{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Scientific Python Quickstart"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### [John Stachurski](http://johnstachurski.net/)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ANU"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a fast-paced, hands-on introduction to scientific computing with Python, contained in a [Jupyter](http://jupyter.org/) notebook.  The main focus will be on introducing Python's four most important scientific libraries: NumPy, Scipy, Pandas and Matplotlib.\n",
    "\n",
    "If you don't know how to use this notebook you need to first work through [this page](http://quant-econ.net/py/getting_started.html).\n",
    "\n",
    "A slower, more detailed and more systematic treatment of Python for scientific applications can be found at [quant-econ.net](http://quant-econ.net/py/index.html).  But this notebook is a good place to start for those who like to learn by doing."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's some information on the version of Python that I'm using:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.5.1 |Anaconda 2.4.1 (64-bit)| (default, Dec  7 2015, 11:16:01) \n",
      "[GCC 4.4.7 20120313 (Red Hat 4.4.7-1)]\n"
     ]
    }
   ],
   "source": [
    "import sys\n",
    "print(sys.version)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Basic NumPy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Perhaps the single most important scientific library for Python is NumPy.  NumPy provides foundational data structures and routines on which many other libraries rely."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.10.2\n"
     ]
    }
   ],
   "source": [
    "import numpy as np     # Import library and give it alias np\n",
    "print(np.__version__)  #  The version I'm using"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "NumPy defines a basic data type called an array (actually a numpy.ndarray)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  0.,  0.])"
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.zeros(3)     # Create an array of zeros\n",
    "a                   # Print a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 244,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.ndarray"
      ]
     },
     "execution_count": 244,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that array data *must be homogeneous*\n",
    "\n",
    "The most important data types are:\n",
    "\n",
    "*    float64: 64 bit floating point number\n",
    "*    float32: 32 bit floating point number\n",
    "*    int64: 64 bit integer\n",
    "*    int32: 32 bit integer\n",
    "*    bool:  8 bit True or False\n",
    "\n",
    "There are also dtypes to represent complex numbers, unsigned integers, etc\n",
    "\n",
    "On most machines, the default dtype for arrays is ``float64`` \n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 245,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.float64"
      ]
     },
     "execution_count": 245,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.zeros(3)\n",
    "type(a[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we create an array such as \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "z = np.zeros(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "``z`` is a \"flat\" array with no dimension--- neither row nor column vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 247,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(10,)"
      ]
     },
     "execution_count": 247,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    " z.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here the shape tuple has only one element, which is the length of the array (tuples with one element end with a comma)\n",
    "\n",
    "To give it dimension, we can change the ``shape`` attribute \n",
    "\n",
    "For example, let's make it a column vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "z.shape = (10, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.],\n",
       "       [ 0.],\n",
       "       [ 0.],\n",
       "       [ 0.],\n",
       "       [ 0.],\n",
       "       [ 0.],\n",
       "       [ 0.],\n",
       "       [ 0.],\n",
       "       [ 0.],\n",
       "       [ 0.]])"
      ]
     },
     "execution_count": 249,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 250,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  0.],\n",
       "       [ 0.,  0.]])"
      ]
     },
     "execution_count": 250,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.zeros(4)\n",
    "z.shape = (2, 2)\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Creating arrays"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Creating empty arrays --- initializing memory:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 251,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  0.,  0.])"
      ]
     },
     "execution_count": 251,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.empty(3)\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These are just garbage numbers --- whatever was in those memory slots\n",
    "\n",
    "Here's how to make a regular gird sequence"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 252,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2. ,  2.5,  3. ,  3.5,  4. ])"
      ]
     },
     "execution_count": 252,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.linspace(2, 4, 5)  # From 2 to 4, with 5 elements\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Creating an array of ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 253,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.,  1.,  1.])"
      ]
     },
     "execution_count": 253,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.ones(3)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 254,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.],\n",
       "       [ 0.,  1.]])"
      ]
     },
     "execution_count": 254,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.identity(2)\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Arrays can be made from Python lists or tuples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 255,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([10, 20])"
      ]
     },
     "execution_count": 255,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.array([10, 20]) \n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 256,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 10.,  20.])"
      ]
     },
     "execution_count": 256,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.array((10, 20), dtype=float) \n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 257,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4]])"
      ]
     },
     "execution_count": 257,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.array([[1, 2], [3, 4]])         # 2D array from a list of lists\n",
    "z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Array indexing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 258,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.  ,  1.25,  1.5 ,  1.75,  2.  ])"
      ]
     },
     "execution_count": 258,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.linspace(1, 2, 5)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 259,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 259,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z[0]  # First element --- Python sequences are zero based, like C, Java, etc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 260,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0"
      ]
     },
     "execution_count": 260,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z[-1]  # Special syntax for last element"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 261,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.  ,  1.25])"
      ]
     },
     "execution_count": 261,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z[0:2]  # Meaning: Two elements, starting from element 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 262,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4]])"
      ]
     },
     "execution_count": 262,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.array([[1, 2], [3, 4]])\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 263,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 263,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z[0, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 264,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2])"
      ]
     },
     "execution_count": 264,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z[0,:]  # First row"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 265,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 3])"
      ]
     },
     "execution_count": 265,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z[:,0]  # First column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 266,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2. ,  2.5,  3. ,  3.5,  4. ])"
      ]
     },
     "execution_count": 266,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.linspace(2, 4, 5)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 267,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([False,  True,  True, False, False], dtype=bool)"
      ]
     },
     "execution_count": 267,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = np.array([0, 1, 1, 0, 0], dtype=bool)\n",
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 268,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2.5,  3. ])"
      ]
     },
     "execution_count": 268,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z[d]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Array methods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 269,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4, 3, 2, 1])"
      ]
     },
     "execution_count": 269,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array((4, 3, 2, 1))\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 270,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A.sort()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 271,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 2, 3, 4])"
      ]
     },
     "execution_count": 271,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 272,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.5"
      ]
     },
     "execution_count": 272,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 273,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 273,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 274,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 274,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 275,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1,  3,  6, 10])"
      ]
     },
     "execution_count": 275,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.cumsum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 276,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.25"
      ]
     },
     "execution_count": 276,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.var()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 277,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4]])"
      ]
     },
     "execution_count": 277,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.shape = (2, 2)\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 278,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 3],\n",
       "       [2, 4]])"
      ]
     },
     "execution_count": 278,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T  # Transpose, equivalent to A.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Operations on arrays"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Standard arithmetic operations on arrays act elementwise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 279,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a = np.array([1, 2, 3, 4])\n",
    "b = np.array([5, 6, 7, 8])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 280,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 6,  8, 10, 12])"
      ]
     },
     "execution_count": 280,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 281,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-4, -4, -4, -4])"
      ]
     },
     "execution_count": 281,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a - b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 282,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([11, 12, 13, 14])"
      ]
     },
     "execution_count": 282,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a + 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 283,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a.shape = 2, 2\n",
    "b.shape = 2, 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 284,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4]])"
      ]
     },
     "execution_count": 284,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 285,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[5, 6],\n",
       "       [7, 8]])"
      ]
     },
     "execution_count": 285,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 286,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5, 12],\n",
       "       [21, 32]])"
      ]
     },
     "execution_count": 286,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * b # Pointwise multiplication!!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 287,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[19, 22],\n",
       "       [43, 50]])"
      ]
     },
     "execution_count": 287,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(a, b) # Matrix multiplication"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For Python $\\geq 3.5$ and NumPy $\\geq 1.1$ the ``@`` operator also works."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 288,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[19, 22],\n",
       "       [43, 50]])"
      ]
     },
     "execution_count": 288,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a @ b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I'll continue to use ``np.dot`` below for the benefit of those who are using older versions.  But in my opinion the ``@`` operator is much nicer."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparisons"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 289,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ True,  True], dtype=bool)"
      ]
     },
     "execution_count": 289,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.array([2, 3])\n",
    "y = np.array([2, 3])\n",
    "z == y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 290,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([False,  True], dtype=bool)"
      ]
     },
     "execution_count": 290,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y[0] = 3\n",
    "z == y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 291,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  0. ,   2.5,   5. ,   7.5,  10. ])"
      ]
     },
     "execution_count": 291,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = np.linspace(0, 10, 5)\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 292,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([False, False,  True,  True,  True], dtype=bool)"
      ]
     },
     "execution_count": 292,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z > 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 293,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  5. ,   7.5,  10. ])"
      ]
     },
     "execution_count": 293,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z[z > 3]  # Conditional extraction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matplotlib"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Matplotlib is an outstanding plotting and visualization library for Python that interacts nicely with NumPy.  Here are a few quick examples.  We'll see more below when we discuss the SciPy library."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 294,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt  # Import main functionality"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Display figures in this browser window rather than having them open up separately:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 295,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create something to plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 296,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = np.linspace(-2, 2, 100)\n",
    "y = x**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 297,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f77a347d048>]"
      ]
     },
     "execution_count": 297,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f77a347f7f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()  # Create axes and figure window\n",
    "ax.plot(x, y, 'b-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a slightly more complex plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 298,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f77a228fcf8>"
      ]
     },
     "execution_count": 298,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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DXMC/tNavnmI9nwr3cqtWmZDfsMEsR0ebM/mRIyXkhfBFa/at4ZH5j5BblEuHxh2YMWRG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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f77a2334630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y3 = x**3\n",
    "fig, ax = plt.subplots()  # Create axes and figure window\n",
    "ax.plot(x, y, 'b-', lw=2, alpha=0.8, label='$x^2$')\n",
    "ax.plot(x, y3, 'g-', lw=2, alpha=0.8, label='$x^3$')\n",
    "ax.legend(loc='lower right')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## SciPy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's just cover some simple examples --- references for further reading are below"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Statistics and distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's use ``scipy.stats`` to generate some data from the Beta distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 299,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy.stats import beta\n",
    "q = beta(5, 5)      # Beta(a, b), with a = b = 5\n",
    "obs = q.rvs(2000)   # 2000 observations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's histogram it and compare it to the original density"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 300,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f77a22d2ef0>]"
      ]
     },
     "execution_count": 300,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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s4ddgkjezFsBvge8CA4EfmtmACE3fds4d6z/09XkTjBs3jo0bNzJkyBBuuumm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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f77a22d2f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.hist(obs, bins=40, normed=True)\n",
    "grid = np.linspace(0.01, 0.99, 100)\n",
    "ax.plot(grid, q.pdf(grid), 'k-', linewidth=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Other methods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 301,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "scipy.stats._distn_infrastructure.rv_frozen"
      ]
     },
     "execution_count": 301,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 302,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['__class__',\n",
       " '__delattr__',\n",
       " '__dict__',\n",
       " '__dir__',\n",
       " '__doc__',\n",
       " '__eq__',\n",
       " '__format__',\n",
       " '__ge__',\n",
       " '__getattribute__',\n",
       " '__gt__',\n",
       " '__hash__',\n",
       " '__init__',\n",
       " '__le__',\n",
       " '__lt__',\n",
       " '__module__',\n",
       " '__ne__',\n",
       " '__new__',\n",
       " '__reduce__',\n",
       " '__reduce_ex__',\n",
       " '__repr__',\n",
       " '__setattr__',\n",
       " '__sizeof__',\n",
       " '__str__',\n",
       " '__subclasshook__',\n",
       " '__weakref__',\n",
       " 'a',\n",
       " 'args',\n",
       " 'b',\n",
       " 'cdf',\n",
       " 'dist',\n",
       " 'entropy',\n",
       " 'expect',\n",
       " 'interval',\n",
       " 'isf',\n",
       " 'kwds',\n",
       " 'logcdf',\n",
       " 'logpdf',\n",
       " 'logpmf',\n",
       " 'logsf',\n",
       " 'mean',\n",
       " 'median',\n",
       " 'moment',\n",
       " 'pdf',\n",
       " 'pmf',\n",
       " 'ppf',\n",
       " 'random_state',\n",
       " 'rvs',\n",
       " 'sf',\n",
       " 'stats',\n",
       " 'std',\n",
       " 'var']"
      ]
     },
     "execution_count": 302,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dir(q)  # Let's see all its methods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 303,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.50000000000000011"
      ]
     },
     "execution_count": 303,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q.cdf(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 304,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.4609375000000009"
      ]
     },
     "execution_count": 304,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q.pdf(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 305,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 305,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Basic linear regression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 306,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gradient = 1.874172371477785\n",
      "intercept = 1.4035329152756753\n"
     ]
    }
   ],
   "source": [
    "from scipy.stats import linregress\n",
    "n = 100\n",
    "alpha, beta, sigma = 1, 2, 1.5\n",
    "x = np.random.randn(n)  # n standard normals\n",
    "y = alpha + beta * x + sigma * np.random.randn(n)\n",
    "beta_hat, alpha_hat, r_value, p_value, std_err = linregress(x, y)\n",
    "print(\"gradient = {}\".format(beta_hat))\n",
    "print(\"intercept = {}\".format(alpha_hat))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot this with data and line of best fit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 307,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f77a21026a0>"
      ]
     },
     "execution_count": 307,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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HlLjUNG0h8BBwGQgAQoBPlVI/veZ+slhLCNHE1cVaTbcwdXSxVmuLv5pbKFRS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\nSpZvAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f77a218d668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 5))\n",
    "ax.plot(x, y, 'bo', alpha=0.6, label='observations')\n",
    "xgrid = np.linspace(-3, 3, 2)\n",
    "ax.plot(xgrid, alpha_hat + beta_hat * xgrid, 'k-', lw=2, alpha=0.8, label='best fit')\n",
    "ax.grid()\n",
    "ax.legend(loc='upper left')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Roots and fixed points"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's choose an arbitrary function to work with"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 308,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f77a2129c18>]"
      ]
     },
     "execution_count": 308,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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KK2H8+HAF0I47xl2RiFSlEJAGsW4d9OsHX34ZxgFstVXcFYnIxqhPQCJXUhLuAWwWpoFW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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f77a2129828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "def f(x):\n",
    "     return np.sin(4 * (x - 0.25)) + x + x**20 - 1\n",
    "x = np.linspace(0, 1, 100)\n",
    "ax.plot(x, f(x))\n",
    "ax.plot(x, 0 * x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 309,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4082935042797544"
      ]
     },
     "execution_count": 309,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.optimize import bisect  # Bisection algorithm --- slow but robust\n",
    "bisect(f, 0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 310,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.40829350427935679"
      ]
     },
     "execution_count": 310,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.optimize import newton  #  Newton's method --- fast but less robust\n",
    "newton(f, 0.2)   # Start the search at initial condition x = 0.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 311,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.70017000000002816"
      ]
     },
     "execution_count": 311,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton(f, 0.7)   # Start the search at x = 0.7 instead "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we see that the algorithm gets it wrong --- ``newton`` is fast but not robust\n",
    "\n",
    "Let's try a hybrid method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 312,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.40829350427936706"
      ]
     },
     "execution_count": 312,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.optimize import brentq\n",
    "brentq(f, 0, 1) # Hybrid method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 313,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10000 loops, best of 3: 64.7 µs per loop\n"
     ]
    }
   ],
   "source": [
    "timeit bisect(f, 0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 314,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100000 loops, best of 3: 14.2 µs per loop\n"
     ]
    }
   ],
   "source": [
    "timeit newton(f, 0.2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 315,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100000 loops, best of 3: 16.1 µs per loop\n"
     ]
    }
   ],
   "source": [
    "timeit brentq(f, 0, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the hybrid method is robust but still quite fast..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Numerical optimization and integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 316,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 316,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.optimize import fminbound\n",
    "fminbound(lambda x: x**2, -1, 2)  # Search in [-1, 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 317,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.33333333333333337"
      ]
     },
     "execution_count": 317,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.integrate import quad\n",
    "integral, error = quad(lambda x: x**2, 0, 1)\n",
    "integral"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear Algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at some of the most common routines from linear and matrix algebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 318,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import scipy.linalg as la"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll experiment with matrices\n",
    "\n",
    "$$ \n",
    "  A = \\begin{bmatrix} 2 & -1 \\\\ 3 & 0 \\end{bmatrix} \n",
    "  \\quad \\text{and} \\quad\n",
    "  b = \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 319,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A = [[2, -1],\n",
    "     [3, 0]]\n",
    "A = np.array(A) # Convert from list to NumPy array\n",
    "b = np.ones((2, 1))  # Shape is 2 x 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 320,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2, -1],\n",
       "       [ 3,  0]])"
      ]
     },
     "execution_count": 320,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 321,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.],\n",
       "       [ 1.]])"
      ]
     },
     "execution_count": 321,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 322,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.33333333]\n",
      " [-0.33333333]]\n"
     ]
    }
   ],
   "source": [
    "x = la.solve(A, b)  # Solve for x in Ax = b\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check that $Ax = b$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 323,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.],\n",
       "       [ 1.]])"
      ]
     },
     "execution_count": 323,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(A, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also invert directly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 324,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.        ,  0.33333333],\n",
       "       [-1.        ,  0.66666667]])"
      ]
     },
     "execution_count": 324,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la.inv(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 325,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.],\n",
       "       [ 0.,  1.]])"
      ]
     },
     "execution_count": 325,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(A, la.inv(A))  # Should be the identity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's compute the eigenvalues and eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 326,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "eigvals, eigvecs = la.eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 327,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "eigenvalues = [ 1.+1.41421356j  1.-1.41421356j]\n"
     ]
    }
   ],
   "source": [
    "print(\"eigenvalues = {}\".format(eigvals))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 328,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "first eigenvector = [ 0.28867513+0.40824829j  0.86602540+0.j        ]\n"
     ]
    }
   ],
   "source": [
    "print(\"first eigenvector = {}\".format(eigvecs[:, 0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### More information"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* linear algebra: http://docs.scipy.org/doc/scipy/reference/linalg.html\n",
    "* numerical integration: http://docs.scipy.org/doc/scipy/reference/integrate.html\n",
    "* interpolation: http://docs.scipy.org/doc/scipy/reference/interpolate.html\n",
    "* optimization: http://docs.scipy.org/doc/scipy/reference/optimize.html\n",
    "* distributions and random number generation: http://docs.scipy.org/doc/scipy/reference/stats.html\n",
    "* signal processing: http://docs.scipy.org/doc/scipy/reference/signal.html\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pandas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pandas is a very popular library for working with data sets.  In pandas, data is held in a dataframe, which is kind of like a spread sheet"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 329,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's start by writing a test data set to the present working directory, so we can read it back in as a dataframe using pandas.  We use an IPython magic to write the data from a cell to a file:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 330,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Overwriting test_data.csv\n"
     ]
    }
   ],
   "source": [
    "%%file test_data.csv\n",
    "\"country\",\"country isocode\",\"year\",\"POP\",\"XRAT\",\"tcgdp\",\"cc\",\"cg\"\n",
    "\"Argentina\",\"ARG\",\"2000\",\"37335.653\",\"0.9995\",\"295072.21869\",\"75.716805379\",\"5.5788042896\"\n",
    "\"Australia\",\"AUS\",\"2000\",\"19053.186\",\"1.72483\",\"541804.6521\",\"67.759025993\",\"6.7200975332\"\n",
    "\"India\",\"IND\",\"2000\",\"1006300.297\",\"44.9416\",\"1728144.3748\",\"64.575551328\",\"14.072205773\"\n",
    "\"Israel\",\"ISR\",\"2000\",\"6114.57\",\"4.07733\",\"129253.89423\",\"64.436450847\",\"10.266688415\"\n",
    "\"Malawi\",\"MWI\",\"2000\",\"11801.505\",\"59.543808333\",\"5026.2217836\",\"74.707624181\",\"11.658954494\"\n",
    "\"South Africa\",\"ZAF\",\"2000\",\"45064.098\",\"6.93983\",\"227242.36949\",\"72.718710427\",\"5.7265463933\"\n",
    "\"United States\",\"USA\",\"2000\",\"282171.957\",\"1\",\"9898700\",\"72.347054303\",\"6.0324539789\"\n",
    "\"Uruguay\",\"URY\",\"2000\",\"3219.793\",\"12.099591667\",\"25255.961693\",\"78.978740282\",\"5.108067988\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 331,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "./test_data.csv\r\n"
     ]
    }
   ],
   "source": [
    "%ls ./*.csv # Check it's there"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 332,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "df = pd.read_csv('./test_data.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 333,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>country</th>\n",
       "      <th>country isocode</th>\n",
       "      <th>year</th>\n",
       "      <th>POP</th>\n",
       "      <th>XRAT</th>\n",
       "      <th>tcgdp</th>\n",
       "      <th>cc</th>\n",
       "      <th>cg</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Argentina</td>\n",
       "      <td>ARG</td>\n",
       "      <td>2000</td>\n",
       "      <td>37335.653</td>\n",
       "      <td>0.999500</td>\n",
       "      <td>295072.218690</td>\n",
       "      <td>75.716805</td>\n",
       "      <td>5.578804</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Australia</td>\n",
       "      <td>AUS</td>\n",
       "      <td>2000</td>\n",
       "      <td>19053.186</td>\n",
       "      <td>1.724830</td>\n",
       "      <td>541804.652100</td>\n",
       "      <td>67.759026</td>\n",
       "      <td>6.720098</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>India</td>\n",
       "      <td>IND</td>\n",
       "      <td>2000</td>\n",
       "      <td>1006300.297</td>\n",
       "      <td>44.941600</td>\n",
       "      <td>1728144.374800</td>\n",
       "      <td>64.575551</td>\n",
       "      <td>14.072206</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Israel</td>\n",
       "      <td>ISR</td>\n",
       "      <td>2000</td>\n",
       "      <td>6114.570</td>\n",
       "      <td>4.077330</td>\n",
       "      <td>129253.894230</td>\n",
       "      <td>64.436451</td>\n",
       "      <td>10.266688</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Malawi</td>\n",
       "      <td>MWI</td>\n",
       "      <td>2000</td>\n",
       "      <td>11801.505</td>\n",
       "      <td>59.543808</td>\n",
       "      <td>5026.221784</td>\n",
       "      <td>74.707624</td>\n",
       "      <td>11.658954</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>South Africa</td>\n",
       "      <td>ZAF</td>\n",
       "      <td>2000</td>\n",
       "      <td>45064.098</td>\n",
       "      <td>6.939830</td>\n",
       "      <td>227242.369490</td>\n",
       "      <td>72.718710</td>\n",
       "      <td>5.726546</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>United States</td>\n",
       "      <td>USA</td>\n",
       "      <td>2000</td>\n",
       "      <td>282171.957</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>9898700.000000</td>\n",
       "      <td>72.347054</td>\n",
       "      <td>6.032454</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>Uruguay</td>\n",
       "      <td>URY</td>\n",
       "      <td>2000</td>\n",
       "      <td>3219.793</td>\n",
       "      <td>12.099592</td>\n",
       "      <td>25255.961693</td>\n",
       "      <td>78.978740</td>\n",
       "      <td>5.108068</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         country country isocode  year          POP       XRAT  \\\n",
       "0      Argentina             ARG  2000    37335.653   0.999500   \n",
       "1      Australia             AUS  2000    19053.186   1.724830   \n",
       "2          India             IND  2000  1006300.297  44.941600   \n",
       "3         Israel             ISR  2000     6114.570   4.077330   \n",
       "4         Malawi             MWI  2000    11801.505  59.543808   \n",
       "5   South Africa             ZAF  2000    45064.098   6.939830   \n",
       "6  United States             USA  2000   282171.957   1.000000   \n",
       "7        Uruguay             URY  2000     3219.793  12.099592   \n",
       "\n",
       "            tcgdp         cc         cg  \n",
       "0   295072.218690  75.716805   5.578804  \n",
       "1   541804.652100  67.759026   6.720098  \n",
       "2  1728144.374800  64.575551  14.072206  \n",
       "3   129253.894230  64.436451  10.266688  \n",
       "4     5026.221784  74.707624  11.658954  \n",
       "5   227242.369490  72.718710   5.726546  \n",
       "6  9898700.000000  72.347054   6.032454  \n",
       "7    25255.961693  78.978740   5.108068  "
      ]
     },
     "execution_count": 333,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try that again but this time using the country as the index column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 334,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "df = pd.read_csv('./test_data.csv', index_col='country')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 335,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>country isocode</th>\n",
       "      <th>year</th>\n",
       "      <th>POP</th>\n",
       "      <th>XRAT</th>\n",
       "      <th>tcgdp</th>\n",
       "      <th>cc</th>\n",
       "      <th>cg</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>country</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Argentina</th>\n",
       "      <td>ARG</td>\n",
       "      <td>2000</td>\n",
       "      <td>37335.653</td>\n",
       "      <td>0.999500</td>\n",
       "      <td>295072.218690</td>\n",
       "      <td>75.716805</td>\n",
       "      <td>5.578804</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Australia</th>\n",
       "      <td>AUS</td>\n",
       "      <td>2000</td>\n",
       "      <td>19053.186</td>\n",
       "      <td>1.724830</td>\n",
       "      <td>541804.652100</td>\n",
       "      <td>67.759026</td>\n",
       "      <td>6.720098</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>India</th>\n",
       "      <td>IND</td>\n",
       "      <td>2000</td>\n",
       "      <td>1006300.297</td>\n",
       "      <td>44.941600</td>\n",
       "      <td>1728144.374800</td>\n",
       "      <td>64.575551</td>\n",
       "      <td>14.072206</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Israel</th>\n",
       "      <td>ISR</td>\n",
       "      <td>2000</td>\n",
       "      <td>6114.570</td>\n",
       "      <td>4.077330</td>\n",
       "      <td>129253.894230</td>\n",
       "      <td>64.436451</td>\n",
       "      <td>10.266688</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Malawi</th>\n",
       "      <td>MWI</td>\n",
       "      <td>2000</td>\n",
       "      <td>11801.505</td>\n",
       "      <td>59.543808</td>\n",
       "      <td>5026.221784</td>\n",
       "      <td>74.707624</td>\n",
       "      <td>11.658954</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>South Africa</th>\n",
       "      <td>ZAF</td>\n",
       "      <td>2000</td>\n",
       "      <td>45064.098</td>\n",
       "      <td>6.939830</td>\n",
       "      <td>227242.369490</td>\n",
       "      <td>72.718710</td>\n",
       "      <td>5.726546</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>United States</th>\n",
       "      <td>USA</td>\n",
       "      <td>2000</td>\n",
       "      <td>282171.957</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>9898700.000000</td>\n",
       "      <td>72.347054</td>\n",
       "      <td>6.032454</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Uruguay</th>\n",
       "      <td>URY</td>\n",
       "      <td>2000</td>\n",
       "      <td>3219.793</td>\n",
       "      <td>12.099592</td>\n",
       "      <td>25255.961693</td>\n",
       "      <td>78.978740</td>\n",
       "      <td>5.108068</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              country isocode  year          POP       XRAT           tcgdp  \\\n",
       "country                                                                       \n",
       "Argentina                 ARG  2000    37335.653   0.999500   295072.218690   \n",
       "Australia                 AUS  2000    19053.186   1.724830   541804.652100   \n",
       "India                     IND  2000  1006300.297  44.941600  1728144.374800   \n",
       "Israel                    ISR  2000     6114.570   4.077330   129253.894230   \n",
       "Malawi                    MWI  2000    11801.505  59.543808     5026.221784   \n",
       "South Africa              ZAF  2000    45064.098   6.939830   227242.369490   \n",
       "United States             USA  2000   282171.957   1.000000  9898700.000000   \n",
       "Uruguay                   URY  2000     3219.793  12.099592    25255.961693   \n",
       "\n",
       "                      cc         cg  \n",
       "country                              \n",
       "Argentina      75.716805   5.578804  \n",
       "Australia      67.759026   6.720098  \n",
       "India          64.575551  14.072206  \n",
       "Israel         64.436451  10.266688  \n",
       "Malawi         74.707624  11.658954  \n",
       "South Africa   72.718710   5.726546  \n",
       "United States  72.347054   6.032454  \n",
       "Uruguay        78.978740   5.108068  "
      ]
     },
     "execution_count": 335,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's drop the year since it's not very informative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 336,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>country isocode</th>\n",
       "      <th>POP</th>\n",
       "      <th>XRAT</th>\n",
       "      <th>tcgdp</th>\n",
       "      <th>cc</th>\n",
       "      <th>cg</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>country</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Argentina</th>\n",
       "      <td>ARG</td>\n",
       "      <td>37335.653</td>\n",
       "      <td>0.999500</td>\n",
       "      <td>295072.218690</td>\n",
       "      <td>75.716805</td>\n",
       "      <td>5.578804</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Australia</th>\n",
       "      <td>AUS</td>\n",
       "      <td>19053.186</td>\n",
       "      <td>1.724830</td>\n",
       "      <td>541804.652100</td>\n",
       "      <td>67.759026</td>\n",
       "      <td>6.720098</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>India</th>\n",
       "      <td>IND</td>\n",
       "      <td>1006300.297</td>\n",
       "      <td>44.941600</td>\n",
       "      <td>1728144.374800</td>\n",
       "      <td>64.575551</td>\n",
       "      <td>14.072206</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Israel</th>\n",
       "      <td>ISR</td>\n",
       "      <td>6114.570</td>\n",
       "      <td>4.077330</td>\n",
       "      <td>129253.894230</td>\n",
       "      <td>64.436451</td>\n",
       "      <td>10.266688</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Malawi</th>\n",
       "      <td>MWI</td>\n",
       "      <td>11801.505</td>\n",
       "      <td>59.543808</td>\n",
       "      <td>5026.221784</td>\n",
       "      <td>74.707624</td>\n",
       "      <td>11.658954</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>South Africa</th>\n",
       "      <td>ZAF</td>\n",
       "      <td>45064.098</td>\n",
       "      <td>6.939830</td>\n",
       "      <td>227242.369490</td>\n",
       "      <td>72.718710</td>\n",
       "      <td>5.726546</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>United States</th>\n",
       "      <td>USA</td>\n",
       "      <td>282171.957</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>9898700.000000</td>\n",
       "      <td>72.347054</td>\n",
       "      <td>6.032454</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Uruguay</th>\n",
       "      <td>URY</td>\n",
       "      <td>3219.793</td>\n",
       "      <td>12.099592</td>\n",
       "      <td>25255.961693</td>\n",
       "      <td>78.978740</td>\n",
       "      <td>5.108068</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              country isocode          POP       XRAT           tcgdp  \\\n",
       "country                                                                 \n",
       "Argentina                 ARG    37335.653   0.999500   295072.218690   \n",
       "Australia                 AUS    19053.186   1.724830   541804.652100   \n",
       "India                     IND  1006300.297  44.941600  1728144.374800   \n",
       "Israel                    ISR     6114.570   4.077330   129253.894230   \n",
       "Malawi                    MWI    11801.505  59.543808     5026.221784   \n",
       "South Africa              ZAF    45064.098   6.939830   227242.369490   \n",
       "United States             USA   282171.957   1.000000  9898700.000000   \n",
       "Uruguay                   URY     3219.793  12.099592    25255.961693   \n",
       "\n",
       "                      cc         cg  \n",
       "country                              \n",
       "Argentina      75.716805   5.578804  \n",
       "Australia      67.759026   6.720098  \n",
       "India          64.575551  14.072206  \n",
       "Israel         64.436451  10.266688  \n",
       "Malawi         74.707624  11.658954  \n",
       "South Africa   72.718710   5.726546  \n",
       "United States  72.347054   6.032454  \n",
       "Uruguay        78.978740   5.108068  "
      ]
     },
     "execution_count": 336,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.drop(['year'], axis=1, inplace=True)\n",
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's add a column for GDP per capita"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 337,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "df['GDP percap'] = df['tcgdp'] / df['POP']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 338,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>country isocode</th>\n",
       "      <th>POP</th>\n",
       "      <th>XRAT</th>\n",
       "      <th>tcgdp</th>\n",
       "      <th>cc</th>\n",
       "      <th>cg</th>\n",
       "      <th>GDP percap</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>country</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Argentina</th>\n",
       "      <td>ARG</td>\n",
       "      <td>37335.653</td>\n",
       "      <td>0.999500</td>\n",
       "      <td>295072.218690</td>\n",
       "      <td>75.716805</td>\n",
       "      <td>5.578804</td>\n",
       "      <td>7.903229</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Australia</th>\n",
       "      <td>AUS</td>\n",
       "      <td>19053.186</td>\n",
       "      <td>1.724830</td>\n",
       "      <td>541804.652100</td>\n",
       "      <td>67.759026</td>\n",
       "      <td>6.720098</td>\n",
       "      <td>28.436433</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>India</th>\n",
       "      <td>IND</td>\n",
       "      <td>1006300.297</td>\n",
       "      <td>44.941600</td>\n",
       "      <td>1728144.374800</td>\n",
       "      <td>64.575551</td>\n",
       "      <td>14.072206</td>\n",
       "      <td>1.717325</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Israel</th>\n",
       "      <td>ISR</td>\n",
       "      <td>6114.570</td>\n",
       "      <td>4.077330</td>\n",
       "      <td>129253.894230</td>\n",
       "      <td>64.436451</td>\n",
       "      <td>10.266688</td>\n",
       "      <td>21.138673</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Malawi</th>\n",
       "      <td>MWI</td>\n",
       "      <td>11801.505</td>\n",
       "      <td>59.543808</td>\n",
       "      <td>5026.221784</td>\n",
       "      <td>74.707624</td>\n",
       "      <td>11.658954</td>\n",
       "      <td>0.425897</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>South Africa</th>\n",
       "      <td>ZAF</td>\n",
       "      <td>45064.098</td>\n",
       "      <td>6.939830</td>\n",
       "      <td>227242.369490</td>\n",
       "      <td>72.718710</td>\n",
       "      <td>5.726546</td>\n",
       "      <td>5.042648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>United States</th>\n",
       "      <td>USA</td>\n",
       "      <td>282171.957</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>9898700.000000</td>\n",
       "      <td>72.347054</td>\n",
       "      <td>6.032454</td>\n",
       "      <td>35.080382</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Uruguay</th>\n",
       "      <td>URY</td>\n",
       "      <td>3219.793</td>\n",
       "      <td>12.099592</td>\n",
       "      <td>25255.961693</td>\n",
       "      <td>78.978740</td>\n",
       "      <td>5.108068</td>\n",
       "      <td>7.843971</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              country isocode          POP       XRAT           tcgdp  \\\n",
       "country                                                                 \n",
       "Argentina                 ARG    37335.653   0.999500   295072.218690   \n",
       "Australia                 AUS    19053.186   1.724830   541804.652100   \n",
       "India                     IND  1006300.297  44.941600  1728144.374800   \n",
       "Israel                    ISR     6114.570   4.077330   129253.894230   \n",
       "Malawi                    MWI    11801.505  59.543808     5026.221784   \n",
       "South Africa              ZAF    45064.098   6.939830   227242.369490   \n",
       "United States             USA   282171.957   1.000000  9898700.000000   \n",
       "Uruguay                   URY     3219.793  12.099592    25255.961693   \n",
       "\n",
       "                      cc         cg  GDP percap  \n",
       "country                                          \n",
       "Argentina      75.716805   5.578804    7.903229  \n",
       "Australia      67.759026   6.720098   28.436433  \n",
       "India          64.575551  14.072206    1.717325  \n",
       "Israel         64.436451  10.266688   21.138673  \n",
       "Malawi         74.707624  11.658954    0.425897  \n",
       "South Africa   72.718710   5.726546    5.042648  \n",
       "United States  72.347054   6.032454   35.080382  \n",
       "Uruguay        78.978740   5.108068    7.843971  "
      ]
     },
     "execution_count": 338,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's sort the whole data frame by GDP per capita"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 339,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "df.sort_values(by='GDP percap', inplace=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 340,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>country isocode</th>\n",
       "      <th>POP</th>\n",
       "      <th>XRAT</th>\n",
       "      <th>tcgdp</th>\n",
       "      <th>cc</th>\n",
       "      <th>cg</th>\n",
       "      <th>GDP percap</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>country</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Malawi</th>\n",
       "      <td>MWI</td>\n",
       "      <td>11801.505</td>\n",
       "      <td>59.543808</td>\n",
       "      <td>5026.221784</td>\n",
       "      <td>74.707624</td>\n",
       "      <td>11.658954</td>\n",
       "      <td>0.425897</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>India</th>\n",
       "      <td>IND</td>\n",
       "      <td>1006300.297</td>\n",
       "      <td>44.941600</td>\n",
       "      <td>1728144.374800</td>\n",
       "      <td>64.575551</td>\n",
       "      <td>14.072206</td>\n",
       "      <td>1.717325</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>South Africa</th>\n",
       "      <td>ZAF</td>\n",
       "      <td>45064.098</td>\n",
       "      <td>6.939830</td>\n",
       "      <td>227242.369490</td>\n",
       "      <td>72.718710</td>\n",
       "      <td>5.726546</td>\n",
       "      <td>5.042648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Uruguay</th>\n",
       "      <td>URY</td>\n",
       "      <td>3219.793</td>\n",
       "      <td>12.099592</td>\n",
       "      <td>25255.961693</td>\n",
       "      <td>78.978740</td>\n",
       "      <td>5.108068</td>\n",
       "      <td>7.843971</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Argentina</th>\n",
       "      <td>ARG</td>\n",
       "      <td>37335.653</td>\n",
       "      <td>0.999500</td>\n",
       "      <td>295072.218690</td>\n",
       "      <td>75.716805</td>\n",
       "      <td>5.578804</td>\n",
       "      <td>7.903229</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Israel</th>\n",
       "      <td>ISR</td>\n",
       "      <td>6114.570</td>\n",
       "      <td>4.077330</td>\n",
       "      <td>129253.894230</td>\n",
       "      <td>64.436451</td>\n",
       "      <td>10.266688</td>\n",
       "      <td>21.138673</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Australia</th>\n",
       "      <td>AUS</td>\n",
       "      <td>19053.186</td>\n",
       "      <td>1.724830</td>\n",
       "      <td>541804.652100</td>\n",
       "      <td>67.759026</td>\n",
       "      <td>6.720098</td>\n",
       "      <td>28.436433</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>United States</th>\n",
       "      <td>USA</td>\n",
       "      <td>282171.957</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>9898700.000000</td>\n",
       "      <td>72.347054</td>\n",
       "      <td>6.032454</td>\n",
       "      <td>35.080382</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              country isocode          POP       XRAT           tcgdp  \\\n",
       "country                                                                 \n",
       "Malawi                    MWI    11801.505  59.543808     5026.221784   \n",
       "India                     IND  1006300.297  44.941600  1728144.374800   \n",
       "South Africa              ZAF    45064.098   6.939830   227242.369490   \n",
       "Uruguay                   URY     3219.793  12.099592    25255.961693   \n",
       "Argentina                 ARG    37335.653   0.999500   295072.218690   \n",
       "Israel                    ISR     6114.570   4.077330   129253.894230   \n",
       "Australia                 AUS    19053.186   1.724830   541804.652100   \n",
       "United States             USA   282171.957   1.000000  9898700.000000   \n",
       "\n",
       "                      cc         cg  GDP percap  \n",
       "country                                          \n",
       "Malawi         74.707624  11.658954    0.425897  \n",
       "India          64.575551  14.072206    1.717325  \n",
       "South Africa   72.718710   5.726546    5.042648  \n",
       "Uruguay        78.978740   5.108068    7.843971  \n",
       "Argentina      75.716805   5.578804    7.903229  \n",
       "Israel         64.436451  10.266688   21.138673  \n",
       "Australia      67.759026   6.720098   28.436433  \n",
       "United States  72.347054   6.032454   35.080382  "
      ]
     },
     "execution_count": 340,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we'll plot per capital GDP using the dataframe's plot method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 341,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.axes._subplots.AxesSubplot at 0x7f77a207cda0>"
      ]
     },
     "execution_count": 341,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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SNsM6n5P2LCVpvqQ9JL2sdas6pk6Snld1DGbdSDpb0jaStgJuAn4m6cNDOXfd27QlfT0i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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f77a20c5d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df['GDP percap'].plot(kind='bar')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are two exercises.  Feel free to consult documentation such as can be found [here](http://docs.scipy.org/doc/scipy/reference/).  The solutions are below.  The cell with \"solution below\" is mean to push them below your line of sight and save you from temptation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercise 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate 10000 data points from the exponential distribution with density\n",
    "\n",
    "$$\n",
    "f(x; \\alpha) = \\alpha \\exp(-\\alpha x)\n",
    "\\qquad\n",
    "(x > 0, \\alpha > 0)\n",
    "$$\n",
    "\n",
    "using `scipy.stats` and taking $\\alpha = 0.5$.  Then, after looking up the maximum likelihood estimator of $\\alpha$, compute the estimate given your data and check that it is in fact close to $\\alpha$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 342,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Put your solution here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Exercise 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the same data set, implement maximum likelihood again, but this time pretending that you don't know the analytical expression for the maximum likelihood estimator.  Set up the log likelihood function and maximize it numerically using a routine from `scipy.optimize`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 343,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Put your solution here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solutions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 344,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "solution below\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Print some nonsense to partially hide solutions\n",
    "filler_text = \"solution below\\n\" * 25\n",
    "print(filler_text)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution to Exercise 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After checking [the docs for the exponential distribution](http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.expon.html) we proceed as follows"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 345,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy.stats import expon\n",
    "alpha = 0.5\n",
    "n = 10000\n",
    "ep = expon(scale=1.0/alpha)  # scale controls the exponential parameter\n",
    "x = ep.rvs(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check we've got the right distribution here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 346,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f77a2118048>"
      ]
     },
     "execution_count": 346,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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TW1zI1q3v+DmliIjUh8q5GTPGcEnbh/DhY+aOcTy2YQSlviI60cLpaCIizZp+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\nRALM/wOHEis7sev3QQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f77a2129b38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8, 5))\n",
    "xmin, xmax = 0.001, 10.0\n",
    "ax.set_xlim(xmin, xmax)\n",
    "ax.hist(x, normed=True, bins=40, alpha=0.3)\n",
    "grid = np.linspace(xmin, xmax, 200)\n",
    "ax.plot(grid, ep.pdf(grid), 'g-', lw=2, label='true density')\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's [well-known](http://en.wikipedia.org/wiki/Exponential_distribution) that the MLE of $\\alpha$ is $1/\\bar x$ where $\\bar x$ is the mean of the sample.  Let's check that it is indeed close to $\\alpha$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 347,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "max likelihood estimate of alpha is 0.5042739418869412\n"
     ]
    }
   ],
   "source": [
    "alpha_mle = 1.0 / x.mean()\n",
    "print(\"max likelihood estimate of alpha is {}\".format(alpha_mle))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 348,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "s = x.sum()\n",
    "def neg_loglike(a):\n",
    "    \"Minus the log likelihood function for exponential\"\n",
    "    return - n * np.log(a) + a * s"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Minimize over a reasonable parameter space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 349,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5042730725954131"
      ]
     },
     "execution_count": 349,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.optimize import fminbound\n",
    "fminbound(neg_loglike, 0.01, 10.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is very close to the analytical value of the max likelihood estimator we got in exercise 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "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": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}