{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# For a lot of applications, you may need a random number generator\n",
    "# They are also useful for simulations and generation of synthetic datasets\n",
    "# Here we will use the random package from NumPy to perform \"random walks\"\n",
    "# Random walks are a sequence of random steps in a random direction\n",
    "# Random walks can be 1-dimensional (along a line), \n",
    "# 2-dimensional (in a plane), or higher dimensional\n",
    "# Following are some simple experiments for a 1-dimensional random walk\n",
    "# Randoom walks can be discrete or continuous.\n",
    "# In a 1-directional discrete random walk, every step is a \"unit\" step \n",
    "# either in the forward or the reverse direction (+1 or -1). \n",
    "# In a 2-directional discrete random walk, every step is a unit step \n",
    "# in one of 4 directions (N, S, E, W). \n",
    "# In a 1-directional continuous random walk, every step is a random step \n",
    "# either in the forward or the reverse direction. These are not unit steps.\n",
    "# The size of the step is a random number from some distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import random\n",
    "position = 0\n",
    "walk = [position] \n",
    "steps = 10\n",
    "for i in range(steps):\n",
    "    # randomly pick either 0 or 1\n",
    "    # This function can pick a discrete random number from a given range, in this case {0,1}\n",
    "    step = 1 if random.randint(0, 1) else -1 \n",
    "    position += step\n",
    "    walk.append(position)\n",
    "walk # of type list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# we redo the above making walks as an NumPy's nd.array \n",
    "nwalks = 5\n",
    "nsteps = 10\n",
    "# This function from NumpY can pick a discrete random number from a given range, \n",
    "# in this case {0,1}\n",
    "draws = np.random.randint(0, 2, size=(nwalks, nsteps)) # 0 or 1\n",
    "draws"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "steps = np.where(draws > 0, 1, -1)\n",
    "steps"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "walks = steps.cumsum(1) # cumulative sum of each row\n",
    "walks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "walks.min()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "walks.max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# how often does it cross 3?\n",
    "hits3 = (np.abs(walks) >= 3).any(1)\n",
    "hits3.sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# how long before it crosses 3?\n",
    "crossing_times = (np.abs(walks[hits3]) >= 3).argmax(1)\n",
    "crossing_times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "crossing_times.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# can also generate random numbers from a normal distribution\n",
    "# These are continuous distributions; the above ones are discrete\n",
    "steps = np.random.normal(loc=0, scale=0.25, size=(nwalks, nsteps))\n",
    "steps"
   ]
  },
  {
   "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": "3.6.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}