Mixture Hidden Markov Models for Sequence Data: The seqHMM Package in R

1. Introduction

• HMMs detect underlying latent structures, while mixture HMMs allow clustering into latent classes, potentially using covariates.The paper frames both as probabilistic approaches to analyzing sequence data.Source: §1 · pp. 1–2
• seqHMM provides R functions for estimating and inferring HMMs and MHMMs, and for visualizing multichannel sequences and models.The package is intended for one or multiple subjects and one or multiple channels.Source: §1 · p. 2
• Existing R packages cover different settings; seqHMM's MHMMs allow emission probabilities to vary between groups and covariates to explain group membership.The introduction contrasts this with LMest's mixed latent Markov models, where the latent process varies across subpopulations.Source: §1 · p. 2
• The paper proceeds from sequence data and HMM/MHMM theory to visualization, package features, and a life-course data example.It also outlines special model cases and an appendix of notation.Source: §1 · p. 2

2.1. Sequences and sequence analysis

• Sequences are ordered categorical states; social sequence analysis describes successions, computes sequence (dis)similarities, and often uses clustering to find trajectory patterns.Source: §2.1 · p. 3

2.2. Hidden Markov models

• HMMs represent observations as probabilistic emissions from hidden states, with time-homogeneous first-order transitions and observations conditionally dependent on the current hidden state.Source: §2.2 · pp. 3–4

HMM for multichannel sequences

• For multichannel data, seqHMM uses one shared latent structure, a transition matrix, and channel-specific emission matrices, assuming channel observations are conditionally independent given the hidden state.Source: HMM for multichannel sequences · pp. 4–5
• Multichannel modeling can retain partially observed channels without combining missing and observed values into a potentially larger, harder-to-interpret state space.Source: HMM for multichannel sequences · p. 5

Log-likelihood and parameter estimation

• Parameters are estimated by maximum likelihood using direct numerical maximization or EM; forward–backward calculations support likelihoods and inference, while many starting values reduce local-maximum risk.Source: Log-likelihood and parameter estimation · pp. 5–6

Inference on hidden states

• Posterior state probabilities give time-specific hidden-state probabilities, whereas the Viterbi algorithm finds the single most probable hidden-state sequence.Source: Inference on hidden states · p. 6

2.3. Clustering by mixture hidden Markov models

• Mixture HMMs assign subjects to clusters probabilistically during joint modeling, avoiding fixed preliminary clusters; covariates can inform membership probabilities.Source: §2.3 · p. 7, Covariates and cluster probabilities · p. 8

Mixture hidden Markov model

• An MHMM can be recast as a larger HMM with block-diagonal transitions that prohibit movement between clusters; seqHMM also includes Markov, mixture Markov, and latent class special cases.Latent class models use one hidden state per cluster and can accommodate multiple channels.Source: Mixture hidden Markov model · pp. 7–8, Markov model · p. 9, Mixture Markov model · p. 9

3. Package features

• seqHMM uses TraMineR’s stslist objects and extends its sequence-analysis features with multichannel plotting and conversion to a single-channel representation.TraMineR supplies sequence construction, summarization, visualization, and data-manipulation features used by seqHMM.Source: §3 · p. 10
• The package covers HMM and MHMM estimation and evaluation, data and model manipulation, visualization, and simulation.Table 1 catalogs the available functions and methods.Source: §3 · p. 10

3.1. Building and fitting models

• Build functions cover HMMs, MHMMs, Markov models, mixture Markov models, and latent class models; zero probabilities are fixed, and covariates must be complete.Users may supply starting values or use random ones; ordinary Markov models receive estimated initial and transition probabilities automatically.Source: §3.1 · p. 11
• Fitting can combine EM, global DNM, and local DNM in that order, with each step generally initializing the next.The package implements EM and gradient calculations in C++ with optional parallel computation between subjects; multiple randomized EM starts and global methods offer alternative searches.Source: §3.1 · pp. 11–12
• No optimization method is universally best: larger unconstrained models increase computation time and the risk of local optima or saddle points, so several settings are advisable.The authors report most success with randomized EM and suggest comparing different optimization strategies.Source: §3.1 · p. 12

State and model inference

• Inference tools compute forward–backward probabilities, posterior probabilities, and most probable hidden paths, while log-likelihood output supports AIC and BIC.MHMM, mixture Markov, and latent class summaries include covariate standard errors and prior and posterior cluster probabilities.Source: State and model inference · pp. 12–13

3.2. Visualizing sequence data

• Stacked sequence plots align channels by subject; ssplot defaults to state distributions and also supports sequence index plots, while gridplot combines views or groups.Grid layouts are configurable, and sequences can be ordered using multidimensional scaling scores based on sequence dissimilarities.Source: §3.2 · pp. 13–14

3.3. Visualizing hidden Markov models

• Directed HMM graphs encode hidden states as pie-chart nodes, emissions as slices, transitions as edges, and initial probabilities beside nodes; MHMM plots can show clusters separately or together.The ssplot method can also display observations, most probable hidden paths, or both; mssplot provides stacked plots by cluster.Source: §3.3 · pp. 14, 16

4. Examples with life course data

• The examples progress from constructing sequence data to visualizing it, fitting single- and multichannel HMMs, clustering with MHMMs, and plotting models.Source: §4 · p. 16

4.1. Sequence data

• The examples use the original biofam data and a three-channel version, defining sequence start ages, state labels, and channel colors for plotting.Source: §4 · p. 16, §4.1 · p. 18

Figure 2: Plotting state distributions

• The default state-distribution plot accepts multichannel data as a list of stslist objects and uses supplied names as channel labels.Source: Figure 2: Plotting state distributions · p. 18

Figure 1: Plotting sequences

• Sequence index plots can show complete multichannel sequences, sorted by the first channel from the beginning, with configurable axes, legends, and title.Source: Figure 1: Plotting sequences · pp. 18–19

Figure 3: Plotting sequence data in a grid

• Sequence plots for women and men can share a grid and legends; plot objects can be updated to specify only differences between panels.Source: Figure 3: Plotting sequence data in a… · p. 19

4.2. Hidden Markov models

• The examples use build_hmm to initialize models and fit_model to estimate parameters; the fitting output includes the estimated model, log-likelihood, and optimization information.Source: §4.2 · pp. 20–21
• For a single-channel HMM, starting probabilities can be initialized from observed state frequencies across age groups, with zero-frequency categories smoothed by adding 0.1 before row normalization.Source: §4.2 · p. 20
• The multichannel example fits a five-state left-to-right model with separate emission matrices per channel and an upper-triangular transition matrix, then estimates parameters using L-BFGS with four threads.Source: §4.2 · p. 22

• The example specifies two clusters with five and four hidden states, uses sex and birth cohort to explain membership, and follows 100 EM restarts with local L-BFGS fitting.Source: §4.3 · p. 23
• The MHMM summary reports covariate estimates and standard errors, log-likelihood and BIC, and cluster probabilities; its classification table gives mean probabilities by most probable cluster.Source: §4.3 · p. 24
• The reported fit has log-likelihood -12969.57 and BIC 26592.66; mean prior cluster probabilities are 0.857 for Cluster 1 and 0.143 for Cluster 2.Source: §4.3 · p. 24

• A basic HMM graph can be drawn with the plot method; the default view is convenient during analysis, while publication may call for a tailored presentation.Source: §4.4 · p. 25

Figure 4: HMM plot with modifications

• Figure 4 customization adjusts vertex size, initial-probability spacing, edge curvature, combined-slice labels, and legend space.Source: Figure 4: HMM plot with modifications · p. 26

Figure 5: HMM plot with a different layout

• The Figure 5 layout uses explicit coordinates and axis limits, with rescaling disabled, and displays every emission probability.It also changes legend placement, label layout, edge widths, arrow sizes, and parameter-label precision.Source: Figure 5: HMM plot with a different… · p. 26
• Figure 5 selects ten colors from palette 14 for its ten combined states.Source: Figure 5: HMM plot with a different… · p. 26
• The ssplot function accepts an HMM object and computes hidden state paths automatically when they are not supplied.Source: Figure 5: HMM plot with a different… · p. 26

• The default interactive plot displays each MHMM cluster model separately, while a combined plot can become difficult to fit with several graphs and legends.Source: §4.5 · p. 27
• Setting interactive = FALSE and nrow = 2 arranges the cluster-model graphs in a two-row grid.Source: §4.5 · p. 27
• The mssplot function shows data and/or hidden paths one cluster at a time, allocating subjects according to their most probable hidden state paths.Source: §4.5 · p. 27
• For more control, separate_mhmm converts an MHMM into separate HMM objects that can be plotted with sequence and grid plotting functions.Source: §4.5 · p. 27

• HMMs can account for observation measurement error, detect unobservable states, or compress information across observation types.Source: §5 · p. 27
• seqHMM handles categorical data from single sequences to multichannel sequences, with HMMs, MHMMs, and restricted variants including Markov and latent class models.Covariates can explain cluster membership in MHMMs.Source: §5 · p. 27
• Because parameter estimation is sensitive to starting values, seqHMM offers fitting options using global and local optimization, direct numerical estimation, and the EM algorithm.Source: §5 · p. 27
• Almost all intensive computations use C++, and the package supports parallel computation.Source: §5 · p. 27
• Planned work includes time-varying covariates in transition and emission matrices, sampling weights, and faster computations for restricted model variants.Source: §5 · p. 29

• Y_i denotes subject i's observation sequences; y_it and y_itc denote observations at time t overall and in channel c, respectively.Source: A. Notations · p. 29
• z_it is the hidden state at time t, and s ranges over the hidden-state space of size S.Source: A. Notations · p. 29
• A gives transition probabilities between hidden states, B_c gives channel-specific emission probabilities, and π gives initial-state probabilities.Source: A. Notations · p. 29
• In the mixture model, M_k denotes cluster k's submodel, w_ik its probability for subject i, and x_i the subject's covariates.The cluster-specific parameter set includes initial probabilities, transition and emission matrices, and regression coefficients.Source: A. Notations · p. 29