This function generates a sample from the posterior distribution of a (sticky) HDP-HMM with a Negative Binomial 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.
HDPHMMnegbin(
formula,
data = parent.frame(),
K = 10,
b0 = 0,
B0 = 1,
a.theta = 50,
b.theta = 5,
a.alpha = 1,
b.alpha = 0.1,
a.gamma = 1,
b.gamma = 0.1,
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,
gamma.start = 0.5,
theta.start = 0.98,
ak.start = 100,
...
)An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
Model formula.
Data frame.
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.
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.
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.
Paramaters for the Beta prior on \(\theta\), which captures the strength of the self-transition bias.
Shape and scale parameters for the Gamma distribution on \(\alpha + \kappa\).
Shape and scale parameters for the Gamma distribution on \(\gamma\).
The hyperprior for the distribution \(\rho\) See details.
The hyperprior for the distribution \(\rho\). See details.
The hyperprior for the distribution \(\rho\). See details.
The number of burn-in iterations for the sampler.
The number of Metropolis iterations for the sampler.
The thinning interval used in the simulation. The number of mcmc iterations must be divisible by this value.
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.
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.
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.
Initial transition matrix between regimes. Should be
a K by K matrix. If not provided, the default value
will be place theta.start along the diagonal and the rest
of the mass even distributed within rows.
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.
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.
The starting values for the random effect, \(\nu\). The default value is a vector of ones.
Scalar starting values for the \(\theta\), \(\alpha + \kappa\), and \(\gamma\) parameters.
further arguments to be passed.
HDPHMMnegbin simulates from the posterior distribution of a
sticky HDP-HMM with a Negative Binomial 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 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-HMM 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:
$$y_t \sim \mathcal{P}oisson(\nu_t\mu_t)$$
$$\mu_t = x_t ' \beta_m,\;\; m = 1, \ldots, M$$
$$\nu_t \sim \mathcal{G}amma(\rho_m, \rho_m)$$
Where \(M\) is an upper bound on the number of states and \(\beta_m\) and \(\rho_m\) are parameters when a state is \(m\) at \(t\).
The transition probabilities between states are assumed to follow a heirarchical Dirichlet process:
$$\pi_m \sim \mathcal{D}irichlet(\alpha\delta_1, \ldots, \alpha\delta_j + \kappa, \ldots, \alpha\delta_M)$$
$$\delta \sim \mathcal{D}irichlet(\gamma/M, \ldots, \gamma/M)$$
The \(\kappa\) value here is the sticky parameter that encourages self-transitions. The sampler follows Fox et al (2011) and parameterizes these priors with \(\alpha + \kappa\) and \(\theta = \kappa/(\alpha + \kappa)\), with the latter representing the degree of self-transition bias. Gamma priors are assumed for \((\alpha + \kappa)\) and \(\gamma\).
We assume Gaussian distribution for prior of \(\beta\):
$$\beta_m \sim \mathcal{N}(b_0,B_0^{-1}),\;\; m = 1, \ldots, M$$
The overdispersion parameters have a prior with the following form:
$$f(\rho_m|e,f,g) \propto \rho^{e-1}(\rho + g)^{-(e+f)}$$
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 \(\nu_i\) are updated their (conjugate) full conditional, which is also Gamma. The states are updated as in Fox et al (2011), supplemental materials.
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. 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.wustl.edu.s3-website-us-east-1.amazonaws.com/.
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>
Emily B. Fox, Erik B. Sudderth, Michael I. Jordan, and Alan S. Willsky. 2011.. ``A sticky HDP-HMM with application to speaker diarization.'' The Annals of Applied Statistics, 5(2A), 1020-1056. <doi:10.1214/10-AOAS395>
MCMCnegbinChange, HDPHMMpoisson
if (FALSE) {
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 <- HDPHMMnegbin(y ~ x1, K = 10, verbose = 1000,
e = 2, f = 2, g = 10,
a.theta = 100, b.theta = 1,
b0 = rep(0, 2), B0 = (1/9) * diag(2),
rho.step = rep(0.75, times = 10),
seed = list(NA, 2),
theta.start = 0.95, gamma.start = 10,
ak.start = 10)
plotHDPChangepoint(posterior, ylab="Density", start=1)
}
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