HDPHMMpoisson  R Documentation 
This function generates a sample from the posterior distribution of a (sticky) HDPHMM with a Poisson outcome distribution (Fox et al, 2011). 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.
HDPHMMpoisson( 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.theta = 50, b.theta = 5, burnin = 1000, mcmc = 1000, thin = 1, verbose = 0, seed = NA, beta.start = NA, P.start = NA, gamma.start = 0.5, theta.start = 0.98, ak.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 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.alpha, b.alpha 
Shape and scale parameters for the Gamma distribution on α + κ. 
a.gamma, b.gamma 
Shape and scale parameters for the Gamma distribution on γ. 
a.theta, b.theta 
Paramaters for the Beta prior on θ, which captures the strength of the selftransition bias. 
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 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 
Initial transition matrix between regimes. Should be
a 
theta.start, ak.start, gamma.start 
Scalar starting values for the θ, α + κ, and γ parameters. 
... 
further arguments to be passed. 
HDPHMMpoisson
simulates from the posterior distribution of a
sticky HDPHMM with a Poisson outcome distribution,
allowing for multiple, arbitrary changepoints in the model. The details of the
model are discussed in Blackwell (2017). 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 HDPHMM 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(μ_t)
μ_t = x_t ' β_m,\;\; m = 1, …, M
Where M is an upper bound on the number of states and β_m are parameters when a state is m at t.
The transition probabilities between states are assumed to follow a heirarchical Dirichlet process:
π_m \sim \mathcal{D}irichlet(αδ_1, …, αδ_j + κ, …, αδ_M)
δ \sim \mathcal{D}irichlet(γ/M, …, γ/M)
The κ value here is the sticky parameter that encourages selftransitions. The sampler follows Fox et al (2011) and parameterizes these priors with α + κ and θ = κ/(α + κ), with the latter representing the degree of selftransition bias. Gamma priors are assumed for (α + κ) and γ.
We assume Gaussian distribution for prior of β:
β_m \sim \mathcal{N}(b_0,B_0^{1}),\;\; m = 1, …, M
The model is simulated via blocked Gibbs conditonal on the states. The β being simulated via the auxiliary mixture sampling method of FuerhwirthSchanetter et al. (2009). The states are updated as in Fox et al (2011), supplemental materials.
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. 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>
Emily B. Fox, Erik B. Sudderth, Michael I. Jordan, and Alan S. Willsky. 2011.. “A sticky HDPHMM with application to speaker diarization.” The Annals of Applied Statistics, 5(2A), 10201056. <doi:10.1214/10AOAS395>
MCMCpoissonChange
, HDPHMMnegbin
## Not run: n < 150 reg < 3 true.s < gl(reg, n/reg, n) b1.true < c(1, 2, 2) x1 < runif(n, 0, 2) mu < exp(1 + x1 * b1.true[true.s]) y < rpois(n, mu) posterior < HDPHMMpoisson(y ~ x1, K = 10, verbose = 1000, a.theta = 100, b.theta = 1, b0 = rep(0, 2), B0 = (1/9) * diag(2), seed = list(NA, 2), theta.start = 0.95, gamma.start = 10, ak.start = 10) plotHDPChangepoint(posterior, ylab="Density", start=1) ## End(Not run)
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