MCMCnegbinChange: Markov Chain Monte Carlo for Negative Binomial Regression...

In MCMCpack: Markov Chain Monte Carlo (MCMC) Package

Markov Chain Monte Carlo for Negative Binomial Regression Changepoint Model

Description

This function generates a sample from the posterior distribution of a Negative Binomial regression model with multiple changepoints. For the changepoints, the sampler 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

MCMCnegbinChange(
  formula,
  data = parent.frame(),
  m = 1,
  fixed.m = TRUE,
  b0 = 0,
  B0 = 1,
  a = NULL,
  b = NULL,
  e = 2,
  f = 2,
  g = 10,
  burnin = 1000,
  mcmc = 1000,
  thin = 1,
  verbose = 0,
  seed = NA,
  beta.start = NA,
  P.start = NA,
  rho.start = NA,
  rho.step,
  nu.start = NA,
  marginal.likelihood = c("none", "Chib95"),
  ...
)

Arguments

formula	Model formula.
data	Data frame.
m	The number of changepoints.
fixed.m	A logical indicator for whether or not the number of changepoints in the sampler should be exactly equal to m or if that is simply an upper bound. Setting fixed.m to FALSE is equivalent to assuming a weak-limit approximation to a Dirichlet process mixture.
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.
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.
e	The hyperprior for the distribution \rho See details.
f	The hyperprior for the distribution \rho. See details.
g	The hyperprior for the distribution \rho. See details.
burnin	The number of burn-in iterations for the sampler.
mcmc	The number of Metropolis 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 current beta vector, and the Metropolis acceptance rate 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 value for the \beta vector. 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 starting value for all of the betas. The default value of NA will use the maximum likelihood estimate of \beta as the starting value for all regimes.
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.
rho.start	The starting value for the \rho variable. This can either be a scalar or a column vector with dimension equal to the number of regimes. If the value is scalar, it will be used for all regimes. The default value is a vector of ones.
rho.step	Tuning parameter for the slice sampling approach to sampling rho. Determines the size of the step-out used to find the correct slice to draw from. Lower values are more accurate, but will take longer (up to a fixed searching limit). Default is 0.1.
nu.start	The starting values for the random effect, \nu. The default value is a vector of ones.
marginal.likelihood	How should the marginal likelihood be calculated? Options are: none in which case the marginal likelihood will not be calculated or Laplace in which case the Laplace approximation (see Kass and Raftery, 1995) is used.
...	further arguments to be passed.

Details

MCMCnegbinChangesimulates from the posterior distribution of a Negative Binomial regression model with multiple changepoints using the methods of Chib (1998) and Fruehwirth-Schnatter et al (2009). The details of the model are discussed in Blackwell (2017).

The model takes the following form:

y_t \sim \mathcal{P}oisson(\nu_t\mu_t)

\mu_t = x_t ' \beta_m,\;\; m = 1, \ldots, M

\nu_t \sim \mathcal{G}amma(\rho_m, \rho_m)

Where M is the number of states and \beta_m and \rho_m are parameters when a state is m at t.

We assume Gaussian distribution for prior of \beta:

\beta_m \sim \mathcal{N}(b_0,B_0^{-1}),\;\; m = 1, \ldots, M

And:

p_{mm} \sim \mathcal{B}eta(a, b),\;\; m = 1, \ldots, M

Where M is the number of states.

The overdispersion parameters have a prior with the following form:

f(\rho_m|e,f,g) \propto \rho^{e-1}(\rho + g)^{-(e+f)}

The model is simulated via blocked Gibbs conditonal on the states. The \beta being simulated via the auxiliary mixture sampling method of Fuerhwirth-Schanetter et al. (2009). The \rho is updated via slice sampling. The \nu_i are updated their (conjugate) full conditional, which is also Gamma. The states are updated as in Chib (1998)

Value

An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.

References

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")}.

Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.wustl.edu.s3-website-us-east-1.amazonaws.com/.

Sylvia Fruehwirth-Schnatter, Rudolf Fruehwirth, Leonhard Held, and Havard Rue. 2009. “Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data”, Statistics and Computing 19(4): 479-492. <doi:10.1007/s11222-008-9109-4>

Matthew Blackwell. 2017. “Game Changers: Detecting Shifts in Overdispersed Count Data,” Political Analysis 26(2), 230-239. <doi:10.1017/pan.2017.42>

See Also

MCMCpoissonChange, plotState, plotChangepoint

Examples

 ## Not run: 
   n <- 150
   reg <- 3
   true.s <- gl(reg, n/reg, n)
   rho.true <- c(1.5, 0.5, 3)
   b0.true <- c(1, 3, 1)
   b1.true <- c(1, -2, 2)
   x1 <- runif(n, 0, 2)
   nu.true <- rgamma(n, rho.true[true.s], rho.true[true.s])
   mu <- nu.true * exp(b0.true[true.s] + x1 * b1.true[true.s])
   y <- rpois(n, mu)

   posterior <- MCMCnegbinChange(y ~ x1, m = 2, verbose = 1000,
                          marginal.likelihood = "Chib95",
                          e = 2, f = 2, g = 10,
                          b0 = rep(0, 2), B0 = (1/9) * diag(2),
                          rho.step = rep(0.75, times = 3),
                          seed = list(NA, 2))

   par(mfrow=c(attr(posterior, "m") + 1, 1), mai=c(0.4, 0.6, 0.3, 0.05))
   plotState(posterior, legend.control = c(1, 0.6))
   plotChangepoint(posterior, verbose = TRUE, ylab="Density",
  start=1, overlay=TRUE)


   open.ended <- MCMCnegbinChange(y ~ x1, m = 10, verbose = 1000,
                          fixed.m = FALSE, mcmc = 2000, burnin = 2000,
                          e = 2, f = 2, g = 10,
                          a = 100, b = 1,
                          b0 = rep(0, 2), B0 = (1/9) * diag(2),
                          rho.step = 0.75,
                          seed = list(NA, 2))

   plotState(open.ended, legend.control = c(1, 0.6))
   
## End(Not run)

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