HDPHSMMnegbin  R Documentation 
This function generates a sample from the posterior distribution of a Hidden SemiMarkov 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.
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,
...
)
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 
b0 
The prior mean of 
B0 
The prior precision of 
a.alpha , b.alpha 
Shape and scale parameters for the Gamma
distribution on 
a.gamma , b.gamma 
Shape and scale parameters for the Gamma
distribution on 
a.omega , b.omega 
Paramaters for the Beta prior on

e 
The hyperprior for the distribution 
f 
The hyperprior for the distribution 
g 
The hyperprior for the distribution 
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 burnin 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 
P.start 
Initial transition matrix between regimes. Should be
a 
rho.start 
The starting value for the 
rho.step 
Tuning parameter for the slice sampling approach to
sampling 
nu.start 
The starting values for the random effect,

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

... 
further arguments to be passed. 
HDPHSMMnegbin
simulates from the posterior distribution of a
HDPHSMM 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 weaklimit approximation, where there is a large, though
finite number of regimes that can be switched between. Unlike other
changepoint models in MCMCpack
, the HDPHSMM approach allows
for the state sequence to return to previous visited states.
The model takes the following form, where we show the fixedlimit 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 HDPHSMM, there is a superstate 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 superstate is drawn from the transition distribution, where selftransitions 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 selftransitions 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_ke,f,g) \propto \rho^{e1}(\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 FuerhwirthSchanetter 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).
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): 121. \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 FruehwirthSchnatter, Rudolf Fruehwirth, Leonhard Held, and Havard Rue. 2009. “Improved auxiliary mixture sampling for hierarchical models of nonGaussian data”, Statistics and Computing 19(4): 479492. <doi:10.1007/s1122200891094>
Matthew Blackwell. 2017. “Game Changers: Detecting Shifts in Overdispersed Count Data,” Political Analysis 26(2), 230239. <doi:10.1017/pan.2017.42>
Matthew J. Johnson and Alan S. Willsky. 2013. “Bayesian Nonparametric Hidden SemiMarkov Models.” Journal of Machine Learning Research, 14(Feb), 673701.
MCMCnegbinChange
,
HDPHMMnegbin
,
## 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)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.