MCMCregressChange: Markov Chain Monte Carlo for a linear Gaussian Multiple...

In MCMCpack: Markov Chain Monte Carlo (MCMC) Package

Markov Chain Monte Carlo for a linear Gaussian Multiple Changepoint Model

Description

This function generates a sample from the posterior distribution of a linear Gaussian model with multiple changepoints. The function uses the Markov chain Monte Carlo method of Chib (1998). The user supplies data and priors, and a sample from the posterior distribution is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.

Usage

MCMCregressChange(
  formula,
  data = parent.frame(),
  m = 1,
  b0 = 0,
  B0 = 0,
  c0 = 0.001,
  d0 = 0.001,
  sigma.mu = NA,
  sigma.var = NA,
  a = NULL,
  b = NULL,
  mcmc = 1000,
  burnin = 1000,
  thin = 1,
  verbose = 0,
  seed = NA,
  beta.start = NA,
  P.start = NA,
  random.perturb = FALSE,
  WAIC = FALSE,
  marginal.likelihood = c("none", "Chib95"),
  ...
)

Arguments

formula	Model formula.
data	Data frame.
m	The number of changepoints.
b0	The prior mean of \beta. This can either be a scalar or a column vector with dimension equal to the number of betas. If this takes a scalar value, then that value will serve as the prior mean for all of the betas.
B0	The prior precision of \beta. This can either be a scalar or a square matrix with dimensions equal to the number of betas. If this takes a scalar value, then that value times an identity matrix serves as the prior precision of beta. Default value of 0 is equivalent to an improper uniform prior for beta.
c0	c_0/2 is the shape parameter for the inverse Gamma prior on \sigma^2 (the variance of the disturbances). The amount of information in the inverse Gamma prior is something like that from c_0 pseudo-observations.
d0	d_0/2 is the scale parameter for the inverse Gamma prior on \sigma^2 (the variance of the disturbances). In constructing the inverse Gamma prior, d_0 acts like the sum of squared errors from the c_0 pseudo-observations.
sigma.mu	The mean of the inverse Gamma prior on \sigma^2. sigma.mu and sigma.var allow users to choose the inverse Gamma prior by choosing its mean and variance.
sigma.var	The variacne of the inverse Gamma prior on \sigma^2. sigma.mu and sigma.var allow users to choose the inverse Gamma prior by choosing its mean and variance.
a	a is the shape1 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.
b	b is the shape2 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.
mcmc	The number of MCMC iterations after burnin.
burnin	The number of burn-in iterations for the sampler.
thin	The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value.
verbose	A switch which determines whether or not the progress of the sampler is printed to the screen. If verbose is greater than 0 the iteration number, the \beta vector, and the error variance are printed to the screen every verboseth iteration.
seed	The seed for the random number generator. If NA, the Mersenne Twister generator is used with default seed 12345; if an integer is passed it is used to seed the Mersenne twister. The user can also pass a list of length two to use the L'Ecuyer random number generator, which is suitable for parallel computation. The first element of the list is the L'Ecuyer seed, which is a vector of length six or NA (if NA a default seed of rep(12345,6) is used). The second element of list is a positive substream number. See the MCMCpack specification for more details.
beta.start	The starting values for the \beta vector. This can either be a scalar or a column vector with dimension equal to the number of betas. The default value of of NA will use the MLE estimate of \beta as the starting value. If this is a scalar, that value will serve as the starting value mean for all of the betas.
P.start	The starting values for the transition matrix. A user should provide a square matrix with dimension equal to the number of states. By default, draws from the Beta(0.9, 0.1) are used to construct a proper transition matrix for each raw except the last raw.
random.perturb	If TRUE, randomly sample hidden states whenever regularly sampled hidden states have at least one single observation state (SOS). SOS is a sign of overfitting in non-ergodic hidden Markov models.
WAIC	Compute the Widely Applicable Information Criterion (Watanabe 2010).
marginal.likelihood	How should the marginal likelihood be calculated? Options are: none in which case the marginal likelihood will not be calculated, and Chib95 in which case the method of Chib (1995) is used.
...	further arguments to be passed

Details

MCMCregressChange simulates from the posterior distribution of the linear regression model with multiple changepoints.

The model takes the following form:

y_t=x_t ' \beta_i + I(s_t=i)\varepsilon_{t},\;\; i=1, \ldots, k

Where k is the number of states and I(s_t=i) is an indicator function that becomes 1 when a state at t is i and otherwise 0.

The errors are assumed to be Gaussian in each regime:

I(s_t=i)\varepsilon_{t} \sim \mathcal{N}(0, \sigma^2_i)

We assume standard, semi-conjugate priors:

\beta_i \sim \mathcal{N}(b_0,B_0^{-1}),\;\; i=1, \ldots, k

And:

\sigma^{-2}_i \sim \mathcal{G}amma(c_0/2, d_0/2),\;\; i=1, \ldots, k

Where \beta_i and \sigma^{-2}_i are assumed a priori independent.

The simulation proper is done in compiled C++ code to maximize efficiency.

Value

An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package. The object contains an attribute prob.state storage matrix that contains the probability of state_i for each period, the log-likelihood of the model (loglike), and the log-marginal likelihood of the model (logmarglike).

References

Jong Hee Park, 2012. “Unified Method for Dynamic and Cross-Sectional Heterogeneity: Introducing Hidden Markov Panel Models.” American Journal of Political Science.56: 1040-1054. <doi: 10.1111/j.1540-5907.2012.00590.x>

Sumio Watanabe. 2010. "Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory" Journal of Machine Learning Research. 11: 3571-3594.

Siddhartha Chib. 1995. "Marginal Likelihood from the Gibbs Output." Journal of the American Statistical Association. 90: 1313-1321. <doi: 10.1016/S0304-4076(97)00115-2>

Siddhartha Chib. 1998. "Estimation and comparison of multiple change-point models." Journal of Econometrics. 86: 221-241. <doi: 10.1080/01621459.1995.10476635>

Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. “MCMCpack: Markov Chain Monte Carlo in R.”, Journal of Statistical Software. 42(9): 1-21. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v042.i09")}.

See Also

plotState, plotChangepoint

Examples

## Not run: 
set.seed(1119)
n <- 100
x1 <- runif(n)
true.beta1 <- c(2, -2)
true.beta2 <- c(0,  2)
true.Sigma <- c(1, 2)
true.s <- rep(1:2, each=n/2)

mu1 <- cbind(1, x1[true.s==1])%*%true.beta1
mu2 <- cbind(1, x1[true.s==2])%*%true.beta2

y <- as.ts(c(rnorm(n/2, mu1, sd=sqrt(true.Sigma[1])), rnorm(n/2, mu2, sd=sqrt(true.Sigma[2]))))
formula=y ~ x1

ols1 <- lm(y[true.s==1] ~x1[true.s==1])
ols2 <- lm(y[true.s==2] ~x1[true.s==2])

## prior
b0 <- 0
B0 <- 0.1
sigma.mu=sd(y)
sigma.var=var(y)

## models
model0 <-  MCMCregressChange(formula, m=0, b0=b0, B0=B0, mcmc=100, burnin=100,
           sigma.mu=sigma.mu, sigma.var=sigma.var, marginal.likelihood="Chib95")
model1 <-  MCMCregressChange(formula, m=1, b0=b0, B0=B0, mcmc=100, burnin=100,
           sigma.mu=sigma.mu, sigma.var=sigma.var, marginal.likelihood="Chib95")
model2 <-  MCMCregressChange(formula, m=2, b0=b0, B0=B0, mcmc=100, burnin=100,
           sigma.mu=sigma.mu, sigma.var=sigma.var, marginal.likelihood="Chib95")
model3 <-  MCMCregressChange(formula, m=3, b0=b0, B0=B0, mcmc=100, burnin=100,
           sigma.mu=sigma.mu, sigma.var=sigma.var, marginal.likelihood="Chib95")
model4 <-  MCMCregressChange(formula, m=4, b0=b0, B0=B0, mcmc=100, burnin=100,
           sigma.mu=sigma.mu, sigma.var=sigma.var, marginal.likelihood="Chib95")
model5 <-  MCMCregressChange(formula, m=5, b0=b0, B0=B0, mcmc=100, burnin=100,
           sigma.mu=sigma.mu, sigma.var=sigma.var, marginal.likelihood="Chib95")

print(BayesFactor(model0, model1, model2, model3, model4, model5))
plotState(model1)
plotChangepoint(model1)


## End(Not run)

MCMCpack documentation built on Sept. 11, 2024, 8:13 p.m.