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@ChadFulton
Last active March 4, 2018 06:10
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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": [
{
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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"
]
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10e120fd0>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
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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"
]
},
{
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x1148f0f10>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
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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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