{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "Sveučilište u Zagrebu
\n", "Fakultet elektrotehnike i računarstva\n", "\n", "# Strojno učenje\n", "\n", "http://www.fer.unizg.hr/predmet/su\n", "\n", "Ak. god. 2015./2016.\n", "\n", "# Bilježnica 3: Osnove vjerojatnosti i statistike\n", "\n", "(c) 2015 Jan Šnajder\n", "\n", "Verzija: 0.8 (2015-11-01)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "source": [ "import scipy as sp\n", "import scipy.stats as stats\n", "import matplotlib.pyplot as plt\n", "from numpy.random import normal\n", "%pylab inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Sadržaj:\n", "\n", "* Vjerojatnost\n", "\n", "* Očekivanje, varijanca i kovarijanca\n", "\n", "* Statistička nezavisnost\n", "\n", "* Matrica kovarijacije\n", "\n", "* Teorijske razdiobe\n", "\n", "* Procjena parametara\n", "\n", "* Procjenitelj MLE\n", "\n", "* Procjenitelj MAP\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Vjerojatnost\n", "\n", "* $X$ je slučajna varijabla, $\\{x_i\\}$ su njezine vrijednosti\n", "\n", "\n", "* Pojednostavljenje notacije: $$P(X=x) \\equiv P(x)$$\n", "\n", "\n", "* $P(x_i)\\geq 0$, $\\sum_i P(x_i)=1$\n", "\n", "\n", "* Distribucija (razdioba) vjerojatnosti\n", "\n", "\n", "* Zajednička (engl. *joint*) distribucija nad $\\{X,Y\\}$: $$P(X=x,Y=y)\\equiv P(x,y)$$\n", " \n", " \n", "* Kontinuirana slučajna varijabla: **funkcija gustoće vjerojatnosti (PDF)**:\n", "\n", "\\begin{eqnarray*}\n", "p(x) & \\geq 0\\\\\n", "\\int_{-\\infty}^{\\infty} p(x)\\,\\textrm{d}x &= 1\\\\\n", "P(a\\leq X\\leq b) &= \\int_a^b p(x)\\,\\mathrm{d}x\n", "\\end{eqnarray*}\n", "\n", "### Dva pravila teorije vjerojatnosti\n", "\n", "* **(1) Pravilo zbroja**\n", "$$P(x)=\\sum_y P(x,y)$$\n", "(Marginalna vjerojatnost varijable $X$)\n", "\n", "\n", "* Uvjetna vjerojatnost:\n", "$$\n", " P(y|x) = \\frac{P(x,y)}{P(x)} \n", "$$\n", "\n", "\n", "* **(2) Pravilo umnoška**\n", "$$P(x,y) = P(y|x) P(x) = P(x|y) P(y)$$\n", "\n", "### Izvedena pravila\n", "\n", "* **Bayesovo pravilo**\n", "$$\n", "P(y|x) = \\frac{P(x|y)P(y)}{P(x)}\n", "= \\frac{P(x|y)P(y)}{\\sum_y P(x,y)}\n", "= \\frac{P(x|y)P(y)}{\\sum_y P(x|y)P(y)}\n", "$$\n", "\n", "\n", "* **Pravilo lanca (engl. *chain rule*)**\n", "$$P(x,y,z) = P(x) P(y|x) P(z|x,y)$$\n", "\n", "* Općenito:\n", "$$\n", "\\begin{align*}\n", "P(x_1,\\dots,x_n) &=\n", "P(x_1)P(x_2|x_1)P(x_3|x_1,x_2)\\cdots P(x_n|x_1,\\dots,x_{n-1})\\\\\n", "&= \\prod_{k=1}^n P(x_k|x_1,\\dots,x_{k-1})\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Očekivanje, varijanca i kovarijanca \n", "\n", "* Očekivanje slučajne varijable:\n", "\n", "\\begin{equation*}\n", "\\mathbb{E}[X]=\\sum_x x P(x)\n", "\\end{equation*}\n", "\n", "$$\n", "\\mathbb{E}[X]=\\int_{-\\infty}^{\\infty} x\\,p(x)\\,\\mathrm{d}x\n", "$$\n", "\n", "* Očekivanje funkcije slučajne varijable:\n", "\n", "\\begin{equation*}\n", "\\mathbb{E}[f]=\\sum_x f(x) P(x)\n", "\\end{equation*}\n", "\n", "* Vrijedi:\n", "\n", "\\begin{align*}\n", "\\mathbb{E}[aX+b] &= a\\mathbb{E}[X]+b\\qquad (a,b\\in\\mathbb{R})\\\\\n", "\\mathbb{E}[X+Y] &= \\mathbb{E}[X] + \\mathbb{E}[Y]\n", "\\end{align*}\n", "\n", "* Varijanca slučajne varijable:\n", "\n", "\\begin{equation*}\n", "\\mathrm{Var}(X) = \\sigma_X^2 = \\mathbb{E}[(X-\\mathbb{E}[X])^2] = \\mathbb{E}[X^2] - \\mathbb{E}[X]^2\n", "\\end{equation*}\n", "\n", "\\begin{equation*}\n", "\\mathrm{Var}(a X) = \\mathbb{E}\\big[(a X)^2\\big] - \\mathbb{E}[a X]^2 = a^2\\mathbb{E}[X^2] - a^2\\mathbb{E}[X]^2 =\n", "a^2\\mathrm{Var}(X)\n", "\\end{equation*}\n", "\n", "* Kovarijanca slučajnih varijabli:\n", "\n", "\\begin{align*}\n", " \\mathrm{Cov}(X,Y) &= \\sigma_{X,Y} = \\mathbb{E}\\big[(X-\\mathbb{E}[X])(Y-\\mathbb{E}[Y])\\big] =\n", " \\mathbb{E}[XY] - \\mathbb{E}[X]\\mathbb{E}[Y]\\\\\n", " \\mathrm{Cov}(X,Y) &=\\mathrm{Cov}(Y, X)\\\\\n", " \\mathrm{Cov}(X,X) &=\\mathrm{Var}(X) =\\sigma^2_X\\\\\n", "\\end{align*}\n", "\n", "* Pearsonov koeficijent korelacije (linearna zavisnost):\n", "$$\n", "\\rho_{X,Y} = \\frac{\\mathrm{Cov}(X,Y)}{\\sigma_X\\sigma_Y}\n", "$$\n", "$\\rho_{X,Y}\\in[-1,+1]$\n", "\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from scipy import stats\n", "\n", "X = sp.random.random(100)\n", "Y0 = sp.random.random(100)\n", "noise = stats.norm.rvs(size=100)\n", "Y1 = X + 0.2 * noise\n", "Y2 = 3 * Y1\n", "Y3 = -Y1\n", "Y4 = 1 - (X - 0.5)**2 + 0.05 * noise" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/usr/local/lib/python2.7/dist-packages/matplotlib/collections.py:590: FutureWarning: elementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparison\n", " if self._edgecolors == str('face'):\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for Y in [Y0, Y1, Y2, Y3, Y4]:\n", " plt.scatter(X,Y, label=\"r = %.3f\" % stats.pearsonr(X, Y)[0])\n", " plt.legend()\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Linearno zavisne varijable imaju $\\rho$ blizu $1$ ili $-1$. Međutim, **nelinearno** zavisne varijable mogu imati $\\rho$ blizu nule!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Statistička nezavisnost\n", "\n", "* Varijable $X$ i $Y$ su **nezavisne** akko:\n", "$$\n", "P(X,Y) = P(X) P(Y)\n", "$$\n", "ili\n", "$$\n", "P(X|Y) = P(X) \\qquad \\text{i} \\qquad P(Y|X) = P(Y)\n", "$$\n", "\n", "\n", "* Znanje o ishodu varijable $Y$ ne utječe na vjerojatnost ishoda varijable $X$ (i obrnuto)\n", "\n", "\n", "* Za nezavisne varijable $X$ i $Y$ vrijedi:\n", "$$\n", "\\begin{align*}\n", "\\mathbb{E}[XY] &= \\mathbb{E}[X]\\, \\mathbb{E}[Y]\\\\\n", "\\mathrm{Var}(X+Y) &= \\mathrm{Var}(X) + \\mathrm{Var}(Y)\\\\\n", "\\mathrm{Cov}(X, Y) &= \\rho_{X,Y} = 0\n", "\\end{align*}\n", "$$\n", "\n", "* Nezavisne varijable su nekorelirane, ali obrat općenito ne vrijedi: nelinarno zavisne varijable mogu imati nisku korelaciju\n", "\n", "\n", "* Varijable $X$ i $Y$ su **uvjetno nezavisne** uz danu varijablu Z, što označavamo kao $X\\bot Y|Z$, akko\n", "$$\n", "P(X|Y,Z) = P(X|Z)\n", "$$\n", "ili\n", "$$\n", "P(X,Y|Z) = P(X|Z) P(Y|Z)\n", "$$\n", "\n", "* Jednom kada nam je\n", "poznat ishod varijable $Z$, znanje o ishodu varijable $Y$ ne utječe na ishod varijable $X$ (i obrnuto)\n", "\n", "* Npr.:\n", " * $X = \\textrm{'Student je primljen na FER'}$\n", " * $Y = \\textrm{'Student je primljen na PMF-MO'}$\n", " * $P(Y|X) \\neq P(Y)$ (varijable nisu nezavisne)\n", " * $Z = \\textrm{'Student je sudjelovao na matematičkim natjecanjima'}$\n", " * $X\\bot Y|Z$\n", " * $P(Y|X,Z) = P(Y|Z)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Matrica kovarijacije\n", "\n", "* $\\mathbf{X} = (X_1,\\dots,X_n)$ je $n$-dimenzijski slučajan vektor\n", "\n", "\n", "* Matrica kovarijacije $\\Sigma$:\n", "$$\n", "\\Sigma_{ij} = \n", "\\mathrm{Cov}(X_i, X_j) =\n", "\\mathbb{E}\\big[(X_i-\\mathbb{E}[X_i])(X_j-\\mathbb{E}[X_j])\\big]\n", "$$\n", "\n", "\n", "* Matrično:\n", "$$\n", "\\begin{align*}\n", "\\Sigma &= \\begin{pmatrix}\n", "\\mathrm{Var}(X_1) & \\mathrm{Cov}(X_1,X_2) & \\dots & \\mathrm{Cov}(X_1, X_n)\\\\\n", "\\mathrm{Cov}(X_2, X_1) & \\mathrm{Var}(X_2) & \\dots & \\mathrm{Cov}(X_2, X_n)\\\\\n", "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n", "\\mathrm{Cov}(X_n, X_1) & \\mathrm{Cov}(X_n, X_2) & \\dots & \\mathrm{Var}(X_n)\\\\\n", "\\end{pmatrix}\n", "\\end{align*}\n", "$$\n", "Simetrična matrica!\n", "\n", "\n", "* Ekvivalentno:\n", "\n", "\\begin{equation*}\n", "\\Sigma = \\mathbb{E}\\Big[(\\textbf{X}-\\mathbb{E}[\\textbf{X}])(\\textbf{X}-\\mathbb{E}[\\textbf{X}])^{\\mathrm{T}}\\Big]\n", "\\end{equation*}\n", "\n", "\n", "* Ako su $X_1...X_n$ međusobno nezavisne, onda $\\Sigma = \\mathrm{diag}(\\sigma_i^2)$\n", "\n", "\n", "* Ako $\\sigma^2_i = \\sigma^2$, onda $\\Sigma = \\sigma^2 \\mathbf{I}$ (izotropna kovarijanca)\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Teorijske razdiobe\n", "\n", "* Diskretna značajka:\n", " * Jednodimenzijska:\n", " * Binarna: **Bernoullijeva razdioba**\n", " * Viševrijednosna: **Kategorička (multinomijalna) razdioba**\n", " * Višedimenzijska:\n", " * Konkatenirani vektor binarnih/viševrijednosnih varijabli\n", "* Kontinuirana značajka:\n", " * Jednodimenzijska: **univarijatna normalna (Gaussova) razdioba**\n", " * Višedimenzijska: **multivarijatna normalna (Gaussova) razdioba**\n", "\n", "### Bernoullijeva razdioba\n", "\n", "\\begin{equation*}\n", "P(X=x | \\mu)=\n", "\\begin{cases}\n", "\\mu & \\text{ako $X=1$}\\\\\n", "1-\\mu & \\text{inače}\n", "\\end{cases}\n", "\\qquad=\n", "\\mu^{x}(1-\\mu)^{1-x}\n", "\\end{equation*}\n", "\n", "\\begin{eqnarray*}\n", "\\mathbb{E}[X] &=& \\mu\\\\\n", "\\mathrm{Var}(X) &=& \\mu(1-\\mu)\n", "\\end{eqnarray*}\n", "\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/usr/local/lib/python2.7/dist-packages/matplotlib/collections.py:650: FutureWarning: elementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparison\n", " if self._edgecolors_original != str('face'):\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "mu = 0.3\n", "p = stats.bernoulli(mu)\n", "xs = sp.array([0,1])\n", "\n", "for x in xs:\n", " plt.plot(x, p.pmf(x), 'bo', ms=8, label='bernoulli pmf')\n", " plt.vlines(x, 0, p.pmf(x), colors='b', lw=5, alpha=0.5)\n", "plt.xlim(xmin=-1, xmax=2)\n", "plt.ylim(ymax=1)\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0,\n", " 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0,\n", " 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1,\n", " 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,\n", " 1, 0, 1, 0, 0, 1, 1, 0])" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X = p.rvs(size=100); X" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.29999999999999999" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.mean(X)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.20999999999999994" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.var(X)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xs = linspace(0,1)\n", "plt.plot(xs, xs * (1-xs));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Kategorička (\"multinomijalna\") razdioba\n", "\n", "* Varijabla koja poprima jednu (i samo jednu) od $K$ mogućih vrijednosti\n", "\n", "\n", "* $\\mathbf{x}=(x_1,x_2,\\dots,x_K)^\\mathrm{T}$ je binaran vektor indikatorskih varijabli\n", " * vektor 1-od-K\n", " * *one-hot encoding*\n", "\n", "* Vjerojatnosti pojedinih vrijednosti: $\\boldsymbol{\\mu}=(\\mu_1,\\dots,\\mu_K)^\\mathrm{T}$, $\\sum_k \\mu_k=1$, $\\mu_k\\geq 0$\n", "\n", "\n", "\\begin{equation*}\n", "P(\\mathbf{X}=\\mathbf{x} | \\boldsymbol{\\mu}) = \\prod_{k=1}^K \\mu_k^{x_k}\n", "\\end{equation*}\n", "\n", "\n", "* Npr. \n", " * $X=x_3\\quad \\Rightarrow\\quad \\mathbf{x} = (0,0,1,0)$\n", " * $\\boldsymbol{\\mu} = (0.2, 0.3, 0.4, 0.1)$\n", " * $P\\big(X = (0,0,1,0)\\big) = \\prod_{k=1}^4 \\mu_k^{x_k} = 1\\cdot 1\\cdot \\mu_3\\cdot 1 = \\mu_3 = 0.4$\n", " \n", "### Gaussova razdioba\n", "\n", "\\begin{equation*}\n", "p(X=x|\\mu,\\sigma^2) =\n", "\\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\Big\\{-\\frac{(x-\\mu)^2}{2\\sigma^2}\\Big\\}\n", "\\end{equation*}\n", "\n", "\\begin{align*}\n", "\\mathbb{E}[X] =& \\mu\\\\\n", "\\mathrm{Var}(X) =& \\sigma^2\n", "\\end{align*}\n", "\n" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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FwAYkoHvcVAFm209TAviviLyfw/lNCd7HbT+9na4/dWXOwDncUuuWy89ffz3M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\np9zcUEtRq2j0OegO7EP3phllf24YMCz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PZ8Zw6VccdJKkpv7KE40kaWIGvCQ1ZcBLUlMG\nvCQ1ZcBLUlMGvCQ1ZcBLUlMGvCQ19QEPiYsZ6yS0+wAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xs = sp.linspace(-5, 5)\n", "for s in range(1, 5):\n", " plt.plot(xs, stats.norm.pdf(xs, 0, s), label='$\\sigma=%d$' % s)\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* $P( \\mu -\\sigma\\leq X \\leq \\mu + \\sigma) = 0.68$\n", "* $P( \\mu -2\\sigma\\leq X \\leq \\mu + 2\\sigma) = 0.95$\n", "* $P( \\mu -3\\sigma\\leq X \\leq \\mu + 3\\sigma) = 0.99.7$" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.682689492137\n", "0.954499736104\n", "0.997300203937\n" ] } ], "source": [ "print stats.norm.cdf(1, 0, 1) - stats.norm.cdf(-1, 0, 1)\n", "print stats.norm.cdf(2, 0, 1) - stats.norm.cdf(-2, 0, 1)\n", "print stats.norm.cdf(3, 0, 1) - stats.norm.cdf(-3, 0, 1)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [], "source": [ "p = stats.norm(loc=5, scale=3)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 3.94363136, 9.6884801 , 6.73686697, 2.23918325,\n", " -0.22621627, 8.0346425 , 7.50790913, 8.16860545,\n", " 6.52408393, 5.5558834 , 3.70190598, 1.00539899,\n", " 5.33881857, 4.42657421, 8.26823883, 4.14257769,\n", " 10.56845245, 6.31164921, 8.72326689, 6.81670834,\n", " 9.6269925 , 9.05832561, 5.89763569, 8.214693 ,\n", " 5.37577967, 2.88912765, 2.47697932, 2.19650056,\n", " 1.90620298, 1.09675323])" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X = p.rvs(size=30); X" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "5.5405217061730685" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.mean(X)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "8.4165690464455274" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.var(X)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Multivarijatna Gaussova razdioba\n", "\n", "\\begin{equation*}\n", "p(\\mathbf{X}=\\mathbf{x}|\\boldsymbol{\\mu},\\mathbf{\\Sigma}) = \n", "\\frac{1}{(2\\pi)^{n/2}|\\mathbf{\\Sigma}|^{1/2}}\n", "\\exp\\Big\\{-\\frac{1}{2}(\\mathbf{x}-\\boldsymbol{\\mu})^\\mathrm{T}\\mathbf{\\Sigma}^{-1}(\\mathbf{x}-\\boldsymbol{\\mu})\\Big\\}\n", "\\end{equation*}\n", "\n", "\n", "* $\\mathbf{\\Sigma}$ mora biti **pozitivno definitna**. Tada (1) matrica je nesingularna i ima inverz te (2) determinanta joj je pozitivna\n", "\n", "\n", "* Kvadratna forma: $\\Delta^2 = (\\mathbf{x}-\\boldsymbol{\\mu})^\\mathrm{T}\\mathbf{\\Sigma}^{-1}(\\mathbf{x}-\\boldsymbol{\\mu})$ je **Mahalanobisova udaljenost** između $\\mathbf{x}$ i $\\boldsymbol{\\mu}$.\n", "\n", "\\begin{align*}\n", "\\mathbb{E}[\\mathbf{X}] =& \\boldsymbol{\\mu}\\\\\n", "\\mathrm{Cov}(X_i, X_j) =& \\mathbf{\\Sigma}_{ij}\n", "\\end{align*}\n", "\n" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [], "source": [ "mu = [0, 1]\n", "covm = sp.array([[1, 1], [1, 3]])\n", "p = stats.multivariate_normal(mu, covm)" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[1 1]\n", " [1 3]]\n" ] } ], "source": [ "print covm" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [], "source": [ "x = np.linspace(-2, 2)\n", "y = np.linspace(-2, 2)\n", "X, Y = np.meshgrid(x, y)\n", "XY = np.dstack((X,Y))" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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mjjpbZsphUX+P618SmW9A0fsOwWEieWvp/1pJSopdnnxylBw4cN3aiURNndO1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\nb2FhYVHI+D/72EXi0mZ7FQAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.contour(X, Y, p.pdf(XY));" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[1 0]\n", " [0 5]]\n" ] } ], "source": [ "covm1 = sp.array([[1, 0], [0, 5]])\n", "print covm1" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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/CQQi+HxDpKrH6Yg/JdkMXLeBtPArFEkk4UfohF05oC2kZHIKofGgvN1BMH3p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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.contour(X, Y, stats.multivariate_normal.pdf(XY, mean=mu, cov=covm1 ));" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[5 0]\n", " [0 5]]\n" ] } ], "source": [ "covm2 = sp.array([[5, 0], [0, 5]])\n", "print covm2" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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INU0/MRLWPgiDF5jC9PPIYzELaUHLEv29U/X4f5AzuI+JCCeuzX6Z84dgYZia\n9Q9ywUteG+f272f500+TlZjIoKlTXfPyXnMNyUePsvLZZ0k+coRBn33m2qm9M87A/O7Q81NoOtZo\nNUViHWuIJ57x3IOH8DB3jz+TDPayx2gZRaNaSxjwq5oIunDYaDWlRq3WrblvzRr6TJnCogceYO64\ncaScPGm0LE0pYUlPZ/XLL/Nd9+406NOHJ/bvd23Tz0mHpcOgzZOmMf1EzrGNrQzl9hJ3kp3K+Ecw\nihUsI400o6UUjaBbofsHsGQwpJ8yWk2pIYSg1Zgx/PPQIWq1acM3HTqw6oUXyEpONlqaxkHkWSxs\n+/xzPg8OJjMhgcf37aPniy/i6eNjtLTSI+8iLB8NtbtAyEtGqykS+eSzgN+5lX5UpnKJj+NUxl+X\nILrQlQXMx4pJomZaToR2z8DCvpB+2mg1pYp3+fL0eeMNHt+/n9ysLD4PDmblc8+RFhtrtDRNCcnJ\nyGDLJ58wtUkTjq9Ywb0rVjBy5kxzl0csCnnZKh1D+ZrQ50swybzFesIpRzk60cWu4ziV8QOEEoYF\nC9swUdWodv+CNv+ERbdChuuboF9gIEO//JLH9+4F4Ks2bVj8j3+QfPSowco0RSUrOZnwKVP4b6NG\nnNm6lfFLlnD3H38Q0L690dJKn3yL6un7VIHbfgQPL6MVFYlTnCKC7YxiNB52Wrfdxi+EGCSEOCyE\nOCqEePk6+0y1Pb9HCHHDdd2eeDKWcaxnHWdJsFde2dH+OWj1iOr5Z8YZraZM8A8KYuAnn/DUkSP4\n1a3LjB49mHfnncRHRhotTXMd0mJjWfncc3weHEzamTM8uHkz4+bMca10Czci3wLLx4B3Rej/s2lM\n34KF+cxlGMPxw/68WnZF9QghPIEo4DYgFtgBjJdSHiqwzxDgSSnlECFEV+C/UspuhRzrb3H8u4hg\nC3/xCI/jjXeJNZY5O9+Hw9/DiLVQqa7RasoUS3o6O7/5hi0ff0zNli3p8MgjtBgxAq9y5YyW5tZI\nq5WY9euJ/PZbji5fTrv776fH88/jHxRktLSyJS8bVt4BHt62HDzm8ZUFzEcgGMnoa54r8+ycQoju\nwGQp5SDb41cApJTvF9jna2CdlPI32+PDQB8p5dmrjvU345dI5vIbPvgU+p91anZ9BPu/hOEroUoz\no9WUOXkWC4d+/53dM2YQHxlJ6/HjCXnwQffpVToJqadOsXvmTHZ//z0+lSoR8uCDtLvvPspXq2a0\ntLInJw2GRDeFAAAgAElEQVSWjVRj+rf9BJ7mmbTeRQSb2cQjPI4v1yaMMyJlQ12g4IzmGdvvbrbP\nTbsaAsEIRnGaU0Sww06ZZUyHF6HT67CgDyTuMlpNmePl60ub8eOZsHo1j0REUKFGDX4bNYqv27dn\n29SpZCSYaAjPZORkZLB/9mx+HjiQaSEhZCQkMG7uXB7bs4duzzzjnqZ/MVGttq/SDPr/YirTj+UM\nq1jJndxdqOmXFHsHuIp6uXB1a1To302ZMuXy/bCwMMLCwhjPPXzHdGoTQD3qlUylEbR6CHyrwpJB\ncNvPUH+A0YoMoUrDhoRNnkyfSZOICQ9n9/ffEz55MrXatKHlmDG0HD2ayvVM9L46IdmpqRxZsoRD\n8+dzYu1agrp3p93Eidy5cCHe5csbLc9YUo7B0iHQ9A6VYsUk0TsAmWQym1+5nRHU4kralPDwcMLD\nw+06tr1DPd2AKQWGev4NWKWUHxTY52sgXEo52/a4SEM9BTnMIZayhMd4gkqYrApW3EZYMRa6vQet\nHjRajVOQZ7EQvXo1h+bPJ2rxYqoFB19uBKo1aWK0PFOQlZRE1OLFHJo/n5MbN9IwLIyWY8bQfPhw\nyletarQ85yBhi4re6TwFWj9qtJpikU8+P/IDQQTRn4E33NeIMX4v1ORuPyAO2M6NJ3e7AZ8VZXL3\natbyJ8c5zkQeNNdkL8CFKNXraDJGNQAeLlCA2kHk5+YSEx7OwXnziFq0CJ+KFWnYty8Nw8Jo2Lcv\n/nXda4L8emSnpnJywwZi1q0jJjyc88eO0WTAAFqNHUvw0KH4+rlWmnC7ifpZ1c/o9wM0HGq0mmIh\nkSxjKUkkMYH7bxq6aUjpRSHEYOAzwBP4Tkr5nhDiUQAp5TTbPl8Ag4BM4AEp5TUD3zczfitW5jEH\ngLHcYXcca5mTnQwr7wThqVI9lHPDsdabIKUk8cABTqxbR8y6dZxcv57y1aurhqBPH+p07ky1Jk0Q\nHiZ770tAenw8cRERnNq4kZh160g6fJi6XbtebhTrdu7s2qtqS4o1D/56GU4sgiELoXproxUVm7/Y\nzC4ieIhHKM/Nh+pcvuZuLrnM5HvqU58BDCojZQ7EmgdbXoHohTBkAVRvY7Qip0ZarZzdt4+Ydes4\ntXEjcRERZKemEtihA3U6dSKwY0fqdOpE1caNTZ0xMiMhgbidO4mLiCDedptvsRDYsSP1evakUd++\n1O3aFS9f1ysB6FBcoHN1kAMsYyn/4BGqULQhO5c3foAsspjONHrQg850LQNlpUDULNj0jFoq3nSc\n0WpMRWZiojLHnTuJj4ggbudOslNSqN6smdqaN6d6s2bUaN6casHBTjMEkmexcCE6muSoKJKPHCEp\nKorztltrXt7fGrI6HTtSuUEDUzdmZU7iblgxGpqMg27vmnI49TSnmcWP3MdE6lwTHHl93ML4Ac6T\nzLd8wwhG0ZwWpayslEjcpSZ9GwyDHh+Cl17kVFIuXrhA8pEjl0318u3Ro/hUqoRfYCB+depQqcBt\npYAAyletik+lStdsNxtCseblkZOZSU5GhtrS08nJyCA7NZXMs2dJj48nIz6e9Li4y7dZyclUrl9f\nNUi2hulSQ1UpIECbfEmREg58A9teh9DPIfguoxWViPMk8x3TGc7IYnua2xg/XGkd72Q8jWhcispK\nEUsKrHtYpXXuPwtqtDVakUshrVYyExP/ZsDpcXFkJCSQkZBAdkrKFfMuYOIAHl6FRzpLqxVrXt61\nDYafH75+flQMCKBS7dr41anz98YmIABPb5MFJTg7WefU9yfjlBraqWrOTmAqKczgW3oRSucSJF9z\nK+MHiCaaucxmPPdQnwalpKyUkRKifoLNz0OHV6D9syBcf/LSmcnPycGan1/oc0IIPH19dQ/daGL+\nUKbf4j7o8papFmUVJJ00vuNbutCVHvQs0THczvgBjnKU35nLvdxH3ZsvCHZe0k7AnxPAwwf6zQQ/\nvahJo7mG3EzY/AKcWq4ya9YJNVpRickkkxl8S1va0YewEh/HiJQNhhNMMCMYxSx+IoF4o+WUHP9G\nMHI9BN0GczuqOGQnaZQ1GqcgYRvM6QB5mXDnHlObfhZZzGQGrbjFLtMvKabv8V9iP/tYxlLuYyIB\nmLyIxLmdsGYiVAqC0C+gsl7NqnFjLKmwbRIcm6MmcE0eCZdFFj/yPY1ozAAG2V1j3C17/JdoTRuG\ncjsz+Z4zmLwSVq2OcMdOqBsG87pCxDsqj7hG405ICUdnw6+tIC8Lxu83vemnk84MvqUxTR1i+iXF\nZXr8l4jiMAv5nTsZT0MaOUCZwaSdhI1PQ0oUhP4P6vUzWpFGU/qkHIH1/4SL56DPVxDYw2hFdpNK\nKj8wg3a0pw9hDjN9t5zcLYzjHGcevzGS0eaN87+aE4tVAxDQA3p+DBVNPpyl0RRGbhbs+gD2/w86\nvgZtnzJNlawbkUQiP/IDXelOT3o59Nja+AtwhtP8ws8MYBDtcZECILmZEPEfOPiNKvDe7llVQk6j\nMTvSCkd+ga2vQkB31bmpZOIovQLEcoZZ/MRtDKADHR1+fG38V3GOc/zMTDrQyaGXVoaTGq1y/iRs\nhk6ToOVDpiojp9FcRko4tQK2/Bs8fZXh13Fsj9hIDnOIRSxgOCNpSatSOYc2/kJIJ41Z/EwtajGc\nkXjZXXvGiTi7A7b+G9JPQdd3oOlYvfhLYx4StqoOTFaCyq/TeJSpCqXcjK38xUY2cDf3luoaI238\n1yGHHOYxh2yyGc89RUp1aipOr1ZfIIRaxdhgsEt9gTQuRtIe2PEmnNsBnSdDi4kuMY5/iXzyWcEy\noonmXu6jahGzbJYUbfw3wIqVVazgCFHcy31Uo3qpncsQpBWOz4eIt9Xq306ToNFw3QBonIdzO9Xn\n8+x2CHkBWj8OXq7VCbNgYS6/kUcedzK+TDqZ2viLwHa2Ec5aRjOWpgSX+vnKHGlVRSh2vA0yHzq8\nrOqNulCPSmMipIS4DSpSJ3mv+jy2+ofLGT5AMkn8yi/Uox7DGI4nZZMaWht/ETnBCebxG53oQh/C\nzFfNqyhICSeXQ+QHag6g/fPqktrHZDWLNebEmq86IJEfqgIpIS9C8/tcNv34QQ6whEX0pR+d6VKm\ngSTa+ItBOmnM4Td88GEM46hAhTI7d5mTsAUiP4LY9dB8ArT5J1RxwasdjfFkJ8PBb2Hfl1Cxjupw\nNB5lysIoRSGffP5kFQfYz52MNyRRpDb+YnLlTTvAndxl7uyeRSH9FOz/Cg59BzU7QpunoMEgHQmk\nsZ/ESNj7OZxYAI1GQJsnoVYno1WVKpc6j954M5Y7DOs8lqnxCyGqAb8BDYAY4A4pZUoh+8UAaUA+\nkCulLLTSgBHGf4kD7GcJi+jHbXQq48s0Q8jLVjlQ9n2uEmC1eUJdCZSvabQyjZnIuwjRC2Df/yDj\ntJqsbfUPt/gcxXCCuU4yXFzWxv8hkCSl/FAI8TJQVUr5SiH7nQA6SinP3+R4hhk/QBJJ/MYv1KAm\ntzPCtYd+LiElnN2qLstjlqg0t83vg0a3q8U0Gs3VSCvEb4LDP0L071CrM9zyqIogc4MAgnzyCWct\nO4lgFGMIppnRksrc+A8DfaSUZ4UQAUC4lPKaxDg24+8kpUy+yfEMNX6AXHJZw2r2sZeRjHaKN7XM\nyElXX+TDP0LSbhUJ1OI+qN1Nh4RqIOWoqhQX9ZNKE9L8fmh2N1QqelFws5PIOeYzl4pUYiSj8cPP\naElA2Rv/BSllVdt9AZy/9Piq/aKBVNRQzzQp5fTrHM9w479ENNEsYD7NaMZABuODOcu6lZj0U3Bk\nFhyeCdZcaDJWbbU66UbAnUg9rtaGHJ8P6THK6JvfBzXau9XnwIqV7WwlnHX0oz+d6OxUw8EON34h\nxGogoJCnXgNmFjR6IcR5KWW1Qo4RKKWMF0LUBFYDT0kpNxayn5w8efLlx2FhYYSFhRXn/+JQLnKR\nZSzlDKcZzTjq4YalEKWEpEjbl3+emhtoMkY1AgHd9KSwK3IhSr3Xx+dBZpyKyGkyFur0cct8UKmk\nsoD55JDDGMZSnRpGSyI8PJzw8PDLj998880yH+oJk1ImCCECgXWFDfVc9TeTgQwp5ceFPOc0Pf6C\n7Gcff7Dk8iSOS+X6KQ5SwvkDVxoBy3loOBzqD4SgW8HH32iFmpKQn6PCfU+tUPM8lgvQeIxq4AN7\nuWwY5s2QSPayhxUsoyvd6E2fMluQVVyMmNxNllJ+IIR4Bahy9eSuEKIC4CmlTBdCVARWAW9KKVcV\ncjynNH5QYVtLWEwSidzOSBq5QoEXe7kQBTFL4fRKZRw1O0D9QaohqNFeXw04M6nR6n07tRJi10GV\nZuq9azAEand1+/fuPMksYTEZZDCK0dTBuecxjAjnnAPUp0A4pxCiDjBdSjlUCNEY+N32J17ALCnl\ne9c5ntMa/yUOcZBlLKUxTRjAICqic+EDqnhG3HrVazy1UvUag26FwN4qUqhaK7c3E0NJP6XSJsRt\nVEafk6Ya6PqDoF5/KG/88IUzkEcem9nIFv6iF6F0p4fT9vILohdwlQEWLKxlDXvZfbnIizNN9DgF\naTGqIYjboLbs81Cn95WGoEY78HSzCfOyQlpVBE78xiuvf95F2+sfCnX7QPU2uiG+ihhiWMJCqlKN\nYdxOlVLOqOlItPGXIXHEsphF+ODD7YygJq6/aKXEZMap3uYlI0qLhmqtoXZnFQdeqzNUae6248kl\nRkq1cOrcDrWd3QGJO8G3ypVGtk6oGspxoyic4pBFFqtYwTGOMpihtOIW03XktPGXMSrMS2X7VAWU\n+7rHwi97yclQ0UIFDeviWajeVjUI1W6xba2gQoA2LQBLCpw/CBcOqkn28wdUmgTh+fcGtGZHqFDL\naLVOTx557GA7GwinNW3px22Uw5wJ5LTxG0QGGaxjDQc5QG/60IWu7hv9U1IsKcrIzh/4u7lJq2oA\nqrYE/ybg3xgqNwb/RuBbzbUahZwMSD8BaSfUVVFqNKQcVq+DJVW9DtVaQVVbo1gzRCVC0xQZieQw\nh1jFCqpSjYEMpja1jZZlF9r4DeYcZ1nJCpJJZgADaUkr0102Oh0XE22NwaG/G2LacfW8f2PwawAV\nApUJVgy03Q+ECnXUxKXRqQSkVU14Z8ZDVnyB2zh1P+O0+n/lpKkGzb+xbWukhsCq3QJ+9fS4vJ3E\nEcsKlpNFJgMZ7DIr87XxOwnHOMoKllOe8gxisOtn/TQCKZWZpkVD+slCTDUesuLUxLJXBfCtqsa+\nfate2bwrqoIgnuUK3JZTt+J68w1SLWTLz1aTpvkF7udlqSsXywXbZrufkwY+fqpButwoFbitVE9d\nxVQI0OZeCqSSyhpWc4yj3Eo/QuhoimidoqKN34mwYmUXO1nHGurTgL7cSi2TX1KaEmlVeYgum3EB\nQ87NLNzA87PV312Pgg3Fpfte5cCzvGpQylUFnyoF7lc2/qrDDckgg01sIJJddKILvQk17Th+YeTm\nQ/JFCPQzufFvOinpWd9oJY4lhxy2sZW/2ERjmhBGX2qiJ980mtIik0w2sZFdRNCGtoQShj+utbL8\ndCrcOR96BMHHA01u/LX+T/JCd3i+O3i42NC4BQvb2MoWNtsagFt1CKhG40AyyWQzm9jJDlrThlD6\nUJkqRstyOMuOwoOL4blu8EIP8PQwufGfTJHcOQ9qVICZI6Ga69VjxoKFrWxhC5tpSjC9CaV2oXnw\nNBpNUUgnjS1sYSc7uIXWhBJGFRc0/DwrvL4WftkPv4yGXrbREZcY48/Jh3+vgXmH4Lcx0M1F50Wz\nyWYH29nCZupQl96E0oCGRsvSaExDMslsZiMH2E9b2tGTXqZacVscYtPgrvlQ0Qd+Ggk1C2SLcQnj\nv8SiKHh4iRr2eaE7eLposEMuuewmks1spCKV6EkvWtDS0FJuGo0zc5pTbGYTMZygM13pRneXzpu1\nOAoeWQpPd4FXel07DO5Sxg9wKhUmLAAJzBwBjVyzMQdUFNAhDrKZTWSRSTe6054OLhWFoNGUlHzy\nOcRBtvAXGaTTnZ6E0AFfXLdEaLoFnl0Fa07AzyO5buCLyxk/QL4VPt0KH/wF7/eDB92g+M8pTrKV\nLRzjKK1pQ1e66XkAjVuSRhoR7GAnO6hGdbrSjZa0cqk4/MLYcBImLoJbG8EnA8D/Bu2bSxr/Jfaf\ngwkLoZ4/TB8GtSuVoTiDSLd96CNsH/oudKUlrXQ6CI1LI5HEEMN2tnKcY7ShLV3o6hadn+w8mLQO\nZu2DaUPh9uY3/xuXNn6AnHx4az18Gwn/GwJjWpaROIO5dJm7nW0kk0QIHQmhA9WpbrQ0jcZhZJLJ\nPvYQQQRW8ulCN9oT4jbDnZHxcN8iaFYNvh769wncG+Hyxn+JLadh4mJoVxumDoIAN+j9X+IcZ9lJ\nBHvZQ3VqEEIHbqG123w5NK5FPvkc5Qi7iSSa4zSjOR3oSCMau02eq+w8eGcDfLMLPh4A97Yp3nC2\n2xg/wMVceGsDfBcJ/7kVHgpxvUVfN+LSFyaSXZwgmua0oD0daEQjHRGkcXrOkkAku9jLHqpSjRA6\n0Jo2bteBWXcCHv0D2to6sXX8in8MtzL+S+w9q0KdvD3gm2HQ0g0Xw2aSyV72EMkuMsngFlrTmrbU\no57b9Jo0zk8SSexnL/vZhwUL7WhPezpQA/cr/ZicBS+shrUx8MWgoo3lXw+3NH5QkT9f74Qp6+GJ\nTvDvXlDOTec/k0hkH/vYz15yyKU1rWlDWwKpoxsBTZlzgQvst30e00nnFtrQmjbUo55bXplKqSZu\nX1gNd7WGt8PAz86I1LIutj4OmAK0ADpLKXddZ79BwGeAJ/CtlPKD6+xnd3bO2DR4agUcSITPB8GA\nJnYdztRIJOc4yz5bDwugBa1oQQvqUd/lw+E0xnDpc3eYQxzmMBc4TytuoTVtaUhDtzT7SxxOgqeW\nQ9JFFZnYyUE1dMra+FsAVmAa8Hxhxi+E8ASigNuAWGAHMF5KeaiQfR2WlnnpEXhmJdxSU02WNK3m\nkMOaFokkgXjbl/EQqaQSTDNa0JKmBLv0IhhN6ZNPPjHEEGUze5C0oCXNaUFDGrl9J+PCRXhzg+rp\nv9YLnuwCXg5s/wwZ6hFCrOP6xt8dmCylHGR7/AqAlPL9QvZ1aD5+Sx78dxt8+Jda9PV66I0XQbgT\nqaQQRRSHOcRpThFEPZoSTFOCqUUtPSSkuSmppHCcYxzjGMc5RjWq04IWtKAltaitP0OopGrf7oLJ\n62FUCzWsU9QQzeLgjMY/FhgopXzY9vheoKuU8qlC9i2VQiwJGfDqWlh+DN7pCxPbuW7en5JgwcJx\n25f3GEfJI48mNKUpwTSmCZVwo1hZzXXJIYcYTtiM/iiZZNKEpjShCcE0w8/F8t3by9oTatShenn4\nbCC0K8W1ZyUx/htOgQohVkOhy+VelVIuKcLxi+XkU6ZMuXw/LCyMsLCw4vx5oQRUghnDISIO/rUC\n/rcDPh0AfRrafWiXwBdfWnELrbgFgPMkc4xjHGAfS1lMZarQkEY0pCENaKgbAjfBgoXTnOYkJ4gh\nhnjiqENdmhLMGMYRQKBbj9dfj2Pn4eU/YVcC/F9/GN3C8SlmwsPDCQ8Pt+sYpd3j7wZMKTDU82/A\nWtgEb1mUXpQSZh9QVwAta8C7t0J7118FXmLyySeWWE4SQwwnOM0pKlGJBrZGoD4NqEpVfVnvAmSQ\nwRlOE0MMJzlBIokEEEgDGtKQRtSnvp4LugFx6fD2Bph7UBVIea572UUWGjnU84KUcmchz3mhJnf7\nAXHAdspgcvdm5OTDNzvhP5sgrAG83VdPABcFK1bOkmBrCGI4zSnyyacuQbafetQliApUMFqq5gbk\nkEM8cZzhDLGc4QxnyOYidQmyGX1D6hKEN95GS3V6LlyEDzbD9Eg1l/hKT6hexh//so7qGQVMBWoA\nqUCklHKwEKIOMF1KOdS232CuhHN+J6V87zrHK/Ni6xk5agL4060wthW8EVqylXPuTBqpfzOQOGKp\nSEUCqUMAAQQQSAAB+FNZXxkYQBZZJJBAAvGcJYF44kgmmVrUsjXY9QgiiGpU10M3xSArF6Zug4+3\nqonbN0IhyKBpDrddwGUvyVmq1f5uNzzQDl7qCbVct65DqWLFShJJxBP3N8PJJ5/atoagFjWpYdsq\nUlE3CA4gm2ySSSLR9nPW9upbyLa97gEFGuJAneG1hFzMhem7VJr4XvXUaEEzg3MlauO3k9g0eH+z\nird9oL2q/BWorwAcQgYZJBBPAgkkco4kkkgiEcDWCNSgBjWpavupQlUqUEE3CgWwYCGFC5d/Lhl9\nEolkk011alDT9lpeamSrUEX35B1AZo5KovZ/W6BzHZgcCiGBRqtSaON3ELFp6g2euQfGt4aXekAD\n16vdbDgSSSaZJJF4uSG4ZGopXMCK9W8NgR/++OOPH36X77vKhGMuuaSTTjpptn/TSCeNFJvVp3CB\nXHKpQhWq2F6TGtSgus3oK1NZG3wpkGZ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.contour(X, Y, stats.multivariate_normal.pdf(XY, mean=mu, cov=covm2 ));" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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i4PsepH5qWKo+9TjOCTLIMKzl6wIPF4PFqf987e1S8z6w2eFUhHptMynUhn//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\n/AaDweBlGOE3GAwGL8MIv8FgMHgZRvgNBoPByzDCbzAYDF6GEX6DwWD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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.contour(X, Y, stats.multivariate_normal.pdf(XY, mean=mu, cov=[[1,0],[0,1]] ));" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from scipy import linalg\n", "x00 = sp.array([0,0])\n", "x01 = sp.array([0,1])\n", "x10 = sp.array([1,0])\n", "x11 = sp.array([1,1])" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1.0" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "linalg.norm(x00 - x01, ord=2)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1.0" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "linalg.norm(x00 - x10, ord=2)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1.4142135623730951" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "linalg.norm(x00 - x11, ord=2)" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1.4142135623730951" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sqrt(sp.dot((x00 - x11),(x00 - x11)))" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def mahalanobis(x1, x2, covm):\n", " return sqrt(sp.dot(sp.dot((x1 - x2), linalg.inv(covm)), (x1 - x2)))\n", "# ili: from scipy.spatial.distance import mahalanobis" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.44721359549995793" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "covm1 = sp.array([[1, 0], [0, 5]])\n", "mahalanobis(x00, x01, covm1)" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1.0" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mahalanobis(x00, x10, covm1)" ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1.0954451150103321" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mahalanobis(x00, x11, covm1)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1.4142135623730951" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mahalanobis(x00, x11, sp.eye(2))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Procjena parametara\n", "\n", "* Ideja: na temelju slučajnog uzorka izračunati procjenu (estimaciju) parametra teorijske razdiobe\n", "\n", "\n", "* Neka je $(X_1,X_2,\\dots,X_n)$ uzorak ($n$-torka slučajnih varijabli koje su iid)\n", "\n", "\n", "* Slučajna varijabla $\\Theta=g(X_1,X_2,\\dots,X_n)$ naziva se **statistika**\n", "\n", "\n", "* Statistika $\\Theta$ je **procjenitelj (estimator)** parametra populacije $\\theta$\n", "\n", "\n", "* Vrijednost procjenitelja $\\hat{\\theta} = g(x_1,x_2,\\dots,x_n)$ naziva se **procjena**\n", "\n", "\n", "* Procjenitelj je slučajna varijable, dakle ima očekivanje i varijancu\n", "\n", "\n", "* [Slika: pristranost i varijanca procjenitelja]\n", "\n", "\n", "* Procjenitelj $\\Theta$ je **nepristran procjenitelj** (engl. *unbiased estimator*) parametra $\\theta$ akko \n", "$$\n", "\\mathbb{E}[\\Theta]=\\theta\n", "$$\n", "\n", "* Pristranost procjenitelj (engl. *estimator bias*):\n", "$$\n", "b_\\theta(\\Theta) = \\mathbb{E}[\\Theta]-\\theta\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Primjer: Procjenitelji srednje vrijednosti i varijance\n", "\n", "* $X$ je slučajna varijabla sa $x\\in\\mathbb{R}$.\n", "\n", "\n", "* Označimo $\\mathbb{E}[X] = \\mu$ (srednja vrijednost) i\n", "$\\mathrm{Var}(X)=\\sigma^2$ (varijanca)\n", "\n", "\n", "* $\\mu$ i $\\sigma^2$ su parametri populacije i oni su nam nepoznati\n", "\n", "\n", "* Parametre $\\mu$ i $\\sigma^2$ možemo ih procijeniti na temelju uzorka $\\{x^{(i)}\\}_{i=1}^N$ pomoću **procjenitelja**\n", "\n", "\n", "* Za procjenitelje možemo upotrijebiti bilo koje statistike. Npr.\n", "$$\n", "\\hat{\\mu}=\\frac{1}{N}\\sum_i x^{(i)}\\qquad\n", "\\hat{\\sigma}^2 = \\frac{1}{N}\\sum_{i=1}^N (x^{(i)}-\\hat{\\mu})^2\n", "$$\n", "\n", "\n", "* Q: Jesu li ovo dobri procjenitelji? (Jesu li nepristrani?)\n", "\n", "\n", "* $\\mathbb{E}[\\hat{\\mu}]=\\mu$ ?\n", "* $\\mathbb{E}[\\hat{\\sigma}^2] = \\sigma^2$ ?\n", "\n" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.042575400812075635" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X = stats.norm.rvs(size=10, loc=0, scale=1) # mean=0, stdev=var=1\n", "sp.mean(X)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Očekivanje procjenitelja:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.0014897662030209388" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mean = 0\n", "n = 10\n", "N = 10000\n", "for i in range(N):\n", " X = stats.norm.rvs(size=n)\n", " mean += sp.sum(X) / len(X)\n", "mean / N" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* $\\mathbb{E}[\\hat{\\mu}]=\\mu$, tj. $\\hat{\\mu}$ je nepristran procjenitelj srednje\n", "vrijednosti\n", "\n", "\n", "* Međutim, $\\mathbb{E}[\\hat{\\sigma}^2] \\neq \\sigma^2$, tj. $\\hat{\\sigma}^2$ **nije\n", "nepristran** procjenitelj varijance!\n", "$$\n", "\\mathbb{E}[\\hat{\\sigma}^2] = \\frac{N-1}{N}\\sigma^2\n", "$$\n", "\n", "\n", "* Pristranost od $\\hat{\\sigma}^2$ je \n", "$$\n", "b(\\hat{\\sigma}^2) = \\frac{N-1}{N}\\sigma^2-\\sigma^2 =\n", "-\\frac{\\sigma^2}{N}\n", "$$\n", "\n", "\n", "* Procjenitelj **podcjenjuje** (engl. *underestimates*) pravu varijancu!\n", "\n", "\n", "* Nepristran procjenitelj varijance:\n", "$$\n", "\\hat{\\sigma}^2_{\\text{nepr.}} = \\frac{1}{N-1}\\sum_{i=1}^N (x^{(i)}-\\hat{\\mu})^2\n", "$$ \n", "\n" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def st_dev(X): \n", " n = len(X)\n", " mean = sp.sum(X) / n\n", " s = 0\n", " for i in range(len(X)):\n", " s += (X[i] - mean)**2\n", " return s / n" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.62610770360872903" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X = stats.norm.rvs(size=10, loc=0, scale=1) # mean=0, stdev=var=1\n", "st_dev(X)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Očekivanje procjenitelja:" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.90251698503102884" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stdev = 0\n", "n = 10\n", "N = 10000\n", "for i in range(N):\n", " X = stats.norm.rvs(size=n)\n", " stdev += st_dev(X)\n", "stdev / N" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.90865744695623196" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stdev = 0\n", "n = 10\n", "N = 10000\n", "for i in range(N):\n", " X = stats.norm.rvs(size=n)\n", " stdev += st_dev(X)\n", "stdev / N" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Kako izvesti procjenitelj za neku teorijsku distribuciju (Bernoullijevu, Gaussovu, ...)?\n", "\n", "\n", "* Tri vrste procjenitelja:\n", " * (1) **Procjenitelj najveće izglednosti** (engl. *maximum likelihood estimator*, MLE)\n", " * (2) **Procjenitelj maximum aposteriori** (MAP)\n", " * (3) **Bayesovski procjenitelj** (engl. *Bayesian estimator*)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Procjenitelj MLE\n", "\n", "* Skup neoznačenih primjera $\\mathcal{D}=\\{\\mathbf{x}^{(i)}\\}_{i=1}^N$ koji su **iid**\n", "\n", "$$\n", "\\mathbf{x}^{(i)} \\sim p(\\mathbf{x} | \\boldsymbol{\\theta})\n", "$$\n", "\n", "\n", "* MLE određuje **najizglednije** parametre $\\boldsymbol{\\theta}$: to su oni parametri koji\n", "izvlačenje uzorka $\\mathcal{D}$ čine **najvjerojatnijim**\n", "$$\n", " p(\\mathcal{D} | \\boldsymbol{\\theta}) = \n", " p(\\mathbf{x}^{(1)},\\dots,\\mathbf{x}^{(N)} | \\mathbf{\\theta}) = \n", "\\prod_{i=1}^N p(\\mathbf{x}^{(i)} | \\mathbf{\\theta})\\ \n", "\\equiv \\color{red}{\\mathcal{L}(\\boldsymbol{\\theta} | \\mathcal{D})}\n", "$$\n", "NB: Druga jednakost vrijedi uz pretpostavku **iid**\n", "\n", "\n", "* **Funkcija izglednosti** $\\mathcal{L} : \\boldsymbol{\\theta}\\mapsto p(\\mathcal{D} | \\boldsymbol{\\theta})$ parametrima pridjeljuje vjerojatnost\n", "\n", "\n", "* $\\mathcal{L}$ nije PDF! Općenito ne vrijedi $\\int_{\\boldsymbol{\\theta}} \\mathcal{L}(\\boldsymbol{\\theta}|\\mathcal{D})\\,\\mathrm{d}\\boldsymbol{\\theta}=1$.\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Primjer: Izglednost Bernoullijeve varijable\n", "\n", "* $\\mathcal{D} \\equiv$ 10 bacanja novčića ($N=10$)\n", "* Glava (H) 8 puta, pismo (T) 2 puta\n", "* $\\mu$ je vjerojatnost da dobijem H\n", "* $P(X=x | \\mu)= \\mu^{x}(1-\\mu)^{1-x}$\n" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def likelihood(mu, m, N):\n", " return mu**m * (1 - mu)**(N - m)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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SgUg97jBxYtik5tFHoW/f2BFJucvHuIGSgUiSnTvhRz+CWbPgn/8M68qLxHbo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\nRK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xs = linspace(0,1)\n", "plt.plot(xs, likelihood(xs, 8, 10));" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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IYYMG+bVxHnggdCQiqXH55dCwoa85S3xKFLKbV17xO9ctWuTfSCK56Isv/FpQ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\nEpMShYiIxKREISIiMSlRiIhITP8H+t+N2VpXXuoAAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xs = linspace(0,1)\n", "plt.plot(xs, likelihood(xs, 5, 10));" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xs = linspace(0,1)\n", "plt.plot(xs, likelihood(xs, 10, 10));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### MLE\n", "\n", "* Nalazi $\\boldsymbol{\\theta}$ koji maksimiziraju funkciju izglednosti: \n", "\n", "$$\n", "\\hat{\\boldsymbol{\\theta}}_{\\mathrm{ML}} = \\mathrm{argmax}_{\\boldsymbol{\\theta}} \\mathcal{L}(\\boldsymbol{\\theta}|\\mathcal{D})\n", "$$\n", "\n", "\n", "* Analitički je jednostavnije maksimizirati **log-izglednost**:\n", "$$\n", "\\ln\\mathcal{L}(\\boldsymbol{\\theta} | \\mathcal{D}) \\ = \\ln p(\\mathcal{D} | \\boldsymbol{\\theta}) =\n", "\\ln \\prod_{i=1}^N p(\\mathbf{x}^{(i)} | \\boldsymbol{\\theta}) = \n", "\\sum_{i=1}^N\\ln p(\\mathbf{x}^{(i)} | \\boldsymbol{\\theta})\n", "$$\n", "\n", "$$\n", "\\hat{\\boldsymbol{\\theta}}_{\\mathrm{ML}} = \\mathrm{argmax}_{\\boldsymbol{\\theta}} \\big(\\ln \\mathcal{L}\\big(\\boldsymbol{\\theta}|\\mathcal{D})\\big)\n", "$$\n", "\n", "* Ako je moguće, maksimizaciju provodimo analitički, inače je provodimo iterativnim metodama" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### MLE za Bernoullijevu razdiobu (parametar: $\\mu$)\n", "\n", "\\begin{align*}\n", " \\ln\\mathcal{L}(\\mu | \\mathcal{D}) &=\n", "\\ln\\prod_{i=1}^N P(x | \\mu) =\n", "\\ln\\prod_{i=1}^N \\mu^{x^{(i)}}(1-\\mu)^{1-x^{(i)}}\\\\\n", "&=\\sum_{i=1}^N x^{(i)}\\ln \\mu + \\Big(N-\\sum_{i=1}^N x^{(i)}\\Big)\\ln(1-\\mu)\n", "\\end{align*}\n", "\n", "$$\n", "\\frac{\\mathrm{d}\\,{\\ln\\mathcal{L}}}{\\mathrm{d}\\mu} =\n", "\\frac{1}{\\mu}\\sum_{i=1}^N x^{(i)} - \\frac{1}{1-\\mu}\\Big(N-\\sum_{i=1}^N x^{(i)}\\Big) = 0\n", "$$\n", "\n", "\\begin{equation*}\n", "\\Rightarrow\\quad \\hat{\\mu}_\\mathrm{ML} = \\frac{1}{N}\\sum_{i=1}^N x^{(i)}\n", "\\end{equation*}\n", "\n", "\n", "* MLE za Bernoullijevu razdiobu je ustvari **relativna frekvencija**\n", "\n", "\n", "* Vrijedi $\\mathbb{E}(\\mu_\\mathrm{ML})=\\mathbb{E}[X]=\\mu$, pa je ovo je nepristran procjenitelj\n", "\n", "### MLE za kategoričku razdiobu (parametri: $\\mu_k$)\n", "\n", "\\begin{align*}\n", "\\ln\\mathcal{L}(\\boldsymbol{\\mu} | \\mathcal{D}) =\n", "\\ln\\prod_{i=1}^N P(\\mathbf{x}^{(i)} | \\boldsymbol{\\mu}) = \n", "\\ln\\prod_{i=1}^N \\color{red}{\\prod_{k=1}^K \\mu_k^{x_k^{(i)}}} =\n", "\\sum_{k=1}^K \\sum_{i=1}^N x_k^{(i)} \\ln \\mu_k\n", "\\end{align*}\n", "\n", "\n", "* Izraz treba maksimizirati prema $\\mu_k$ uz **ograničenje** $\\sum_{k=1}^K\\mu_k=1$.\n", "\n", "\n", "* Primjenom **metode Lagrangeovih multiplikatora** dobivamo:\n", "$$\n", "\\hat{\\mu}_{k,\\mathrm{ML}} = \\frac{1}{N}\\sum_{i=1}^N x_k^{(i)} = \\frac{N_k}{N}\n", "$$\n", "$N_k$ je broj nastupanja k-te vrijednosti\n", "\n", "\n", "### MLE za Gaussovu razdiobu (parametri: $\\mu, \\sigma^2$)\n", "\n", "\\begin{align*}\n", "\\ln\\mathcal{L}(\\mu,\\sigma^2 | \\mathcal{D}) &= \n", "\\ln\\prod_{i=1}^N\n", " \\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\Big\\{-\\frac{(x^{(i)}-\\mu)^2}{2\\sigma^2}\\Big\\} \\\\\n", "&= -\\frac{N}{2}\\ln(2\\pi) \n", " - N\\ln\\sigma \n", " - \\frac{\\sum_i(x^{(i)}-\\mu)^2}{2\\sigma^2}\\\\\n", "\\end{align*}\n", "\n", "\\begin{align*}\n", "\\nabla\\ln\\mathcal{L}(\\mu,\\sigma^2 | \\mathcal{D})&=0\\\\\n", "\\vdots\\\\\n", "\\hat{\\mu}_\\mathrm{ML} &= \\frac{1}{N}\\sum_{i=1}^N x^{(i)}\\\\\n", "\\hat{\\sigma}^2_\\mathrm{ML} &= \\frac{1}{N}\\sum_{i=1}^N(x^{(i)}-\\hat{\\mu}_\\mathrm{ML})^2\n", "\\end{align*}\n", "\n", "\n", "* NB: Procjenitelj $\\hat{\\sigma}^2_\\mathrm{ML}$ je pristran!\n", "\n", "\n", "* MLE ne mora nužno biti nepristran!" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "p = stats.norm(5, 2)\n", "X = sort(p.rvs(30))\n", "plt.scatter(X, sp.zeros(len(X)));" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "4.8057053896461506" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mean_mle = sp.mean(X); mean_mle" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "3.1111795224611711" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "var_mle = np.var(X, axis=0, ddof=1); var_mle" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false }, "outputs": [], "source": [ "p_mle = stats.norm(mean_mle, sqrt(var_mle))" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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Tz/Hll19StWoMpUvXKOowi92IEQ8wf/6vtGjxI/7+Kc6jnvc+1bU0GSgvF5Rp+6frtk5P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\nmDx5Mtu3b6dx48Y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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.scatter(X, p_mle.pdf(X))\n", "plt.plot(X, p.pdf(X), c='gray');\n", "plt.plot(X, p_mle.pdf(X), c='blue', linewidth=2)\n", "plt.vlines(X, 0, p_mle.pdf(X), colors='b', lw=2, alpha=0.2)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### MLE za multivarijatnu Gaussovu razdiobu\n", "\n", "\\begin{align*}\n", "\\ln\\mathcal{L}(\\boldsymbol{\\mu},\\boldsymbol{\\Sigma}|\\mathcal{D}) &=\n", "\\ln\\prod_{i=1}^N\n", "p(\\mathbf{x}^{(i)}|\\boldsymbol{\\mu},\\boldsymbol{\\Sigma})\\\\\n", "&=\n", "-\\frac{n N}{2}\\ln(2\\pi)-\\frac{N}{2}|\\boldsymbol{\\Sigma}|\n", "-\\frac{1}{2}\\sum_{i=1}^N(\\boldsymbol{x}^{(i)}-\\boldsymbol{\\mu})^\\mathrm{T}\\boldsymbol{\\Sigma}^{-1}(\\mathbf{x}^{(i)}-\\boldsymbol{\\mu})\n", "\\end{align*}\n", "\n", "\\begin{align*}\n", " \\nabla\\ln\\mathcal{L}(\\boldsymbol{\\mu},\\boldsymbol{\\Sigma} | \\mathcal{D})&=0\\\\\n", "\\vdots\\\\\n", " \\hat{\\boldsymbol{\\mu}}_\\mathrm{ML} &= \\frac{1}{N}\\sum_{i=1}^N\\mathbf{x}^{(i)}\\\\\n", " \\hat{\\boldsymbol{\\Sigma}}_\\mathrm{ML} &= \\frac{1}{N}\\sum_{i=1}^N\n", "(\\mathbf{x}^{(i)}-\\hat{\\boldsymbol{\\mu}}_\\mathrm{ML})(\\mathbf{x}^{(i)}-\\hat{\\boldsymbol{\\mu}}_\\mathrm{ML})^\\mathrm{T}\n", "\\end{align*}\n" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false }, "outputs": [], "source": [ "mu = [3, 2]\n", "covm = sp.array([[5, 2], [2, 10]])\n", "p = stats.multivariate_normal(mu, covm)" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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ThsYF7GqB/hI4tsuwiTvu3DNTD9ZRowREmUObAk6ZyNX25ImWpk2tr++b15Ei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\n7L9vpYEyTTqahgAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "x = np.linspace(-10, 10)\n", "y = np.linspace(-10, 10)\n", "X, Y = np.meshgrid(x, y)\n", "XY = np.dstack((X,Y))\n", "plt.contour(X, Y, p.pdf(XY))\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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CM2fOBPBlENuwYSE2bFiIRYtuwuOPP45JkyZh9uzZcc0jjx8/HkePHo26j8Fg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\nSUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "D = p.rvs(100)\n", "plt.contour(X, Y, p.pdf(XY), cmap='binary', alpha=0.5)\n", "plt.scatter(D[:,0], D[:,1])\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 3.10940042, 2.13853873])" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mean_mle = sp.mean(D, axis=0); mean_mle" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "collapsed": false }, "outputs": [], "source": [ "cov_mle = 0\n", "s = 0\n", "for x in D:\n", " s += sp.outer(x - mean_mle, x - mean_mle)\n", "cov_mle = s / len(D)" ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([[ 5.36556926, 1.61872328],\n", " [ 1.61872328, 9.05445152]])" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "cov_mle" ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([[ 5.41976693, 1.63507402],\n", " [ 1.63507402, 9.14591063]])" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.cov(D, rowvar=0, bias=0)" ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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HFyygdIMGdPzkE6r37IlBZbdq6QYNqPX00wBEnTvHsQUL+L5hQyq2bUuPOXPw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\nsjkEzIxSOSIiUgedalhEpBlpEt01IiLSNCjkRUQ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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "p_mle = stats.multivariate_normal(mean_mle, cov_mle)\n", "plt.contour(X, Y, p_mle.pdf(XY));" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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d3TxyyMSAXsmvJj38xa0Xo6uJvO2iKAqXXnopzzzzDJ/eeBFbX+7nb/7nCD9f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\nMHSec19csEhBl5zzFJbE65oGVmhqbp5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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.contour(X, Y, p.pdf(XY), cmap='binary', alpha=0.5)\n", "plt.scatter(D[:,0], D[:,1], c='gray', alpha=0.5)\n", "plt.contour(X, Y, p_mle.pdf(XY), cmap='Blues', linewidths=2);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Procjenitelj MAP\n", "\n", "* MLE lako dovodi do **prenaučenosti** modela\n", " * Npr. za skup primjera za koji $\\forall x^{(i)}\\in \\mathcal{D}. x^{(i)}=0$, procjena je $\\hat{\\mu}_\\mathrm{ML}=0$\n", "\n", "\n", "* Ideja: nisu sve vrijednosti za $\\mu$ jednako vjerojatne!\n", "\n", "\n", "* Definiramo **apriornu razdiobu parametra** $p(\\boldsymbol{\\theta})$ i zatim maksimiziramo \n", "aposteriornu vjerojatnost:\n", "$$\n", " p(\\boldsymbol{\\theta}|\\mathcal{D}) =\n", "\\frac{p(\\mathcal{D}|\\boldsymbol{\\theta}) P(\\boldsymbol{\\theta})} {p(\\mathcal{D})}\n", "$$\n", "\n", "\n", "* MLE:\n", "$$\n", "\\hat{\\boldsymbol{\\theta}} = \\mathrm{argmax}_{\\boldsymbol{\\theta}}\\ \\mathcal{L}(\\boldsymbol{\\theta}|\\mathcal{D})\n", "$$\n", "\n", "\n", "* MAP:\n", "$$\n", " \\hat{\\mathbf{\\theta}}_\\mathrm{MAP} = \\mathrm{argmax}_{\\boldsymbol{\\theta}} \\ p(\\boldsymbol{\\theta}|\\mathcal{D}) =\n", " p(\\mathcal{D}|\\boldsymbol{\\theta})\\,\\color{red}{p(\\boldsymbol{\\theta})}\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Procjenitelj MAP za Bernoullijevu varijablu" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [ "TODO" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "xs = sp.linspace(0,1)\n", "beta = stats.beta(1,1)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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2k+hWq5tJZ2xkSol3YCB1ihZlRl4Lv4WE6HqW7t0LjbULK+WGW/G3mHViFn/5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\nSkGbUDUUREhaI9jSlt3yeTHTUkvpTtvd2XX4nuR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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(xs,stats.beta.pdf(xs,1,1), label='a=1,b=1')\n", "plt.plot(xs,stats.beta.pdf(xs,2,2), label='a=2,b=2')\n", "plt.plot(xs,stats.beta.pdf(xs,4,2), label='a=4,b=2')\n", "plt.plot(xs,stats.beta.pdf(xs,2,4), label='a=2,b=4')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Laplaceovo zaglađivanje\n", "\n", "TODO" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Sažetak\n", "\n", "* Za strojno učenje posebno su važne **Bernoullijeva**, **kategorička** i **Gaussova** razdioba\n", "\n", "\n", "* **Procjenitelj** je statistika (slučajna varijabla izračunata iz uzorka) kojom se procjenjuju parametri neke teorijske razdiobe\n", "\n", "\n", "* Dobri procjenitelju su **nepristrani**\n", "\n", "\n", "* **Procjenitelj najveće izglednosti (MLE)** odabire parametre koji maksimiziraju vjerojatnost realizacije uzorka (tzv. izglednost)\n", "\n", "\n", "* MLE procjenitelj nije uvijek nepristran i sklon je prenaučenosti\n", "\n", "\n", "* **MAP-procjenitelj ** dodatno koristi apriornu razdiobu parametara i maksimizira aposteriornu vjerojatnost parametara\n", " \n", " \n", "* MAP-procjenitelj na taj način ugrađuje apriorno znanje te izbjegava prenaučenost samo na temelju podataka" ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.6" } }, "nbformat": 4, "nbformat_minor": 0 }