source-code/numpy/indexing_arrays.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Array indexing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T14:46:55.303275Z",
     "start_time": "2019-11-20T14:46:54.888648Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ellipsis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a 2 by 3 by 4 array with consecutive numbers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-19T14:49:35.474148Z",
     "start_time": "2019-11-19T14:49:35.465949Z"
    },
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "A = np.arange(0.0, 23.5, 1.0).reshape((2, 3, 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2D subarrays can be selected by specifying an index for a single dimension, while representing the others with an ellipsis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-19T14:49:46.502761Z",
     "start_time": "2019-11-19T14:49:46.462592Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  4.,  8.],\n",
       "       [12., 16., 20.]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[..., 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-19T14:50:20.941044Z",
     "start_time": "2019-11-19T14:50:20.916155Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  1.,  2.,  3.],\n",
       "       [ 4.,  5.,  6.,  7.],\n",
       "       [ 8.,  9., 10., 11.]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[0, ...]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# `np.newaxis`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a 400 by 500 of random numbers.  The array should subsequently be normalized along the rows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:08:29.975760Z",
     "start_time": "2019-11-20T15:08:29.961298Z"
    }
   },
   "outputs": [],
   "source": [
    "A = np.random.uniform(size=(400, 500))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The normalization could be donw in an iteration over the rows."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:32:30.698012Z",
     "start_time": "2019-11-20T15:32:27.885755Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.99 ms ± 22.3 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "B = np.empty_like(A)\n",
    "for row in range(A.shape[0]):\n",
    "    B[row, :] = A[row, :]/A[row, :].sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to work with `B`, we have to do the computation without using `%%timeit`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:32:30.698012Z",
     "start_time": "2019-11-20T15:32:27.885755Z"
    }
   },
   "outputs": [],
   "source": [
    "B = np.empty_like(A)\n",
    "for row in range(A.shape[0]):\n",
    "    B[row, :] = A[row, :]/A[row, :].sum()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify that `B` is normalized."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:33:19.294992Z",
     "start_time": "2019-11-20T15:33:19.269023Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9999999999999998, 1.0000000000000002)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.sum(axis=1).min(), B.sum(axis=1).max()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A somewhat faster alternative would be the following code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:34:35.570287Z",
     "start_time": "2019-11-20T15:34:33.333112Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.1 ms ± 6.66 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "B = np.empty_like(A)\n",
    "for row, total in enumerate(A.sum(axis=1)):\n",
    "    B[row, :] = A[row, :]/total"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:34:35.570287Z",
     "start_time": "2019-11-20T15:34:33.333112Z"
    }
   },
   "outputs": [],
   "source": [
    "B = np.empty_like(A)\n",
    "for row, total in enumerate(A.sum(axis=1)):\n",
    "    B[row, :] = A[row, :]/total"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:35:26.347457Z",
     "start_time": "2019-11-20T15:35:26.333023Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9999999999999998, 1.0000000000000002)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.sum(axis=1).min(), B.sum(axis=1).max()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, using broadcasting would be most efficient.  For this to work the array containing the sums of the rows must be a 2D array, which can be created from the original using `np.newaxis`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:36:49.645156Z",
     "start_time": "2019-11-20T15:36:48.770936Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "235 µs ± 1.85 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "B = A/A.sum(axis=1)[..., np.newaxis]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:36:49.645156Z",
     "start_time": "2019-11-20T15:36:48.770936Z"
    }
   },
   "outputs": [],
   "source": [
    "B = A/A.sum(axis=1)[..., np.newaxis]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:36:57.214506Z",
     "start_time": "2019-11-20T15:36:57.203685Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9999999999999998, 1.0000000000000002)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.sum(axis=1).min(), B.sum(axis=1).max()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integer indexing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a 4 by 6 2D array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:50:03.470986Z",
     "start_time": "2019-11-20T15:50:03.448749Z"
    }
   },
   "outputs": [],
   "source": [
    "A = np.arange(0.0, 23.5, 1.0).reshape((4, -1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:50:40.285773Z",
     "start_time": "2019-11-20T15:50:40.251872Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  1.,  2.,  3.,  4.,  5.],\n",
       "       [ 6.,  7.,  8.,  9., 10., 11.],\n",
       "       [12., 13., 14., 15., 16., 17.],\n",
       "       [18., 19., 20., 21., 22., 23.]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can select elements in this 2D array using lists of indices for each dimension."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T16:56:47.840833Z",
     "start_time": "2019-11-20T16:56:47.703511Z"
    }
   },
   "outputs": [],
   "source": [
    "i0, i1 = [0, 1, 2, 1, 0], [0, 1, 2, 3, 4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T16:57:02.722103Z",
     "start_time": "2019-11-20T16:57:02.645186Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  7., 14.,  9.,  4.])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[i0, i1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Boolean indexing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a 4 by 6 2D array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T15:50:03.470986Z",
     "start_time": "2019-11-20T15:50:03.448749Z"
    }
   },
   "outputs": [],
   "source": [
    "A = np.arange(0.0, 23.5, 1.0).reshape((4, -1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T17:04:41.619371Z",
     "start_time": "2019-11-20T17:04:41.587593Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  1.,  2.,  3.,  4.,  5.],\n",
       "       [ 6.,  7.,  8.,  9., 10., 11.],\n",
       "       [12., 13., 14., 15., 16., 17.],\n",
       "       [18., 19., 20., 21., 22., 23.]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Evaluating a Boolean condidtion taht involves an numpy array yields a Boolean array with the same shape."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T17:05:06.788576Z",
     "start_time": "2019-11-20T17:05:06.759976Z"
    }
   },
   "outputs": [],
   "source": [
    "i = A > 8.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T17:05:09.143539Z",
     "start_time": "2019-11-20T17:05:09.071613Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[False, False, False, False, False, False],\n",
       "       [False, False, False,  True,  True,  True],\n",
       "       [ True,  True,  True,  True,  True,  True],\n",
       "       [ True,  True,  True,  True,  True,  True]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This array can be used to index an array of the same shape, e.g., `A`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-11-20T17:06:35.344750Z",
     "start_time": "2019-11-20T17:06:35.320187Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 9., 10., 11., 12., 13., 14., 15., 16., 17., 18., 19., 20., 21.,\n",
       "       22., 23.])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[i]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the shape and even the number of dimensions of the resulting array is not necessarily that of the original array."
   ]
  }
 ],
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