numpy-random.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# _NumPy_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### About\n",
    "\n",
    "NumPy is a numerical package for Python. It is the fundamental package for scientific computing with Python. It contains among other things:\n",
    "\n",
    "- a powerful N-dimensional array object\n",
    "- sophisticated (broadcasting) functions\n",
    "- tools for integrating C/C++ and Fortran code\n",
    "- useful linear algebra, Fourier transform, and random number capabilities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### This Notebook\n",
    "\n",
    "The main focus for this notebook will be of a subpackage within the NumPy package itself ([random](https://docs.scipy.org/doc/numpy/reference/routines.random.html)). This notebook will highlight some of the features of the NumPy.random subpackage with various explanations and references and adaptations to code snippets, which can be found above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Aims & Objectives\n",
    "\n",
    "This notebook will aim to provide a clear understanding of the various functions and concepts within the NumPy.random subpackage. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can import the entire module with a short name as shown above. This enables you to work with all the functions present in the module."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Table of Contents\n",
    "-  [Simple Random Data](#SimpleRandomData)\n",
    "    -  [random.rand](#random.rand)\n",
    "    -  [random.randn](#random.randn)\n",
    "    -  [random.randint](#random.randint)    \n",
    "    -  [random.random_sample](#random.random_sample)\n",
    "    -  [random.choice](#random.choice)\n",
    "    -  [random.bytes](#random.bytes)\n",
    "-  [Permutations](#Permutations)\n",
    "    -  [random.shuffle](#random.shuffle)\n",
    "    -  [random.permutation](#random.permutation)\n",
    "-  [Distributions](#Distributions)\n",
    "    -  [random.normal](#random.normal)\n",
    "    -  [random.binomial](#random.binomial)\n",
    "    -  [random.exponential](#random.exponential)\n",
    "    -  [random.poisson](#random.poisson)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import scipy as sp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_This notebook also contains references to other packages such as:_\n",
    "\n",
    "- [mathplotlib.pyplot](https://matplotlib.org/api/api_overview.html)\n",
    "- [scipy.special](https://docs.scipy.org/doc/scipy/reference/special.html)\n",
    "\n",
    "**Please Note** that these packages will not be the referenced throughtout the rest of this notebook."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simple random data <a id=\"SimpleRandomData\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.rand](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.rand.html#numpy.random.rand)  <a id=\"random.rand\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.78135128, 0.2379166 ],\n",
       "       [0.93429122, 0.08457713],\n",
       "       [0.26237834, 0.98203204],\n",
       "       [0.19913545, 0.22706284],\n",
       "       [0.66077598, 0.74930319]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.random.rand(5,2)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above creates an array of the given shape (5 rows, 2 columns) and populates it with random samples from a uniform distribution over (0, 1). The dimensions of the returned array, should all be positive. If no argument is given a single Python float is returned."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.randn](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.randn.html#numpy.random.randn)  <a id=\"random.randn\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.random.randn(3,4)\n",
    "y = np.sin(x)\n",
    "plt.plot(x, y, 'b.')\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "np.random.randn() allows you to sample from the normal distribution. It can take an integer or a shape for its input."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.randint](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.randint.html#numpy.random.randint)  <a id=\"random.randint\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 1, 0, 1, 0, 0, 1, 0, 1])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.randint(2, size=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example II__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 2, 0, 3],\n",
       "       [3, 0, 2, 4]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.randint(5, size=(2, 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Return random integers from low (inclusive) to high (exclusive). This function returns an int or array of ints, depending on if the size is specified."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.random_sample](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.random_sample.html#numpy.random.random_sample)  <a id=\"random.random_sample\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Return random floats in the half-open interval [0.0, 1.0).\n",
    "\n",
    "Results are from the “continuous uniform” distribution over the stated interval. To sample _Unif(a, b), b > a_ multiply the output of random_sample by (b-a) and add a:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`(b - a) * random_sample() + a`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "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": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "out = 12* np.random.random_sample((5, 2)) - 5\n",
    "count, bins, ignored = plt.hist(out,5, density=True)\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.choice](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.choice.html#numpy.random.choice)  <a id=\"random.choice\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function generates a random sample from a given 1-D array."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The below code snippet will generate a uniform random sample from np.arange(2) of size 10:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1, 0, 0, 1, 1, 0, 0, 0, 0, 1])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.choice(2, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example II__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(['Gal', 'Mayo', 'Gal', 'Dub', 'Dub', 'Mayo', 'Ros', 'Mayo', 'Leit',\n",
       "       'Sligo'], dtype='<U5')"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "counties = ['Gal', 'Mayo', 'Ros', 'Sligo', 'Leit', 'Dub']\n",
    "np.random.choice(counties, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Permutations <a id=\"Permutations\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.shuffle](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.shuffle.html#numpy.random.shuffle)  <a id=\"random.shuffle\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5, 9, 3, 6, 7, 1, 2, 0, 4, 8])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr = np.arange(10)\n",
    "np.random.shuffle(arr)\n",
    "arr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.permutation](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.permutation.html#numpy.random.permutation)  <a id=\"random.permutation\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function is different from random.shuffle, where, if passed an __array__, it will return a shuffled copy of the array. If passed an __integer__ it will return a shuffled range."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4, 7, 9, 2, 8, 3, 6, 0, 1, 5])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.permutation(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 4, 12, 15,  1,  9])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.permutation([1, 4, 9, 12, 15])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Distributions <a id=\"Distributions\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.normal](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.normal.html#numpy.random.normal)  <a id=\"random.normal\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function returns an n dimensional array or a scalar."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The parameters of this function are as follows:\n",
    "\n",
    "_loc_ (float or array_like of floats):\n",
    ">Mean (“centre”) of the distribution.\n",
    "\n",
    "_scale_ (float or array_like of floats):\n",
    ">Standard deviation (spread or “width”) of the distribution.\n",
    "\n",
    "_size_ (int or tuple of ints, optional):\n",
    ">Output shape. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.31157538, -0.46368461,  0.37575444,  0.18047139, -0.81326928,\n",
       "        0.6872668 , -0.48796218, -1.40251454, -1.2381321 , -0.46049123])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.random.normal(loc=0.0, scale=1.0, size=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.binomial](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.binomial.html#numpy.random.binomial) <a id=\"random.binomial\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A binomial distribution represents the number of successes in a sequence of n __independent__ trials, each of which yields success with probability p."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So what would be the probability of getting 5 heads out of 10 flips of a coin?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The probability of getting a head is 50%.\n",
    "This means that on average you can expect to get a head 5 out of 10 flips."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[47.2 48.6 45.8 50.8 48.8 48.4 51.  50.2 48.2 48.6]\n"
     ]
    }
   ],
   "source": [
    "n = 500 #(samples or trials)\n",
    "p = 0.5 #(probability of success)\n",
    "size = 10 #(number of experiments)\n",
    "\n",
    "tests = np.random.binomial(n, p, size)\n",
    "\n",
    "print(tests/n*100)\n",
    "# The result is 10 randomly selected \n",
    "# experiments from the distribution\n",
    "# [241, 262, ..., 248, 255, 235]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.exponential](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.exponential.html#numpy.random.exponential) <a id=\"random.exponential\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The exponential distribution represents a process in which events occur continuously and independently at a constant average rate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The general formula for the probability density function of the exponential distribution is:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "f(x) = \\frac 1\\beta \\epsilon^{-(x-\\mu)/\\beta} \\quad x\\geq\\mu;\\beta\\gt0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\mu$ is the location parameter and $\\beta$ is the scale parameter (the scale parameter is often referred to as $\\lambda$ which equals $\\frac 1\\beta$). The case where $\\mu$ = 0 and $\\beta$ = 1 is called the standard exponential distribution. The equation for the standard exponential distribution is:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "f(x) = \\epsilon^{-x}\\quad \\text{for}\\quad x \\geq 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*__Example__*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show(*args, **kw)>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lambd = 0.5\n",
    "x = np.arange(0, 15, 0.1)\n",
    "y = lambd * np.exp(-lambd * x)\n",
    "plt.plot(x,y)\n",
    "plt.title('Exponential: $\\lambda$ =%.2f' % lambd)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('Probability Density')\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [random.poisson](https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.random.poisson.html#numpy.random.poisson) <a id=\"random.poisson\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\"The Poisson distribution is the limit of the binomial distribution for large N.\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Poisson distribution is used to model the number of events occurring within a given time interval."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The formula for the Poisson probability mass function is:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "p(x; \\lambda) = \\frac{\\epsilon^{-\\lambda}\\lambda^{x}}{x!} \\text{for}\\quad x=0,1,2... \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\lambda$ is the shape parameter which indicates the average number of events in the given time interval."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=sp.random.poisson(lam=1, size=100)\n",
    "a = 5. # shape \n",
    "n = 1000 \n",
    "s = np.random.power(a, n) \n",
    "count, bins, ignored = plt.hist(s, bins=30) \n",
    "x = np.linspace(0, 1, 100) \n",
    "y = a*x**(a-1.) \n",
    "normed_y = n*np.diff(bins)[0]*y \n",
    "plt.title(\"Poisson distribution\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.xlabel(\"x\")\n",
    "plt.plot(x, normed_y) \n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}