From a80fdc9a380360eccde3109a1cdccd9b1e9bc08a Mon Sep 17 00:00:00 2001
From: Soh Yee Lee <51136409+SohYeeLee@users.noreply.github.com>
Date: Wed, 12 Jun 2019 17:46:05 +0200
Subject: [PATCH] Add files via upload

---
 3_MonteCarlo_Approx Integration.ipynb | 117 ++++++++++++++++++++++++++
 1 file changed, 117 insertions(+)
 create mode 100644 3_MonteCarlo_Approx Integration.ipynb

diff --git a/3_MonteCarlo_Approx Integration.ipynb b/3_MonteCarlo_Approx Integration.ipynb
new file mode 100644
index 0000000..eb00b04
--- /dev/null
+++ b/3_MonteCarlo_Approx Integration.ipynb	
@@ -0,0 +1,117 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This exercise shows how to use Monte Carlo to approximate integration, $F = \\int_{0}^{1} sin (x) dx$ and $F = \\int_{2}^{4} sin (x) dx$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import chaospy as cp\n",
+    "from matplotlib.pyplot import *\n",
+    "\n",
+    "np.random.seed(11)\n",
+    "# declare function to integrate via Monte Carlo sampling\n",
+    "func = lambda x: np.sin(x)\n",
+    "\n",
+    "# declare vector with number of samples\n",
+    "N = [10, 100, 1000, 10000, 100000, 1000000, 10000000] #, 10000000, 100000000\n",
+    "\n",
+    "I_hat       = np.zeros(len(N))\n",
+    "I           = np.cos(2) - np.cos(4)\n",
+    "est_std_dev = np.zeros(len(N))\n",
+    "rms = np.zeros(len(N))\n",
+    "\n",
+    "log_dec = np.zeros(len(N))\n",
+    "sqrt_dec = np.zeros(len(N))\n",
+    "linear_dec = np.zeros(len(N))\n",
+    "quad_dec = np.zeros(len(N))\n",
+    "\n",
+    "# define new bounds\n",
+    "a = 2\n",
+    "b = 4\n",
+    "\n",
+    "zeroone_sampling = True\n",
+    "# for each N, perform Monte Carlo integration\n",
+    "for i, n in enumerate(N):\n",
+    "\n",
+    "    if zeroone_sampling:\n",
+    "        # draw uniform samples in [0, 1]\n",
+    "        distr_w = cp.Uniform(0, 1)\n",
+    "        samples_01  = distr_w.sample(size=n)\n",
+    "        # transform to [a, b]\n",
+    "        samples     = a + (b-a)*samples_01\n",
+    "        # approximate the integral as the expectation of the underlying function w.r.t. the uniform distribution\n",
+    "        # write the sample in [a, b] in terms of the generated sampled in [0, 1]\n",
+    "        I_hat[i]    = (b-a)*np.mean(func(samples))\n",
+    "        # approximate the standard deviation of the estimation\n",
+    "        est_std_dev[i]  = np.sqrt(np.var(func(samples), ddof=1))/np.sqrt(n)\n",
+    "    else:\n",
+    "        distr_w = cp.Uniform(2, 4)\n",
+    "        samples     = distr_w.sample(size=n)\n",
+    "        I_hat[i]    = (b-a)*np.mean(func(samples))\n",
+    "        est_std_dev[i] = np.sqrt( np.var(func(samples), ddof=1))/np.sqrt(n)\n",
+    "\n",
+    "    rms[i] =  np.abs(I_hat[i] - I)\n",
+    "\n",
+    "    if i == 0:\n",
+    "        log_dec[0] = est_std_dev[0]\n",
+    "        sqrt_dec[0] = est_std_dev[0]\n",
+    "        linear_dec[0] = est_std_dev[0]\n",
+    "        quad_dec[0] = est_std_dev[0]\n",
+    "    else:\n",
+    "        log_dec[i] = log_dec[i-1] / np.log(10)\n",
+    "        sqrt_dec[i] = sqrt_dec[i-1] / np.sqrt(10)\n",
+    "        linear_dec[i] = linear_dec[i-1] / 10\n",
+    "        quad_dec[i] = quad_dec[i-1] / (10**2)\n",
+    "\n",
+    "    print (I, I_hat[i], rms[i], est_std_dev[i])\n",
+    "\n",
+    "# plot results\n",
+    "loglog(N, est_std_dev, 'r', label='standard error decay')\n",
+    "loglog(N, rms, 'b', label=r'$\\epsilon$')\n",
+    "loglog(N, log_dec, 'k--', label=r'log')\n",
+    "loglog(N, sqrt_dec, 'k-*', label=r'sqrt')\n",
+    "#loglog(N, linear_dec, 'k-.', label=r'linear')\n",
+    "#loglog(N, quad_dec, 'k-', label=r'linear')\n",
+    "legend(loc='best')\n",
+    "show()"
+   ]
+  },
+  {
+   "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": "3.7.3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}