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 },
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Week 4: Predicting the Weather"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The opener for Week 4 discusses an example in which tomorrow's weather is predicted from today's weather. This is done by assigning probabilities to how today's weather transitions to tomorrow's weather."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The data"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<img src=\"https://studio.edx.org/c4x/UTAustinX/UT.5.01x/asset/4.1.1WeatherPicture.png\" alt=\"Transition Table\" width=50%>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The transition matrix, $ P $"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "\n",
      "P = np.matrix( '0.4, 0.3, 0.1; \\\n",
      "                0.4, 0.3, 0.6; \\\n",
      "                0.2, 0.4, 0.3' )\n",
      "\n",
      "print( P )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Today is cloudy"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Vector $ x $ consists of three components: \n",
      "$$ \\left( \\begin{array}{c} \\chi_s \\\\ \\chi_c \\\\ \\chi_r \\end{array} \\right). $$\n",
      "If we want to use $ x^{(0)} $ to express that on day $ 0 $ it is cloudy, we set\n",
      "$$ \\left( \\begin{array}{c} \\chi_s^{(0)} \\\\ \\chi_c^{(0)} \\\\ \\chi_r^{(0)} \\end{array} \\right) = \n",
      "\\left( \\begin{array}{r} 0 \\\\ 1 \\\\ 0 \\end{array} \\right). $$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x0 = np.matrix( '0; \\\n",
      "                 1; \\\n",
      "                 0' )\n",
      "print( x0 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Predicting the weather a week from today"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The weather for day $ 1 $ is predicted by $ x^{(1)} = P x^{(0)} $.  Notice that if <code> P  </code> and <code> x0 </code> are numpy matrices, then <code> P * x0 </code> computes the product of matrix <code> P </code> times vector <code> x0 </code>."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x1 = P * x0 \n",
      "print( x1 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Predict the weather for day 2 from day 1."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x2 = P * x1\n",
      "print( x2 )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Predict the weather for day 3 from day 2."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Predict the weather for day 4 from day 3."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Predict the weather for day 5 from day 4."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Predict the weather for day 6 from day 5."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Predict the weather for day 7 from day 6."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What are the probabilities it is sunny, cloudy, rainy on day 7 (given that it was cloudy on day 0)?  (Just think about this...)"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Predicting the next 20 days"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is best done with a loop, where $ x $ simply takes on the next value everytime through the loop."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = x0\n",
      "for i in range( 21 ):      # you saw range( n ) in Week 1\n",
      "    print( 'Day', i )        # print the predictions for day i\n",
      "    print( x )\n",
      "    x = P * x              # compute the predictions for day i+1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Predicting the weather for a year from now"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Modify the above loop to compute the prediction for Day 365."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Compare Day 365 to Day 20.  Notice that they are almost the same.  Go back and start with x0 = \"today is sunny\".  Go back and start with x0 = \"today is rainy\" Notice that no matter what the weather is today, eventually the prediction becomes \"the typical forecast for this location\"."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}