dmm creates an object of class dmm, a
dependent mixture model.
lca creates an object of class dmm,lca, a
latent class model or an independent mixture model.
dmm(nstates, itemtypes, modname = NULL, fixed = NULL,
stval = NULL, conrows = NULL, conpat = NULL, tdfix =
NULL, tdst = NULL, linmat = NULL, snames = NULL,
inames = NULL)
# S3 method for dmm
summary(object, specs=FALSE, precision=3, se=NULL, ...) lca(nclasses, itemtypes, modname = NULL, fixed = NULL,
stval = NULL, conrows = NULL, conpat = NULL,
linmat = NULL, snames = NULL, inames = NULL)
The number of latent states/classes of the model.
The number of classes of an lca model, ie the number of
states in a dmm model. They are now called classes because they do
not change over time.
A vector of length nitems providing the type of
measurement, 1 for gaussian data, 2 for a binary item, n>3 for
categorical items with n answer possibilities. Answer categories are
assumed to be unordered categorical. Ordinal responses can be implemented
using inequality and/or linear constraints.
A character string with the name of the model, good when fitting many models. Components of mixture models keep their own names. Names are printed in the summary. Boring default names are provided.
A vector of length the number of parameters of the model idicating whether parameters are fixed (0) or not (>0). This may be identical to conpat (see below).
Start values of the parameters. These will be random if not specified. Start values must be specified (for all parameters) if there are fixed parameters.
Argument conrows can be used to specify general
constraints between parameters. See details below.
Argument conpat can be used to specify fixed
parameters and equality constraints. It can not be used in conjuction
with fixed. See details below.
The first is a logical vector indicating (with 1's) which parameters are dependent on covariates (it should have length npars). tdst provides the starting values for the regression parameters. Using tdcov=TRUE in fitdmm will actually fit the regression parameters. The covariate itself has to be specified in the data as "covariate" (see help on markovdata) and should be scaled to 0-1.
A complete matrix of linear constraints. This argument
is intended for internal use only, it is used by the fit routine to
re-create the model with the fitted parameter values. Warning: use of
this argument results in complete replacement of the otherwise created
matrix A, which contains e.g. sum contraints for transition matrix
parameters. If linmat is provided, make sure it is correct,
otherwise strange results may occur in fitting models.
Names for the states may be provided in snames. Defaults are State1, State2 etc. These are printed in the summary.
Names for items may be provided in inames. Defaults are Item1, Item2 etc. They are printed in the summary.
Object of class dmm.
Precision sets the number of digits to be printed in the summary functions.
Vector with standard errors, these are passed on from the summary.fit function if and when ses are available.
Internal use.
An object of class dmm.
dmm returns an object of class dmm which has its own summary
method. This will print the parameter values, itemtypes, number of (free)
parameters, and the number of states. There is no print method. Using
print will print all fields of the model which is a list of the following:
See above.
See above
See above.
The number of items(=length(itemtypes)).
See above.
See above.
The total parameter count of the model.
The total number of parameters of when the covariate parameters are included.
The number of freely estimated parameters (it is computed as sum(as.logical(fixed))-rank(qr(A)).
The number of freely estimated parameters (it is computed as sum(as.logical(fixed))-rank(qr(A)); this version without the covariate parameters.
A vector of length npars containing parameter values.
fixed is a (logical) vector of length npars
specifying which parameters are fixed and which are not.
The matrix A contains the general linear constraints of the model. nrow(A) is the number of linear constraints. A starts with a number of rows for the sum constraints for the transition, observation and initial state parameters, after which the user provided constraints are added.
bu and bl represent the upper and lower bounds of the parameters and the constraints. These vectors are each of length npars + nrow(A).
The lower and upper bounds of the linear constraints.
Logicals indicating whehter there covariates, in which parameters they are, and whether they are estimated or not (the latter is used to decide whether to print those values or not).
Logical indicating whether the model has user specified starting values.
The function dmm creates an object of class dmm and sets
random initial parameter values if these are not provided. Even though
dmm is not a mixture of Markov models, the mixture parameter is
is included in the parameter vector. This is important when specifying
constraints. Parameters are ordered as follows: the first parameter(s)
are the mixing proportions of the mixture of Markov and/or latent class
models. I.e., when a single latent class model or a single Markov
chain is fitted, this mixture proportion has value 1.0 and is it is
fixed in estimation. After the mixing proportions, the next parameters
in the parameter vector are the transition matrix parameters, the
square of nstates in row-major order. That is, first the transition
probabilities from state 1 to all the other states are given, then the
probabilities from state 2 to all the other states etc. Next are the
observation matrix parameters. These are provided consecutively for
each state/class. Ie a trichtomous item model with two states has 6
observation parameters; the first three are the probabilities of
observing category 1, 2 and 3 respectively in state 1 (which sum to
one), and then similarly for state 2. As another example: suppose we
have model for one binary item and one gaussian item, in that order, we
would have 4 observation parameters for each state, first the
probabilities of observing a symbol from category 1 or 2 in state 1,
the two parameters, the mean and standard deviation for state 1, and
then the same state 2 (see the example in fitdmm with data from rudy).
Finally the initial state probabilities are provided, in the order of
the states. In the case of a latent class model or a finite mixture
model, these parameters are usually denote as the mixture proportions.
Linear constraints can be set using arguments conrows and
conpat. conrows must be contain nc by npars values, in
row major order, with nc the number of contraints to be specified.
conrows is used to define general linear constraints. A row of
conrows must contain the partial derivatives of a general linear
constraint with respect to each of the parameters. Suppose we want the
constraint x1 -2*x2=0, one row of conrows should contain a 1 in
position one and -2 in position and zeroes in the remaining positions.
In the function mixdmm conrows is understood to specify
linear constraints on the mixing proportions only. As a consequence,
it is not possible to easily constrain parameters between components of
a mixture model.
conpat can be used as a shortcut for both fixed and conrows. It
must be a single vector of length npars contaning 0's (zeroes) for
fixed parameters, 1's (ones) for free parameters and higher numbers for
possibly equality constrained parameters. E.g.
conpat=c(1,1,0,2,2,3,3,3) would indicate that pars 1 and 2 are
freely estimated, par 3 is fixed at its startvalue (which must be
provided in this case), par 4 and 5 are to estimated equal and pars 6,
7 and 8 are also to be estimated equal.
On hidden Markov models: Lawrence R. Rabiner (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of IEEE, 77-2, p. 267-295.
On latent class models: A. L. McCutcheon (1987). Latent class analysis. Sage Publications.
mixdmm on defining mixtures of dmm's,
mgdmm for defining multi group models, and
generate for generating data from models.
# NOT RUN {
# create a 2 state model with one continuous and one binary response
# with start values provided in st
st <- c(1,0.9,0.1,0.2,0.8,2,1,0.7,0.3,5,2,0.2,0.8,0.5,0.5)
mod <- dmm(nsta=2,itemt=c(1,2), stval=st)
summary(mod)
# 2 class latent class model with equal conditional probabilities in each class
stv=c(1,rep(c(0.9,0.1),5),rep(c(0.1,0.9),5),0.5,0.5)
# here the conditional probs of the first item are set equal to those in
# the subsequent items
conpat=c(1,rep(c(2,3),5),rep(c(4,5),5),1,1)
lc=lca(ncl=2,itemtypes=rep(2,5),conpat=conpat,stv=stv)
summary(lc)
# }
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