depmixS4 (version 1.5-0)

posterior: Posterior state/class probabilities and classification

Description

Return posterior state classifications and/or probabilities for a fitted (dep-)mix object. In the case of a latent class or mixture model, states refer to the classes/mixture components.

There are different ways to define posterior state probabilities and the resulting classifications. The 'type' argument can be used to specify the desired definition. The default is currently set to 'viterbi'. Other options are 'global' and 'local' for state classification, and 'filtering' and 'smoothing' for state probabilities. See Details for more information.

Usage

# S4 method for depmix
posterior(object, type = c("viterbi", "global", "local", "filtering", "smoothing"))
	# S4 method for depmix.fitted
posterior(object, type = c("viterbi", "global", "local", "filtering", "smoothing"))
	# S4 method for mix
posterior(object, type = c("viterbi", "global", "local", "filtering", "smoothing"))
	# S4 method for mix.fitted
posterior(object, type = c("viterbi", "global", "local", "filtering", "smoothing"))

Arguments

object

A (fitted)(dep-)mix object.

type

character, partial matching allowed. The type of classification or posterior probability desired.

Value

The return value of posterior depends on the value of the type argument:

type = 'viterbi'

Returns a data.frame with nstates(object) + 1 columns; the first column contains the states decoded through the Viterbi algorithm, the remaining columns contain the (normalized) delta probabilities.

type = 'global'

Returns a vector which contains the states decoded through the Viterbi algorithm.

type = 'local'

Returns a vector which contains the states decoded as the maximum of the smoothing probabilities.

type = 'filtering'

Returns a matrix which contains the posterior probabilities of each state, conditional upon the responses observed thus far.

type = 'smoothing'

Returns a matrix which contains the posterior probabilities of each state, conditional upon all the responses observed.

See Details for more information.

Details

After fitting a mix or depmix model, one is often interested in determining the most probable mixture components or hidden states at each time-point t. This is also called decoding the hidden states from the observed data. There are at least two general ways to consider state classification: 'global' decoding means determining the most likely state sequence, whilst 'local' decoding means determining the most likely state at each time point whilst not explicitly considering the identity of the hidden states at other time points. For mixture models, both forms of decoding are identical.

Global decoding is based on the conditional probability \(p(S_1, \ldots, S_T \mid Y_1, \ldots, Y_T)\), and consists of determining, at each time point \(t = 1, \ldots, T\): $$s*_t = \arg \max_{i=1}^N p(S_1 = s*_1, \ldots, S_{t-1} = s*_{t-1}, S_t = i, S_{t+1} = s*_{t+1}, \ldots, S_T = s*_{T} \mid Y_1, \ldots, Y_T)$$ where N is the total number of states. These probabilities and the resulting classifications, are computed through the viterbi algorithm. Setting type = 'viterbi' returns a data.frame with the Viterbi-decoded global state sequence in the first column, and the normalized "delta" probabilities in the remainining columns. These "delta" probabilities are defined as the joint probability of the most likely state sequence ending in state i at time t, and all the observations up to time t. The normalization of these joint probabilities is done on a time-point basis (i.e., dividing the delta probability by the sum of the delta probabilities for that time point for all possible states j (including state i)). These probabilities are not straightforward to interpret. Setting type = "global" returns just a vector with the Viterbi-decoded global state sequence.

Local decoding is based on the smoothing probabilities \(p(S_t \mid Y_1, \ldots, Y_T)\), which are the "gamma" probabilities computed with the forwardbackward algorithm. Local decoding then consists of determining, at each time point \(t = 1, \ldots, T\) $$s*_t = \arg \max_{i=1}^N p(S_t = i \mid Y_1, \ldots, Y_T)$$ where N is the total number of states. Setting type = "local" returns a vector with the local decoded states. Setting type = "smoothing" returns the smoothing probabilities which underlie this classification. When considering the posterior probability of each state, the values returned by type = "smoothing" are most likely what is wanted by the user.

The option type = "filtering" returns a matrix with the so-called filtering probabilities, defined as \(p(S_t \mid Y_1, \ldots, Y_t)\), i.e. the probability of a hidden state at time t considering the observations up to and including time t.

See the fit help page for an example.

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

# NOT RUN {
data(speed)

# 2-state model on rt and corr from speed data set 
# with Pacc as covariate on the transition matrix
# ntimes is used to specify the lengths of 3 separate series
mod <- depmix(list(rt~1,corr~1),data=speed,transition=~Pacc,nstates=2,
	family=list(gaussian(),multinomial("identity")),ntimes=c(168,134,137))
fmod <- fit(mod)

# Global decoding:
pst_global <- posterior(fmod, type = "global")

# Local decoding:
pst_local <- posterior(fmod,type="local")

# Global and local decoding provide different results:
identical(pst_global, pst_local)

# smoothing probabilities are used for local decoding, and may be used as 
# easily interpretable posterior state probabilities
pst_prob <- posterior(fmod, type = "smoothing")

# "delta" probabilities from the Viterbi algorithm
pst_delta <- posterior(fmod, type="viterbi")[,-1]

# The smoothing and "delta" probabilities are different:
identical(pst_prob, pst_delta)

# Filtering probabilities are an alternative to smoothing probabilities:
pst_filt <- posterior(fmod, type = "filtering")

# The smoothing and filtering probabilities are different:
identical(pst_prob, pst_filt)
# }