# HDPHSMMnegbin: Markov Chain Monte Carlo for HDP-HSMM with a Negative... In MCMCpack: Markov Chain Monte Carlo (MCMC) Package

 HDPHSMMnegbin R Documentation

## Markov Chain Monte Carlo for HDP-HSMM with a Negative Binomial outcome distribution

### Description

This function generates a sample from the posterior distribution of a Hidden Semi-Markov Model with a Heirarchical Dirichlet Process and a Negative Binomial outcome distribution (Johnson and Willsky, 2013). 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

HDPHSMMnegbin(
formula,
data = parent.frame(),
K = 10,
b0 = 0,
B0 = 1,
a.alpha = 1,
b.alpha = 0.1,
a.gamma = 1,
b.gamma = 0.1,
a.omega,
b.omega,
e = 2,
f = 2,
g = 10,
r = 1,
burnin = 1000,
mcmc = 1000,
thin = 1,
verbose = 0,
seed = NA,
beta.start = NA,
P.start = NA,
rho.start = NA,
rho.step,
nu.start = NA,
omega.start = NA,
gamma.start = 0.5,
alpha.start = 100,
...
)


### Arguments

 formula Model formula. data Data frame. K The number of regimes under consideration. This should be larger than the hypothesized number of regimes in the data. Note that the sampler will likely visit fewer than K regimes. 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.alpha, b.alpha Shape and scale parameters for the Gamma distribution on \alpha. a.gamma, b.gamma Shape and scale parameters for the Gamma distribution on \gamma. a.omega, b.omega Paramaters for the Beta prior on \omega, which determines the regime length distribution, which is Negative Binomial, with parameters r and omega. 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. r Parameter of the Negative Binomial prior for regime durations. It is the target number of successful trials. Must be strictly positive. Higher values increase the variance of the duration distributions. 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 Initial transition matrix between regimes. Should be a K by K matrix. If not provided, the default value will be uniform transition distributions. 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. omega.start A vector of starting values for the probability of success parameter in the Negative Binomial distribution that governs the duration distributions. alpha.start, gamma.start Scalar starting values for the \alpha, and \gamma parameters. ... further arguments to be passed.

### Details

HDPHSMMnegbin simulates from the posterior distribution of a HDP-HSMM with a Negative Binomial outcome distribution, allowing for multiple, arbitrary changepoints in the model. The details of the model are discussed in Johnson & Willsky (2013). The implementation here is based on a weak-limit approximation, where there is a large, though finite number of regimes that can be switched between. Unlike other changepoint models in MCMCpack, the HDP-HSMM approach allows for the state sequence to return to previous visited states.

The model takes the following form, where we show the fixed-limit version:

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

\mu_t = x_t ' \beta_k,\;\; k = 1, \ldots, K

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

Where K is an upper bound on the number of states and \beta_k and \rho_k are parameters when a state is k at t.

In the HDP-HSMM, there is a super-state sequence that, for a given observation, is drawn from the transition distribution and then a duration is drawn from a duration distribution to determin how long that state will stay active. After that duration, a new super-state is drawn from the transition distribution, where self-transitions are disallowed. The transition probabilities between states are assumed to follow a heirarchical Dirichlet process:

\pi_k \sim \mathcal{D}irichlet(\alpha\delta_1, \ldots , \alpha\delta_K)

\delta \sim \mathcal{D}irichlet(\gamma/K, \ldots, \gamma/K)

In the algorithm itself, these \pi vectors are modified to remove self-transitions as discussed above. There is a unique duration distribution for each regime with the following parameters:

D_k \sim \mathcal{N}egBin(r, \omega_k)

\omega_k \sim \mathcal{B}eta(a_{\omega,k}, b_{\omega, k})

We assume Gaussian distribution for prior of \beta:

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

The overdispersion parameters have a prior with the following form:

f(\rho_k|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_t are updated their (conjugate) full conditional, which is also Gamma. The states and their durations are drawn as in Johnson & Willsky (2013).

### 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.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>

Matthew J. Johnson and Alan S. Willsky. 2013. “Bayesian Nonparametric Hidden Semi-Markov Models.” Journal of Machine Learning Research, 14(Feb), 673-701.

MCMCnegbinChange, HDPHMMnegbin,

### Examples


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

posterior <- HDPHSMMnegbin(y ~ x1, K = 10, verbose = 1000,
e = 2, f = 2, g = 10,
b0 = 0, B0 = 1/9,
a.omega = 1, b.omega = 100, r = 1,
rho.step = rep(0.75, times = 10),
seed = list(NA, 2),
omega.start = 0.05, gamma.start = 10,
alpha.start = 5)

plotHDPChangepoint(posterior, ylab="Density", start=1)

## End(Not run)



MCMCpack documentation built on May 29, 2024, 11:23 a.m.