HMMHSMM (version 0.1.0)

HSMMVarianceMatrix: Variance-Covariance Matrix for Hidden Semi-Markov Models

Description

Computes an approximate variance-covariance matrix of the parameter estimates from a fitted Hidden Semi-Markov Model (HSMM) using a numerical Hessian of the log-likelihood. The Hessian is inverted and adjusted to ensure positive-definiteness, providing an estimate of parameter uncertainty.

Usage

HSMMVarianceMatrix(x, HSMM, M = NA, obsdist, dwelldist,
                   h = 1e-5, method = "central", verbose = TRUE)

Value

A variance-covariance matrix of the HSMM parameter estimates, computed as the inverse of the observed information matrix (negative Hessian of the log-likelihood), adjusted to be positive-definite.

Arguments

x

Numeric vector. The observed data sequence.

HSMM

A fitted HSMM object (typically the output from findmleHSMM), containing estimated model parameters Pi, delta, observation parameters, and dwell-time parameters.

M

Integer. Maximum dwell time to consider for semi-Markov states. If NA, defaults to min(length(x), 1000).

obsdist

Character string. Observation distribution. Supported distributions: "norm", "pois", "weibull", "nbinom", "exp", "gamma", "lnorm", "gev", "ZInormal", "ZIgamma".

dwelldist

Character string. Dwell-time distribution. Supported distributions: "pois", "nbinom", "geom".

h

Numeric. Step size for finite-difference approximation of the Hessian. Default is 1e-5.

method

Character string. Method for numerical Hessian computation. Options are "central" (default, central differences) or "forward" (forward differences).

verbose

Logical, if TRUE (default), progress messages are printed to the console. Set to FALSE to suppress informational output.

Author

Aimee Cody

Details

This function approximates the observed Fisher information for an HSMM by numerically differentiating the log-likelihood with respect to the model parameters. The resulting Hessian is inverted to obtain a variance-covariance matrix. Since numerical Hessians can yield non-positive-definite matrices due to rounding error or flat likelihood surfaces, the eigenvalues are adjusted to enforce positive-definiteness.

See Also

generateHSMM, for simulating HSMM data. findmleHSMM, for fitting Hidden Semi-Markov Models. HMMVarianceMatrix, for numerically computing the variance-covariance matrix of a fitted hidden Markov model.

Examples

# Example with 2-state Normal HSMM
J <- 2
Pi <- matrix(c(0, 1,
               1, 0), nrow = 2, byrow = TRUE)
obspar <- list(mean = c(0, 3), sd = c(1, 1))
dwellpar <- list(lambda = c(5, 7))
# Simulate data
sim_data <- generateHSMM(n = 100, J = J, obsdist = "norm",
                         dwelldist = "pois", obspar = obspar,
                         dwellpar = dwellpar, Pi = Pi)
# Fit HSMM
HSMM_fit <- findmleHSMM(x = sim_data$x, J = J, M = 50,
                        obsdist = "norm", dwelldist = "pois",
                        obspar = obspar, dwellpar = dwellpar, Pi = Pi)
# Compute variance-covariance matrix
vcov_matrix <- HSMMVarianceMatrix(x = sim_data$x, HSMM = HSMM_fit,
                                  obsdist = "norm", dwelldist = "pois",
                                  M = 50)
vcov_matrix