Example.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# How to use the generator\n",
    "\n",
    "This notebook shows how to use the library to generate sparse stochastic processes.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "First `import` the necessary files :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from lib.lspline import L_spline\n",
    "from lib.loperator import Operator\n",
    "from lib.white_noise import white_noise"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setting the white noise\n",
    "\n",
    "\n",
    "In the library, there is the possiblity to choose among the four following white noise processes :\n",
    "<ul>\n",
    "    <li> Gaussian </li>\n",
    "    <li> Alpha-Stable </li>\n",
    "    <li> Laplace </li>\n",
    "    <li> Gamma </li>\n",
    "</ul>\n",
    " \n",
    "\n",
    "The following block of code shows how to contruct all four types of white noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "w_gauss = white_noise('gaussian', params=(0,1))\n",
    "w_alpha_stable = white_noise('alpha_stable', params=(1, 0, 0, 1))\n",
    "w_laplace = white_noise('laplace', params=(0,1))\n",
    "w_gamma = white_noise('gamma', params=(1, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The parametrization of the laws follows the one provided on the Wikipedia page of each law."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Defining the operator\n",
    "\n",
    "The framework allows for rational operators of the form\n",
    "$$\n",
    "P(\\mathrm{D})Q(\\mathrm{D})^{-1}\n",
    "$$\n",
    "The polynomials $P$ and $Q$ are defined through the list of their coefficients :\n",
    "* $P(\\mathrm{D}) = p_0 \\mathrm{D}^n + p_1 \\mathrm{D}^{n-1} + ... + p_n$ is represented as  ``` P = [p_0, ..., p_n ]```\n",
    "* $Q(\\mathrm{D}) = q_0 \\mathrm{D}^m + q_1 \\mathrm{D}^{m-1} + ... + q_m$ is represented as  ``` Q = [q_0, ..., q_m ]```\n",
    "\n",
    "\n",
    "Defining the operator consists of providing both lists `P` and `Q`. For example to define the operator\n",
    "$$\n",
    "\\mathrm{L} = P(\\mathrm{D})Q(\\mathrm{D})^{-1} \\text{ with } P(\\mathrm{D}) = {\\mathrm{D}^2 + 4\\mathrm{Id}} \\quad \\text{ and } Q(\\mathrm{D}) = \\mathrm{D},\n",
    "$$\n",
    "write :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "L = Operator(P = [1, 0, 4], Q = [1, 0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Defining the sparse stochastic process solving $\\mathrm{L} s = w$\n",
    "\n",
    "\n",
    "Here is how to define $s$ for the different types of innovations defined previously :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "s_gauss = L_spline(L, w_gauss)\n",
    "s_alpha_stable = L_spline(L, w_alpha_stable)\n",
    "s_laplace = L_spline(L, w_laplace)\n",
    "s_gamma = L_spline(L, w_gamma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Setting the approximation quality with $n$:\n",
    "\n",
    "The compound Poisson Approximation is chosen to have the rate $\\lambda = n$. \n",
    "\n",
    "Set it as follows :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "s_gauss.set_lambda(100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Sampling and evaluating the process :\n",
    "\n",
    "Pick an interval $[0, T]$ on which to sample the process and sample it as follows :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "s_gauss.sample(T=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can then evaluate `s_gauss` at any point $t\\in[0,T]$ like so :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7903192539229389"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s_gauss.eval(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualizing the process on a grid\n",
    "\n",
    "This section describes how to use the library to visualize the process on the interval `[0, T]` sampled on a regular grid with step size `h`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "T = 1\n",
    "h = 0.01"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since there should be at least one impulse per bin, set $\\lambda = n$ as follows :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "s_gauss.set_lambda(2 * 1/h)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then sample the process on `[0, T]` and obtain its grid values like so :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "s_gauss.sample(T)\n",
    "grid_samples = s_gauss.get_grid_samples(T, h)"
   ]
  }
 ],
 "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
}