Markov Chain Monte Carlo for Negative Binomial Regression Changepoint Model
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.
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"),
...
)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 |
b0 |
The prior mean of β. 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 β. 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 ρ See details. |
f |
The hyperprior for the distribution ρ. See details. |
g |
The hyperprior for the distribution ρ. 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 |
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
|
beta.start |
The starting value for the β 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 β 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 |
rho.start |
The starting value for the ρ 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, ν. The default value is a vector of ones. |
marginal.likelihood |
How should the marginal likelihood be calculated?
Options are: |
... |
further arguments to be passed. |
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(ν_tμ_t)
μ_t = x_t ' β_m,\;\; m = 1, …, M
ν_t \sim \mathcal{G}amma(ρ_m, ρ_m)
Where M is the number of states and β_m and ρ_m are parameters when a state is m at t.
We assume Gaussian distribution for prior of β:
β_m \sim \mathcal{N}(b_0,B_0^{-1}),\;\; m = 1, …, M
And:
p_{mm} \sim \mathcal{B}eta(a, b),\;\; m = 1, …, M
Where M is the number of states.
The overdispersion parameters have a prior with the following form:
f(ρ_m|e,f,g) \propto ρ^{e-1}(ρ + g)^{-(e+f)}
The model is simulated via blocked Gibbs conditonal on the states. The β being simulated via the auxiliary mixture sampling method of Fuerhwirth-Schanetter et al. (2009). The ρ is updated via slice sampling. The ν_i are updated their (conjugate) full conditional, which is also Gamma. The states are updated as in Chib (1998)
An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
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. https://www.jstatsoft.org/v42/i09/.
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.lsa.umich.edu.
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>
## 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.