Nonhomogeneous HSMM-VAR Regime-Switching Model

• The paper introduces a nonhomogeneous HSMM-VAR framework that models explicit state-specific dwell times and covariate-dependent transition hazards.
• It integrates hidden semi-Markov dynamics with regime-specific vector autoregression to capture temporal autocorrelation and cross-variable dependencies.
• Regularized likelihood inference via a tailored EM algorithm ensures accurate state detection, lag selection, and control of overfitting.

Searching arXiv for papers on nonhomogeneous HSMMs and VAR-HSMMs to ground the article. Nonhomogeneous Hidden Semi-Markov Vector Auto-Regression denotes a class of regime-switching models for multivariate time series in which the latent regime process is semi-Markov, the observation process is state-dependent vector autoregressive, and exogenous covariates may affect both observation dynamics and state persistence. In the environmental formulation introduced for multivariate air pollution data, the hidden process represents unobserved environmental conditions, the vector autoregressive component accounts for temporal autocorrelation and cross-variable dependence, and time-varying conditions influence both pollutant levels and the duration of transient states (Mingione et al., 17 Sep 2025). The model sits at the intersection of hidden semi-Markov modeling, switching VAR processes, regularized likelihood inference, and covariate-dependent duration analysis.

1. Conceptual structure

A nonhomogeneous HSMM-VAR combines two layers. The first is a hidden state process over $K$ latent regimes, represented by indicators $u_{tk} \in \{0,1\}$ with $\sum_k u_{tk}=1$. The second is a multivariate observation process $\mathbf{y}_t \in \mathbb{R}^p$ whose conditional distribution depends on the active regime and on lagged observations. A convenient hierarchical representation is

$$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$

where $f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$ is the regime-specific observation model and $p(\mathbf{u}; \boldsymbol{\eta})$ is the semi-Markov state model (Mingione et al., 17 Sep 2025).

The “hidden” component indicates that the regimes are unobserved; the “semi-Markov” component indicates that state durations are modeled explicitly rather than being forced to be geometric; the “vector auto-regression” component indicates that each regime induces its own multivariate linear dependence on lagged observations. The nonhomogeneous qualifier refers to time-varying or covariate-dependent behavior in the state process and, in some formulations, in the observation means as well.

This architecture is distinct from a standard HMM with autoregressive emissions. In a standard HMM, dwell times are geometric because persistence is encoded only through self-transition probabilities. In an HSMM, regime duration has an explicit state-specific distribution, so persistence is modeled directly rather than indirectly. A central consequence is that non-geometric and covariate-modulated persistence can be represented without forcing unrealistic memoryless state occupancy.

2. Semi-Markov state dynamics and nonhomogeneity

For a current state $k$ and dwell time $d_{t-1}$, one formulation of the transition kernel is

$$\gamma_{kj}(d_{t-1}) = \begin{cases} 1 - q_k(d_{t-1}) & \text{if } j=k,\ q_k(d_{t-1}) \cdot \omega_{kj} & \text{if } j\neq k, \end{cases}$$

where $u_{tk} \in \{0,1\}$0 is a discrete hazard for exiting state $u_{tk} \in \{0,1\}$1, and $u_{tk} \in \{0,1\}$2 is the probability of moving from $u_{tk} \in \{0,1\}$3 to $u_{tk} \in \{0,1\}$4 conditional on leaving $u_{tk} \in \{0,1\}$5 (Mingione et al., 17 Sep 2025). The associated sojourn distribution is

$u_{tk} \in \{0,1\}$6

Nonhomogeneity is introduced by allowing the hazard to depend on time-varying covariates: $u_{tk} \in \{0,1\}$7 with $u_{tk} \in \{0,1\}$8 used as a link function, often complementary log-log in the discrete-hazard setting (Mingione et al., 17 Sep 2025). This means that external conditions can modulate the probability of exiting a regime at each time point, not merely at regime entry.

A more general inhomogeneous HSMM framework extends the standard HMM-with-extended-state-space construction so that covariates influence all aspects of the state process model, especially the dwell-time distributions (Koslik, 2024). In that representation, each substantive state $u_{tk} \in \{0,1\}$9 is replaced by an aggregate $\sum_k u_{tk}=1$0 of latent substates, and the time-varying transition matrix $\sum_k u_{tk}=1$1 is organized in blocks whose diagonal parts govern within-state sojourn behavior and whose off-diagonal parts govern state changes. The dwell-time hazard for a stay of length $\sum_k u_{tk}=1$2 initiated at time $\sum_k u_{tk}=1$3 is written

$\sum_k u_{tk}=1$4

The periodic special case is analytically important. If $\sum_k u_{tk}=1$5 and $\sum_k u_{tk}=1$6 vary with period $\sum_k u_{tk}=1$7, then one can define a periodically varying unconditional state distribution and an overall dwell-time distribution. The latter has a mixture representation,

$\sum_k u_{tk}=1$8

which separates within-cycle duration variation from the frequency with which new visits to state $\sum_k u_{tk}=1$9 begin at different times of the cycle (Koslik, 2024). This suggests that “nonhomogeneous” in HSMMs is not confined to transition matrices; dwell-time laws themselves may be time- or covariate-indexed.

3. State-dependent vector autoregression

Conditional on regime $\mathbf{y}_t \in \mathbb{R}^p$0, the observation process in HSMM-VAR is Gaussian with regime-specific VAR structure: $\mathbf{y}_t \in \mathbb{R}^p$1 where $\mathbf{y}_t \in \mathbb{R}^p$2 is a baseline mean vector, $\mathbf{y}_t \in \mathbb{R}^p$3 is the lag-$\mathbf{y}_t \in \mathbb{R}^p$4 autoregression matrix for state $\mathbf{y}_t \in \mathbb{R}^p$5, and $\mathbf{y}_t \in \mathbb{R}^p$6 is the innovation covariance matrix (Mingione et al., 17 Sep 2025). If exogenous covariates $\mathbf{y}_t \in \mathbb{R}^p$7 are included in the observation model, the baseline can be written

$\mathbf{y}_t \in \mathbb{R}^p$8

This structure separates three types of dependence. Serial within-variable dependence is encoded by diagonal and same-variable lag effects in the matrices $\mathbf{y}_t \in \mathbb{R}^p$9. Cross-variable lag dependence is encoded by off-diagonal elements of those matrices. Contemporaneous dependence is encoded by $$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$0. Because all of these are state-specific, the multivariate dependence pattern is allowed to change abruptly when the latent regime changes.

Closely related Bayesian work adopts the same regime-specific VAR principle but imposes sparsity through an $$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$1-ball projection prior. In that framework,

$$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$2

and the vectorized VAR coefficients are mapped onto an $$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$3-ball so that exact zeros occur with positive probability, enabling state-specific variable selection and Hamiltonian Monte Carlo inference (Hadj-Amar et al., 2023). The Bayesian sparse VAR-HSMM and the regularized nonhomogeneous HSMM-VAR share the goal of controlling high-dimensional regime-specific dynamics, but they differ in inferential strategy and in the way dwell-time modeling enters the specification.

4. Likelihood, estimation, and regularization

Direct likelihood maximization is difficult because the latent state process is unobserved and duration dependence breaks the simple first-order Markov property. The nonhomogeneous HSMM-VAR therefore uses a tailored Expectation-Maximization algorithm (Mingione et al., 17 Sep 2025). In the E-step, the required posterior quantities include univariate occupancies such as $$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$4 and bivariate transition posteriors $$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$5. These are computed with forward-backward recursions, using an HMM approximation of the HSMM when necessary.

The M-step separates hidden-process and observation-process updates. For the hidden process, weighted multinomial and binomial regressions update conditional transition probabilities and covariate-dependent hazards. For the observation process, each response coordinate and each regime are fitted by weighted regression using the posterior regime probabilities as soft weights. To counter overfitting and to perform lag selection, the state-specific autoregressive coefficients are penalized with an $$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$6 term: $$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$7 where $$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$8 is a diagonal weight matrix and $$f(\mathbf{Y};\, \boldsymbol{\theta}, \boldsymbol{\eta}) = \sum_{\mathbf{u}} f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta}) \cdot p(\mathbf{u}; \boldsymbol{\eta}),$$9 is a state-specific penalty (Mingione et al., 17 Sep 2025). The tuning rule

$f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$0

adapts regularization strength to the effective sample size of each regime. Covariance matrices are then estimated from weighted residuals.

Model selection is conducted with ICL for both the number of hidden states $f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$1 and the regularization level $f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$2, and uncertainty quantification is handled through parametric bootstrap with debiasing for the LASSO estimates (Mingione et al., 17 Sep 2025). Simulation results reported for this framework state that ICL successfully recovers the true number of hidden states and optimal regularization, that true nonzero coefficients are selected while noise lags and covariates are shrunk to zero, and that the method works well for $f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$3–$f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$4 state settings with up to $f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$5 observed variables and lag orders up to $f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$6 (Mingione et al., 17 Sep 2025).

Alternative inferential schemes exist. Sequential Bayesian learning for HSMMs uses particle filters, Particle MCMC, and SMC$f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$7 to obtain exact inference up to Monte Carlo error in a sequential setting, with AR emissions as the main illustration and multivariate VAR and nonhomogeneous transitions discussed as extensions (Aschermayr et al., 2023). This suggests that online inference for nonhomogeneous HSMM-VAR is methodologically plausible, although the cited sequential framework does not present that full model as its primary case.

5. Relation to adjacent model classes and implementations

The HSMM-VAR belongs to a broader family of switching autoregressive latent-state models. Hidden hybrid Markov/semi-Markov models permit a mixture of Markovian and semi-Markovian states, including absorbing or macro-states, and support left-to-right and series/parallel state-network structures (Amini et al., 2021). The hhsmm R package implements functions for initializing, fitting, and prediction in such models, including Markov/semi-Markov switching regression, auto-regressive HHSMMs, nonparametric estimation of emission distributions using penalized B-splines, future-state prediction, and residual useful lifetime estimation (Amini et al., 2021). These capabilities place auto-regressive semi-Markov switching models in an applied software ecosystem, although the package description is broader than the specific nonhomogeneous penalized HSMM-VAR framework.

The state-process side of HSMM-VAR is closely connected to extended-state-space constructions used to represent HSMMs as HMMs with carefully structured transition matrices (Koslik, 2024). The same idea appears in Bayesian sparse VAR-HSMM work, where a generic state distribution is embedded in a special transition matrix structure, yielding efficient likelihood evaluation with arbitrary approximation accuracy (Hadj-Amar et al., 2023). These representations are especially important when duration distributions are not geometric.

The observation side is also extensible. Generalizations of auto-regressive HMMs replace the linear autoregressive map by a linear combination of non-linear basis functions,

$f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$8

or define linear dynamics directly in unit quaternion space for orientation data (Ginesi et al., 2023). The same source states that these emission constructions are modular and can be extended to Auto-Regressive Hidden semi-Markov Models and to Nonhomogeneous Hidden Semi-Markov Vector Auto-Regression. A plausible implication is that the linear Gaussian VAR kernel in HSMM-VAR can serve as a baseline rather than a structural limitation.

A bibliographic complication arises with “A Regularized Vector Autoregressive Hidden Semi-Markov Model, with Application to Multivariate Financial Data” (Xu et al., 2018). The supplied record identifies the title and abstract, but the provided document is described as containing only AAAI Press formatting and submission instructions, with no technical material on VAR-HSMMs. Consequently, that source is relevant as a citation record but not as a usable technical exposition in the present context.

6. Applications, interpretation, and recurrent misconceptions

In environmental risk assessment, the nonhomogeneous HSMM-VAR was applied to daily concentrations of NO, NO$f(\mathbf{Y} \mid \mathbf{u}; \boldsymbol{\theta})$9, PM$p(\mathbf{u}; \boldsymbol{\eta})$0, PM$p(\mathbf{u}; \boldsymbol{\eta})$1, and PM$p(\mathbf{u}; \boldsymbol{\eta})$2 recorded over three years in Bergen, Norway, with meteorological covariates including temperature, precipitation, and wind (Mingione et al., 17 Sep 2025). The fitted models considered up to $p(\mathbf{u}; \boldsymbol{\eta})$3 states and lag order $p(\mathbf{u}; \boldsymbol{\eta})$4, and two regimes were selected as optimal: a high-pollution regime and a low-pollution regime. The high-pollution regime exhibited stronger cross-pollutant correlation and autocorrelation, while precipitation and wind promoted exit from the high-pollution state, thereby shortening pollution episodes (Mingione et al., 17 Sep 2025).

The same application coupled latent-state inference with a Shapley value-based decomposition for multivariate environmental risk. Using multivariate risk measures such as MCoVaR and MCoES, the procedure attributed marginal risk contributions among pollutants. Reported findings were that NO and NO$p(\mathbf{u}; \boldsymbol{\eta})$5 are mutually reinforcing and function as key risk drivers, particulate measures cluster together with their own mutual influence, and PM$p(\mathbf{u}; \boldsymbol{\eta})$6 behaves more independently with seasonally shifting associations (Mingione et al., 17 Sep 2025). This shows that HSMM-VAR can be used not only for segmentation and prediction but also for regime-resolved attribution.

Adjacent applications in the hhsmm ecosystem illustrate the breadth of semi-Markov autoregressive modeling rather than the exact same specification. A left-to-right HHSMM with gamma sojourn distributions was used for residual useful lifetime estimation in C-MAPSS turbofan engine data, and a regime-switching additive regression HHSMM with penalized B-splines was applied to Spain’s energy demand and price data (Amini et al., 2021). These examples do not define the nonhomogeneous HSMM-VAR itself, but they show how duration dependence, regime switching, and flexible emissions recur across reliability and energy problems.

Several misconceptions recur in discussions of the topic. One is that a semi-Markov model differs trivially from a hidden Markov model; in fact, the distinction is substantive because HMM dwell times are geometric, whereas HSMMs admit arbitrary state-specific duration laws (Koslik, 2024). Another is that nonhomogeneity concerns only conditional transition probabilities; the cited work shows that covariates may enter dwell-time distributions directly, including through duration-dependent hazards and periodically varying dwell models (Koslik, 2024). A third is that regularization is merely a numerical convenience. In penalized HSMM-VAR, state-specific $p(\mathbf{u}; \boldsymbol{\eta})$7 shrinkage is part of the inferential design, used to control overfitting and automatically select relevant temporal lags in regime-specific multivariate dynamics (Mingione et al., 17 Sep 2025).