{ "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "ec7d5a70-51ca-4c9c-a4bb-dcfc293e1e47", "metadata": {}, "source": [ "(MvGaussianRandomWalk)=\n", "# Multivariate Gaussian Random Walk\n", ":::{post} Feb 2, 2023\n", ":tags: linear model, regression, time series \n", ":category: beginner\n", ":author: Lorenzo Itoniazzi, Chris Fonnesbeck\n", ":::" ] }, { "cell_type": "code", "execution_count": 1, "id": "7461c2a1-9e07-4a60-8df0-5f8e70a9d4f5", "metadata": {}, "outputs": [], "source": [ "import arviz as az\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import pymc as pm\n", "import pytensor\n", "\n", "from scipy.linalg import cholesky\n", "\n", "%matplotlib inline" ] }, { "cell_type": "code", "execution_count": 2, "id": "f1dbc074-72ff-436b-8063-682779163661", "metadata": {}, "outputs": [], "source": [ "RANDOM_SEED = 8927\n", "rng = np.random.default_rng(RANDOM_SEED)\n", "az.style.use(\"arviz-darkgrid\")" ] }, { "cell_type": "markdown", "id": "20d86071-b975-430d-973d-87295a17c8ac", "metadata": {}, "source": [ "This notebook shows how to [fit a correlated time series](https://en.wikipedia.org/wiki/Curve_fitting) using multivariate [Gaussian random walks](https://en.wikipedia.org/wiki/Random_walk#Gaussian_random_walk) (GRWs). In particular, we perform a Bayesian [regression](https://en.wikipedia.org/wiki/Regression_analysis) of the time series data against a model dependent on GRWs.\n", "\n", "We generate data as the 3-dimensional time series\n", "\n", "$$\n", "\\mathbf y = \\alpha_{i[\\mathbf t]} +\\beta_{i[\\mathbf t]} *\\frac{\\mathbf t}{300} +\\xi_{\\mathbf t},\\quad \\mathbf t = [0,1,...,299], \n", "$$ (eqn:model)\n", "\n", "where \n", "- $i\\mapsto\\alpha_{i}$ and $i\\mapsto\\beta_{i}$, $i\\in\\{0,1,2,3,4\\}$, are two 3-dimensional Gaussian random walks for two correlation matrices $\\Sigma_\\alpha$ and $\\Sigma_\\beta$,\n", "- we define the index \n", "$$\n", "i[t]= j\\quad\\text{for}\\quad t = 60j,60j+1,...,60j+59, \\quad\\text{and}\\quad j = 0,1,2,3,4,\n", "$$ \n", "- $*$ means that we multiply the $j$-th column of the $3\\times300$ matrix with the $j$-th entry of the vector for each $j=0,1,...,299$, and \n", "- $\\xi_{\\mathbf t}$ is a $3\\times300$ matrix with iid normal entries $N(0,\\sigma^2)$.\n", "\n", "\n", "So the series $\\mathbf y$ changes due to the GRW $\\alpha$ in five occasions, namely steps $0,60,120,180,240$. Meanwhile $\\mathbf y$ changes at steps $1,60,120,180,240$ due to the increments of the GRW $\\beta$ and at every step due to the weighting of $\\beta$ with $\\mathbf t/300$. Intuitively, we have a noisy ($\\xi$) system that is shocked five times over a period of 300 steps, but the impact of the $\\beta$ shocks gradually becomes more significant at every step. \n", "\n", "## Data generation\n", "\n", "Let's generate and plot the data." ] }, { "cell_type": "code", "execution_count": 3, "id": "4c329308-80b9-4e88-9698-199bdab66b68", "metadata": {}, "outputs": [ { "data": { "image/png": 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