Multivariate and Multichannel Discrete Hidden Markov Models for Categorical Sequences
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README.md

Build Status downloads cran version

seqHMM: Hidden Markov Models for Life Sequences and Other Multivariate, Multichannel Categorical Time Series

The seqHMM package is designed for fitting hidden (or latent) Markov models (HMMs) and mixture hidden Markov models (MHMMs) for social sequence data and other categorical time series. Also some more restricted versions of these type of models are now available: Markov models, mixture Markov models, and latent class models.

The package supports models for one or multiple subjects with one or multiple parallel sequences (channels). External covariates can be added to explain cluster membership in mixture models. The package provides functions for evaluating and comparing models, as well as functions for easy plotting of multichannel sequence data and hidden Markov models.

Maximum likelihood estimation via EM algorithm and direct numerical maximization with analytical gradients is supported. All main algorithms are written in C++ with parallel computation support via OpenMP.

We would be happy to hear feedback! If you have any questions, comments, or wishes, please contact Satu Helske or Jouni Helske, firstname.lastname@liu.se. Or more preferably, you can also add a new issue here in GitHub.

The package is available on CRAN. Istall it via

install.packages("seqHMM")

If you want to try the development version of the seqHMM package, install it from Github using the devtools package:

install.packages("devtools")
library(devtools)
install_github("helske/seqHMM")

Preview of the seqHMM package

This example uses the biofam data from the TraMineR package. The data consist of a sample of 2000 individuals born between 1909 and 1972 constructed from the Swiss Household Panel (SHP) survey in 2002. The sequences consist of family life states from age 15 to 30 (in columns 10 to 25).

  • 0 = "living with parents"
  • 1 = "left home"
  • 2 = "married"
  • 3 = "left home+married"
  • 4 = "child"
  • 5 = "left home+child"
  • 6 = "left home+married+child"
  • 7 = "divorced"

For the functions of the seqHMM package, sequence data is given as a state sequence object (stslist) using the seqdef function in the TraMineR package. To show a more complex example, the original data is split into three separate channels. This data is pre-generated and stored as biofam3c. It contains a list of three sequence data sets and a data frame including the covariates. Find more information and the code for the conversion by typing help(biofam3c).

library(seqHMM)
data(biofam3c)

# Building sequence objects (starting at age 15)
marr.seq <- seqdef(biofam3c$married, start = 15)
child.seq <- seqdef(biofam3c$children, start = 15)
left.seq <- seqdef(biofam3c$left, start = 15)

# Choosing colours for states
attr(marr.seq, "cpal") <- c("#AB82FF", "#E6AB02", "#E7298A")
attr(child.seq, "cpal") <- c("#66C2A5", "#FC8D62")
attr(left.seq, "cpal") <- c("#A6CEE3", "#E31A1C")

Plotting multichannel sequence data

Multichannel sequence data are easily plotted using the ssplot function (ssplot for Stacked Sequence Plot). It can plot two types of plots: sequences themselves (index plots) or state distributions at each time point. Plotting index plots with large sequence data can take time, especially if the sequences are also sorted at the same time, so by default the function plots states distributions.

# Plotting state distribution plots of observations
ssplot(
  list(marr.seq, child.seq, left.seq), title = "State distribution plots")

ssp1

Multiple ssp objects can also be plotted together in a grid.

# Preparing plots for women's state distributions
# Sorting by scores from multidimensional scaling
ssp_f2 <- ssp(
  list(marr.seq[biofam3c$covariates$sex == "woman",], 
       child.seq[biofam3c$covariates$sex == "woman",],
       left.seq[biofam3c$covariates$sex == "woman",]),
  type = "d", plots = "obs", border = NA, withlegend = FALSE,
  title = "State distributions for women", title.n = FALSE,
  ylab = c("Married", "Parenthood", "Left home"), ylab.pos = c(1, 2, 1),
  xlab = "Age", xtlab = 15:30)

# Same plot, but sequences instead of state distributions
ssp_f3 <- update(
  ssp_f2, type = "I", sortv = "mds.obs", title = "Sequences for women")

# State distributions with men's data
ssp_m2 <- update(
  ssp_f2, x = list(marr.seq[biofam3c$covariates$sex == "man",], 
                   child.seq[biofam3c$covariates$sex == "man",], 
                   left.seq[biofam3c$covariates$sex == "man",]),
  title = "State distributions for men")

# Men's sequences
ssp_m3 <- update(
  ssp_m2, type = "I", sortv = "mds.obs", title = "Sequences for men")

# Plotting state distributions and index plots of observations for women and men 
gridplot(
  list(ssp_f2, ssp_f3, ssp_m2, ssp_m3), ncol = 2, byrow = TRUE, 
  row.prop = c(0.42, 0.42, 0.16), legend.pos2 = "top")

gridplot

Building models

A model is first constructed using an appropriate build function. There are several such functions available: build_hmm for hidden Markov models, build_mhmm for mixture hidden Markov models, build_mm for Markov models, build_mmm for mixture Markov models, and build_lcm for latent class models.

Build functions check that the data and matrices are of the right form and create an object of class hmm (for HMMs and MMs) or mhmm (for MHMMs, MMMs, and LCMs). For the latter, covariates can be omitted or added with the usual formula argument using symbolic formulas familiar from e.g. the lm function. Even though missing observations are allowed in sequence data, covariates must be completely observed.

Fitting hidden Markov models

When fitting Hidden Markov models (HMMs), starting values for initial, transition, and emission probabilities are given in the build_hmm function. After that, parameters are estimated with the fit_model function. The fitting function provides three estimation steps: 1) EM algorithm, 2) global optimization, and 3) local optimization. By default, only the EM step is performed. The results from a former step are used as starting values in a latter.

In order to reduce the risk of being trapped in a poor local maximum, a large number of initial values should be tested. If global step is chosen, by default the fit_model function uses the multilevel single-linkage method (MLSL) with the LDS modification. The MLSL draws multiple random starting values and performs local optimization (L-BFGS by default) from each starting point. The LDS modification uses low-discrepancy sequences instead of random numbers as starting points and should improve convergence rate.

In order to reduce computation time spent on non-global optima, the convergence tolerance of the local optimizer is set relatively large. At step 3, a local optimization (again L-BFGS by default) is run with a lower tolerance to find the optimum with high precision.

There are some theoretical guarantees that the MLSL method shoud find all local optima in a finite number of local optimizations. Of course, it might not always succeed in a reasonable time. The EM algorithm can help in finding good boundaries for the search, especially with good starting values, but in some cases it can mislead. A good strategy is to try a couple of different fitting options with different combinations of the methods: e.g. all steps, only global and local steps, and a few evaluations of EM followed by global and local optimization.

It is also possible to run the EM algorithm several times with random starting values. This is done by setting the value restarts in the control_em argument. Although not done by default, this method seems to perform very well as EM algorithm is relatively fast compared to direct numerical estimation.

# Initial values for emission matrices
emiss_marr <- matrix(NA, nrow=4, ncol=3)
emiss_marr[1,] <- seqstatf(marr.seq[, 1:4])[, 2] + 0.1
emiss_marr[2,] <- seqstatf(marr.seq[, 5:8])[, 2] + 0.1
emiss_marr[3,] <- seqstatf(marr.seq[, 9:12])[, 2] + 0.1
emiss_marr[4,] <- seqstatf(marr.seq[, 13:16])[, 2] + 0.1
emiss_marr <- emiss_marr / rowSums(emiss_marr)

emiss_child <- matrix(NA, nrow=4, ncol=2)
emiss_child[1,] <- seqstatf(child.seq[, 1:4])[, 2] + 0.1
emiss_child[2,] <- seqstatf(child.seq[, 5:8])[, 2] + 0.1
emiss_child[3,] <- seqstatf(child.seq[, 9:12])[, 2] + 0.1
emiss_child[4,] <- seqstatf(child.seq[, 13:16])[, 2] + 0.1
emiss_child <- emiss_child / rowSums(emiss_child)

emiss_left <- matrix(NA, nrow=4, ncol=2)
emiss_left[1,] <- seqstatf(left.seq[, 1:4])[, 2] + 0.1
emiss_left[2,] <- seqstatf(left.seq[, 5:8])[, 2] + 0.1
emiss_left[3,] <- seqstatf(left.seq[, 9:12])[, 2] + 0.1
emiss_left[4,] <- seqstatf(left.seq[, 13:16])[, 2] + 0.1
emiss_left <- emiss_left / rowSums(emiss_left)

# Initial values for transition matrix
trans <- matrix(
  c(0.90, 0.06, 0.03, 0.01,
       0, 0.90, 0.07, 0.03,
       0,    0, 0.90, 0.10,
       0,    0,    0,    1), 
  nrow = 4, ncol = 4, byrow = TRUE)

# Initial values for initial state probabilities
initial_probs <- c(0.9, 0.07, 0.02, 0.01)

# Building the hidden Markov model with initial parameter values 
bhmm <- build_hmm(
  observations = list(marr.seq, child.seq, left.seq),
  initial_probs = initial_probs, transition_probs = trans, 
  emission_probs = list(emiss_marr, emiss_child, emiss_left),
  channel_names = c("Marriage", "Parenthood", "Residence"))

# Fitting the HMM:
# step 1) EM algorithm
hmm <- fit_model(bhmm)
hmm$logLik
# -16854.16

# EM + 50 restarts with random starting values for emission probabilities
hmm2 <- fit_model(bhmm, 
  control_em = list(restart = list(times = 50, transition = FALSE, emission = TRUE)))
hmm2$logLik
# -16854.16

# Using all three steps:
# step 1) EM algorithm
# step 2) global optimization (default: MLSL_LDS with LBFGS as local optimizer)
# step 3) local optimization (default: LBFGS) for "final polishing"
# Note: By default, estimation time limited to 60 seconds in step 2.
# Setting 3000 evaluations with unlimited time
hmm3 <- fit_model(bhmm, global_step = TRUE, local_step = TRUE, control_global = list(maxeval = 3000, maxtime = 0))
hmm3$logLik
# -16854.16

# Only global optimization (3000 iterations, unlimited time)
hmm4 <- fit_model(bhmm, em_step = FALSE, global_step = TRUE, local_step = FALSE,
                control_global = list(maxeval = 3000, maxtime = 0))
hmm4$logLik
# -16856.78


Plotting hidden Markov models

A simple plot method is used to show an hmm object as a graph. It shows hidden states as pie charts (vertices), with emission probabilities as slices and transition probabilities as arrows (edges). By default, initial probabilities are shown below the pies.

# Plot HMM
plot(hmm$model)

HMMdefault


# A prettier version
plot(
  hmm$model,
  # larger vertices
  vertex.size = 45,
  # varying curvature of edges
  edge.curved = c(0, -0.7, 0.6, 0, -0.7, 0),
  # legend with two columns and less space
  ncol.legend = 2, legend.prop = 0.4,
  # new label for combined slice
  combined.slice.label = "States with probability < 0.05")

HMM

The ssplot function can also be used for plotting the observed states and/or the most probable paths of hidden states of a HMM.

# Plotting observations and hidden states
ssplot(hmm$model, plots = "both", type = "I")

sspboth_default

# Prettier version
ssplot(
  hmm$model, type = "I", plots = "both",
  # Sorting subjects according to multidimensional
  # scaling scores of the most probable hidden state paths
  sortv = "mds.hidden", 
  # Naming the channels
  ylab = c("Children", "Married", "Residence"), 
  # Title for the plot
  title = "Observed sequences and the 
most probable paths of hidden states",
  # Labels for hidden states (most common states)
  hidden.states.labels = c("1: Childless single, with parents", 
                           "2: Childless single, left home",
                           "3: Married without children",
                           "4: Married parent, left home"),
  # Colours for hidden states
  hidden.states.colors = c("olivedrab", "bisque", "plum", "indianred"),
  # Labels for x axis
  xtlab = 15:30, xlab = "Age",
  # Proportion for legends
  legend.prop = 0.45)

sspboth

Computing likelihood and BIC

The logLik and BIC functions are used for model comparison with the log-likelihood or the Bayesian information criterion (BIC).

# Likelihood
logLik(hmm$model)
# 'log Lik.' -16854.16 (df=25)

# BIC
BIC(hmm$model)
# 33967.66

Trimming HMMs

The trim_hmm function can be used to trim models by setting small probabilities to zero. Here the trimmed model led to model with slightly improved likelihood, so probabilities less than 0.001 could be set to zero.

trimmed_hmm <- trim_model(hmm$model, maxit = 100, zerotol = 1e-03)
# "1 iteration(s) used."
# "Trimming improved log-likelihood, ll_trim-ll_orig = 4.28e-05"

# Transition probabilities of the original HMM
hmm$model$emission_probs
# $Marriage
#            symbol_names
# state_names     divorced      married       single
#           1 0.000000e+00 3.026210e-22 1.000000e+00
#           2 1.793831e-50 2.301489e-09 1.000000e+00
#           3 4.713737e-02 9.528626e-01 6.419152e-14
#           4 1.747152e-02 9.497448e-01 3.278367e-02
# 
# $Parenthood
#            symbol_names
# state_names    childless     children
#           1 9.988180e-01 1.181965e-03
#           2 1.000000e+00 2.390547e-09
#           3 1.000000e+00 5.123136e-09
#           4 1.715309e-14 1.000000e+00
# 
# $`Residence`
#            symbol_names
# state_names    left home with parents
#           1 8.126227e-11 1.000000e+00
#           2 1.000000e+00 2.145277e-16
#           3 6.916852e-01 3.083148e-01
#           4 1.000000e+00 5.190911e-50

# Emission probabilities of the trimmed HMM
trimmed_hmm$emiss
# $Marriage
#            symbol_names
# state_names   divorced   married     single
#           1 0.00000000 0.0000000 1.00000000
#           2 0.00000000 0.0000000 1.00000000
#           3 0.04713737 0.9528626 0.00000000
#           4 0.01747154 0.9497448 0.03278367
# 
# $Parenthood
#            symbol_names
# state_names childless   children
#           1  0.998818 0.00118196
#           2  1.000000 0.00000000
#           3  1.000000 0.00000000
#           4  0.000000 1.00000000
# 
# $`Residence`
#            symbol_names
# state_names left home with parents
#           1 0.0000000    1.0000000
#           2 1.0000000    0.0000000
#           3 0.6916852    0.3083148
#           4 1.0000000    0.0000000

Converting multichannel to single channel models and data

The mc_to_sc function converts a multichannel model into a single channel representation. E.g. the plot function for hmm objects uses this type of conversion. The seqHMM package also includes a similar function mc_to_sc_data for merging multiple state sequence objects.

sc_hmm <- mc_to_sc(hmm$model)

ssplot(sc_hmm, plots = "both", type = "I", sortv = "from.end", sort.channel = 0, 
       xtlab = 15:30, legend.prop = 0.5)

scssp

Mixture hidden Markov models

A mixture hidden Markov model (MHMM) is, by definition, a mixture of HMMs that are fitted together. These are fitted and plotted with similar functions to ones presented before. Starting values are given as a list consisting of the parameter values for each cluster. The build_mhmm function checks that the model is properly constructed before estimating parameters with the fit_model function.

# Starting values for emission probabilities

# Cluster 1
alphabet(child.seq) # Checking for the order of observed states
emiss_1_child <- matrix(
  c(0.99, 0.01, # High probability for childless
    0.99, 0.01,
    0.99, 0.01,
    0.99, 0.01), 
  nrow = 4, ncol = 2, byrow = TRUE)

alphabet(marr.seq)
emiss_1_marr <- matrix(
  c(0.01, 0.01, 0.98, # High probability for single
    0.01, 0.01, 0.98,
    0.01, 0.98, 0.01, # High probability for married
    0.98, 0.01, 0.01), # High probability for divorced
  nrow = 4, ncol = 3, byrow = TRUE)

alphabet(left.seq)
emiss_1_left <- matrix(
  c(0.01, 0.99, # High probability for living with parents
    0.99, 0.01, # High probability for having left home
    0.99, 0.01,
    0.99, 0.01), 
  nrow = 4, ncol = 2, byrow = TRUE)

# Cluster 2
emiss_2_child <- matrix(
  c(0.99, 0.01, # High probability for childless
    0.99, 0.01,
    0.99, 0.01,
    0.01, 0.99), 
  nrow = 4, ncol = 2, byrow = TRUE)

emiss_2_marr <- matrix(
  c(0.01, 0.01, 0.98, # High probability for single
    0.01, 0.01, 0.98,
    0.01, 0.98, 0.01, # High probability for married
    0.29, 0.7, 0.01), 
  nrow = 4, ncol = 3, byrow = TRUE)

emiss_2_left <- matrix(
  c(0.01, 0.99, # High probability for living with parents
    0.99, 0.01,
    0.99, 0.01,
    0.99, 0.01), 
  nrow = 4, ncol = 2, byrow = TRUE)

# Cluster 3
emiss_3_child <- matrix(
  c(0.99, 0.01, # High probability for childless
    0.99, 0.01,
    0.01, 0.99,
    0.99, 0.01,
    0.01, 0.99,
    0.01, 0.99), 
  nrow = 6, ncol = 2, byrow = TRUE)

emiss_3_marr <- matrix(
  c(0.01, 0.01, 0.98, # High probability for single
    0.01, 0.01, 0.98,
    0.01, 0.01, 0.98,
    0.01, 0.98, 0.01, # High probability for married
    0.01, 0.98, 0.01,
    0.98, 0.01, 0.01), # High probability for divorced
  nrow = 6, ncol = 3, byrow = TRUE)

emiss_3_left <- matrix(
  c(0.01, 0.99, # High probability for living with parents
    0.99, 0.01,
    0.50, 0.50,
    0.01, 0.99,
    0.99, 0.01,
    0.99, 0.01), 
  nrow = 6, ncol = 2, byrow = TRUE)

# Starting values for transition matrices
trans_1 <- matrix(
  c(0.80, 0.16, 0.03, 0.01,
       0, 0.90, 0.07, 0.03,
       0,    0, 0.90, 0.10,
       0,    0,    0,    1),
  nrow = 4, ncol = 4, byrow = TRUE)

trans_2 <- matrix(
  c(0.80, 0.10, 0.05, 0.03, 0.01, 0.01,
       0, 0.70, 0.10, 0.10, 0.05, 0.05,
       0,    0, 0.85, 0.01, 0.10, 0.04,
       0,    0,    0, 0.90, 0.05, 0.05,
       0,    0,    0,    0, 0.90,  0.1,
       0,    0,    0,    0,    0,    1),
  nrow = 6, ncol = 6, byrow = TRUE)

# Starting values for initial state probabilities
initial_probs_1 <- c(0.9, 0.07, 0.02, 0.01)
initial_probs_2 <- c(0.9, 0.04, 0.03, 0.01, 0.01, 0.01)

# Birth cohort
biofam3c$covariates$cohort <- cut(
  biofam3c$covariates$birthyr, c(1908, 1935, 1945, 1957))
biofam3c$covariates$cohort <- factor(
  biofam3c$covariates$cohort, labels=c("1909-1935", "1936-1945", "1946-1957")
  )

# Build MHMM
init_mhmm <- build_mhmm(
  observations = list(marr.seq, child.seq, left.seq),
  transition_probs = list(trans_1, trans_1, trans_2),
  emission_probs = list(list(emiss_1_marr, emiss_1_child, emiss_1_left), 
                        list(emiss_2_marr, emiss_2_child, emiss_2_left),
                        list(emiss_3_marr, emiss_3_child, emiss_3_left)),
  initial_probs = list(initial_probs_1, initial_probs_1, initial_probs_2),
  formula = ~sex * cohort, data = biofam3c$covariates, 
  cluster_names = c("Cluster 1", "Cluster 2", "Cluster 3"),
  channel_names = c("Marriage", "Parenthood", "Left home")
  )

mhmm_fit <- fit_model(init_mhmm)

Summary of MHMM

The summary method computes summaries of the MHMM, e.g. standard errors for covariates and prior and posterior probabilities for subjects. A print method shows some summaries of these: estimates and standard errors for covariates, log-likelihood and BIC, information on most probable clusters and prior and posterior probabilities. Parameter estimates for transitions, emissions, and initial probabilities are omitted by default.

The classification table shows the mean probabilities of belonging to each cluster by the most probable cluster. The most probable cluster is determined by the posterior probabilities. A good model shoud have high proportions in the diagonal. Here, for individuals assigned to cluster 1, the average probability for cluster 1 is 0.84, 0.16 for cluster 2, and close to 0 for cluster 3. The highest probability for the assigned cluster is 0.93 for cluster 3.

summ_mhmm <- summary(mhmm_fit$model)

names(summ_mhmm)
# [1] "logLik"                          "BIC"                            
# [3] "most_probable_cluster"           "coefficients"                   
# [5] "vcov"                            "prior_cluster_probabilities"    
# [7] "posterior_cluster_probabilities" "classification_table"           
# [9] "model" 

summ_mhmm
# Covariate effects :
# Cluster 1 is the reference.
# 
# Cluster 2 :
#                           Estimate  Std. error
# (Intercept)                 1.1400       0.176
# sexwoman                   -0.2150       0.241
# cohort1936-1945             0.0829       0.239
# cohort1946-1957             0.1420       0.218
# sexwoman:cohort1936-1945    0.2960       0.329
# sexwoman:cohort1946-1957    0.0715       0.295
# 
# Cluster 3 :
#                           Estimate  Std. error
# (Intercept)                  0.430       0.197
# sexwoman                     0.149       0.263
# cohort1936-1945             -0.647       0.290
# cohort1946-1957             -0.899       0.269
# sexwoman:cohort1936-1945     0.212       0.387
# sexwoman:cohort1946-1957    -0.122       0.356
# 
# Log-likelihood: -12712.65   BIC: 26524.88 
# 
# Means of prior cluster probabilities :
# Cluster 1 Cluster 2 Cluster 3 
#     0.191     0.622     0.187 
# 
# Most probable clusters :
#             Cluster 1  Cluster 2  Cluster 3
# count             310       1364        326
# proportion      0.155      0.682      0.163
# 
# Classification table :
# Mean cluster probabilities (in columns) by the most probable cluster (rows)
# 
#           Cluster 1 Cluster 2 Cluster 3
# Cluster 1    0.8391    0.1606  0.000274
# Cluster 2    0.0848    0.8634  0.051819
# Cluster 3    0.0169    0.0499  0.933151

Plotting MHMMs

Also MHMMs are plotted with the plot function. The user can choose between an interactive mode (interactive = TRUE), where the model for each cluster is plotted separately, and a combined plot with all models at once.

# Plot mixture hidden Markov model
# Interactive plot, one cluster at a time
plot(mhmm_fit$model, interactive = TRUE)

mixHMM1 mixHMM2 mixHMM3

# Plotting observed sequences and most probable hidden states
# Interactive plot, one cluster at a time
mssplot(
  mhmm_fit$model, plots = "both", type = "I", sortv = "from.end", sort.channel = 1, 
  xtlab = 15:30, xlab = "Age")

mssplot1 mssplot2 mssplot3