depmixS4 (version 0.2-1)

fit: Fit 'depmix' or 'mix' models

Description

fit optimizes parameters of depmix or mix models, optionally subject to general linear (in)equality constraints.

Usage

## S3 method for class 'depmix':
fit(object, fixed=NULL, equal=NULL, conrows=NULL,
		conrows.upper=0, conrows.lower=0, method=NULL,...)
	
	## S3 method for class 'depmix.fitted':
summary(object)
	## S3 method for class 'mix':
fit(object, fixed=NULL, equal=NULL, conrows=NULL,
		conrows.upper=0, conrows.lower=0, method=NULL,...)
	
	## S3 method for class 'mix.fitted':
summary(object)

Arguments

object

An object of class (dep-)mix.

fixed

Vector of mode logical indicating which parameters should be fixed.

equal

Vector indicating equality constraints; see details.

conrows

Rows of a general linear constraint matrix; see details.

conrows.upper, conrows.lower

Upper and lower bounds for the linear constraints; see details.

method

The optimization method; mostly determined by constraints.

...

Further arguments passed on to the optimization methods.

Value

• fit returns an object of class depmix.fitted which contains the original depmix object, and further has slots:
• message
  : Convergence information.
• conMat
  : The constraint matrix A, see details.
• posterior
  : The posterior state sequence (computed with the viterbi algorithm), and the posterior probabilities (delta probabilities in Rabiner, 1989, notation).
• The print method shows the message along with the likelihood and AIC and BIC; the summary method prints the parameter estimates.

  Posterior densities and the viterbi state sequence can be accessed through posterior. As fitted models are depmixS4 models, they can be used as starting values for new fits, for example with constraints added.

Details

Models are fitted by the EM algorithm if there are no constraints on the parameters. Otherwise the general optimizer donlp2 from the package Rdonlp2 is used which handles general linear (in-)equality constraints. Three types of constraints can be specified on the parameters: fixed, equality, and general linear (in-)equality constraints. Constraint vectors should be of length npar(object). See help on getpars and setpars about the ordering of parameters. The equal argument is used to specify equality constraints: parameters that get the same integer number in this vector are estimated to be equal. Any integers can be used in this way except 0 and 1, which indicate fixed and free parameters, respectively.

Using the donlp2 optimizer a Newton-Raphson scheme is employed to estimate parameters subject to linear constraints by imposing: bl <= a*x="" <="bu," where="" x="" is="" the="" parameter="" vector,="" bl="" a="" vector="" of="" lower="" bounds,="" bu="" upper="" and="" constraint="" matrix.<="" p="">

The conrows argument is used to specify rows of A directly, and the conrows.lower and conrows.upper arguments to specify the bounds on the constraints. conrows is a matrix of npar(object) columns and one row for each constraint (a vector in the case of a single constraint). Examples of these three ways of constraining parameters are provided below. llratio performs a log-likelihood ratio test on two fit'ted models; the first object should have the largest degrees of freedom (find out by using freepars).

References

Lawrence R. Rabiner (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of IEEE, 77-2, p. 267-295.

Examples

data(speed)

# 2-state model on rt and corr from speed data set with Pacc as covariate on the transition matrix
# starting values for the transition pars (without those EM does not get off the ground)
set.seed(1)
tr=runif(6)
trst=c(tr[1:2],0,tr[3:5],0,tr[6])
mod1 <- depmix(list(rt~1,corr~1),data=speed,transition=~Pacc,nstates=2,family=list(gaussian(),multinomial()),
	trstart=trst)
# fit the model
fmod1 <- fit(mod1)
fmod1 # to see the logLik and optimization information
# to see the parameters
summary(fmod1)

# NOTE: this requires Rdonlp2 package to be installed

# FIX SOME PARAMETERS

# get the starting values of this model to the optimized 
# values of the previously fitted model to speed optimization

pars <- c(unlist(getpars(fmod1)))

# constrain the initial state probs to be 0 and 1 
# also constrain the guessing probs to be 0.5 and 0.5 
# (ie the probabilities of corr in state 1)
# change the ones that we want to constrain
pars[2]=+Inf # this means the process will always start in state 2
pars[14]=0 # the corr parameters in state 1 are now both 0, corresponding the 0.5 prob
mod2 <- setpars(mod1,pars)

# fix the parameters by setting: 
free <- c(0,0,rep(c(0,1),4),1,1,0,0,1,1,0,1)
# fit the model
fmod2 <- fit(mod2,fixed=!free)

# likelihood ratio insignificant, hence fmod2 better than fmod1
llratio(fmod1,fmod2)

# NOW ADD SOME GENERAL LINEAR CONSTRAINTS

# set the starting values of this model to the optimized 
# values of the previously fitted model to speed optimization

pars <- c(unlist(getpars(fmod2)))
mod3 <- setpars(mod2,pars)

# start with fixed and free parameters
conpat <- c(0,0,rep(c(0,1),4),1,1,0,0,1,1,0,1)
# constrain the beta's on the transition parameters to be equal
conpat[4] <- conpat[8] <- 2
conpat[6] <- conpat[10] <- 3

fmod3 <- fit(mod3,equal=conpat)

llratio(fmod2,fmod3)

# above constraints can also be specified using the conrows argument as follows
conr <- matrix(0,2,18)
# pars 4 and 8 have to be equal, otherwise stated, their diffence should be zero
conr[1,4] <- 1
conr[1,8] <- -1
conr[2,6] <- 1
conr[2,10] <- -1

# note here that we use the fitted model fmod2 as that has appropriate 
# starting values
fmod3b <- fit(fmod2,conrows=conr,fixed=!free) # using free defined above

data(balance)
# four binary items on the balance scale task

instart=c(0.5,0.5)
set.seed(1)
respstart=runif(16)
# note that ntimes argument is used to make this a mixture model
mod4 <- mix(list(d1~1,d2~1,d3~1,d4~1), data=balance, nstates=2,
	family=list(multinomial(),multinomial(),multinomial(),multinomial()),
	respstart=respstart,instart=instart)

fmod4 <- fit(mod4)

# add age as covariate on class membership by using the prior argument
instart=c(0.5,0.5,0,0) # we need the initial probs and the coefficients of age 
set.seed(2)
respstart=runif(16)
mod5 <- mix(list(d1~1,d2~1,d3~1,d4~1), data=balance, nstates=2,
	family=list(multinomial(),multinomial(),multinomial(),multinomial()),
	instart=instart, respstart=respstart, prior=~age, initdata=balance)

fmod5 <- fit(mod5)

# check the likelihood ratio; adding age significantly improves the goodness-of-fit
llratio(fmod5,fmod4)