Adhikari-Chandra_Class-3_NumPy-20211229.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:3.75em;color:purple; font-style:bold\"><br>\n",
    "Introduction to numpy:\n",
    "</p><br>\n",
    "\n",
    "Chandra Mani Adhikari, PhD\n",
    "\n",
    "Dec 2021, Fayettevilllle, NC\n",
    "\n",
    "<p style=\"font-family: Arial; font-size:1.05em;color:#2462C0; font-style:bold\"><br>\n",
    "NumPy= Numerical Python. \n",
    "    \n",
    "Numpy provides us with high performance multi-dimensional arrays which we can use as vectors or matrices.  \n",
    "\n",
    "<p style=\"font-family: Arial; font-size:1.05em;color:#2462C0; font-style:bold\"><br>\n",
    "The key features of numpy:\n",
    "\n",
    "- ndarrays: n-dimensional arrays of the same data type which are fast and space-efficient.  There are a number of built-in methods for ndarrays which allow for rapid processing of data without using loops (e.g., compute the mean).\n",
    "- Broadcasting: a useful tool which defines implicit behavior between multi-dimensional arrays of different sizes.\n",
    "- Vectorization: enables numeric operations on ndarrays.\n",
    "- Input/Output: simplifies reading and writing of data from/to file.\n",
    "\n",
    "<p style=\"font-family: Arial; font-size:1.05em;color:#2462C0; font-style:bold\"><br>\n",
    "Why is NumPy Fast?\n",
    "\n",
    "NumPy has a vectorized code. It is densely packed in memory due to its homogeneous type. It frees the memory faster. \n",
    "\n",
    "\n",
    "<b>Additional Resources:</b><br>\n",
    "<a href=\"https://docs.scipy.org/doc/numpy/reference/\">Numpy Documentation</a><br>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.75em;color:purple; font-style:bold\"><br>\n",
    "\n",
    "Getting started with ndarray<br><br></p>\n",
    "\n",
    "**ndarrays** are time and space-efficient multidimensional arrays at the core of numpy. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "\n",
    "How to create Rank 1 numpy arrays:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'numpy.ndarray'>\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "an_array = np.array([1, 11, 111, 1111])  # Create a rank 1 array\n",
    "\n",
    "print(type(an_array))              # The type of an ndarray is: \"<class 'numpy.ndarray'>\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(4,)\n"
     ]
    }
   ],
   "source": [
    "# test the shape of the array we just created, it should have just one dimension (Rank 1)\n",
    "print(an_array.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11\n"
     ]
    }
   ],
   "source": [
    "print(an_array[1]) #In Python indexing begin from 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 11 111 1111\n"
     ]
    }
   ],
   "source": [
    "# because this is a 1-rank array, we need only one index to accesss each element\n",
    "print(an_array[0], an_array[1], an_array[2],an_array[3]) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "an_array[0] =888            # ndarrays are mutable, here we change an element of the array\n",
    "\n",
    "print(an_array)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "\n",
    "How to create a Rank 2 numpy array:</p>\n",
    "\n",
    "A rank 2 **ndarray** is one with two dimensions.  Notice the format below of [ [row] , [row] ].  2 dimensional arrays are great for representing matrices which are often useful in data science."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  2  3]\n",
      " [11 12 13]]\n",
      "The shape is 2 rows, 3 columns:  (2, 3)\n",
      "Accessing elements [0,0], [0,1], and [1,0] of the ndarray:  1 ,  2 ,  11\n"
     ]
    }
   ],
   "source": [
    "another = np.array([[1,2,3],[11,12,13]])   # Create a rank 2 array\n",
    "\n",
    "print(another)  # print the array\n",
    "\n",
    "print(\"The shape is 2 rows, 3 columns: \", another.shape)  # rows x columns                   \n",
    "\n",
    "print(\"Accessing elements [0,0], [0,1], and [1,0] of the ndarray: \", another[0, 0], \", \",another[0, 1],\", \", another[1, 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Joining NumPy Arrays**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 6,  8, 10, 12])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#What does array1 + array2 do?\n",
    "array1 + array2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 2 3 4 5 6 7 8]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "array1 = np.array([1,2,3,4])\n",
    "array2 = np.array([5,6,7,8])\n",
    "array=np.concatenate((array1,array2))\n",
    "print(array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2]\n",
      " [3 4]\n",
      " [5 6]\n",
      " [7 8]]\n"
     ]
    }
   ],
   "source": [
    "#How to join 2-D arrays?\n",
    "ar1= np.array([[1,2],[3,4]])\n",
    "ar2=np.array([[5,6],[7,8]])\n",
    "arC=np.concatenate((ar1,ar2),axis=0)#axis=0 means joining along the columns\n",
    "print(arC)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 5 6]\n",
      " [3 4 7 8]]\n"
     ]
    }
   ],
   "source": [
    "arR=np.concatenate((ar1,ar2),axis=1) # axis=1 means along the rows\n",
    "print(arR)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.25em;color:#2462C0; font-style:bold\"><br>\n",
    "\n",
    "There are many way to create numpy arrays:\n",
    "</p>\n",
    "\n",
    "Here we create a number of different size arrays with different shapes and different pre-filled values.  numpy has a number of built in methods which help us quickly and easily create multidimensional arrays."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0. 0. 0.]\n",
      " [0. 0. 0.]\n",
      " [0. 0. 0.]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "# create a 3x3 array of zeros\n",
    "ex1 = np.zeros((3,3))      \n",
    "print(ex1)                              "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[5. 5. 5. 5.]\n",
      " [5. 5. 5. 5.]\n",
      " [5. 5. 5. 5.]\n",
      " [5. 5. 5. 5.]]\n"
     ]
    }
   ],
   "source": [
    "# create a 4x4 array filled with 5.0\n",
    "ex2 = np.full((4,4), 5.0)  \n",
    "print(ex2)   "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 0. 0. 0.]\n",
      " [0. 1. 0. 0.]\n",
      " [0. 0. 1. 0.]\n",
      " [0. 0. 0. 1.]]\n"
     ]
    }
   ],
   "source": [
    "# create a 4x4 matrix with the diagonal 1s and the others 0\n",
    "ex3 = np.eye(4,4)\n",
    "print(ex3)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 1. 1. 1. 1.]\n",
      " [1. 1. 1. 1. 1.]]\n"
     ]
    }
   ],
   "source": [
    "# create a 2D-array of ones\n",
    "ex4 = np.ones((2,5))\n",
    "print(ex4)    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.67100029 0.57413541 0.08096473 0.09774681]\n",
      " [0.61373744 0.6408935  0.80814513 0.3656748 ]\n",
      " [0.87678313 0.76569919 0.90353169 0.32644243]\n",
      " [0.23077947 0.48192698 0.33620231 0.30436702]]\n"
     ]
    }
   ],
   "source": [
    "# create an array of random floats between 0 and 1\n",
    "ex5 = np.random.random((4,4))\n",
    "print(ex5)    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.75em;color:purple; font-style:bold\"><br>\n",
    "Array Indexing\n",
    "<br><br></p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Slice indexing:\n",
    "</p>\n",
    "\n",
    "Similar to the use of slice indexing with lists and strings, we can use slice indexing to pull out sub-regions of ndarrays."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  2  3  4  5]\n",
      " [11 12 13 14 15]\n",
      " [21 22 23 24 25]\n",
      " [31 32 33 34 35]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "# Rank 2 array of shape (4, 5)\n",
    "an_array = np.array([[1,2,3,4,5], [11,12,13,14,15], [21,22,23,24,25],[31,32,33,34,35]])\n",
    "print(an_array)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use array slicing to get a subarray consisting of the first 3 rows x 3 columns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2  3  4]\n",
      " [12 13 14]\n",
      " [22 23 24]]\n"
     ]
    }
   ],
   "source": [
    "a_slice = an_array[:3, 1:4]\n",
    "print(a_slice)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When you modify a slice, you actually modify the underlying array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Before: 2\n",
      "After: 1000\n"
     ]
    }
   ],
   "source": [
    "print(\"Before:\", an_array[0, 1])   #inspect the element at 0, 1  \n",
    "a_slice[0, 0] = 1000    # a_slice[0, 0] is the same piece of data as an_array[0, 1]\n",
    "print(\"After:\", an_array[0, 1])    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Use both integer indexing & slice indexing\n",
    "</p>\n",
    "\n",
    "We can use combinations of integer indexing and slice indexing to create different shaped matrices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[11 12 13 14]\n",
      " [21 22 23 24]\n",
      " [31 32 33 34]]\n"
     ]
    }
   ],
   "source": [
    "# Create a Rank 2 array of shape (3, 4)\n",
    "an_array = np.array([[11,12,13,14], [21,22,23,24], [31,32,33,34]])\n",
    "print(an_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[21 22 23 24] (4,)\n"
     ]
    }
   ],
   "source": [
    "# Using both integer indexing & slicing generates an array of lower rank\n",
    "row_rank1 = an_array[1, :]    # Rank 1 view \n",
    "\n",
    "print(row_rank1, row_rank1.shape)  # notice only a single []"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[21 22 23 24]] (1, 4)\n"
     ]
    }
   ],
   "source": [
    "# Slicing alone: generates an array of the same rank as the an_array\n",
    "row_rank2 = an_array[1:2, :]  # Rank 2 view \n",
    "\n",
    "print(row_rank2, row_rank2.shape)   # Notice the [[ ]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "[12 22 32] (3,)\n",
      "\n",
      "[[12]\n",
      " [22]\n",
      " [32]] (3, 1)\n"
     ]
    }
   ],
   "source": [
    "#We can do the same thing for columns of an array:\n",
    "\n",
    "print()\n",
    "col_rank1 = an_array[:, 1]\n",
    "col_rank2 = an_array[:, 1:2]\n",
    "\n",
    "print(col_rank1, col_rank1.shape)  # Rank 1\n",
    "print()\n",
    "print(col_rank2, col_rank2.shape)  # Rank 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: We can also use negative indexing. -1 being the last element. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([31, 32, 33, 34])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "an_array[-1,:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Array Indexing for changing elements:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sometimes it's useful to use an array of indexes to access or change elements."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Original Array:\n",
      "[[11 12 13 14]\n",
      " [21 22 23 24]\n",
      " [31 32 33 34]\n",
      " [41 42 43 44]]\n"
     ]
    }
   ],
   "source": [
    "# Create a new array\n",
    "an_array = np.array([[11,12,13,14], [21,22,23,24], [31,32,33,34], [41,42,43,44]])\n",
    "\n",
    "print('Original Array:')\n",
    "print(an_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Col indices picked :  [0 1 2 3]\n",
      "\n",
      "Rows indices picked :  [0 1 2 3]\n"
     ]
    }
   ],
   "source": [
    "# Create an array of indices\n",
    "col_indices = np.array([0, 1, 2, 3])\n",
    "print('\\nCol indices picked : ', col_indices)\n",
    "\n",
    "row_indices = np.arange(4)\n",
    "print('\\nRows indices picked : ', row_indices)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 ,  0\n",
      "1 ,  1\n",
      "2 ,  2\n",
      "3 ,  3\n"
     ]
    }
   ],
   "source": [
    "# Examine the pairings of row_indices and col_indices.  These are the elements we'll change next.\n",
    "for row,col in zip(row_indices,col_indices):\n",
    "    print(row, \", \",col)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Values in the array at those indices:  [11 22 33 44]\n"
     ]
    }
   ],
   "source": [
    "# Select one element from each row\n",
    "print('Values in the array at those indices: ',an_array[row_indices, col_indices])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Changed Array:\n",
      "[[100011     12     13     14]\n",
      " [    21 100022     23     24]\n",
      " [    31     32 100033     34]\n",
      " [    41     42     43 100044]]\n"
     ]
    }
   ],
   "source": [
    "# Change one element from each row using the indices selected\n",
    "an_array[row_indices, col_indices] += 100000\n",
    "\n",
    "print('\\nChanged Array:')\n",
    "print(an_array)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.75em;color:purple; font-style:bold\"><br>\n",
    "Boolean Indexing\n",
    "<br><br></p>\n",
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Array Indexing for changing elements:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[11 12]\n",
      " [21 22]\n",
      " [31 32]]\n"
     ]
    }
   ],
   "source": [
    "# create a 3x2 array\n",
    "an_array = np.array([[11,12], [21, 22], [31, 32]])\n",
    "print(an_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[False, False],\n",
       "       [ True,  True],\n",
       "       [ True,  True]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# create a filter which will be boolean values for whether each element meets this condition\n",
    "filter = (an_array > 15)\n",
    "filter"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that the filter is a same size ndarray as an_array which is filled with True for each element whose corresponding element in an_array which is greater than 15 and False for those elements whose value is less than 15."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[21 22 31 32]\n"
     ]
    }
   ],
   "source": [
    "# we can now select just those elements which meet that criteria\n",
    "print(an_array[filter])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([12, 22, 32])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# For short, we could have just used the approach below without the need for the separate filter array.\n",
    "\n",
    "an_array[(an_array % 2 == 0)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is particularly useful is that we can actually change elements in the array applying a similar logical filter.  Let's add 100 to all the even values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 11 112]\n",
      " [ 21 122]\n",
      " [ 31 132]]\n"
     ]
    }
   ],
   "source": [
    "an_array[an_array % 2 == 0] +=100\n",
    "print(an_array)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.75em;color:purple; font-style:bold\"><br>\n",
    "Datatypes and Array Operations\n",
    "<br><br></p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Datatypes:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ex1 = np.array([11, 12]) # Python assigns the  data type\n",
    "print(ex1.dtype)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ex2 = np.array([11.0, 12.0]) # Python assigns the  data type\n",
    "print(ex2.dtype)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ex3 = np.array([11, 21], dtype=np.int64) #You can also tell Python the  data type\n",
    "print(ex3.dtype)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# you can use this to force floats into integers (using floor function)\n",
    "ex4 = np.array([11.1,12.7], dtype=np.int64)\n",
    "print(ex4.dtype)\n",
    "print()\n",
    "print(ex4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# you can use this to force integers into floats if you anticipate\n",
    "# the values may change to floats later\n",
    "ex5 = np.array([11, 21], dtype=np.float64)\n",
    "print(ex5.dtype)\n",
    "print()\n",
    "print(ex5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Arithmetic Array Operations:\n",
    "\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([[111,112],[121,122]], dtype=np.int)\n",
    "y = np.array([[211.1,212.1],[221.1,222.1]], dtype=np.float64)\n",
    "\n",
    "print(x)\n",
    "print()\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# add\n",
    "print(x + y)         # The plus sign works\n",
    "print()\n",
    "print(np.add(x, y))  # so does the numpy function \"add\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# subtract\n",
    "print(x - y)\n",
    "print()\n",
    "print(np.subtract(x, y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# multiply\n",
    "print(x * y)\n",
    "print()\n",
    "print(np.multiply(x, y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# divide\n",
    "print(x / y)\n",
    "print()\n",
    "print(np.divide(x, y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# square root\n",
    "print(np.sqrt(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# exponent (e ** x)\n",
    "print(np.exp(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.25em;color:purple; font-style:bold\"><br>\n",
    "Statistical Methods, Sorting, and Set Operations:\n",
    "<br><br>\n",
    "</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Basic Statistical Operations:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  2.06392169 -11.47083777  -7.48943027 -16.19493525  -3.47418929]\n",
      " [ -3.49336514  11.6867531   -3.11243149  12.11429351 -25.05963762]]\n"
     ]
    }
   ],
   "source": [
    "# setup a random 2 x 4 matrix\n",
    "arr = 10 * np.random.randn(2,5)\n",
    "print(arr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-4.442985852094031\n"
     ]
    }
   ],
   "source": [
    "# compute the mean for all elements\n",
    "print(arr.mean())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-7.31309418 -1.57287753]\n"
     ]
    }
   ],
   "source": [
    "# compute the means by row\n",
    "print(arr.mean(axis = 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ -0.71472172   0.10795767  -5.30093088  -2.04032087 -14.26691345]\n"
     ]
    }
   ],
   "source": [
    "# compute the means by column\n",
    "print(arr.mean(axis = 0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-44.429858520940314\n"
     ]
    }
   ],
   "source": [
    "# sum all the elements\n",
    "print(arr.sum())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-7.48943027 -3.11243149]\n"
     ]
    }
   ],
   "source": [
    "# compute the medians\n",
    "print(np.median(arr, axis = 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Sorting:\n",
    "</p>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.92964256 -0.25912623 -0.44998102  0.73604048 -2.33207984 -0.04760347\n",
      " -0.88446465  0.85168097 -0.69869518 -1.4345007 ]\n"
     ]
    }
   ],
   "source": [
    "# create a 10 element array of randoms\n",
    "unsorted = np.random.randn(10)\n",
    "\n",
    "print(unsorted)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-2.33207984 -1.4345007  -0.88446465 -0.69869518 -0.44998102 -0.25912623\n",
      " -0.04760347  0.73604048  0.85168097  0.92964256]\n",
      "\n",
      "[ 0.92964256 -0.25912623 -0.44998102  0.73604048 -2.33207984 -0.04760347\n",
      " -0.88446465  0.85168097 -0.69869518 -1.4345007 ]\n"
     ]
    }
   ],
   "source": [
    "# create copy and sort\n",
    "sorted = np.array(unsorted)\n",
    "sorted.sort()\n",
    "\n",
    "print(sorted)\n",
    "print()\n",
    "print(unsorted)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-2.33207984 -1.4345007  -0.88446465 -0.69869518 -0.44998102 -0.25912623\n",
      " -0.04760347  0.73604048  0.85168097  0.92964256]\n"
     ]
    }
   ],
   "source": [
    "# inplace sorting\n",
    "unsorted.sort() \n",
    "\n",
    "print(unsorted)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Finding Unique elements:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "array = np.array([1,2,1,4,2,1,4,2])\n",
    "\n",
    "print(np.unique(array))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Set Operations with np.array data type:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['desk' 'chair' 'bulb'] ['lamp' 'bulb' 'chair']\n"
     ]
    }
   ],
   "source": [
    "s1 = np.array(['desk','chair','bulb'])\n",
    "s2 = np.array(['lamp','bulb','chair'])\n",
    "print(s1, s2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['bulb' 'chair']\n"
     ]
    }
   ],
   "source": [
    "print( np.intersect1d(s1, s2) ) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['bulb' 'chair' 'desk' 'lamp']\n"
     ]
    }
   ],
   "source": [
    "print( np.union1d(s1, s2) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['desk']\n"
     ]
    }
   ],
   "source": [
    "print( np.setdiff1d(s1, s2) )# elements in s1 that are not in s2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False  True  True]\n"
     ]
    }
   ],
   "source": [
    "print( np.in1d(s1, s2) )#which element of s1 is also in s2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.75em;color:purple; font-style:bold\"><br>\n",
    "Broadcasting:\n",
    "<br><br>\n",
    "</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Introduction to broadcasting. <br>\n",
    "For more details, please see: <br>\n",
    "https://docs.scipy.org/doc/numpy-1.10.1/user/basics.broadcasting.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0. 0. 0.]\n",
      " [0. 0. 0.]\n",
      " [0. 0. 0.]\n",
      " [0. 0. 0.]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "start = np.zeros((4,3))\n",
    "print(start)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 0 2]\n"
     ]
    }
   ],
   "source": [
    "# create a rank 1 ndarray with 3 values\n",
    "add_rows = np.array([1, 0, 2])\n",
    "print(add_rows)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 0. 2.]\n",
      " [1. 0. 2.]\n",
      " [1. 0. 2.]\n",
      " [1. 0. 2.]]\n"
     ]
    }
   ],
   "source": [
    "y = start + add_rows  # add to each row of 'start' using broadcasting\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0]\n",
      " [1]\n",
      " [2]\n",
      " [3]]\n"
     ]
    }
   ],
   "source": [
    "# create an ndarray which is 4 x 1 to broadcast across columns\n",
    "add_cols = np.array([[0,1,2,3]])\n",
    "add_cols = add_cols.T\n",
    "\n",
    "print(add_cols)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0. 0. 0.]\n",
      " [1. 1. 1.]\n",
      " [2. 2. 2.]\n",
      " [3. 3. 3.]]\n"
     ]
    }
   ],
   "source": [
    "# add to each column of 'start' using broadcasting\n",
    "y = start + add_cols \n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 1. 1.]\n",
      " [1. 1. 1.]\n",
      " [1. 1. 1.]\n",
      " [1. 1. 1.]]\n"
     ]
    }
   ],
   "source": [
    "# this will just broadcast in both dimensions\n",
    "add_scalar = np.array([1])  \n",
    "print(start+add_scalar)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  2  3  4]\n",
      " [ 5  6  7  8]\n",
      " [ 9 10 11 12]]\n"
     ]
    }
   ],
   "source": [
    "# create our 3x4 matrix\n",
    "arrA = np.array([[1,2,3,4],[5,6,7,8],[9,10,11,12]])\n",
    "print(arrA)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 1, 0, 2]\n"
     ]
    }
   ],
   "source": [
    "# create our 4x1 array\n",
    "arrB = [0,1,0,2]\n",
    "print(arrB)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  3  3  6]\n",
      " [ 5  7  7 10]\n",
      " [ 9 11 11 14]]\n"
     ]
    }
   ],
   "source": [
    "# add the two together using broadcasting\n",
    "print(arrA + arrB)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.75em;color:purple; font-style:bold\"><br>\n",
    "Speedtest: ndarrays vs lists\n",
    "<br><br>\n",
    "</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First setup paramaters for the speed test. We'll be testing time to sum elements in an ndarray versus a list."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import arange\n",
    "from timeit import Timer\n",
    "\n",
    "size    = 1000000\n",
    "timeits = 1000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'numpy.ndarray'>\n"
     ]
    }
   ],
   "source": [
    "# create the ndarray with values 0,1,2...,size-1\n",
    "nd_array = arange(size)\n",
    "print( type(nd_array) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time taken by numpy ndarray: 0.000521 seconds\n"
     ]
    }
   ],
   "source": [
    "# timer expects the operation as a parameter, \n",
    "# here we pass nd_array.sum()\n",
    "timer_numpy = Timer(\"nd_array.sum()\", \"from __main__ import nd_array\")\n",
    "\n",
    "print(\"Time taken by numpy ndarray: %f seconds\" % \n",
    "      (timer_numpy.timeit(timeits)/timeits))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'list'>\n"
     ]
    }
   ],
   "source": [
    "# create the list with values 0,1,2...,size-1\n",
    "a_list = list(range(size))\n",
    "print (type(a_list) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time taken by list:  0.031250 seconds\n"
     ]
    }
   ],
   "source": [
    "# timer expects the operation as a parameter, here we pass sum(a_list)\n",
    "timer_list = Timer(\"sum(a_list)\", \"from __main__ import a_list\")\n",
    "\n",
    "print(\"Time taken by list:  %f seconds\" % \n",
    "      (timer_list.timeit(timeits)/timeits))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.75em;color:purple; font-style:bold\"><br>\n",
    "Read or Write to Disk:\n",
    "<br><br>\n",
    "</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.3em;color:#2462C0; font-style:bold\"><br>\n",
    "Binary Format:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([ 23.23, 24.24] )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.save('an_array', x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([23.23, 24.24])"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.load('an_array.npy')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.3em;color:#2462C0; font-style:bold\"><br>\n",
    "Text Format:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.savetxt('array.txt', X=x, delimiter=',')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.323000000000000043e+01\n",
      "2.423999999999999844e+01\n"
     ]
    }
   ],
   "source": [
    "!cat array.txt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([23.23, 24.24])"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.loadtxt('array.txt', delimiter=',')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.75em;color:purple; font-style:bold\"><br>\n",
    "Additional Common ndarray Operations\n",
    "<br><br></p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Dot Product on Matrices and Inner Product on Vectors:\n",
    "\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[4 4]\n",
      " [4 4]]\n",
      "\n",
      "[[4 4]\n",
      " [4 4]]\n"
     ]
    }
   ],
   "source": [
    "# determine the dot product of two matrices\n",
    "x2d = np.array([[1,1],[1,1]])\n",
    "y2d = np.array([[2,2],[2,2]])\n",
    "\n",
    "print(x2d.dot(y2d))\n",
    "print()\n",
    "print(np.dot(x2d, y2d))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "180\n",
      "\n",
      "180\n"
     ]
    }
   ],
   "source": [
    "# determine the inner product of two vectors\n",
    "a1d = np.array([9 , 9 ])\n",
    "b1d = np.array([10, 10])\n",
    "\n",
    "print(a1d.dot(b1d))\n",
    "print()\n",
    "print(np.dot(a1d, b1d))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[18 18]\n",
      "\n",
      "[18 18]\n"
     ]
    }
   ],
   "source": [
    "# dot produce on an array and vector\n",
    "print(x2d.dot(a1d))\n",
    "print()\n",
    "print(np.dot(x2d, a1d))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Inverse Matrix:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3,  2,  3],\n",
       "       [ 4, 16,  2],\n",
       "       [17,  3,  9]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A=np.array([[3,2,3],[4,16,2],[17,3,9]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.37297297  0.02432432  0.11891892]\n",
      " [ 0.00540541  0.06486486 -0.01621622]\n",
      " [ 0.7027027  -0.06756757 -0.10810811]]\n"
     ]
    }
   ],
   "source": [
    "print(np.linalg.inv(A)) #inverse of A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "CW: Find inverse of the following matrix\n",
    "\n",
    "$\\begin{equation*}\n",
    "B = \n",
    "\\begin{pmatrix}\n",
    "1 & 1  \\\\\n",
    "1 & 2  \\\\\n",
    "\\end{pmatrix}\n",
    "\\end{equation*}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Sum:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "66\n"
     ]
    }
   ],
   "source": [
    "# sum elements in the array\n",
    "ex1 = np.array([[11,12],[21,22]])\n",
    "\n",
    "print(np.sum(ex1))          # add all members"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[32 34]\n"
     ]
    }
   ],
   "source": [
    "print(np.sum(ex1, axis=0))  # columnwise sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[23 43]\n"
     ]
    }
   ],
   "source": [
    "print(np.sum(ex1, axis=1))  # rowwise sum"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Element-wise Functions: </p>\n",
    "\n",
    "For example, let's compare two arrays values to get the maximum of each."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.23284402, -0.63041143,  0.22763238,  0.15232129,  1.25849663,\n",
       "        0.55876665,  0.62720116,  0.24619595])"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# random array\n",
    "x = np.random.randn(8)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.04705588,  0.2855565 , -0.50242668,  1.20492901,  1.36098074,\n",
       "        0.78693659,  0.10101255,  0.4215848 ])"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# another random array\n",
    "y = np.random.randn(8)\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.04705588, 0.2855565 , 0.22763238, 1.20492901, 1.36098074,\n",
       "       0.78693659, 0.62720116, 0.4215848 ])"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# returns element wise maximum between two arrays\n",
    "\n",
    "np.maximum(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Reshaping array:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19]\n"
     ]
    }
   ],
   "source": [
    "# grab values from 0 through 19 in an array\n",
    "arr = np.arange(20)\n",
    "print(arr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0,  1,  2,  3,  4],\n",
       "       [ 5,  6,  7,  8,  9],\n",
       "       [10, 11, 12, 13, 14],\n",
       "       [15, 16, 17, 18, 19]])"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# reshape to be a 4 x 5 matrix\n",
    "arr.reshape(4,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Transpose:\n",
    "\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[11, 21],\n",
       "       [12, 22]])"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# transpose\n",
    "ex1 = np.array([[11,12],[21,22]])\n",
    "\n",
    "ex1.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Indexing using where():</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_1 = np.array([1,2,3,4,5])\n",
    "\n",
    "y_1 = np.array([11,22,33,44,55])\n",
    "\n",
    "filter = np.array([True, False, True, False, True])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1 22  3 44  5]\n"
     ]
    }
   ],
   "source": [
    "out = np.where(filter, x_1, y_1)\n",
    "print(out)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.68790132, 0.32078411, 0.77000088, 0.60542305, 0.57696129],\n",
       "       [0.89863451, 0.34835134, 0.85371216, 0.10266528, 0.88060263],\n",
       "       [0.44324867, 0.29055933, 0.33697253, 0.86111342, 0.87614499],\n",
       "       [0.15898321, 0.7998803 , 0.16988014, 0.64498052, 0.64311993],\n",
       "       [0.72981852, 0.44852495, 0.91335728, 0.15805574, 0.32859936]])"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mat = np.random.rand(5,5)\n",
    "mat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1000,   -1, 1000, 1000, 1000],\n",
       "       [1000,   -1, 1000,   -1, 1000],\n",
       "       [  -1,   -1,   -1, 1000, 1000],\n",
       "       [  -1, 1000,   -1, 1000, 1000],\n",
       "       [1000,   -1, 1000,   -1,   -1]])"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.where( mat > 0.5, 1000, -1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "\"any\" or \"all\" conditionals:</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr_bools = np.array([ True, False, True, True, False ])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr_bools.any()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr_bools.all()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Random Number Generation:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.51504003 -1.11717734  0.25005086  0.84789693  1.90686578]\n"
     ]
    }
   ],
   "source": [
    "Y = np.random.normal(size = (1,5))[0]\n",
    "print(Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[38 37 31 25]\n"
     ]
    }
   ],
   "source": [
    "Z = np.random.randint(low=2,high=50,size=4)\n",
    "print(Z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([31, 38, 37, 25])"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.permutation(Z) #return a new ordering of elements in Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.50630635, 0.87404887, 0.19221168, 0.61075033])"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.uniform(size=4) #uniform distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.31115148, -0.29792705,  0.539375  ,  0.48554358])"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.normal(size=4) #normal distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Merging data sets:\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 5 27]\n",
      " [47 12]]\n",
      "\n",
      "[[27 13]\n",
      " [29 35]]\n"
     ]
    }
   ],
   "source": [
    "K = np.random.randint(low=2,high=50,size=(2,2))\n",
    "print(K)\n",
    "\n",
    "print()\n",
    "M = np.random.randint(low=2,high=50,size=(2,2))\n",
    "print(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5, 27],\n",
       "       [47, 12],\n",
       "       [27, 13],\n",
       "       [29, 35]])"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.vstack((K,M))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5, 27, 27, 13],\n",
       "       [47, 12, 29, 35]])"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.hstack((K,M))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5, 27],\n",
       "       [47, 12],\n",
       "       [27, 13],\n",
       "       [29, 35]])"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.concatenate([K, M], axis = 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5, 27, 27, 29],\n",
       "       [47, 12, 13, 35]])"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.concatenate([K, M.T], axis = 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:2.75em;color:purple; font-style:bold\"><br>\n",
    "Projects:\n",
    "<br><br>\n",
    "</p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Recap: Plotting\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1fc580b4cd0>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.plot([0,0.25,0.5,0.75,1,1.25,1.5, 1.75, 2],[0,0.0125,0.25,0.75**2,1,1.25**2,1.5**2,1.75**2,4],\"rd-\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.plot([0,0.25,0.5,0.75,1,1.25,1.5, 1.75, 2],[0,0.0125,0.25,0.75**2,1,1.25**2,1.5**2,1.75**2,4],\"rd-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Recap: Random Number\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import random \n",
    "random.choice(range(0,10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2, 5, 4, 3])"
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "x = np.array([1,2,5,4,3,6])\n",
    "x[1:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.choice(list([1,2,3,4]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "47"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(random.choice(range(10)) for i in range(10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "45"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(random.sample(range(10),10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 4, 1, 6, 2, 7, 9, 3, 5, 8]"
      ]
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.sample(range(10),10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[9, 4, 8, 4, 5, 8, 0, 8, 1, 4]"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "list(random.choice(range(10)) for i in range(10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import random \n",
    "random.choice([1,2,3,4,5,6])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Roll a dice\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "rolls=[]\n",
    "for i in range(100):\n",
    "    rolls.append(random.choice([1,2,3,4,5,6]))\n",
    "plt.hist(rolls, bins=np.linspace(0.5,6.5,12));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "rolls=[]\n",
    "for i in range(50000):\n",
    "    rolls.append(random.choice([1,2,3,4,5,6]))\n",
    "plt.hist(rolls, bins=np.linspace(0.5,6.5,12));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy \n",
    "numpy.random.normal?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.28438369, 0.82201781, 0.06121663],\n",
       "       [0.97318597, 0.60260568, 0.46855158],\n",
       "       [0.17463329, 0.80796803, 0.38415704],\n",
       "       [0.24227777, 0.41904316, 0.12786771],\n",
       "       [0.59706206, 0.22558619, 0.69714711]])"
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.random((5,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Normal Distribution\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.29043733,  1.03674076,  1.37992621, -0.28955312,  0.09952414])"
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.normal(0,1,5) #(mean, standard deviation, length of list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.55594758, -1.27167292, -1.13362447, -0.03608246,  1.35271953],\n",
       "       [ 2.15786508,  1.10372163,  1.33411421, -1.77539929, -0.37820308]])"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.normal(0,1,(2,5)) #(mean, standard deviation, length of list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[6, 5, 6, 2, 5, 5],\n",
       "       [6, 1, 3, 3, 3, 1],\n",
       "       [1, 4, 2, 2, 1, 5],\n",
       "       [4, 2, 6, 2, 3, 1],\n",
       "       [5, 2, 2, 6, 1, 3],\n",
       "       [4, 2, 5, 2, 5, 2],\n",
       "       [6, 2, 5, 3, 6, 3],\n",
       "       [2, 3, 1, 6, 3, 2],\n",
       "       [2, 5, 1, 5, 5, 4],\n",
       "       [2, 3, 2, 6, 1, 1]])"
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X=np.random.randint(1,7,(10,6))\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.sum?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 4, 1, 6, 3, 6],\n",
       "       [5, 2, 3, 5, 4, 6],\n",
       "       [4, 4, 3, 4, 2, 3],\n",
       "       [2, 3, 2, 5, 3, 6],\n",
       "       [3, 5, 2, 2, 3, 1],\n",
       "       [1, 1, 3, 4, 4, 3],\n",
       "       [1, 5, 3, 6, 4, 1],\n",
       "       [6, 5, 2, 4, 2, 5],\n",
       "       [4, 6, 6, 1, 2, 5],\n",
       "       [4, 3, 6, 1, 4, 5]])"
      ]
     },
     "execution_count": 162,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X=np.random.randint(1,7,(10,6))\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([39, 41, 33, 33, 30, 23])"
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(X,axis=0)# Summing up rows"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([22, 25, 20, 21, 16, 16, 20, 24, 24, 23])"
      ]
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sum(X,axis=1) # Summing up Columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 6, 6, 6, 1, 6, 6, 5, 4, 6],\n",
       "       [1, 4, 5, 3, 2, 1, 1, 2, 1, 1],\n",
       "       [2, 4, 1, 1, 5, 1, 6, 1, 3, 3],\n",
       "       [4, 5, 4, 2, 5, 2, 3, 1, 5, 5],\n",
       "       [6, 2, 4, 2, 2, 4, 5, 6, 4, 2],\n",
       "       [2, 3, 5, 3, 4, 3, 4, 4, 5, 3],\n",
       "       [3, 4, 5, 3, 4, 5, 1, 4, 4, 2],\n",
       "       [1, 2, 6, 5, 5, 3, 5, 6, 1, 1],\n",
       "       [3, 5, 3, 2, 1, 6, 6, 3, 5, 2],\n",
       "       [1, 5, 1, 6, 1, 1, 1, 2, 6, 1],\n",
       "       [2, 3, 1, 2, 4, 5, 1, 4, 6, 6],\n",
       "       [3, 2, 2, 6, 6, 3, 5, 2, 3, 6],\n",
       "       [2, 5, 3, 5, 4, 2, 2, 6, 1, 1],\n",
       "       [2, 5, 3, 1, 5, 4, 1, 6, 2, 1],\n",
       "       [5, 2, 5, 1, 4, 3, 6, 3, 2, 6],\n",
       "       [5, 6, 5, 6, 4, 6, 3, 3, 5, 5],\n",
       "       [3, 1, 2, 5, 4, 5, 2, 4, 4, 6],\n",
       "       [4, 1, 4, 3, 2, 3, 6, 2, 2, 4],\n",
       "       [1, 1, 5, 6, 6, 1, 1, 3, 6, 2],\n",
       "       [4, 1, 6, 2, 6, 4, 5, 5, 1, 3],\n",
       "       [1, 6, 5, 2, 3, 1, 2, 5, 3, 2],\n",
       "       [5, 3, 6, 3, 4, 6, 5, 4, 6, 5],\n",
       "       [2, 4, 6, 2, 3, 2, 5, 4, 2, 1],\n",
       "       [4, 6, 5, 4, 5, 3, 1, 6, 6, 2],\n",
       "       [1, 5, 6, 3, 1, 1, 6, 2, 5, 4],\n",
       "       [1, 2, 4, 2, 2, 1, 2, 4, 2, 1],\n",
       "       [3, 2, 5, 4, 3, 3, 5, 1, 6, 3],\n",
       "       [2, 5, 3, 3, 3, 3, 6, 3, 5, 2],\n",
       "       [2, 1, 4, 4, 4, 5, 1, 6, 4, 2],\n",
       "       [6, 6, 5, 5, 4, 2, 2, 3, 5, 4],\n",
       "       [3, 3, 5, 2, 4, 4, 5, 6, 2, 6],\n",
       "       [1, 3, 3, 4, 2, 4, 1, 3, 5, 1],\n",
       "       [2, 4, 4, 5, 2, 4, 6, 4, 5, 3],\n",
       "       [4, 6, 3, 2, 6, 1, 3, 1, 5, 4],\n",
       "       [3, 2, 1, 6, 4, 6, 2, 1, 3, 3],\n",
       "       [4, 3, 4, 2, 3, 6, 3, 1, 1, 3],\n",
       "       [1, 1, 6, 6, 5, 3, 5, 1, 6, 4],\n",
       "       [2, 4, 3, 4, 1, 4, 4, 5, 5, 4],\n",
       "       [3, 5, 6, 5, 1, 2, 1, 3, 2, 3],\n",
       "       [3, 6, 1, 6, 6, 5, 4, 2, 3, 6],\n",
       "       [4, 2, 2, 2, 6, 2, 5, 6, 2, 4],\n",
       "       [3, 5, 4, 1, 3, 1, 1, 3, 1, 1],\n",
       "       [3, 3, 2, 1, 6, 3, 4, 2, 2, 3],\n",
       "       [5, 1, 3, 3, 2, 3, 4, 3, 3, 3],\n",
       "       [2, 1, 4, 3, 2, 6, 6, 2, 5, 1],\n",
       "       [6, 5, 3, 3, 5, 6, 5, 6, 5, 1],\n",
       "       [3, 5, 6, 5, 5, 6, 4, 1, 1, 3],\n",
       "       [4, 3, 3, 1, 3, 5, 2, 6, 5, 4],\n",
       "       [1, 1, 4, 4, 2, 4, 6, 3, 5, 3],\n",
       "       [2, 4, 2, 1, 4, 5, 5, 4, 2, 3],\n",
       "       [3, 3, 6, 1, 4, 5, 3, 3, 5, 1],\n",
       "       [2, 5, 5, 3, 1, 4, 4, 1, 5, 1],\n",
       "       [5, 5, 6, 2, 3, 6, 3, 1, 2, 2],\n",
       "       [1, 4, 1, 4, 3, 3, 1, 2, 5, 2],\n",
       "       [1, 6, 6, 1, 5, 2, 2, 1, 3, 5],\n",
       "       [4, 6, 2, 3, 5, 3, 1, 5, 1, 4],\n",
       "       [3, 2, 2, 4, 6, 2, 4, 3, 6, 5],\n",
       "       [3, 4, 4, 2, 1, 1, 4, 6, 5, 5],\n",
       "       [6, 2, 6, 6, 1, 4, 6, 2, 5, 2],\n",
       "       [1, 2, 2, 2, 5, 2, 6, 6, 6, 3],\n",
       "       [4, 3, 3, 5, 2, 1, 6, 4, 1, 1],\n",
       "       [3, 5, 4, 6, 2, 2, 3, 4, 4, 5],\n",
       "       [5, 4, 5, 3, 2, 6, 2, 6, 1, 5],\n",
       "       [3, 6, 2, 1, 2, 5, 3, 2, 3, 1],\n",
       "       [2, 4, 5, 5, 4, 6, 4, 1, 1, 5],\n",
       "       [4, 6, 3, 4, 1, 6, 6, 6, 3, 4],\n",
       "       [4, 6, 2, 1, 2, 6, 6, 2, 1, 1],\n",
       "       [3, 2, 1, 6, 1, 4, 1, 1, 3, 6],\n",
       "       [6, 4, 6, 4, 5, 3, 4, 6, 5, 5],\n",
       "       [5, 4, 2, 6, 1, 3, 2, 2, 5, 4],\n",
       "       [3, 1, 3, 6, 1, 2, 4, 5, 4, 1],\n",
       "       [5, 1, 2, 2, 6, 2, 2, 4, 3, 2],\n",
       "       [4, 6, 4, 5, 1, 3, 2, 1, 3, 1],\n",
       "       [6, 3, 1, 4, 2, 1, 4, 3, 5, 6],\n",
       "       [6, 5, 3, 1, 3, 4, 2, 3, 6, 6],\n",
       "       [6, 6, 1, 5, 4, 4, 6, 1, 3, 4],\n",
       "       [6, 2, 5, 5, 5, 6, 1, 5, 4, 2],\n",
       "       [6, 1, 4, 3, 6, 5, 4, 5, 6, 4],\n",
       "       [2, 1, 6, 1, 6, 3, 6, 2, 6, 2],\n",
       "       [4, 6, 6, 4, 1, 5, 5, 6, 6, 4],\n",
       "       [3, 6, 1, 3, 4, 2, 2, 4, 3, 1],\n",
       "       [3, 6, 4, 1, 5, 6, 3, 2, 5, 2],\n",
       "       [4, 6, 2, 3, 4, 3, 1, 4, 1, 1],\n",
       "       [2, 2, 3, 1, 4, 3, 4, 3, 6, 2],\n",
       "       [6, 6, 3, 6, 1, 4, 2, 5, 6, 5],\n",
       "       [4, 6, 1, 1, 3, 5, 6, 5, 3, 6],\n",
       "       [3, 1, 5, 3, 6, 2, 2, 1, 6, 6],\n",
       "       [1, 1, 2, 1, 3, 2, 5, 5, 2, 3],\n",
       "       [1, 5, 2, 2, 5, 1, 3, 2, 2, 5],\n",
       "       [4, 4, 4, 6, 2, 1, 5, 3, 4, 1],\n",
       "       [6, 5, 1, 2, 3, 3, 6, 2, 2, 3],\n",
       "       [6, 6, 4, 4, 6, 1, 1, 2, 2, 3],\n",
       "       [5, 3, 2, 4, 2, 4, 3, 1, 4, 1],\n",
       "       [1, 2, 6, 5, 3, 6, 3, 4, 6, 2],\n",
       "       [4, 4, 2, 6, 5, 2, 1, 6, 1, 5],\n",
       "       [4, 5, 5, 6, 1, 2, 4, 4, 1, 2],\n",
       "       [6, 6, 2, 2, 3, 3, 6, 4, 6, 4],\n",
       "       [5, 2, 4, 6, 2, 4, 3, 6, 4, 2],\n",
       "       [2, 5, 6, 3, 6, 5, 6, 5, 5, 5],\n",
       "       [1, 3, 2, 5, 4, 5, 2, 5, 5, 4]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X=np.random.randint(1,7,(100,10))\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([48, 21, 27, 36, 37, 36, 35, 35, 36, 25, 34, 38, 31, 30, 37, 48, 36,\n",
       "       31, 32, 37, 30, 47, 31, 42, 34, 21, 35, 35, 33, 42, 40, 27, 39, 35,\n",
       "       31, 30, 38, 36, 31, 42, 35, 23, 29, 30, 32, 45, 39, 36, 33, 32, 34,\n",
       "       31, 35, 26, 32, 34, 37, 35, 40, 35, 30, 38, 39, 28, 37, 43, 31, 28,\n",
       "       48, 34, 30, 29, 30, 35, 39, 40, 41, 44, 35, 47, 29, 37, 29, 30, 44,\n",
       "       40, 35, 25, 28, 34, 33, 35, 29, 38, 36, 34, 42, 38, 48, 36])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y=np.sum(X,axis=1)\n",
    "Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 1.,  2.,  9., 11., 15., 29., 16., 10.,  4.,  3.]),\n",
       " array([20., 23., 26., 29., 32., 35., 38., 41., 44., 47., 50.]),\n",
       " <BarContainer object of 10 artists>)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "X=np.random.randint(1,7,(100,10))\n",
    "Y=np.sum(X,axis=1)\n",
    "plt.hist(Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.34271609999996144\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import time \n",
    "start_time=time.perf_counter() #time.clock() has been deprecated\n",
    "X=np.random.randint(1,7,(1000000,10)) # matix of sizes 1000000 by 10\n",
    "Y=np.sum(X,axis=1) # summing over rows\n",
    "plt.hist(Y,np.linspace(1,70,69));\n",
    "end_time=time.perf_counter()\n",
    "print(end_time-start_time)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Random Walk\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00225513,  0.2347021 ,  1.26917254, -1.61026875,  0.58173896,\n",
       "         0.58038764, -2.44735424,  0.71052789, -0.27109953],\n",
       "       [-2.06050629, -2.15493917,  0.17201829, -1.3417805 , -1.61515607,\n",
       "         1.94310586, -0.58679995, -0.14964036,  0.02476626]])"
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "delta_X=np.random.normal(0,1,(2,9))\n",
    "delta_X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1f5b7e8f220>]"
      ]
     },
     "execution_count": 178,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(delta_X[0],delta_X[1],\"go\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "delta_X=np.random.normal(0,1,(2,20))\n",
    "X=np.cumsum(delta_X,axis=1)\n",
    "plt.plot(X[0], X[1], \"bs\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p style=\"font-family: Arial; font-size:1.75em;color:#2462C0; font-style:bold\"><br>\n",
    "Begin random walk from origin\n",
    "</p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.        ,  0.08794916, -0.51530824, -1.17405348, -0.59676864,\n",
       "        -0.77251259,  0.70308557,  0.35060047,  1.02921601, -0.5104622 ,\n",
       "        -0.33011257, -0.58228312, -0.83742138, -0.47719654,  0.45513244,\n",
       "        -0.19240951,  0.68907212,  0.48939035,  0.80719795,  1.41656159,\n",
       "         2.13064784],\n",
       "       [ 0.        , -0.78664376,  0.23226694, -0.0555344 , -0.38675443,\n",
       "        -0.29626394, -1.62974351, -4.06043569, -3.60164845, -5.84952231,\n",
       "        -7.06616426, -6.67160633, -7.66269465, -7.8423965 , -7.38678161,\n",
       "        -6.31952138, -4.51491077, -2.94886356, -3.79014123, -3.67670627,\n",
       "        -4.82715233]])"
      ]
     },
     "execution_count": 181,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_0=np.array([[0],[0]])\n",
    "delta_x=np.random.normal(0,1,(2,5))\n",
    "X_1=np.cumsum(delta_X, axis=1)\n",
    "X=np.concatenate((X_0, X_1),axis=1)\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_0=np.array([[0],[0]])\n",
    "delta_X=np.random.normal(0,1,(2,5))\n",
    "X_1=np.cumsum(delta_X, axis=1)\n",
    "X=np.concatenate((X_0, X_1),axis=1)\n",
    "plt.plot(X[0],X[1],'rd--')\n",
    "plt.savefig(\"rw.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_0=np.array([[0],[0]])\n",
    "delta_X=np.random.normal(0,1,(2,1000))\n",
    "X_1=np.cumsum(delta_X, axis=1)\n",
    "X=np.concatenate((X_0, X_1),axis=1)\n",
    "plt.plot(X[0],X[1],'rd--')\n",
    "plt.savefig(\"rw1.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_0=np.array([[0],[0]])\n",
    "delta_X=np.random.normal(0,1,(2,10000))\n",
    "X_1=np.cumsum(delta_X, axis=1)\n",
    "X=np.concatenate((X_0, X_1),axis=1)\n",
    "plt.plot(X[0],X[1],'rd--')\n",
    "plt.savefig(\"rw2.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_0=np.array([[0],[0]])\n",
    "delta_X=np.random.normal(0,1,(2,10000))\n",
    "X_1=np.cumsum(delta_X, axis=1)\n",
    "X=np.concatenate((X_0, X_1),axis=1)\n",
    "plt.plot(X[0],X[1],'go--')\n",
    "plt.savefig(\"rw3.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAX8AAAD5CAYAAADP2jUWAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuNCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8QVMy6AAAACXBIWXMAAAsTAAALEwEAmpwYAAAcGUlEQVR4nO3dfYwdV3kG8OfZDZhKkBLqZe3ace1AUjWGytQXCwShlLiNSVNMUoUGIaBCqklEKiit2gQLCSEhlQQKpbRJnUJLBDSlmC8RkImrJqhVPribOI6NMdghKE68y0KkEqnI1N63f9yZ7Ox45t65d+bMnDPz/KSr3Z379e7s3mfOnDlzhmYGERHplqmmCxARkfop/EVEOkjhLyLSQQp/EZEOUviLiHSQwl9EpIPOcfniJM8HcDuANQCWAOwxs78l+QEAfwJgMXro+8zsG8Nea/Xq1bZx40aH1YqItM/c3NxPzGwmvdxp+AM4DeDPzexBks8DMEfyrui+j5nZR4q+0MaNG9Hv950UKSLSViR/lLXcafib2UkAJ6PvnyZ5BMA6l+8pIiKj1dbnT3IjgJcBuD9adD3JgyQ/TfK8uuoQEZGawp/kcwHsBfAeM/sZgFsAvAjAFgz2DD6a87xdJPsk+4uLi1kPERGRCTgPf5LPwiD4P2dmXwIAM1swszNmtgTgNgDbsp5rZnvMrGdmvZmZs45XiIjIhJyGP0kC+BSAI2b2N4nlaxMPuxLAIZd1iIjISq5H+7wKwFsBPELyQLTsfQDeTHILAAPwGIB3Oq5DREZYswZYWDh7+ewsMD9ffz3iluvRPv8FgBl3DR3TL9KErodf1u8+bLmETWf4SqesWQOQ2bcuh9+aNcPvj9fRqMdJOBT+0il1BHneBsan4EzXWHS9dGFD2BUKf5GK+b4Hkde9Jd2i8BcpIK+rKL5NTy+3pofxofVfNvh9+B2kPIW/tFq6e8OVpaViodqGFncbfgdR+EvLKajc8O0YhoxP4S9Ss3GD09cDyNqwhk3hL9KAhYXBcYKij81bHh9vaIpPGyMZj8JfpCFLS8PvL3IAOX6dcUJ4drZ4jePQnkBYXE/vICJDJMM9DuWyITrq+fHZyi4OgMev2ZWzokOmlr+IJxYWqms9N90Fo70A/yn8pdVcdXH4Lhm+eQeMXWt6AyTDqdtHWi3Z9VBH4PmoqVa4Wv9+U8tfpKXqauEXqUEjgvyj8BeR2mhvwB8KfxGRDlL4i4h0kMJfRKSDFP4iIh2k8JfO6OqYf5/ob+CPxsKf5A6SR0keI3lDU3VId2i6gebpb+CPRsKf5DSAvwfwegAXA3gzyYubqEW6RS1PkYGmWv7bABwzs0fN7BcA7gCws6FapEPm5wGzpqsQaV5T4b8OwOOJn09Ey1YguYtkn2R/cXGxtuJExjE7qz0KCU9T4Z910vlZ7TEz22NmPTPrzczM1FCWSHGzs4O9iPl59WUXoQ2kX5oK/xMAzk/8vB7Akw3VIpJpWFg1OV99vNEJqftK8/v7p6nw/w6AC0luIvlsANcA+FpDtUgHjWqFxmEVh2z61mSQLSyENUFa0+tLsjUypbOZnSZ5PYB9AKYBfNrMDjdRi3RT6GEUX7/Xd1NTo+vUXkEzGpvP38y+AeAbTb2/SNVmZ9s1a2UVl5UcdZ3isq8vk9PFXERKmp4uFnK+qvLYQQh7IzKg6R1ExpB1SUSfg18jbCSPwl9kDKF1USwsrBwdlHUrK7lBlHAo/EVS8i54Hmq4uR4dFNoGUQYU/tI5w8KdbGeYhTY8tIxRf9+urIdRFP7SOW0M9yLi4aFVBqBvQbpmzei/b3I9dHnjoPAX6aBx9gSGtaR925BWUU/WxqGNGwSFv0hHjQrKOPTrCHjfw9W3jVwVFP4iHeZLq37cLqm8ul1q216Awl+kw3xs0RbZEDRZt4/rbBIKf+kcnfgUDl+Dtg3HAxT+0jmaREyqlD5AHMrGQOEvIlIhX/dW0hT+IiIdpPCXTlK/v3SdpnSWVipypqeEIdQ5lXynlr+0koJfRpnqePp1/NeXNtHUwlJEVVNZ5wmlS1HhL63RdGs/lA99bNQ8//EttN9rmPh3qfLqa1NTK9dXKEOJFf4iFZmfDyssi4bU/Hw4v1NaekM2Pz/YQ6zy6mtnzlT3WnVyFv4kbyb5PZIHSX6Z5POj5RtJ/pzkgeh2q6saROqS7D+ONwJVXi3LhXFOSkr/TiFreg/RFy5b/ncBeImZ/SaA7wO4MXHfcTPbEt2udViDiBPpLpNQW39Aty704kKo685Z+JvZt8zsdPTjfQDWu3ovkbrEoR9Kv25R8RQFoQZZk0LdeNbV5/8OAN9M/LyJ5EMk7yF5Sd6TSO4i2SfZX1xcdF+lBC3UfmmfLCwMDoa2QZ3/D3FX0rApsn3bQNBKdOCR3A8g61fabWZfjR6zG0APwFVmZiRXAXiumf2U5FYAXwGw2cx+Nuy9er2e9fv9iWuVbnE53HOSj0xow09nZ4fv3YTw++T9nZqsvYnjJSTnzKyXXl7qDF8z2z7iTd8O4AoAl1q0lTGzUwBORd/PkTwO4CIASnaZSN1n8ybDY1RIJh8X0oHGkGrNor3A0ZxN70ByB4C/AvDbZva/ieUzAJ4yszMkLwBwIYBHXdUhYcsL9qmpaofrTapoSM7Ph9FaDl3RjbG4ndvnkwBWAbiLg//6+6KRPa8B8EGSpwGcAXCtmT3lsA4JWF64+hD84/Ctv7dNQhp6OsleoyvOwt/MXpyzfC+Ava7eV6RpmlTOT779XeIRVkVVvbHQrJ4iJXWxOye0YxhAePWmVV2/pncQkbOMOmAaUr96PPxSVlL4i0jmHDij+DCipkgNobf4k6o8dqRuH5GOmzTE4w1EU63qkA70VqXKDZla/iIdlJybqGwXThN7AF2/EEsVtArFaz50LQArw7INquyzr6P/Pz1nfsgT6flC4S9eKzKXfBzMrjYUbZvIzcV6qvo1Q5s1NVmvLw2WURT+4r2s+fGzDk6mH+fqQ+jbhzuvnrwrdbnYkFV1wRcXs6a6/D/IWqfx/6HvFP7SWnkbjXHCIOuxPl2sJT7xp66QH6bs+p6acrdhctEoiE/Syru5UOWGTOEvnTNqT2KSAK3qQ5ns2x72miFdV6Do+q6rayeEdZanytoV/iIVqOL6vbOzKwNwWGiGHGAymaq7rxT+IhUaFtij+uYV6DJM1f8fOslLpCYKd3+EODdR1dTyF5HOqaKbLnRq+YtIZ43aG2vzhHBq+YuI5GjzNBIt/tVERMo5c6a93UPq9hERyeDTlb9cbHwU/iIi8Cvsk1ydQa5uHxER+Bn8LrubnIU/yQ+QfILkgeh2eeK+G0keI3mU5GWuahARCUXdZ3K77vb5mJl9JLmA5MUArgGwGcCvAthP8iIz83zSVhERN5o4oNxEn/9OAHeY2SkAPyR5DMA2APc2UIuIiDNTU/5ei8B1n//1JA+S/DTJ86Jl6wA8nnjMiWjZWUjuItkn2V9cXHRcqoiEbM2a7KmVq7zoeRHJef59DX6gZPiT3E/yUMZtJ4BbALwIwBYAJwF8NH5axktlHs82sz1m1jOz3szMTJlSRaTl8g7YJufdn56u9j2zLpgTyhxOpbp9zGx7kceRvA3A16MfTwA4P3H3egBPlqlDRKSIpaVqp2yINyzDxBfc8Y3L0T5rEz9eCeBQ9P3XAFxDchXJTQAuBPCAqzpERJrk4xBSwO0B35tIbsGgS+cxAO8EADM7TPILAL4L4DSAd2mkj4iUUXe/fhs4C38ze+uQ+z4E4EOu3ltE2s3Xs3FDojN8RSQ4Cv7yFP4iIh2k8BcRccjX6aAV/iIijvg6zBNQ+IuIOOFz8AMKfxGpSRXTL8SvUZfkGbzjdN/4HvyALuYiIo4UHY45zsidukf5JAPc9zAfl1r+IlJaVqu+6qBu4kSuJiaGq4vCX0RKqeKEq6yATW9Qmhrb39ZzCtTtIyKlVBGORSZIa1KythD684tQy1+cyjvIl3WrerpdcSP9N+2atuwJKPzFiTggxvmgLC1V//6Tjiyp8sIgvlxkpKxJ/qbiL4W/VKpsQFQRiMP6oOPuhfT7FO1fLvp7TU8Xe62QNgAK/XZR+Esl4rArGxCTXnUpGd6TDC8cp+4iXVhF92LS7+vTXkJyA9bF7p220wFfqUSVXTajXrOq6XxJPy6wHQf7sN8p74BolfVrmuRuUfiL11y3OF1stMZVJnCrrF/BX4yvE7WNS+EvnRd6lwbZnuGHPjNruoJqqc9fnpHX35zu+w11tEqblTnArj790drS2k9S+Msz8gIkr2sheXBWwjHuwfFJxROiZd18DdPkRG7JWxv3qtTtIwDUim+DkDbCTR9fUDeZw5Y/yX8jeSC6PUbyQLR8I8mfJ+671VUNUkzbRnmkW2/SDaP2KpL/F10PfsBhy9/M/ij+nuRHAfxP4u7jZrbF1XvLcG0L+zTf54npgia7dRTsxTjv9iFJAG8C8DrX7yX52h74Up+pqezjQOpKCUsdB3wvAbBgZj9ILNtE8iGS95C8JO+JJHeR7JPsLy4uuq+0pRT8fguta2ppKfvAqII/LKXCn+R+kocybjsTD3szgH9N/HwSwAYzexmA9wL4PMlzs17fzPaYWc/MejMzM2VKDd44s2O6vqiGVMfXUS+jJEd6TTLst+rfO9T12KRS3T5mtn3Y/STPAXAVgK2J55wCcCr6fo7kcQAXAeiXqaXtFODhS3aLxHtjbTk+Me7/Z3ovoeg6UNdSdVx3+2wH8D0zOxEvIDlDcjr6/gIAFwJ41HEdQdM89+FLd4u0cWNe5uS/oi13BX91XIf/NVjZ5QMArwFwkOTDAL4I4Foze8pxHcGanvZj/hmZXNe6JCbZsM3Pj15PXVuPrjkd7WNmf5yxbC+AvS7fN1Q6MNtObenacU2t+nppegePKPhFpC4KfxGRDlL4i5SgKSQkVAp/kQlN6dOTSQdmw6BZPUUmtLSkA7lp2gsKh9ouHlGLSUTqovD3iIa6ScjUeAmLwt8z+gBJqNR4CYv6/D2T/ACpP1lEXFH4e0ihL6HRHmt4FP4iMhHNsBk29fl7Ijlfv0gIFPxhU/h7QvP6SGgmmbpZ/KHwl0aprzhcarCETeEvjVpY0DQJIStzARdplj520rilpZUXAtfeQHi0FxAehb94IXkxcGDlxkDCoD2AsCj8PaHW7rJ0K1LrJhzaAwiHwt8T8/MrW7tdb/EmW5BFru8qIuNR+Husy4GXbkGqRSlSrVLhT/JqkodJLpHspe67keQxkkdJXpZYvpXkI9F9nyB1WlNS8mQvBZ6IuFK25X8IwFUAvp1cSPJiANcA2AxgB4B/IDkd3X0LgF0ALoxuO0rW0CoKfBGpQ6nwN7MjZnY0466dAO4ws1Nm9kMAxwBsI7kWwLlmdq+ZGYDbAbyxTA1topESEroud1WGxtXEbusA3Jf4+US07P+i79PLM5HchcFeAjZs2FB9lTWbnh6MaW8rM81N1EWa4C1MI8Of5H4AWW3S3Wb21bynZSyzIcszmdkeAHsAoNfrBT/+pc3BD1Qb/MkWpPaI/KPAD9/I8Dez7RO87gkA5yd+Xg/gyWj5+ozlrdfmAKu6xZ8e5qrjIH5R8LeDq6GeXwNwDclVJDdhcGD3ATM7CeBpkq+IRvm8DUDe3kOrtDnA1NXTHQr+9ijV50/ySgB/B2AGwJ0kD5jZZWZ2mOQXAHwXwGkA7zKzM9HTrgPwLwB+CcA3o5uIeK7rJx62DS2Qv2iv17N+v990GRNT67i49L+k1p0fAokKSSE5Z2a99HJdxlG8o7D3j4Zwto/CX0RWUL9+N2hunwkkp2AocqtzpM/sbLE58XUBFclipuDvCrX8JzDuyB2XI33y+mHTH+A1a1bW0fZzDkRkOLX/atJ0n+mwDVBoU0g3vS5F2kDhXxOfd6VDOwGtzedMiNRF4V8TFyNYqmoBLyxohI1I16jPf0w+tJKHddGk+/ZFRLIo/MfkQ7BmtdKnpnQQV8ojNdSzK9Tt0xIKfqmKDw0cca8T4T9sXH6Rbpzp6eXHi4i0QevDf1Qf+MLC8gYgbyOhVrV0Td0nJ0r9Wh/+RXZh49Eu2t2tXpGzjcVP+jy0W6vDXy2XasVTR6RveeEeL9fBQxH/tHa0j4Y8Vi8vxBXuIuFpbcu/TcE/O9t810nZ92+i/qbXmYjPWtvyb4P0eOsm92bKtu7j57seMZVeZxqhJZKttS3/NkgH/fx8sb52n1VZc9YxiK53QVU5SV+I/19SnFr+nku2XJOt2lCPaaTDeXp6/KG0Ic1A2pRJzvjWeu0WtfwDEg9JbdOw1DNnwt6T8UnyAj3jrlet++4pFf4kryZ5mOQSyV5i+e+SnCP5SPT1dYn77iZ5lOSB6PbCMjXk0T9zeObnhw8bnaRLo43/B3lDbs+cyX58ursw69b17rIuKtvtcwjAVQD+MbX8JwD+wMyeJPkSAPsArEvc/xYz65d876Hm58PtGumyqkMo9P8DdcWIK6Va/mZ2xMyOZix/yMyejH48DOA5JFeVea9JxC2eNrb+6hT69X5DPVAeSp0Spjo+1n8I4CEzO5VY9s9Rl8/7yfzBeCR3keyT7C8uLk5cwKjWZDIY5Gxtm9vI9y4OdcVIHUaGP8n9JA9l3HYWeO5mAB8G8M7E4reY2UsBXBLd3pr3fDPbY2Y9M+vNzMyM/m2GGDUFgYgP9P8odRnZ529m2yd5YZLrAXwZwNvM7Hji9Z6Ivj5N8vMAtgG4fZL3GEfRVtTsbLj9w1KMz3M+qbUvdXHS7UPy+QDuBHCjmf13Yvk5JFdH3z8LwBUYHDT2Rtw/3KTQ+9h918TGXcMtxTdlh3peSfIEgFcCuJPkvuiu6wG8GMD7U0M6VwHYR/IggAMAngBwW5kaXKnrg5g1bC85RlvCF5+cp+GW4hNaIAnT6/Ws33c6OjTXqKGCw655mvfcotdJ9WFumrZd07WOdRrIx0o6gOScmfXSyzW9QwFlgi/00Gxb8NdB3TcSAvUue8yHA5MK/tGmpsp13wy7xvS415sWKUrh7zGNOiovK1irVuQ8iGEBX/TvrP8HqZK6faTVfAjMSWYuFXFNLX9ppelpPw6WAwp+8ZPCX1pJgSsynMJfcmnUSnF5B2inp5uuTCSbwt9jTYav6yGeeQdA2zSiZWGh2j0QbYylSgp/jyXPCq3zg1/H2aZ5B2KTVysremvTBiNLfBa4ht1KlTTaJwB1XowkxNZlvMFoI51kJ64o/APgOvinpvIvASjN0RQR4pK6fTpqdna5lb+01K1uFBFRy7+zhu1N+HBilIi4pZa/ZHK5F6A9C5HmqeUvQ1W1F1DnQWsRGU0t/wA0PQJnkuGV6XH8Cv7xNP03l/ZT+AcgfRUoH4JhVJgr7CejMf1SF3X7BCgOBp/GtqtbpxgN3xRfqOUfsKb3AMjB3DUK/pXy/i5N/71EkspewP1qkodJLpHsJZZvJPnzxMXbb03ct5XkIySPkfwE6VP7NSx5FwWv09KSgj8t7++irhzxSdmW/yEAVwH4dsZ9x81sS3S7NrH8FgC7AFwY3XaUrEFERMZUKvzN7IiZHS36eJJrAZxrZveamQG4HcAby9QgZ2tz90J8vVxftXndS7u47PPfRPIhkveQvCRatg7AicRjTkTLpELJboe2iH+feA4iX0NWXTsSipGjfUjuB5A1qnu3mX0152knAWwws5+S3ArgKyQ3A8jq38+NKJK7MOgiwoYNG0aVKhlmZ0f3yadnjvTtKExW0Cfr1QFnkfGNDH8z2z7ui5rZKQCnou/nSB4HcBEGLf31iYeuB/DkkNfZA2APAPR6vRa1Y+sTh2ReQPo2ZfAkeyt1D32dmsq+SIuveyMiWZx0+5CcITkdfX8BBgd2HzWzkwCeJvmKaJTP2wDk7T1IheoagRKfpDTJSWlTgQw8PnNGo3kkfGWHel5J8gSAVwK4k+S+6K7XADhI8mEAXwRwrZk9Fd13HYB/AnAMwHEA3yxTg1RvnBZsOuyzAjBvw5O+lb2mQB0tb7XupS1ogRwV7PV61u/3my5DAlD1MQDfusZExkFyzsx66eWB7GiLFFdkT2PYWbjqzpEu0Nw+0kkKdOk6tfxFRDpI4S8i0kEKfxGRDlL4i4h0kMJfRKSDghnnT3IRwI/GeMpqAD9xVE7VVKsbodQaSp2AanXFZa2/ZmYz6YXBhP+4SPazTmzwkWp1I5RaQ6kTUK2uNFGrun1ERDpI4S8i0kFtDv89TRcwBtXqRii1hlInoFpdqb3W1vb5i4hIvja3/EVEJEfw4U/yapKHSS6R7KXuu5HkMZJHSV6WWL6V5CPRfZ+ILixTd91bSN5H8gDJPslto+puEsk/jeo5TPKmxHLvagUAkn9B0kiuTizzqlaSN5P8HsmDJL9M8vmJ+7yqFQBI7ojqOUbyhqbrSSJ5Psn/JHkk+h99d7T8BSTvIvmD6Ot5TdcKACSno2ucfz36uf46zSzoG4DfAPDrAO4G0EssvxjAwwBWAdiEwYVjpqP7HsDgAjTE4GIyr2+g7m/F7wvgcgB3j6q7wXX8OwD2A1gV/fxCX2uN6jofwD4MzgtZ7WutAH4PwDnR9x8G8GGPa52O6rgAwLOj+i5u+m+dqG8tgN+Kvn8egO9H6/EmADdEy2+I13HTNwDvBfB5AF+Pfq69zuBb/mZ2xMyOZty1E8AdZnbKzH6IwZXDtpFcC+BcM7vXBmv6dgBvrK/iZxiAc6PvfxnL1zLOrLuB+pKuA/DXNrg2M8zsx9FyH2sFgI8B+EsM1nHMu1rN7Ftmdjr68T4sX9/au1qj9z9mZo+a2S8A3IFBnV4ws5Nm9mD0/dMAjgBYh0GNn4ke9hk081lfgeR6AL+PwRUNY7XXGXz4D7EOwOOJn09Ey9ZF36eX1+09AG4m+TiAjwC4MVqeV3eTLgJwCcn7Sd5D8uXRcu9qJfkGAE+Y2cOpu7yrNeUdWL6kqY+1+lhTJpIbAbwMwP0AZm1w7XBEX1/YYGmxj2PQOFlKLKu9ziAu5kJyP4A1GXftNrO8C8Bn9ePbkOWVG1Y3gEsB/JmZ7SX5JgCfArC9zvqSRtR6DoDzALwCwMsBfIHkBfCz1vdh0J1y1tMyljVaa/y/S3I3gNMAPhc/LePxTQ/L87Gms5B8LoC9AN5jZj9r4HDeUCSvAPBjM5sj+domawki/M1s+wRPO4FB329sPQZdKyewvHudXF65YXWTvB3Au6Mf/x3Lu4B5dTs1otbrAHwp6iZ7gOQSBnOReFUryZdi0Ef+cPShXw/gwehgule1xki+HcAVAC6N1i/QUK0j+FjTCiSfhUHwf87MvhQtXiC51sxORl2+P85/hVq8CsAbSF4O4DkAziX5WTRRZ9MHPio8gHI3Vh7w3YyVB80exfIB3+9g0IqND/he3kC9RwC8Nvr+UgBzo+pucN1eC+CD0fcXYbD7Tx9rTdX9GJYP+HpXK4AdAL4LYCa13Mdaz4nq2ITlA76bm/4bJ+ojBsfvPp5afjNWHki9qelaE7W9FssHfGuvs/EVUMEKvBKDVskpAAsA9iXu243BCIWjSIzoAdADcCi675OITnarue5XA5iLPkT3A9g6qu4G1/GzAXw2WmcPAnidr7Wm6n4m/H2sFYMDuY8DOBDdbvW11qimyzEYRXMcg26rxmtK1PZqDLqhDibW5+UAfgXAfwD4QfT1BU3Xmqg5Gf6116kzfEVEOqjNo31ERCSHwl9EpIMU/iIiHaTwFxHpIIW/iEgHKfxFRDpI4S8i0kEKfxGRDvp/kZHjR5A+B3MAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_0=np.array([[0],[0]])\n",
    "delta_X=np.random.normal(0,1,(2,10000))\n",
    "X_1=np.cumsum(delta_X, axis=1)\n",
    "X=np.concatenate((X_0, X_1),axis=1)\n",
    "plt.plot(X[0],X[1],'bs--')\n",
    "plt.savefig(\"rw4.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_0=np.array([[0],[0]])\n",
    "delta_X=np.random.normal(0,1,(2,10000))\n",
    "X_1=np.cumsum(delta_X, axis=1)\n",
    "X=np.concatenate((X_0, X_1),axis=1)\n",
    "plt.plot(X[0],X[1],'kd--')\n",
    "plt.savefig(\"rw5.pdf\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}