Distribution Basics and Sampling.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import norm as gaussian"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Probability Density Function\n",
    "- PDF a.k.a. Probability Distribution Density Function\n",
    "- Probabiliy of $x$ between $a$ and $b$ for any $(a,b)$ is\n",
    "\n",
    "> $\\displaystyle P_{ab} = \\int_a^b p(x)\\,dx$\n",
    "\n",
    "- Always \n",
    "\n",
    "> $\\displaystyle  \\int_{-\\infty}^{\\infty} p(x)\\,dx = 1$\n",
    "\n",
    "\n",
    "- Example 1: uniform distribution on $(a,b)$\n",
    "\n",
    "> $\\displaystyle  U(x;a,b) = \\frac{\\pmb{1}_{ab}(x)}{b\\!-\\!a} $,\n",
    "> where $\\pmb{1}_{ab}(x)$ is 1 between $a$ and $b$, but 0 otherwise\n",
    "\n",
    "- Example 2: Gaussian or normal distribution\n",
    "\n",
    "> $\\displaystyle  G(x;\\mu,\\sigma) = \\frac{{1}}{\\sqrt{2\\pi\\sigma^2}}\\ \\exp\\left[{-\\frac{(x\\!-\\!\\mu)^2}{2 \\sigma^2} }\\right]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cummulative Distribution Function\n",
    "- Integral up to given $x$: prob of being less than $x$\n",
    "\n",
    "> $\\displaystyle \\mathrm{CDF}(x) = \\int_{-\\infty}^{x} p(t)\\,dt$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Characterization of PDFs\n",
    "\n",
    "- Expectation value of $X$\n",
    "\n",
    "> $\\displaystyle \\mu = \\mathbb{E}[X] = \\int_{-\\infty}^{\\infty}\\! x\\ p(x)\\,dx$\n",
    "\n",
    "- Expectation value of any $f(X)$\n",
    "\n",
    "> $\\displaystyle \\mathbb{E}[f(X)] = \\int_{-\\infty}^{\\infty}\\! f(x)\\,p(x)\\,dx$\n",
    "\n",
    "- Moments \n",
    "\n",
    "> $\\displaystyle \\mathbb{E}[X^k]$\n",
    "    \n",
    "- Central moments \n",
    "\n",
    "> $\\displaystyle \\mathbb{E}\\big[(X\\!-\\!\\mu)^k\\big]$\n",
    "\n",
    "- Variance\n",
    "\n",
    "> $\\displaystyle \\mathbb{Var}[X] = \\mathbb{E}\\big[(X\\!-\\!\\mu)^2\\big]$\n",
    "\n",
    "- Standard deviation\n",
    "\n",
    "> $\\displaystyle \\sigma = \\sqrt{\\mathbb{Var}[X]}$\n",
    "\n",
    "- Normalized moments\n",
    "\n",
    "> $\\displaystyle \\mathbb{E}\\left[\\left(\\frac{X\\!-\\!\\mu}{\\sigma}\\right)^k\\right]$\n",
    "\n",
    "- Skewness\n",
    "\n",
    "> 3rd normalized moment ($k$=3)\n",
    "\n",
    "- Kurtosis\n",
    "\n",
    "> 4th normalized moment ($k$=4)\n",
    " "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"http://www.astroml.org/_images/fig_kurtosis_skew_1.png\" width=400 align=left>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 2, array(0.), array(4.), array(0.), array(0.))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mu, sig = 0, 2\n",
    "mean, var, skew, kurt = gaussian.stats(mu, sig, moments='mvsk');\n",
    "mu, sig, mean, var, skew, kurt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Density Estimation\n",
    "- Histograms\n",
    "    - Width of bins, $h$\n",
    "    - Start of bin boundary, $x_0$\n",
    "\n",
    " >$\\displaystyle \\mathrm{Hist}(x) = \\frac{1}{N}\\sum_i \\pmb{1}_{\\mathrm{bin}(x_i;x_0,h)}(x)$\n",
    "        \n",
    "- Kernel Density Estimation (KDE)\n",
    "    - Bandwidth $h$\n",
    "   \n",
    " >$\\displaystyle \\mathrm{KDE}(x) = \\frac{1}{N}\\sum_i K_h(x\\!-\\!x_i) = \\frac{1}{Nh}\\sum_i K\\left(\\frac{x\\!-\\!x_i}{h}\\right)$\n",
    "   \n",
    "    - Can use different $K(\\cdot)$ kernel functions\n",
    "        - E.g., Uniform, Triangular, Gauss, Epanechnikov\n",
    "\n",
    "See animations at\n",
    "http://www.mglerner.com/blog/?p=28"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Kernel Function\n",
    "- Finite vs Infinite support\n",
    "- Numerical evaluations\n",
    "- Frequently used kernels\n",
    "\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/4/47/Kernels.svg\" alt=\"All of the above kernels in a common coordinate system\"  width=\"350\" align=left>\n",
    "\n",
    "Learn more about KDE \n",
    "[here](https://jakevdp.github.io/blog/2013/12/01/kernel-density-estimation/) and also check out Bayesian Blocks \n",
    "[here](https://jakevdp.github.io/blog/2012/09/12/dynamic-programming-in-python/)\n",
    "<br>\n",
    "&mdash; tutorials by Jake Vanderplas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Detour: Dirac delta\n",
    "\n",
    "- In the limit of $h\\rightarrow{}0$, the kernel will become strange:\n",
    "\n",
    "<img src=\"https://upload.wikimedia.org/wikipedia/commons/4/48/Dirac_distribution_PDF.svg\" align=right width=250>\n",
    "\n",
    "> **Dirac's $\\delta$** \"function\" is 0 everywhere except at 0 such that\n",
    "\n",
    "> $\\displaystyle \\int \\delta(x)\\,dx = 1$\n",
    "\n",
    "- Interesting properties, e.g., \n",
    "\n",
    "> $\\displaystyle \\int f(x)\\,\\delta(x\\!-\\!a)\\,dx = f(a)$\n",
    "\n",
    "-  See **distribution theory** and **functionals** for more background"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### An interesting result \n",
    "\n",
    "- Bad density estimation but if...\n",
    "\n",
    "> $\\displaystyle p(x) = \\frac{1}{N} \\sum_{i=1}^N \\delta(x\\!-\\!x_i)$\n",
    "\n",
    "- The expectation value\n",
    "\n",
    "> $\\displaystyle \\mathbb{E}[X] =  \\int x\\, \\frac{1}{N} \\sum_{i=1}^N  \\delta(x\\!-\\!x_i) \\,dx$\n",
    "\n",
    "> $\\displaystyle \\mathbb{E}[X] = \\frac{1}{N} \\sum_{i=1}^N \\int x\\, \\delta(x\\!-\\!x_i) \\,dx$\n",
    "\n",
    "> $\\displaystyle \\mathbb{E}[X] = \\frac{1}{N} \\sum_{i=1}^N x_i$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Excercise\n",
    "\n",
    "0. Sample from a mixture of two Gaussians using uniform random numbers in the [0,1) interval. Try different $(\\mu_1, \\sigma_1)$ and $(\\mu_2,\\sigma_2$) values!\n",
    "0. Build different density estimators and compare to the original PDF. Try histogramming and KDE with different parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import norm as gaussian\n",
    "from scipy.stats import uniform as uni\n",
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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       "        12., 11., 17., 14., 12., 18., 28., 28., 30., 38., 46., 25., 34.,\n",
       "        30., 44., 39., 47., 35., 28., 38., 47., 24., 24., 24., 26., 15.,\n",
       "        14., 19., 15., 11., 17.,  8., 10.,  8.,  6.,  7., 10.,  4.,  2.,\n",
       "         7.,  3.,  7.,  3.,  6.,  9.,  7.,  9.,  4.,  5.,  3.,  2.,  3.,\n",
       "         1.,  6.,  6.,  1.,  3.,  3.,  2.,  1.,  1.,  0.,  1.,  1.,  1.,\n",
       "         3.,  3.,  2.,  0.,  0.,  0.,  0.,  4.,  0.,  0.,  0.,  0.,  1.,\n",
       "         0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  1.]),\n",
       " array([-3.24126734, -3.12445128, -3.00763523, -2.89081917, -2.77400311,\n",
       "        -2.65718706, -2.540371  , -2.42355494, -2.30673889, -2.18992283,\n",
       "        -2.07310677, -1.95629072, -1.83947466, -1.7226586 , -1.60584255,\n",
       "        -1.48902649, -1.37221043, -1.25539438, -1.13857832, -1.02176226,\n",
       "        -0.90494621, -0.78813015, -0.67131409, -0.55449804, -0.43768198,\n",
       "        -0.32086592, -0.20404987, -0.08723381,  0.02958225,  0.14639831,\n",
       "         0.26321436,  0.38003042,  0.49684648,  0.61366253,  0.73047859,\n",
       "         0.84729465,  0.9641107 ,  1.08092676,  1.19774282,  1.31455887,\n",
       "         1.43137493,  1.54819099,  1.66500704,  1.7818231 ,  1.89863916,\n",
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       "         2.5995355 ,  2.71635155,  2.83316761,  2.94998367,  3.06679972,\n",
       "         3.18361578,  3.30043184,  3.41724789,  3.53406395,  3.65088001,\n",
       "         3.76769606,  3.88451212,  4.00132818,  4.11814423,  4.23496029,\n",
       "         4.35177635,  4.4685924 ,  4.58540846,  4.70222452,  4.81904057,\n",
       "         4.93585663,  5.05267269,  5.16948874,  5.2863048 ,  5.40312086,\n",
       "         5.51993691,  5.63675297,  5.75356903,  5.87038509,  5.98720114,\n",
       "         6.1040172 ,  6.22083326,  6.33764931,  6.45446537,  6.57128143,\n",
       "         6.68809748,  6.80491354,  6.9217296 ,  7.03854565,  7.15536171,\n",
       "         7.27217777,  7.38899382,  7.50580988,  7.62262594,  7.73944199,\n",
       "         7.85625805,  7.97307411,  8.08989016,  8.20670622,  8.32352228,\n",
       "         8.44033833]),\n",
       " <a list of 100 Patch objects>)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Traditional Method\n",
    "mu1, sigma1, n1 = 3, 2, 200\n",
    "mu2, sigma2, n2 = 0, 1, 800\n",
    "np.random.seed(seed=42)\n",
    "x = gaussian.rvs(mu1,sigma1,n1)\n",
    "y = gaussian.rvs(mu2,sigma2,n2)\n",
    "z = np.hstack([x,y])\n",
    "plt.hist(z,bins=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 1.,  2.,  2.,  6.,  1.,  6.,  6.,  6.,  8., 15., 19., 17., 20.,\n",
       "        24., 23., 24., 33., 32., 35., 32., 35., 45., 45., 40., 43., 46.,\n",
       "        29., 36., 35., 33., 25., 28., 19., 17., 14., 11., 14.,  8.,  5.,\n",
       "        12.,  5.,  5.,  8.,  5., 11.,  3.,  6.,  5.,  4.,  3.,  4.,  4.,\n",
       "         4., 10.,  6.,  6.,  6.,  4.,  1.,  2.,  4.,  4.,  1.,  2.,  1.,\n",
       "        12.,  0.,  2.,  1.,  1.,  0.,  4.,  1.,  0.,  1.,  0.,  1.,  4.,\n",
       "         1.,  2.,  0.,  0.,  0.,  2.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,\n",
       "         0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.]),\n",
       " array([-2.72467263, -2.6025164 , -2.48036018, -2.35820395, -2.23604772,\n",
       "        -2.1138915 , -1.99173527, -1.86957904, -1.74742281, -1.62526659,\n",
       "        -1.50311036, -1.38095413, -1.25879791, -1.13664168, -1.01448545,\n",
       "        -0.89232923, -0.770173  , -0.64801677, -0.52586054, -0.40370432,\n",
       "        -0.28154809, -0.15939186, -0.03723564,  0.08492059,  0.20707682,\n",
       "         0.32923304,  0.45138927,  0.5735455 ,  0.69570172,  0.81785795,\n",
       "         0.94001418,  1.06217041,  1.18432663,  1.30648286,  1.42863909,\n",
       "         1.55079531,  1.67295154,  1.79510777,  1.91726399,  2.03942022,\n",
       "         2.16157645,  2.28373268,  2.4058889 ,  2.52804513,  2.65020136,\n",
       "         2.77235758,  2.89451381,  3.01667004,  3.13882626,  3.26098249,\n",
       "         3.38313872,  3.50529495,  3.62745117,  3.7496074 ,  3.87176363,\n",
       "         3.99391985,  4.11607608,  4.23823231,  4.36038853,  4.48254476,\n",
       "         4.60470099,  4.72685721,  4.84901344,  4.97116967,  5.0933259 ,\n",
       "         5.21548212,  5.33763835,  5.45979458,  5.5819508 ,  5.70410703,\n",
       "         5.82626326,  5.94841948,  6.07057571,  6.19273194,  6.31488817,\n",
       "         6.43704439,  6.55920062,  6.68135685,  6.80351307,  6.9256693 ,\n",
       "         7.04782553,  7.16998175,  7.29213798,  7.41429421,  7.53645043,\n",
       "         7.65860666,  7.78076289,  7.90291912,  8.02507534,  8.14723157,\n",
       "         8.2693878 ,  8.39154402,  8.51370025,  8.63585648,  8.7580127 ,\n",
       "         8.88016893,  9.00232516,  9.12448139,  9.24663761,  9.36879384,\n",
       "         9.49095007]),\n",
       " <a list of 100 Patch objects>)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Inverse Transform\n",
    "U = uni.rvs(size=200)\n",
    "x = gaussian.ppf(U, mu1, sigma1)\n",
    "U = uni.rvs(size=800)\n",
    "y = gaussian.ppf(U, mu2, sigma2)\n",
    "z = np.hstack([x,y])\n",
    "plt.hist(z,bins=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#KDE\n",
    "kernel=stats.gaussian_kde(z)\n",
    "xx = np.linspace(-7,10,201)\n",
    "kdepdf = kernel.evaluate(xx)\n",
    "plt.plot(xx, kdepdf, color=\"g\")\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}