliku za pomocą numpy wersja macierzowa\n",
"data = np.loadtxt(\"data01_train.csv\", delimiter=\",\")\n",
"m, n_plus_1 = data.shape\n",
"n = n_plus_1 - 1\n",
"Xn = data[:, 0:n].reshape(m, n)\n",
"\n",
"# Dodaj kolumnę jedynek do macierzy\n",
"XMx = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
"yMx = np.matrix(data[:, 1]).reshape(m, 1)\n",
"\n",
"thetaStartMx = np.zeros((2, 1))\n",
"\n",
"fig = regdotsMx(XMx, yMx)\n",
"theta_e1, history1 = GDMx(\n",
" JMx, dJMx, thetaStartMx, XMx, yMx, alpha=0.01, eps=0.01\n",
") # niebieska linia\n",
"reglineMx(fig, hMx, theta_e1, XMx)\n",
"theta_e2, history2 = GDMx(\n",
" JMx, dJMx, thetaStartMx, XMx, yMx, alpha=0.01, eps=0.000001\n",
") # pomarańczowa linia\n",
"reglineMx(fig, hMx, theta_e2, XMx)\n",
"legend(fig)\n"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\theta_{10^{-2}} = \\left[\\begin{array}{r}0.0531 \\\\ 0.8365 \\\\ \\end{array}\\right]\\quad\\theta_{10^{-6}} = \\left[\\begin{array}{r}-3.4895 \\\\ 1.1786 \\\\ \\end{array}\\right]$"
],
"text/plain": [
"<IPython.core.display.Math object>"
]
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"metadata": {},
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"source": [
"display(\n",
" Math(\n",
" r\"\\theta_{10^{-2}} = \"\n",
" + LatexMatrix(theta_e1)\n",
" + r\"\\quad\\theta_{10^{-6}} = \"\n",
" + LatexMatrix(theta_e2)\n",
" )\n",
")\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Długość kroku ($\\alpha$)"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [],
"source": [
"# Jak zmienia się koszt w kolejnych krokach w zależności od alfa\n",
"\n",
"\n",
"def costchangeplot(history):\n",
" fig = plt.figure(figsize=(16 * 0.6, 9 * 0.6))\n",
" ax = fig.add_subplot(111)\n",
" fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
" ax.set_xlabel(\"krok\")\n",
" ax.set_ylabel(r\"$J(\\theta)$\")\n",
"\n",
" X = np.arange(0, 500, 1)\n",
" Y = [history[step][0] for step in X]\n",
" ax.plot(X, Y, linewidth=\"2\", label=(r\"$J(\\theta)$\"))\n",
" return fig\n",
"\n",
"\n",
"def slide7(alpha):\n",
" best_theta, history = gradient_descent(\n",
" h, J, [0.0, 0.0], x, y, alpha=alpha, eps=0.0001\n",
" )\n",
" fig = costchangeplot(history)\n",
" legend(fig)\n",
"\n",
"\n",
"sliderAlpha1 = widgets.FloatSlider(\n",
" min=0.01, max=0.03, step=0.001, value=0.02, description=r\"$\\alpha$\", width=300\n",
")\n"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "52b0d91e39104f4facbb7f57819aae0c",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"interactive(children=(FloatSlider(value=0.02, description='$\\\\alpha$', max=0.03, min=0.01, step=0.001), Button…"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"<function __main__.slide7(alpha)>"
]
},
"execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"widgets.interact_manual(slide7, alpha=sliderAlpha1)\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## 3.3. Normalizacja danych"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"source": [
"Normalizacja danych to proces, który polega na dostosowaniu danych wejściowych w taki sposób, żeby ułatwić działanie algorytmowi gradientu prostego.\n",
"\n",
"Wyjaśnię to na przykladzie."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"Użyjemy danych z „Gratka flats challenge 2017”.\n",
"\n",
"Rozważmy model $h(x) = \\theta_0 + \\theta_1 x_1 + \\theta_2 x_2$, w którym cena mieszkania prognozowana jest na podstawie liczby pokoi $x_1$ i metrażu $x_2$:"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
"text/html": [
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" 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>price</th>\n",
" <th>rooms</th>\n",
" <th>sqrMetres</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
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" <th>0</th>\n",
" <td>476118.00</td>\n",
" <td>3</td>\n",
" <td>78</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>459531.00</td>\n",
" <td>3</td>\n",
" <td>62</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>411557.00</td>\n",
" <td>3</td>\n",
" <td>15</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>496416.00</td>\n",
" <td>4</td>\n",
" <td>14</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>406032.00</td>\n",
" <td>3</td>\n",
" <td>15</td>\n",
" </tr>\n",
" <tr>\n",
" <th>5</th>\n",
" <td>450026.00</td>\n",
" <td>3</td>\n",
" <td>80</td>\n",
" </tr>\n",
" <tr>\n",
" <th>6</th>\n",
" <td>571229.15</td>\n",
" <td>2</td>\n",
" <td>39</td>\n",
" </tr>\n",
" <tr>\n",
" <th>7</th>\n",
" <td>325000.00</td>\n",
" <td>3</td>\n",
" <td>54</td>\n",
" </tr>\n",
" <tr>\n",
" <th>8</th>\n",
" <td>268229.00</td>\n",
" <td>2</td>\n",
" <td>90</td>\n",
" </tr>\n",
" <tr>\n",
" <th>9</th>\n",
" <td>604836.00</td>\n",
" <td>4</td>\n",
" <td>40</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" price rooms sqrMetres\n",
"0 476118.00 3 78\n",
"1 459531.00 3 62\n",
"2 411557.00 3 15\n",
"3 496416.00 4 14\n",
"4 406032.00 3 15\n",
"5 450026.00 3 80\n",
"6 571229.15 2 39\n",
"7 325000.00 3 54\n",
"8 268229.00 2 90\n",
"9 604836.00 4 40"
]
},
"execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Wczytanie danych przy pomocy biblioteki pandas\n",
"import pandas\n",
"\n",
"alldata = pandas.read_csv(\n",
" \"data_flats.tsv\", header=0, sep=\"\\t\", usecols=[\"price\", \"rooms\", \"sqrMetres\"]\n",
")\n",
"alldata[:10]\n"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [],
"source": [
"# Funkcja, która pokazuje wartości minimalne i maksymalne w macierzy X\n",
"\n",
"\n",
"def show_mins_and_maxs(XMx):\n",
" mins = np.amin(XMx, axis=0).tolist()[0] # wartości minimalne\n",
" maxs = np.amax(XMx, axis=0).tolist()[0] # wartości maksymalne\n",
" for i, (xmin, xmax) in enumerate(zip(mins, maxs)):\n",
" display(Math(r\"${:.2F} \\leq x_{} \\leq {:.2F}$\".format(xmin, i, xmax)))\n"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [],
"source": [
"# Przygotowanie danych\n",
"\n",
"import numpy as np\n",
"\n",
"%matplotlib inline\n",
"\n",
"data2 = np.matrix(alldata[['rooms', 'sqrMetres', 'price']])\n",
"\n",
"m, n_plus_1 = data2.shape\n",
"n = n_plus_1 - 1\n",
"Xn = data2[:, 0:n]\n",
"\n",
"XMx2 = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n_plus_1)\n",
"yMx2 = np.matrix(data2[:, -1]).reshape(m, 1) / 1000.0"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Cechy w danych treningowych przyjmują wartości z zakresu:"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.00 \\leq x_0 \\leq 1.00$"
],
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"<IPython.core.display.Math object>"
]
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{
"data": {
"text/latex": [
"$\\displaystyle 2.00 \\leq x_1 \\leq 7.00$"
],
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"<IPython.core.display.Math object>"
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{
"data": {
"text/latex": [
"$\\displaystyle 12.00 \\leq x_2 \\leq 196.00$"
],
"text/plain": [
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]
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"metadata": {},
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],
"source": [
"show_mins_and_maxs(XMx2)\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Jak widzimy, $x_2$ przyjmuje wartości dużo większe niż $x_1$.\n",
"Powoduje to, że wykres funkcji kosztu jest bardzo „spłaszczony” wzdłuż jednej z osi:"
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"outputs": [],
"source": [
"def contour_plot(X, y, rescale=10**8):\n",
" theta0_vals = np.linspace(-100000, 100000, 100)\n",
" theta1_vals = np.linspace(-100000, 100000, 100)\n",
"\n",
" J_vals = np.zeros(shape=(theta0_vals.size, theta1_vals.size))\n",
" for t1, element in enumerate(theta0_vals):\n",
" for t2, element2 in enumerate(theta1_vals):\n",
" thetaT = np.matrix([1.0, element, element2]).reshape(3, 1)\n",
" J_vals[t1, t2] = JMx(thetaT, X, y) / rescale\n",
"\n",
" plt.figure()\n",
" plt.contour(theta0_vals, theta1_vals, J_vals.T, np.logspace(-2, 3, 20))\n",
" plt.xlabel(r\"$\\theta_1$\")\n",
" plt.ylabel(r\"$\\theta_2$\")\n"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
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],
"source": [
"contour_plot(XMx2, yMx2, rescale=10**10)\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "notes"
}
},
"source": [
"Jeżeli funkcja kosztu ma kształt taki, jak na powyższym wykresie, to łatwo sobie wyobrazić, że znalezienie minimum lokalnego przy użyciu metody gradientu prostego musi stanowć nie lada wyzwanie: algorytm szybko znajdzie „rynnę”, ale „zjazd” wzdłuż „rynny” w poszukiwaniu minimum będzie odbywał się bardzo powoli.\n",
"\n",
"Jak temu zaradzić?\n",
"\n",
"Spróbujemy przekształcić dane tak, żeby funkcja kosztu miała „ładny”, regularny kształt."
]
},
{
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"metadata": {
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"slide_type": "slide"
}
},
"source": [
"### Skalowanie"
]
},
{
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"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"Będziemy dążyć do tego, żeby każda z cech przyjmowała wartości w podobnym zakresie.\n",
"\n",
"W tym celu przeskalujemy wartości każdej z cech, dzieląc je przez wartość maksymalną:\n",
"\n",
"$$ \\hat{x_i}^{(j)} := \\frac{x_i^{(j)}}{\\max_j x_i^{(j)}} $$"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle 1.00 \\leq x_0 \\leq 1.00$"
],
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],
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],
"source": [
"XMx2_scaled = XMx2 / np.amax(XMx2, axis=0)\n",
"\n",
"show_mins_and_maxs(XMx2_scaled)\n"
]
},
{
"cell_type": "code",
"execution_count": 88,
"metadata": {
"slideshow": {
"slide_type": "subslide"
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"source": [
"### Normalizacja średniej"
]
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"metadata": {
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"source": [
"Będziemy dążyć do tego, żeby dodatkowo średnia wartość każdej z cech była w okolicach $0$.\n",
"\n",
"W tym celu oprócz przeskalowania odejmiemy wartość średniej od wartości każdej z cech:\n",
"\n",
"$$ \\hat{x_i}^{(j)} := \\frac{x_i^{(j)} - \\mu_i}{\\max_j x_i^{(j)}} $$"
]
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{
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"execution_count": 89,
"metadata": {
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"$\\displaystyle 0.00 \\leq x_0 \\leq 0.00$"
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"$\\displaystyle -0.23 \\leq x_2 \\leq 0.70$"
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"source": [
"XMx2_norm = (XMx2 - np.mean(XMx2, axis=0)) / np.amax(XMx2, axis=0)\n",
"\n",
"show_mins_and_maxs(XMx2_norm)\n"
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{
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"execution_count": 90,
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