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State space models in Python: Bayesian and Classical estimation
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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "# State space models in Python: Metropolis-Hastings\n", | |
| "\n", | |
| "See this notebook as a blog post: [http://pages.uoregon.edu/cfulton/posts/state_space_mh.html](http://pages.uoregon.edu/cfulton/posts/state_space_mh.html), along with other state space resources.\n", | |
| "\n" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "%matplotlib inline\n", | |
| "\n", | |
| "import numpy as np\n", | |
| "import pandas as pd\n", | |
| "import pymc as mc\n", | |
| "from scipy import signal\n", | |
| "import statsmodels.api as sm\n", | |
| "import matplotlib.pyplot as plt\n", | |
| "\n", | |
| "np.set_printoptions(precision=4, suppress=True, linewidth=120)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Suppose we have a time series $Y_T \\equiv \\{ y_t \\}_{t=0}^T$ which we model as local level process:\n", | |
| "\n", | |
| "$$\n", | |
| "\\begin{align}\n", | |
| "y_t & = \\mu_t + \\varepsilon_t, \\qquad \\varepsilon_t \\sim N(0, \\sigma_\\varepsilon^2) \\\\\n", | |
| "\\mu_{t+1} & = \\mu_t + \\eta_t, \\qquad \\eta_t \\sim N(0, \\sigma_\\eta^2) \\\\\n", | |
| "\\end{align}\n", | |
| "$$\n", | |
| "\n", | |
| "In this model, there are two unknown parameters, which we will collected in a vector $\\psi$, so that: $\\psi = (\\sigma_\\varepsilon^2, \\sigma_\\eta^2)$; let's set their true values as follows (denoted with the subscript 0):\n", | |
| "\n", | |
| "$$\n", | |
| "\\psi_0 = (\\sigma_{\\varepsilon,0}^2, \\sigma_{\\eta,0}^2) = (3, 10)\n", | |
| "$$\n", | |
| "\n", | |
| "Finally, we also must specify the prior $\\mu_0 \\sim N(m_0, P_0)$ to initialize the Kalman filter.\n", | |
| "\n", | |
| "Set $T = 1000$." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x1148b8990>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# True values\n", | |
| "T = 1000\n", | |
| "sigma2_eps0 = 3\n", | |
| "sigma2_eta0 = 10\n", | |
| "\n", | |
| "# Simulate data\n", | |
| "np.random.seed(1234)\n", | |
| "eps = np.random.normal(scale=sigma2_eps0**0.5, size=T)\n", | |
| "eta = np.random.normal(scale=sigma2_eta0**0.5, size=T)\n", | |
| "mu = np.cumsum(eta)\n", | |
| "y = mu + eps\n", | |
| "\n", | |
| "# Plot the time series\n", | |
| "fig, ax = plt.subplots(figsize=(13,2))\n", | |
| "ax.plot(y);\n", | |
| "ax.set(xlabel='$T$', title='Simulated series');" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "It turns out it will be convenient to write the model in terms of the precision of $\\varepsilon$, defined to be $h^{-1} \\equiv \\sigma_\\varepsilon^2$, and the ratio of the variances: $q \\equiv \\sigma_\\eta^2 / \\sigma_\\varepsilon^2$ so that $q h^{-1} = \\sigma_\\eta^2$.\n", | |
| "\n", | |
| "Then our error terms can be written:\n", | |
| "\n", | |
| "$$\n", | |
| "\\varepsilon_t \\sim N(0, h^{-1}) \\\\\n", | |
| "\\eta_t \\sim N(0, q h^{-1}) \\\\\n", | |
| "$$\n", | |
| "\n", | |
| "And the true values are:\n", | |
| "\n", | |
| "- $h_0^{-1} = 1 / 3 = 0.33$\n", | |
| "- $q = 10 / 3 = 3.33$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "To take a Bayesian approach to this problem, we assume that $\\psi$ is a random variable, and we want to learn about the values of $\\psi$ based on the data $Y_T$; in fact we want a density $p(\\psi | Y_T)$. To do this, we use Bayes rule to write:\n", | |
| "\n", | |
| "$$\n", | |
| "p(\\psi | Y_T) = \\frac{p(Y_T | \\psi) p(\\psi)}{p(Y_T)}\n", | |
| "$$\n", | |
| "\n", | |
| "or\n", | |
| "\n", | |
| "$$\n", | |
| "\\underbrace{p(\\psi | Y_T)}_{\\text{posterior}} \\propto \\underbrace{p(Y_T | \\psi)}_{\\text{likelihood}} \\underbrace{p(\\psi)}_{\\text{prior}}\n", | |
| "$$\n", | |
| "\n", | |
| "The object of interest is the posterior; to achieve it we need to specify a prior density for the unknown parameters and the likelihood function of the model." | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "### Prior\n", | |
| "\n", | |
| "We will use the following priors:\n", | |
| "\n", | |
| "#### Precision\n", | |
| "\n", | |
| "Since the precision must be positive, but has no theoretical upper bound, we use a Gamma prior:\n", | |
| "\n", | |
| "$$h \\sim \\text{Gamma}(\\alpha_h, \\beta_h)$$\n", | |
| "\n", | |
| "to be specific, the density is written:\n", | |
| "\n", | |
| "$$p(h) = \\frac{\\beta_h^{\\alpha_h}}{\\Gamma(\\alpha)} h^{\\alpha_h-1}e^{-\\beta_h h}$$\n", | |
| "\n", | |
| "and we set the hyperparameters as $\\alpha_h = 2, \\beta_h = 2$. In this case, we have $E(h) = \\alpha_h / \\beta_h = 1$ and also $E(h^{-1}) = E(\\sigma_\\varepsilon^2) = 1$.\n", | |
| "\n", | |
| "#### Ratio of variances\n", | |
| "\n", | |
| "Similarly, the ratio of variances must be positive, but has no theoretical upper bound, so we again use an (independent) Gamma prior:\n", | |
| "\n", | |
| "$$q \\sim \\text{Gamma}(\\alpha_q, \\beta_q)$$\n", | |
| "\n", | |
| "and we set the same hyperparameters, so $\\alpha_q = 2, \\beta_q = 2$. Since $E(q) = 1$, our prior is of equal variances. We then have $E(\\sigma_\\eta^2) = E(q h^{-1}) = E(q) E(h^{-1}) = 1$.\n", | |
| "\n", | |
| "#### Initial state prior\n", | |
| "\n", | |
| "As noted above, the Kalman filter must be initialized with $\\mu_0 \\sim N(m_0, P_0)$. We will use the following approximately diffuse prior:\n", | |
| "\n", | |
| "$$\\mu_0 \\sim N(0, 10^6)$$" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "### Likelihood\n", | |
| "\n", | |
| "For given parameters, likelihood of this model can be calculated via prediction error decomposition using an application of the Kalman filter iterations." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "# Priors\n", | |
| "precision = mc.Gamma('precision', 2, 4)\n", | |
| "ratio = mc.Gamma('ratio', 2, 1)\n", | |
| "\n", | |
| "# Likelihood calculated using the state-space model\n", | |
| "class LocalLevel(sm.tsa.statespace.MLEModel):\n", | |
| " def __init__(self, endog):\n", | |
| " # Initialize the state space model\n", | |
| " super(LocalLevel, self).__init__(endog, k_states=1,\n", | |
| " initialization='approximate_diffuse',\n", | |
| " loglikelihood_burn=1)\n", | |
| " \n", | |
| " # Initialize known components of the state space matrices\n", | |
| " self.ssm['design', :] = 1\n", | |
| " self.ssm['transition', :] = 1\n", | |
| " self.ssm['selection', :] = 1\n", | |
| " \n", | |
| " @property\n", | |
| " def start_params(self):\n", | |
| " return [1. / np.var(self.endog), 1.]\n", | |
| " \n", | |
| " @property\n", | |
| " def param_names(self):\n", | |
| " return ['h_inv', 'q']\n", | |
| " \n", | |
| " def update(self, params, transformed=True):\n", | |
| " params = super(LocalLevel, self).update(params, transformed)\n", | |
| " \n", | |
| " h, q = params\n", | |
| " sigma2_eps = 1. / h\n", | |
| " sigma2_eta = q * sigma2_eps\n", | |
| " \n", | |
| " self.ssm['obs_cov', 0, 0] = sigma2_eps\n", | |
| " self.ssm['state_cov', 0, 0] = sigma2_eta\n", | |
| "\n", | |
| "# Instantiate the local level model with our simulated data\n", | |
| "ll_mod = LocalLevel(y)\n", | |
| "\n", | |
| "# Create the stochastic (observed) component\n", | |
| "@mc.stochastic(dtype=LocalLevel, observed=True)\n", | |
| "def local_level(value=ll_mod, h=precision, q=ratio):\n", | |
| " return value.loglike([h, q], transformed=True)\n", | |
| "\n", | |
| "# Create the PyMC model\n", | |
| "ll_mc = mc.Model((precision, ratio, local_level))\n", | |
| "\n", | |
| "# Create a PyMC sample\n", | |
| "ll_sampler = mc.MCMC(ll_mc)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Bayesian Estimation: Metropolis-Hastings" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 4, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| " [-----------------100%-----------------] 10000 of 10000 complete in 19.8 secPlotting ratio\n", | |
| "Plotting precision\n" | |
| ] | |
| }, | |
| { | |
| "data": { | |
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| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x10e120fd0>" | |
| ] | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x114aa3a10>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# Sample\n", | |
| "ll_sampler.sample(iter=10000, burn=1000, thin=10)\n", | |
| "\n", | |
| "# Plot traces\n", | |
| "mc.Matplot.plot(ll_sampler)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Classical Estimation\n", | |
| "\n", | |
| "For comparison, here is the output from maximum likelihood estimation:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 5, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| " Statespace Model Results \n", | |
| "==============================================================================\n", | |
| "Dep. Variable: y No. Observations: 1000\n", | |
| "Model: LocalLevel Log Likelihood -2789.529\n", | |
| "Date: Fri, 31 Jul 2015 AIC 5583.058\n", | |
| "Time: 12:33:08 BIC 5592.873\n", | |
| "Sample: 0 HQIC 5586.788\n", | |
| " - 1000 \n", | |
| "Covariance Type: opg \n", | |
| "==============================================================================\n", | |
| " coef std err z P>|z| [0.025 0.975]\n", | |
| "------------------------------------------------------------------------------\n", | |
| "h_inv 0.3045 0.049 6.196 0.000 0.208 0.401\n", | |
| "q 2.9588 0.690 4.289 0.000 1.607 4.311\n", | |
| "==============================================================================\n", | |
| "\n", | |
| "Warnings:\n", | |
| "[1] Covariance matrix calculated using the outer product of gradients.\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "ll_res = ll_mod.fit()\n", | |
| "print ll_res.summary()" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Alternate Model: ARIMA(0,1,1)\n", | |
| "\n", | |
| "It can be shown (see e.g. Harvey (1989), among many others) that the local level model is equivalent to an ARIMA(0,1,1) model, which is written:\n", | |
| "\n", | |
| "$$y_t = \\theta \\zeta_{t-1} + \\zeta_t, \\qquad \\zeta_t \\sim N(0, \\sigma^2)$$\n", | |
| "\n", | |
| "It is important to note that $\\sigma_\\zeta^2$ here is not equivalent to $\\sigma_\\varepsilon^2$ from the local level model. However, we can connect the models through $q = - (1 + \\theta)^2 / \\theta$. Note that to generate a valid $q$, $\\theta$ must be negative.\n", | |
| "\n", | |
| "We will use the same Gamma prior as above on the precision $h^{-1} = \\sigma_\\eta^2$, but we must specify a new prior for $\\theta$. We will specify that $-\\theta \\equiv \\gamma \\sim \\text{Beta}(\\alpha_\\theta, \\beta_\\theta)$. Since the entire support for the Beta distribution falls between 0 and 1, this prior implies $\\theta \\in [-1, 0]$.\n", | |
| "\n", | |
| "We would expect that $\\hat \\theta \\approx -0.21$, since then $q = \\frac{-(1 - 0.21)^2}{-0.21} \\approx 3$ (and so $\\hat \\gamma \\approx 0.21$)." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 24, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "# Priors\n", | |
| "arima_precision = mc.Gamma('arima_precision', 2, 4)\n", | |
| "gamma = mc.Beta('gamma', 2, 1)\n", | |
| "\n", | |
| "# Instantiate the ARIMA model with our simulated data\n", | |
| "arima_mod = sm.tsa.SARIMAX(y, order=(0,1,1))\n", | |
| "\n", | |
| "# Create the stochastic (observed) component\n", | |
| "@mc.stochastic(dtype=sm.tsa.SARIMAX, observed=True)\n", | |
| "def arima(value=arima_mod, h=arima_precision, gamma=gamma):\n", | |
| " # Rejection sampling\n", | |
| " if gamma < 0 or h < 0:\n", | |
| " return 0\n", | |
| " return value.loglike(np.r_[-gamma, 1./h], transformed=True)\n", | |
| "\n", | |
| "# Create the PyMC model\n", | |
| "arima_mc = mc.Model((arima_precision, gamma, arima))\n", | |
| "\n", | |
| "# Create a PyMC sample\n", | |
| "arima_sampler = mc.MCMC(arima_mc)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Bayesian Estimation: Metropolis-Hastings" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 25, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| " [-----------------100%-----------------] 10000 of 10000 complete in 22.0 secPlotting arima_precision\n", | |
| "Plotting gamma\n" | |
| ] | |
| }, | |
| { | |
| "data": { | |
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| "text/plain": [ | |
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| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x114df3690>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "# Sample\n", | |
| "arima_sampler.sample(iter=10000, burn=1000, thin=10)\n", | |
| "\n", | |
| "# Plot traces\n", | |
| "mc.Matplot.plot(arima_sampler)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "## Classical Estimation" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 26, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "name": "stdout", | |
| "output_type": "stream", | |
| "text": [ | |
| " Statespace Model Results \n", | |
| "==============================================================================\n", | |
| "Dep. Variable: y No. Observations: 1000\n", | |
| "Model: SARIMAX(0, 1, 1) Log Likelihood -2789.529\n", | |
| "Date: Fri, 31 Jul 2015 AIC 5583.058\n", | |
| "Time: 12:37:46 BIC 5592.873\n", | |
| "Sample: 0 HQIC 5586.788\n", | |
| " - 1000 \n", | |
| "Covariance Type: opg \n", | |
| "==============================================================================\n", | |
| " coef std err z P>|z| [0.025 0.975]\n", | |
| "------------------------------------------------------------------------------\n", | |
| "ma.L1 -0.2106 0.032 -6.576 0.000 -0.273 -0.148\n", | |
| "sigma2 15.5913 0.667 23.386 0.000 14.285 16.898\n", | |
| "==============================================================================\n", | |
| "\n", | |
| "Warnings:\n", | |
| "[1] Covariance matrix calculated using the outer product of gradients.\n" | |
| ] | |
| } | |
| ], | |
| "source": [ | |
| "arima_res = arima_mod.fit()\n", | |
| "print arima_res.summary()" | |
| ] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "display_name": "Python 2", | |
| "language": "python", | |
| "name": "python2" | |
| }, | |
| "language_info": { | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 2 | |
| }, | |
| "file_extension": ".py", | |
| "mimetype": "text/x-python", | |
| "name": "python", | |
| "nbconvert_exporter": "python", | |
| "pygments_lexer": "ipython2", | |
| "version": "2.7.6" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 0 | |
| } |
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