hidden {repeated}

Discrete-time Hidden Markov Chain Models

Description

hidden fits a two or more state hidden Markov chain model with a variety of distributions. All series on different individuals are assumed to start at the same time point. Time points are equal, discrete steps.

Usage

hidden(
  response = NULL,
  totals = NULL,
  distribution = "Bernoulli",
  mu = NULL,
  cmu = NULL,
  tvmu = NULL,
  pgamma,
  pmu = NULL,
  pcmu = NULL,
  ptvmu = NULL,
  pshape = NULL,
  pfamily = NULL,
  par = NULL,
  pintercept = NULL,
  delta = NULL,
  envir = parent.frame(),
  print.level = 0,
  ndigit = 10,
  gradtol = 1e-05,
  steptol = 1e-05,
  fscale = 1,
  iterlim = 100,
  typsize = abs(p),
  stepmax = 10 * sqrt(p %*% p)
)

Arguments

response

A list of two or three column matrices with counts or category indicators, times, and possibly totals (if the distribution is binomial), for each individual, one matrix or dataframe of counts, or an object of class, response (created by restovec) or repeated (created by rmna or lvna). If the repeated data object contains more than one response variable, give that object in envir and give the name of the response variable to be used here. If there is only one series, a vector of responses may be supplied instead.

Multinomial and ordinal categories must be integers numbered from 0.

totals

If response is a matrix, a corresponding matrix of totals if the distribution is binomial. Ignored if response has class, response or repeated.

distribution

Bernoulli, Poisson, multinomial, proportional odds, continuation ratio, binomial, exponential, beta binomial, negative binomial, normal, inverse Gauss, logistic, gamma, Weibull, Cauchy, Laplace, Levy, Pareto, gen(eralized) gamma, gen(eralized) logistic, Hjorth, Burr, gen(eralized) Weibull, gen(eralized) extreme value, gen(eralized) inverse Gauss, power exponential, skew Laplace, or Student t. (For definitions of distributions, see the corresponding [dpqr]distribution help.)

mu

A general location function with two possibilities: (1) a list of formulae (with parameters having different names) or functions (with one parameter vector numbering for all of them) each returning one value per observation; or (2) a single formula or function which will be used for all states (and all categories if multinomial) but with different parameter values in each so that pmu must be a vector of length the number of unknowns in the function or formula times the number of states (times the number of categories minus one if multinomial).

cmu

A time-constant location function with three possibilities: (1) a list of formulae (with parameters having different names) or functions (with one parameter vector numbering for all of them) each returning one value per individual; (2) a single formula or function which will be used for all states (and all categories if multinomial) but with different parameter values in each so that pcmu must be a vector of length the number of unknowns in the function or formula times the number of states (times the number of categories minus one if multinomial); or (3) a function returning an array with one row for each individual, one column for each state of the hidden Markov chain, and, if multinomial, one layer for each category but the last. If used, this function or formula should contain the intercept. Ignored if mu is supplied.

tvmu

A time-varying location function with three possibilities: (1) a list of formulae (with parameters having different names) or functions (with one parameter vector numbering for all of them) each returning one value per time point; (2) a single formula or function which will be used for all states (and all categories if multinomial) but with different parameter values in each so that ptvmu must be a vector of length the number of unknowns in the function or formula times the number of states (times the number of categories minus one if multinomial); or (3) a function returning an array with one row for each time point, one column for each state of the hidden Markov chain, and, if multinomial, one layer for each category but the last. This function or formula is usually a function of time; it is the same for all individuals. It only contains the intercept if cmu does not. Ignored if mu is supplied.

pgamma

A square mxm matrix of initial estimates of the hidden Markov transition matrix, where m is the number of hidden states. Rows must sum to one. If the matrix contains zeroes or ones, these are fixed and not estimated. (Ones cannot appear on the diagonal.) If a 1x1 matrix or a scalar value of 1 is given, the independence model is fitted.

pmu

Initial estimates of the unknown parameters in mu.

pcmu

Initial estimates of the unknown parameters in cmu.

ptvmu

Initial estimates of the unknown parameters in tvmu.

pshape

Initial estimate(s) of the dispersion parameter, for those distributions having one. This can be one value or a vector with a different value for each state.

pfamily

Initial estimate of the family parameter, for those distributions having one.

par

Initial estimate of the autoregression parameter.

pintercept

For multinomial, proportional odds, and continuation ratio models, p-2 initial estimates for intercept contrasts from the first intercept, where p is the number of categories.

delta

Scalar or vector giving the unit of measurement (always one for discrete data) for each response value, set to unity by default. For example, if a response is measured to two decimals, delta=0.01. If the response is transformed, this must be multiplied by the Jacobian. For example, with a log transformation, delta=1/response. Ignored if response has class, response or repeated.

envir

Environment in which model formulae are to be interpreted or a data object of class, repeated, tccov, or tvcov; the name of the response variable should be given in response. If response has class repeated, it is used as the environment.

print.level

Arguments for nlm.

ndigit

Arguments for nlm.

gradtol

Arguments for nlm.

steptol

Arguments for nlm.

fscale

Arguments for nlm.

iterlim

Arguments for nlm.

typsize

Arguments for nlm.

stepmax

Arguments for nlm.

Details

To fit an ‘observed’ Markov chain, as well, with Bernoulli or multinomial responses, use the lagged response as a time-varying covariate. For quantitative responses, specifying par allows an 'observed" autoregression to be fitted as well as the hidden Markov chain.

All functions and formulae for the location parameter are on the (generalized) logit scale for the Bernoulli, binomial, and multinomial distributions.

If cmu and tvmu are used, these two mean functions are additive so that interactions between time-constant and time-varying variables are not possible.

The object returned can be plotted to give the probabilities of being in each hidden state at each time point. For distributions other than the multinomial, proportional odds, and continuation ratio, the (recursive) predicted values can be plotted using mprofile and iprofile.

See MacDonald, I.L. and Zucchini, W. (1997) Hidden Markov and Other Models for Discrete-valued Time Series. Chapman and Hall.

Value

A list of classes hidden and recursive (unless multinomial, proportional odds, or continuation ratio) is returned that contains all of the relevant information calculated, including error codes.

Author(s)

J.K. Lindsey and P.J. Lindsey

References

MacDonald, I.L. and Zucchini, W. (1997) Hidden Markov and other Models for Discrete-valued Time Series. Chapman & Hall.

Examples

# generate two random Poisson sequences with change-points
set.seed(8)
y <- rbind(c(rpois(5,1), rpois(15,5)), c(rpois(15,1), rpois(5,5)))
print(z <- hidden(y,dist="Poisson", cmu=~1, pcmu=c(1,5),
                  pgamma=matrix(c(0.95,0.05,0.05,0.95),ncol=2)))
# or equivalently
mu <- function(p) array(rep(p[1:2],rep(2,2)), c(2,2))
print(z <- hidden(y,dist="Poisson", cmu=mu, pcmu=c(1,5),
                  pgamma=matrix(c(0.95,0.05,0.05,0.95),ncol=2)))
# or
# param nind For plotting: numbers of individuals to plot.
# param state For plotting: states to plot.
print(z <- hidden(y,dist="Poisson", mu=~rep(a,40), pmu=c(1,5),
                  pgamma=matrix(c(0.95,0.05,0.05,0.95),ncol=2)))
par(mfrow=c(3,2))
plot(z, nind=1:2)
plot(z, nind=1:2, smooth=TRUE)
plot(iprofile(z), lty=2)
plot(mprofile(z), add=TRUE)
plot(iprofile(z), nind=2, lty=2)
plot(mprofile(z), nind=2, add=TRUE)