intro to probability density estimation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Source Code owners: Jason Brownlee, machinelearningmastery\n",
    "\n",
    "Referance: \n",
    "1. https://machinelearningmastery.com/probability-density-estimation/   \n",
    "2. https://en.wikipedia.org/wiki/Kernel_density_estimation \n",
    "3. https://mathisonian.github.io/kde/\n",
    "4. http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/AV0405/MISHRA/kde.html\n",
    "\n",
    "\n",
    "1. Histogram plots provide a fast and reliable way to visualize the probability density of a data sample.\n",
    "2. Parametric probability density estimation involves selecting a common distribution and estimating the parameters for the density function from a data sample.\n",
    "3. Nonparametric probability density estimation involves using a technique to fit a model to the arbitrary distribution of the data, like kernel density estimation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Parametric probability density estimation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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PEH7SpQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# example of parametric probability density estimation\n",
    "from matplotlib import pyplot\n",
    "from numpy.random import normal\n",
    "from numpy import mean\n",
    "from numpy import std\n",
    "from scipy.stats import norm\n",
    "# generate a sample\n",
    "sample = normal(loc=50, scale=5, size=1000)\n",
    "pyplot.hist(sample, bins=50)\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean=50.127, Standard Deviation=4.914\n"
     ]
    }
   ],
   "source": [
    "# calculate parameters\n",
    "sample_mean = mean(sample)\n",
    "sample_std = std(sample)\n",
    "print('Mean=%.3f, Standard Deviation=%.3f' % (sample_mean, sample_std))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "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": [
    "# define the distribution\n",
    "dist = norm(sample_mean, sample_std)\n",
    "# sample probabilities for a range of outcomes\n",
    "values = [value for value in range(30, 70)]\n",
    "probabilities = [dist.pdf(value) for value in values]\n",
    "# plot the histogram and pdf\n",
    "pyplot.hist(sample, bins=50, density=True)\n",
    "pyplot.plot(values, probabilities)\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Nonparametric probability density estimation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Kernel Density Estimation*: Nonparametric method for using a dataset to estimating probabilities for new points\n",
    "\n",
    "*Smoothing Parameter (bandwidth)*: Parameter that controls the number of samples or window of samples used to estimate the probability for a new point.\n",
    "\n",
    "*Basis Function (kernel)*: The function chosen used to control the contribution of samples in the dataset toward estimating the probability of a new point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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": [
    "# example of kernel density estimation for a bimodal data sample\n",
    "from matplotlib import pyplot\n",
    "from numpy.random import normal\n",
    "from numpy import hstack\n",
    "from numpy import asarray\n",
    "from numpy import exp\n",
    "from sklearn.neighbors import KernelDensity\n",
    "# generate a sample\n",
    "sample1 = normal(loc=20, scale=5, size=300)\n",
    "sample2 = normal(loc=40, scale=5, size=700)\n",
    "sample = hstack((sample1, sample2))\n",
    "# plot the histogram\n",
    "pyplot.hist(sample, bins=50)\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "KernelDensity(algorithm='auto', atol=0, bandwidth=1, breadth_first=True,\n",
       "              kernel='gaussian', leaf_size=40, metric='euclidean',\n",
       "              metric_params=None, rtol=0)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# fit density\n",
    "model = KernelDensity(bandwidth=1, kernel='gaussian')\n",
    "sample = sample.reshape((len(sample), 1))\n",
    "model.fit(sample)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# sample probabilities for a range of outcomes\n",
    "values = asarray([value for value in range(1, 60)])\n",
    "values = values.reshape((len(values), 1))\n",
    "probabilities = model.score_samples(values)\n",
    "probabilities = exp(probabilities)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "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": [
    "# plot the histogram and pdf\n",
    "pyplot.hist(sample, bins=50, density=True)\n",
    "pyplot.plot(values[:], probabilities)\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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
}