UQ/Monte_Carlo/MC.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3ecc320d",
   "metadata": {},
   "source": [
    "# 1. Monte Carlo Methods\n",
    "\n",
    "The Monte Carlo method is a computational algorithm that relies on repeated random sampling to obtain numerical results. The basic idea is to use randomness to solve problems that might be deterministic in principle. Monte Carlo methods are used in various fields such as physics, finance, and engineering for numerical integration, optimization, and simulation.\n",
    "\n",
    "## Key Concepts\n",
    "1. **Random Sampling:**  Randomly generating inputs to a function to simulate a process.\n",
    "2. **Statistical Analysis:** Using the results of random sampling to infer properties of the function or process.\n",
    "3. **Law of Large Numbers:** The more samples you use, the closer the estimate will be to the true value.\n",
    "\n",
    "Given i.i.d. random variables $X_1, \\ldots, X_n$. We define the mean\n",
    "\\begin{equation*}\n",
    "\\bar{X}:= \\frac{1}{n} \\sum_{i=1}^{n} X_i.\n",
    "\\end{equation*}\n",
    "\n",
    "By the strong law of large numbers it holds:\n",
    "\\begin{equation}\n",
    "\\bar{X} \\rightarrow \\mathbb{E}[X] \\ \\ \\textit{a.s.}\n",
    "\\end{equation}\n",
    "\n",
    "A basic **Monte Carlo (MC) simualtion** for $n$ i.i.d. copies of a given random variable $X$ uses the estimator $\\bar{X}$ for estimating the mean $\\mathbb{E}[X]$.\n",
    "\n",
    "Let us consider a function $f:\\mathbb{R}^n \\rightarrow \\mathbb{R}^m$ (measurable) that represents the model that we are interested in, i.e. the source location of a tsunami. If we like to estimate $\\mathbb{E}[f(X)]$ the MC estimator is given by \n",
    "\\begin{equation}\n",
    "\\hat{F}^{MC} = \\frac{1}{n} \\sum_{i=1}^{n} f(X_i).\n",
    "\\end{equation}\n",
    "\n",
    "**UM-Bridge context:** The model function $f$ represents the **server** side while we refer to the MC estimator $\\hat{F}^{MC}$ as the **client**.\n",
    "\n",
    "---\n",
    "\n",
    "## 1.1 First Example\n",
    "In the following we give some basic MC simulation for a sample of normal distributed random variables.\n",
    "\n",
    "We fix the sample size $N=1000$ and sample $N$ random variables $X_1,..,X_N$ from $\\mathcal{N}(0.5, 0.1)$. The MC estimator to estimate the expectation value is given by the mean: $\\frac{1}{N} \\sum_{i=1}^{N} x_i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "cfcf6857",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Calculated mean: 0.4966874536822834\n",
      "Expected mean: 0.5\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Set the number of random variables\n",
    "N = 1000\n",
    "\n",
    "# Generate N normal distributed random variables between 0 and 1\n",
    "X = np.random.normal(loc=0.5, scale=0.1, size=N)\n",
    "\n",
    "# Calculate the mean of the random variables\n",
    "mean_X = np.mean(X)\n",
    "\n",
    "# Expected value for a normal distribution with mean = 0.5\n",
    "expected_value = 0.5\n",
    "\n",
    "print(f\"Calculated mean: {mean_X}\")\n",
    "print(f\"Expected mean: {expected_value}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "c8ef15b4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 1000x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot\n",
    "plt.figure(figsize=(10, 6))\n",
    "plt.hist(X, bins=30, alpha=0.7, color='skyblue', edgecolor='black')\n",
    "plt.axvline(mean_X, color='red', linestyle='dashed', linewidth=2, label=f'Calculated Mean: {mean_X:.2f}')\n",
    "plt.axvline(expected_value, color='green', linestyle='solid', linewidth=2, label='Expected Mean: 0.5')\n",
    "plt.title('Histogram of Normally Distributed Random Variables')\n",
    "plt.xlabel('Value')\n",
    "plt.ylabel('Frequency')\n",
    "plt.legend()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f4de982",
   "metadata": {},
   "source": [
    "---\n",
    "## Tasks\n",
    "1. **Change the Sample Size**: Vary the value of $N$. How does increasing or decreasing $N$ affect the calculated mean? \n",
    "2. **Change the Distribution**:\n",
    "    - Modify the distribution of the random variables. For example, use a uniform distribution or an exponential distribution instead of the normal distribution.\n",
    "     - `X = np.random.uniform(low=0, high=1, size=N)`\n",
    "     - `X = np.random.exponential(scale=1.0, size=N)`\n",
    "     \n",
    "    How does the change in distribution affect the mean and the histogram? Do different distributions converge to their expected values as $N$ increases? Compare the spread (variance) of the distributions.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "02f42a10",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.12.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}