{
 "cells": [
  {
   "cell_type": "markdown",
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   "source": [
    "<!-- dom:TITLE: Polynomial Chaos with the Ishigami Function -->\n",
    "# Polynomial Chaos with the Ishigami Function\n",
    "<!-- dom:AUTHOR: Jacob Sturdy at Department of Structural Engineering, NTNU -->\n",
    "<!-- Author: -->  \n",
    "**Jacob Sturdy**, Department of Structural Engineering, NTNU\n",
    "\n",
    "Date: **Jul 13, 2018**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# ipython magic\n",
    "%matplotlib notebook\n",
    "%load_ext autoreload\n",
    "%autoreload 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# plot configuration\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use(\"ggplot\")\n",
    "# import seaborn as sns # sets another style\n",
    "matplotlib.rcParams['lines.linewidth'] = 3\n",
    "fig_width, fig_height = (7.0,5.0)\n",
    "matplotlib.rcParams['figure.figsize'] = (fig_width, fig_height)\n",
    "\n",
    "# font = {'family' : 'sans-serif',\n",
    "#         'weight' : 'normal',\n",
    "#         'size'   : 18.0}\n",
    "# matplotlib.rc('font', **font)  # pass in the font dict as kwar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import chaospy as cp\n",
    "import numpy as np\n",
    "from numpy import linalg as LA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction\n",
    "<div id=\"sec:introduction\"></div>\n",
    "\n",
    "The Ishigami function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- Equation labels as ordinary links -->\n",
    "<div id=\"_auto1\"></div>\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "y = \\sin(z_1) + a \\sin^2(z_2) + b z_3^4 \\sin(z_1)\n",
    "\\label{_auto1} \\tag{1}\n",
    "\\end{equation}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "is a commonly used benchmark function for sensitivity analysis. You will now get the chance to evaluate polynomial chaos on this function.\n",
    "\n",
    "Each component of $\\mathbf{Z}$ is distributed uniformly over the range $(-\\pi, \\pi)$ and $a=7$ while $b=0.1$.\n",
    "\n",
    "These two functions implement the function in python as well as the exact solutions for its mean, variance and sensitivities."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def ishigami_function(sample):\n",
    "    q1 = sample[0]\n",
    "    q2 = sample[1]\n",
    "    q3 = sample[2]\n",
    "    a = 7.\n",
    "    b = 0.1\n",
    "    return np.sin(q1) + a*np.sin(q2)**2 + b* q3**4 * np.sin(q1)\n",
    "\n",
    "def ishigami_analytic():\n",
    "    #Analytical values\n",
    "    #Total variance\n",
    "    measures = {}\n",
    "    a = 7.\n",
    "    measures[\"mean\"] = a/2.0\n",
    "    b = 0.1\n",
    "    D = a**2./8 + b*np.pi**4./5 + b**2*np.pi**8./18 + 1./2\n",
    "    measures \n",
    "    measures[\"var\"] = D\n",
    "    # Conditional variances\n",
    "    D1 = b*np.pi**4./5 + b**2*np.pi**8./50. + 1./2\n",
    "    D2 = a**2/8.\n",
    "    D3 = 0\n",
    "    \n",
    "    D12  = 0\n",
    "    D13  = b**2. * np.pi**8 / 18 - b**2*np.pi**8./50.\n",
    "    D23  = 0\n",
    "    D123 = 0\n",
    "    \n",
    "    # Main and total sensitivity indices\n",
    "    measures[\"sens_m\"] = {}\n",
    "    measures[\"sens_m\"][0] = D1/D\n",
    "    measures[\"sens_m\"][1] = D2/D\n",
    "    measures[\"sens_m\"][2] = D3/D\n",
    "   \n",
    "\n",
    "    measures[\"sens_t\"] = {}\n",
    "    measures[\"sens_t\"][0] = (D1 + D12 + D13 + D123)/D\n",
    "    measures[\"sens_t\"][1] = (D2 + D12 + D23 + D123)/D\n",
    "    measures[\"sens_t\"][2] = (D3 + D13 + D23 + D123)/D\n",
    "    return measures"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare the results you get for different methods of performing polynomial chaos with this function. What sampling methods work best?"
   ]
  }
 ],
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