hidden {repeated} R Documentation

## 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.

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.

### 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=0.00001, steptol=0.00001, fscale=1, iterlim=100,
typsiz=abs(p), stepmax=10*sqrt(p%*%p))
plot.hidden(z, nind=1, state=NULL, smooth=FALSE, main=NULL,
ylab=NULL, xlab="Time", xlim=NULL, ...)
```

### 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. `nind` For plotting: numbers of individuals to plot. `state` For plotting: states to plot. `others` Arguments controlling `nlm`.

### 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.

`chidden`, `gar`, `gnlmm`, `iprofile`, `kalcount`, `mprofile`, `nbkal`, `read.list`, `restovec`, `rmna`.

### Examples

```# generate two random Poisson sequences with change-points
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(0.9,0.2,0.1,0.8),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(0.9,0.2,0.1,0.8),ncol=2)))
# or
print(z <- hidden(y,dist="Poisson", mu=~rep(a,40), pmu=c(1,5),
pgamma=matrix(c(0.9,0.2,0.1,0.8),ncol=2)))
par(mfrow=c(3,2))
plot(z, nind=1:2)
plot(z, nind=1:2, smooth=TRUE)
plot(iprofile(z), lty=2)