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Title:

Parsimonious Families of Hidden Markov Models for Matrix-Variate Longitudinal Data

Version:

1.0.0

Description:

Implements three families of parsimonious hidden Markov models (HMMs) for matrix-variate longitudinal data using the Expectation-Conditional Maximization (ECM) algorithm. The package supports matrix-variate normal, t, and contaminated normal distributions as emission distributions. For each hidden state, parsimony is achieved through the eigen-decomposition of the covariance matrices associated with the emission distribution. This approach results in a comprehensive set of 98 parsimonious HMMs for each type of emission distribution. Atypical matrix detection is also supported, utilizing the fitted (heavy-tailed) models.

License:

GPL (≥ 3)

Encoding:

UTF-8

RoxygenNote:

7.3.2

Imports:

data.table, doSNOW, foreach, LaplacesDemon, mclust, progress, snow, tensor, tidyr, withr

Depends:

R (≥ 2.10)

LazyData:

true

NeedsCompilation:

Packaged:

2024-08-22 16:58:17 UTC; Daniele

Author:

Salvatore D. Tomarchio [aut, cre]

Maintainer:

Salvatore D. Tomarchio <daniele.tomarchio@unict.it>

Repository:

CRAN

Date/Publication:

2024-08-28 08:00:06 UTC

Fitting Parsimonious Hidden Markov Models for Matrix-Variate Longitudinal Data

Description

Fits parsimonious Hidden Markov Models for matrix-variate longitudinal data using ECM algorithms. The models are based on the matrix-variate normal, matrix-variate t, and matrix-variate contaminated normal distributions. Parallel computing is implemented and highly recommended for faster model fitting.

Usage

Eigen.HMM_fit(
  Y,
  init.par = NULL,
  tol = 0.001,
  maxit = 500,
  nThreads = 1,
  verbose = FALSE
)

Arguments

Y

An array with dimensions p x r x num x t, where p is the number of variables in the rows of each data matrix, r is the number of variables in the columns of each data matrix, num is the number of data observations, and t is the number of time points.

init.par

A list of initial values for starting the algorithms, as generated by the Eigen.HMM_init() function.

tol

A numeric value specifying the tolerance level for the ECM algorithms' convergence.

maxit

A numeric value specifying the maximum number of iterations for the ECM algorithms.

nThreads

A positive integer indicating the number of cores to use for parallel processing.

verbose

A logical value indicating whether to display the running output.

Value

A list containing the following elements:

results

A list of the results from the fitted models.

c.time

A numeric value providing information on the computational time required to fit all models for each state.

models

A data frame listing the models that were fitted.

Examples

data(simData)
Y <- simData$Y
init <- Eigen.HMM_init(Y = Y, k = 2, density = "MVT", mod.row = "EEE", mod.col = "EE", nstartR = 10)
fit <- Eigen.HMM_fit(Y = Y, init.par = init, nThreads = 1)

Initialization for ECM Algorithms in Matrix-Variate Hidden Markov Models

Description

Initializes the ECM algorithms used for fitting parsimonious matrix-variate Hidden Markov Models (HMMs). Parallel computing is implemented and highly recommended for faster computations.

Usage

Eigen.HMM_init(
  Y,
  k,
  density,
  mod.row = "all",
  mod.col = "all",
  nstartR = 50,
  nThreads = 1,
  verbose = FALSE,
  seed = 3
)

Arguments

Y

k

An integer or vector indicating the number of states in the model(s).

density

A character string specifying the distribution to use in the HMM. Possible values are: "MVN" for the matrix-variate normal distribution, "MVT" for the matrix-variate t-distribution, and "MVCN" for the matrix-variate contaminated normal distribution.

mod.row

A character string indicating the parsimonious structure of the row covariance (or scale) matrices. Possible values are: "EII", "VII", "EEI", "VEI", "EVI", "VVI", "EEE", "VEE", "EVE", "EEV", "VVE", "VEV", "EVV", "VVV", or "all". When "all" is specified, all 14 parsimonious structures are considered.

mod.col

A character string indicating the parsimonious structure of the column covariance (or scale) matrices. Possible values are: "II", "EI", "VI", "EE", "VE", "EV", "VV", or "all". When "all" is specified, all 7 parsimonious structures are considered.

nstartR

An integer specifying the number of random starts to consider.

nThreads

A positive integer indicating the number of cores to use for parallel processing.

verbose

A logical value indicating whether to display the running output.

seed

A positive integer specifying the seed for random generation.

Value

A list containing the following elements:

results

A list of the results from the initialization.

k

The number of states fitted in each model.

req.model

A data frame listing the models that were initialized.

init.used

A data frame listing the initializations used for the required models.

index

A numeric vector to be used by the Eigen.HMM_fit() function.

dens

The density used for the HMMs.

Examples

data(simData)
Y <- simData$Y
init <- Eigen.HMM_init(Y = Y, k = 2, density = "MVT", mod.row = "EEE", mod.col = "EE", nstartR = 10)

Atypical Detection Points Using Matrix-Variate Contaminated Normal Hidden Markov Models

Description

Detects atypical matrices via matrix-variate contaminated normal Hidden Markov Models.

Usage

atp.MVCN(Y, pgood, class)

Arguments

Y

pgood

An array with dimensions num x t x k containing the estimated probability of being typical for each point, given the time and state.

class

An num x t matrix containing the state memberships.

Value

An num x t matrix containing, for each observation and time, a 0 if it that matrix is typical and 1 otherwise.

Examples

data("simData2")
Y <- simData2$Y
init <- Eigen.HMM_init(Y = Y, k = 2, density = "MVCN", mod.row = "EEE", mod.col = "EE", nstartR = 1)
fit <- Eigen.HMM_fit(Y = Y, init.par = init, nThreads = 1)
atp <- atp.MVCN(Y = Y,
               pgood = fit[["results"]][[1]][[1]][[1]][["pgood"]],
               class = fit[["results"]][[1]][[1]][[1]][["class"]])
which(atp==1)
which(simData2[["atp.tr"]]==1)

Atypical Detection Points Using Matrix-Variate t Hidden Markov Models

Description

Detects atypical matrices via matrix-variate t Hidden Markov Models given a specified value of epsilon.

Usage

atp.MVT(Y, M, U, V, class, epsilon)

Arguments

Y

M

An array with dimensions p x p x k, where k is the number of states, containing the mean matrices.

U

An array with dimensions p x p x k, where k is the number of states, containing the row covariance (scale) matrices.

V

An array with dimensions r x r x k, where k is the number of states, containing the column covariance (scale) matrices.

class

An num x t matrix containing the state memberships.

epsilon

A numeric value specifying the selected percentile of the chi-squared distribution with pr degrees of freedom.

Value

An num x t matrix containing, for each observation and time, a 0 if it that matrix is typical and 1 otherwise.

Examples

data("simData2")
Y <- simData2$Y
init <- Eigen.HMM_init(Y = Y, k = 2, density = "MVT", mod.row = "EEE", mod.col = "EE", nstartR = 1)
fit <- Eigen.HMM_fit(Y = Y, init.par = init, nThreads = 1)
atp <- atp.MVT(Y = Y, M = fit[["results"]][[1]][[1]][[1]][["M"]],
              U = fit[["results"]][[1]][[1]][[1]][["U"]],
              V = fit[["results"]][[1]][[1]][[1]][["V"]],
              class = fit[["results"]][[1]][[1]][[1]][["class"]],
              epsilon = 0.99)
which(atp==1)
which(simData2[["atp.tr"]]==1)

Selection of the best fitting model(s)

Description

This functions extracts the best fitting model(s) according to the Bayesian information criterion (BIC).

Usage

extract.bestM(results, top = 1)

Arguments

results

The output of the Eigen.HMM_fit() function.

top

Integer. Specifies the number of top-ranked models to display based on the Bayesian Information Criterion (BIC).

Value

A list containing the required best fitting model(s).

Examples

data(simData)
Y <- simData$Y
init <- Eigen.HMM_init(Y = Y, k = 2, density = "MVT", mod.row = "EEE", mod.col = "EE", nstartR = 10)
fit <- Eigen.HMM_fit(Y = Y, init.par = init, nThreads = 1)
win <- extract.bestM(results = fit, top = 1)

Random Number Generation for Matrix-Variate Hidden Markov Models

Description

Generates random numbers for matrix-variate Hidden Markov Models (HMMs) based on matrix-variate normal, t, and contaminated normal distributions.

Usage

r.HMM(density, num, t, PI, M, U, V, IP, nu, alpha, eta)

Arguments

density

A character string specifying the distribution to use for the HMM. Possible values are: "MVN" for the matrix-variate normal distribution, "MVT" for the matrix-variate t-distribution, and "MVCN" for the matrix-variate contaminated normal distribution.

num

An integer specifying the number of random matrices to generate.

t

An integer specifying the number of time points.

PI

A matrix representing the transition probability matrix.

M

An array with dimensions p x r x k, where k is the number of states, containing the mean matrices.

U

An array with dimensions p x p x k, where k is the number of states, containing the row covariance (scale) matrices.

V

An array with dimensions r x r x k, where k is the number of states, containing the column covariance (scale) matrices.

IP

A numeric vector of length k containing the initial probability weights.

nu

A numeric vector of length k containing the degrees of freedom for each state in the MVT distribution.

alpha

A numeric vector of length k containing the proportion of typical points in each state for the MVCN distribution.

eta

A numeric vector of length k containing the inflation parameters for each state in the MVCN distribution.

Value

A list containing the following elements:

Y

An array with dimensions p x r x num x t containing the generated data.

obs.states

An num x t matrix containing the state memberships.

Examples

p <- 2
r <- 3
num <- 50
t <- 3
k <- 2
IP <- c(0.5, 0.5)
PI <- matrix(c(0.9, 0.1, 0.3, 0.7), nrow = k, ncol = k, byrow = TRUE)
M <- array(NA, dim = c(p, r, k))
M[,,1]<- matrix(c(0,1,1,
                 -1,-1.5,-1),nrow = p, ncol = r, byrow = TRUE)
M[,,2]<- M[,,1]+3
U <- array(NA, dim = c(p, p, k))
V <- array(NA, dim = c(r, r, k))
U[, , 1] <- U[, , 2] <- matrix(c(1.73, -0.59, -0.59, 2.52), nrow = p, ncol = p, byrow = TRUE)
V[, , 1] <- V[, , 2] <- matrix(c(0.69, 0.23, -0.03,
                                 0.23, 0.48,  0.16,
                                -0.03, 0.16,  0.88), nrow = r, ncol = r, byrow = TRUE)
nu <- c(4.5, 6.5)
simData <- r.HMM(density = "MVT", num = num, t = t, PI = PI,
                M = M, U = U, V = V, IP = IP, nu = nu)

A Simulated Dataset from a Matrix-Variate t Hidden Markov Model

Description

A simulated dataset generated from a matrix-variate t Hidden Markov Model with 2 states and an EE - EE covariance structure.

Usage

data(simData)

Format

A list containing two elements:

1): An array with p = 2 variables in the rows, r = 3 variables in the columns, num = 50 matrices, and t = 3 time points.
2): An num x t matrix containing the state memberships.

A Simulated Dataset with Atypical Matrices

Description

A simulated dataset containing atypical matrices. The data are initially generated from a matrix-variate normal Hidden Markov Model with 2 states and an EE - EE covariance structure. Atypical matrices are then introduced by randomly replacing some of the original matrices with values from a uniform distribution.

Usage

data(simData2)

Format

A list containing three elements:

1): An array with p = 2 variables in the rows, r = 3 variables in the columns, num = 50 matrices, and t = 3 time points.
2): An num x t matrix containing the state memberships.
3): An num x t matrix identifying the atypical matrices, where atypical matrices are coded with a 1.