Comprehensive Guide to Fitting Negative Binomial Hidden Markov Models in R

Unlock the power of advanced statistical modeling for real-world count data analysis.

Key Takeaways

• Select the appropriate R package based on your modeling needs and data characteristics.
• Proper data preparation is crucial for accurate Hidden Markov Model (HMM) fitting.
• Model diagnostics ensure the reliability and validity of your fitted HMM.

Introduction

Hidden Markov Models (HMMs) are powerful statistical tools used to model systems that exhibit underlying, unobservable states influencing observable data. When dealing with count data that display overdispersion, the Negative Binomial distribution serves as a robust choice for modeling emission probabilities within HMMs. This guide provides an in-depth approach to fitting Negative Binomial Hidden Markov Models (NB-HMMs) in R using real data, synthesizing best practices and methodologies from leading sources.

Required Packages

Several R packages facilitate the fitting of NB-HMMs, each offering unique features catering to different modeling needs:

• hmmTMB: Provides flexible HMM fitting with automatic differentiation, supporting various distributions including Negative Binomial.
• depmixS4: A versatile package for HMMs that can be extended to handle Negative Binomial distributions through custom functions.
• RIPSeeker: Specialized in handling Negative Binomial HMMs with functions like nbh_init and nbh_em.
• mHMMbayes: Facilitates Bayesian estimation of multilevel HMMs, suitable for complex data structures.

Installation

Ensure that all necessary packages are installed and loaded before proceeding:

# Install required packages
install.packages(c("hmmTMB", "depmixS4", "MASS", "RIPSeeker", "mHMMbayes"))

# Load the packages
library(hmmTMB)
library(depmixS4)
library(MASS)
library(RIPSeeker)
library(mHMMbayes)

Data Preparation

Proper data preparation is essential for accurate model fitting. Your count data should be organized in a vector or data frame, ensuring that it reflects the temporal sequence inherent to HMMs.

Example Data Structure

# Example count data
observations <- c(5, 3, 6, 2, 7, 8, 4, 5, 3, 6, 9, 2, 4, 5, 7)

Model Specification

Defining the structure of your NB-HMM involves specifying the number of hidden states, the distribution of the observations, and initializing the model parameters. Below, we explore different approaches using the aforementioned packages.

Using hmmTMB

The hmmTMB package offers a streamlined approach to fitting NB-HMMs with automatic parameter estimation.

# Define the model with hmmTMB
model <- hmmTMB(data = observations,
               n_states = 2,  # Number of hidden states
               dists = list(count = "negbin"),
               par0 = list(count = list(size = 1, mu = 1)))

# Fit the model
fit_model <- fit(model)

# Summary of the fitted model
summary(fit_model)

Using depmixS4 with Custom Negative Binomial Family

The depmixS4 package does not directly support the Negative Binomial distribution. However, by defining a custom family, you can extend its capabilities.

# Define a custom Negative Binomial family
NBfamily <- function(fixed.theta) {
  ret <- list()
  ret$density <- function(y, eta) {
    dnbinom(y, mu = eta[1], size = fixed.theta, log = FALSE)
  }
  ret$loglik <- function(y, eta) {
    dnbinom(y, mu = eta[1], size = fixed.theta, log = TRUE)
  }
  ret$simulate <- function(eta, nsim) {
    rnbinom(nsim, mu = eta[1], size = fixed.theta)
  }
  ret$family <- "negbin"
  ret
}

# Initialize fixed theta
fixed.theta <- 5

# Create response models for each state
resp1 <- response(formula = observations ~ 1, family = NBfamily(fixed.theta))
resp2 <- response(formula = observations ~ 1, family = NBfamily(fixed.theta))

# Build the Dependent Mixture Model
mod <- depmix(response = list(resp1, resp2),
             data = data.frame(observations),
             nstates = 2,
             family = list(NBfamily(fixed.theta), NBfamily(fixed.theta)))

# Fit the model
fm <- fit(mod, verbose = FALSE)

# Summary of the fitted model
summary(fm)

Using RIPSeeker

The RIPSeeker package provides specialized functions for initializing and fitting NB-HMMs.

# Initialize NB-HMM parameters
nbhInit <- nbh_init(count_data = observations, K = 2)

# Fit the NB-HMM
nbhModel <- nbh_em(count_data = observations,
                  TRANS0 = nbhInit$TRANS,
                  alpha0 = nbhInit$alpha,
                  beta0 = nbhInit$beta)

# Summary of the fitted model
print(nbhModel)

Using mHMMbayes

For Bayesian estimation of multilevel HMMs, mHMMbayes is the preferred package.

# Define the model using mHMMbayes
bayes_model <- mHMMbayes(count_data = observations,
                        n_states = 2,
                        distribution = "nbinom",
                        n_iter = 10000,  # Number of iterations
                        burn_in = 2000)   # Burn-in period

# Summary of the fitted model
summary(bayes_model)

Parameter Estimation

Accurate parameter estimation is vital for capturing the underlying state dynamics and ensuring model efficacy. Each package offers different mechanisms for estimating parameters such as transition probabilities and state-dependent emission parameters.

Transition Matrix and Emission Parameters

The transition matrix defines the probabilities of moving from one hidden state to another, while emission parameters characterize the distribution of observations within each state.

Example: Transition Matrix from hmmTMB

# Extract transition probabilities
trans_probs <- fit_model$model$transitions
print(trans_probs)

Example: Emission Parameters from hmmTMB

# Extract emission parameters
emission_params <- fit_model$model$parameters$par
print(emission_params)

Model Diagnostics

After fitting the model, it is essential to perform diagnostics to assess the model's fit and validate its assumptions.

Residual Analysis

Residual analysis helps identify patterns that the model may have missed. Plotting residuals versus fitted values can reveal systematic deviations.

# Calculate residuals
residuals <- observations - fitted(fit_model)

# Plot residuals
plot(residuals, type = "h", main = "Residuals vs Fitted Values", xlab = "Observation", ylab = "Residuals")

State Decoding

Decoding the most probable sequence of hidden states provides insights into the system's behavior over time. The Viterbi algorithm is commonly used for this purpose.

# Decode states using Viterbi algorithm with hmmTMB
decoded_states <- decode(fit_model, method = "viterbi")

# Plot observed data with decoded states
plot(observations, type = "l", col = "blue", main = "Observed Data with Decoded States", ylab = "Count")
lines(decoded_states, col = "red")
legend("topright", legend = c("Observed", "Decoded States"), col = c("blue", "red"), lty = 1)

Model Selection Criteria

Utilize information criteria such as Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC) to compare models with differing numbers of states or distributional assumptions.

# Extract AIC and BIC
aic <- AIC(fit_model)
bic <- BIC(fit_model)

print(paste("AIC:", aic))
print(paste("BIC:", bic))

Advanced Features

Beyond the basic model fitting, advanced features can enhance the robustness and applicability of your NB-HMM.

Incorporating Covariates

Including covariates in the transition probabilities allows the model to account for external factors influencing state transitions.

# Example of including a covariate in hmmTMB
covariate <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)  # Example covariate

model_with_cov <- hmmTMB(data = observations,
                       n_states = 2,
                       dists = list(count = "negbin"),
                       formula = list(count ~ 1, transition = ~ covariate),
                       par0 = list(count = list(size = 1, mu = 1),
                                   transition = list(coefficients = c(0.5, 0.5))))

fit_model_cov <- fit(model_with_cov)
summary(fit_model_cov)

Bayesian Estimation

Bayesian methods offer a probabilistic framework for parameter estimation, providing credible intervals and incorporating prior knowledge.

# Example using mHMMbayes for Bayesian estimation
bayes_model <- mHMMbayes(count_data = observations,
                        n_states = 2,
                        distribution = "nbinom",
                        n_iter = 10000,
                        burn_in = 2000)

# Summary of Bayesian model
summary(bayes_model)

# Plot posterior distributions
plot(bayes_model)

Comparison of R Packages for NB-HMM

Conclusion

Fitting Negative Binomial Hidden Markov Models in R is a robust approach for analyzing count data with overdispersion and underlying state dynamics. By selecting the appropriate package and meticulously preparing your data, you can effectively model complex systems and derive meaningful insights. Advanced features like covariate inclusion and Bayesian estimation further enhance the model's applicability, ensuring that it aligns with the specific requirements of your research context.

References

github.com

hmmTMB Package Documentation

rdrr.io

RIPSeeker Package Vignettes

cran.r-project.org

depmixS4 Package Documentation