3b-exercise_matplotlib.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Matplotlib Exercise: Visualizing Correlated Gaussian Random Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we know how to generate correlated random variables, let's visualize them.\n",
    "\n",
    "### Exercise\n",
    "\n",
    "Make the following plots. All plots must have x and y labels, titles, and legends if there is more than one dataset in the same axes.\n",
    "\n",
    "1. Overlaid histograms of your samples of uncorrelated random variables with 30 bins (use `histtype='step'`)\n",
    "2. A scatterplot of $X_2$ vs $X_1$ with marker size equal to 2. Overlay the the theoretical line ($y=x$) in a black, dashed line.\n",
    "3. Overlaid histograms of your samples of correlated random variables with 30 bins (use `histtype='step'`)\n",
    "\n",
    "### Hints\n",
    "\n",
    "- In the arrays of random variables, each row `i` corresponds to a *sample* of random variable `i` (just FYI).\n",
    "- Google is your friend :)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt  # need to import matplotlib, of course\n",
    "import numpy as np  # import any needed modules here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n_real = int(1E6)  # number of realizations\n",
    "n_vars = 3  # number of random variables we want to correlate\n",
    "cov = np.array([[ 1. ,  0.2,  0.4], [ 0.2,  0.8,  0.3], [ 0.4,  0.3,  1.1]])  # covariance matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "unc_vars = np.random.randn(n_vars, n_real)  # create [n_vars x n_real] array of uncorrelated (unc) normal random variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "chol_mat = np.linalg.cholesky(cov)  # calculate the cholesky decomposition of the covariance matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cor_vars = chol_mat @ unc_vars  # [n_vars x n_real] array of correlated (cor) random variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cor_vars.var(axis=1)  # calculate variances of each sample of random variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "np.corrcoef(cor_vars[0, :], cor_vars[1, :])  # calculate the correlation coefficient between the first and second random samples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot 1: Histogram of Uncorrelated Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make a plot with overlaid histograms of your samples of uncorrelated random variables with 30 bins (use histtype='step')."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "  # insert code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot 2: Scatterplot of X2 vs. X1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make a scatterplot of $X_2$ vs $X_1$ with marker size equal to 2. Overlay the the theoretical line ($y=x$) in a black, dashed line."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "  # insert code here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plot 3: Histogram of Correlated Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make a plot with overlaid histograms of your samples of uncorrelated random variables with 30 bins (use histtype='step')."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "  # insert code here"
   ]
  }
 ],
 "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": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}