Markov Chain Monte Carlo for HDP-HSMM with a Negative Binomial outcome distribution
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.
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 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. |
a.gamma, b.gamma |
Shape and scale parameters for the Gamma distribution on γ. |
a.omega, b.omega |
Paramaters for the Beta prior on
ω, which determines the regime length distribution,
which is Negative Binomial, with parameters |
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. |
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 |
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 |
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. |
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 α, and γ parameters. |
... |
further arguments to be passed. |
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(ν_tμ_t)
μ_t = x_t ' β_k,\;\; k = 1, …, K
ν_t \sim \mathcal{G}amma(ρ_k, ρ_k)
Where K is an upper bound on the number of states and β_k and ρ_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:
π_k \sim \mathcal{D}irichlet(αδ_1, … , αδ_K)
δ \sim \mathcal{D}irichlet(γ/K, …, γ/K)
In the algorithm itself, these π 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, ω_k)
ω_k \sim \mathcal{B}eta(a_{ω,k}, b_{ω, k})
We assume Gaussian distribution for prior of β:
β_k \sim \mathcal{N}(b_0,B_0^{-1}),\;\; m = 1, …, K
The overdispersion parameters have a prior with the following form:
f(ρ_k|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_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): 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>
Matthew J. Johnson and Alan S. Willsky. 2013. “Bayesian Nonparametric Hidden Semi-Markov Models.” Journal of Machine Learning Research, 14(Feb), 673-701.
## 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.