Dynamic Structural Equation Models (DSEM)

• DSEM is a modeling framework that integrates time-indexed structural dependencies with latent variable decomposition for intensive longitudinal data.
• It unifies latent measurement, autoregression, and hierarchical modeling to capture within-person, between-person, and time-specific variations.
• Recent advances leverage state-space reformulations and online algorithms to boost computational efficiency and extend applications to binomial outcomes, functional predictors, and dynamic network inference.

Searching arXiv for papers on Dynamic Structural Equation Models and closely related formulations. Dynamic structural equation models (DSEM) are structural equation models for time-dependent data in which structural dependencies are coupled to explicit temporal dynamics. In recent arXiv literature, the term is used in at least two closely related but non-identical senses: as a multilevel latent-variable framework for intensive longitudinal data, and as a dynamic network or topology model in which SEM coefficients vary over time and are inferred from sequential observations (Sørensen, 4 Mar 2026, Baingana et al., 2013, Lin et al., 2023). Across these usages, DSEM generalizes static SEM by allowing contemporaneous effects, lagged effects, between-unit heterogeneity, and time-specific variation to coexist within a single probabilistic system.

1. Scope and meanings of the term

The most established contemporary usage treats DSEM as a Bayesian framework for intensive longitudinal data in which within-person processes evolve over time, between-person differences are modeled hierarchically, and latent variables may be measured by multiple indicators (Faleh et al., 18 Aug 2025). A second usage, prominent in network science and signal processing, uses dynamic SEM to denote models of time-varying adjacency or influence matrices, typically with sparse regularization and online updating (Zaman et al., 2020). A closely related line of work in high-dimensional time series studies structural vector autoregressions with contemporaneous DAG structure and lagged dynamics; this work is described as being “very close in spirit” to what is often called DSEM (Lin et al., 2023).

Formulation in the literature	Canonical equation	Primary inferential target
Intensive longitudinal latent-variable DSEM	$$\mathbf y_{it}=\mathbf y_{1,it}+\mathbf y_{2,i}+\mathbf y_{3,t}$$	Within-person dynamics, between-person heterogeneity, latent measurement
Dynamic topology SEM	$$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$	Time-varying graph topology and exogenous effects
High-dimensional SVAR/SEM with lag dynamics	$$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$	Contemporaneous DAG and temporal structure discovery

This multiplicity of meanings is not a terminological accident. The common core is the replacement of static structural coefficients by explicitly time-indexed or lag-indexed structural relations. What differs is the inferential emphasis: latent psychological or behavioral processes in intensive longitudinal analysis, versus changing network topology or contemporaneous causal structure in dynamical systems.

A recurrent misconception is that DSEM denotes a single standardized model class. The literature instead shows a family resemblance. Some formulations are measurement-heavy and multilevel; others are sparse, online, and topology-oriented. This suggests that DSEM is best understood as an umbrella category defined by dynamic structural dependence rather than by one fixed parameterization.

2. Canonical multilevel latent-variable architecture

In the intensive longitudinal formulation, DSEM decomposes an observed response vector for person $i$ at time $t$ into within-person, between-person, and between-timepoint components:

$$\mathbf y_{it}=\mathbf y_{1,it}+\mathbf y_{2,i}+\mathbf y_{3,t}.$$

Here, $\mathbf y_{1,it}$ is within-person time-varying variation, $\mathbf y_{2,i}$ is stable between-person variation, and $\mathbf y_{3,t}$ is timepoint-specific systematic variation shared across participants (Sørensen, 4 Mar 2026).

The within-level component is the dynamic core. In a general formulation,

$$\begin{aligned} \mathbf y_{1,it} &= \boldsymbol{\nu}_{1,it} +\boldsymbol{\Lambda}_{1,it}(L)\boldsymbol{\eta}_{1,it} +\mathbf R_{it}(L)\mathbf y_{1,it} +\mathbf K_{1,it}\mathbf X_{1,it} +\boldsymbol{\epsilon}_{1,it},\ \boldsymbol{\eta}_{1,it} &= \boldsymbol{\alpha}_{1,it} +\mathbf B_{1,it}(L)\boldsymbol{\eta}_{1,it} +\mathbf Q_{it}(L)\mathbf y_{1,it} +\boldsymbol{\Gamma}_{1,it}\mathbf X_{1,it} +\boldsymbol{\xi}_{1,it}, \end{aligned}$$

with Gaussian measurement error and process noise in the latent-variable setting (Sørensen, 4 Mar 2026). This specification admits autoregressive and cross-lagged effects, factor-analytic measurement structure, observed-variable lag effects, and time-varying coefficients.

The tutorial literature presents the same architecture as the conjunction of confirmatory factor analysis, time-series analysis, and multilevel models. In that framing, DSEM “can be viewed as a combination of time-series models and multilevel SEM,” with CFA supplying the measurement model, time series supplying lagged dependence, and multilevel modeling supplying between-person and between-time variation (Faleh et al., 18 Aug 2025). The tutorial formulation also emphasizes centered autoregressive specifications to reduce Nickell bias, random intercepts and random slopes for temporal persistence, and measurement invariance across people and time in the main longitudinal examples.

This architecture is scientifically consequential because it separates measurement from dynamics. Rather than fitting autoregression to manifest scores alone, DSEM can let a latent construct evolve across time while indicators remain noisy and partially unreliable. That distinction is central in applications where construct validity matters, such as anxiety, alliance, affect, or other latent psychological processes (Faleh et al., 18 Aug 2025).

3. Estimation, state-space reformulation, and computational scaling

A central development in recent DSEM research is the shift from brute-force latent-state sampling to state-space marginalization. Earlier Mplus-style Metropolis-within-Gibbs algorithms suffer from three limitations: poor scaling because they sample $$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$0 latent states or auxiliary parameters each iteration, conjugacy restrictions on priors and model extensions, and slow mixing caused by highly correlated latent variables updated one-at-a-time or in small blocks (Sørensen et al., 31 Mar 2026).

The key 2026 result is that the within-level part of any DSEM can be rewritten exactly as a linear Gaussian state space model by augmenting the state vector to include current and lagged latent states and lagged observed components (Sørensen, 4 Mar 2026). Once reformulated, the latent within-person states can be integrated out analytically with the Kalman filter. The log marginal likelihood is then computed from Kalman innovations rather than from sampled state trajectories, and the remaining parameters can be estimated efficiently with HMC, specifically NUTS in Stan (Sørensen, 4 Mar 2026).

This reparameterization changes the effective sampling dimension from latent-state scaling of order $$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$1 to order $$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$2 for the latent-state part (Sørensen, 4 Mar 2026). In simulation, the computational effect is substantial. For a scalar latent AR(1) model, NUTS-Kalman was reported to be $$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$3 more efficient than Metropolis-within-Gibbs and $$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$4 more efficient than a brute-force NUTS sampler over all states; it also achieved $$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$5 in almost all runs (Sørensen, 4 Mar 2026). The performance gain is attributed specifically to marginalizing the latent states, not merely to changing software.

A second major extension addresses categorical outcomes. For binomial data with a logit link, DSEM estimation can be made conditionally Gaussian through Pólya–Gamma augmentation, where

$$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$6

Conditioning on $$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$7 yields Gaussian-like pseudo-observations, which then enter the same state-space machinery (Sørensen et al., 31 Mar 2026). The resulting sampler alternates one Gibbs step for latent responses with one NUTS transition for global and higher-level parameters, using the Kalman filter to marginalize the within-level latent states exactly (Sørensen et al., 31 Mar 2026).

The reported efficiency improvements are again large. In a participant-invariant AR(1) model with five binary indicators, the hybrid sampler was about $$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$8 more efficient than the next best method under probit and $$\mathbf Y^t=\mathbf A^t\mathbf Y^t+\mathbf B^t\mathbf X+\mathbf E^t$$9 under logit for $$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$0; for $$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$1, it was about $$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$2 better under probit and up to $$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$3 better under logit (Sørensen et al., 31 Mar 2026). In a nine-indicator VAR(1) with binomial logit outcomes, the hybrid sampler showed about $$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$4 bulk-efficiency gain and about $$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$5 tail-efficiency gain over pure NUTS (Sørensen et al., 31 Mar 2026). These results place state-space marginalization at the center of current Bayesian DSEM computation.

4. Time-varying topology inference and online dynamic SEM

In network-oriented work, DSEM is formulated as a time-varying linear SEM:

$$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$6

Here, $$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$7 is a directed adjacency matrix with zero diagonal, $$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$8 is diagonal and captures exogenous effects, and the same topology is shared across contagions or interactions at time $$X_t=\mu + A X_t + B_1 X_{t-1}+\cdots+B_d X_{t-d}+\boldsymbol\epsilon_t$$9 (Baingana et al., 2013). The objective is not latent-state recovery but topology inference: if $i$0, node $i$1 has directed influence on node $i$2 at time $i$3 (Baingana et al., 2013).

The 2013 formulation estimates $i$4 by minimizing a sparsity-regularized exponentially weighted least-squares criterion with forgetting factor $i$5:

$i$6

The paper develops proximal gradient/ISTA, accelerated proximal gradient/FISTA, and SGD variants, with row-wise decomposition and recursive exponentially weighted summary statistics (Baingana et al., 2013). FISTA has the same per-iteration order as ISTA but a faster worst-case convergence rate, and inexact online FISTA performed best in the reported synthetic studies (Baingana et al., 2013).

The 2020 online-tracking work sharpens this topology-inference perspective by studying a time-varying SEM with sparse exponentially weighted least squares, proximal online gradient descent, and a dynamic regret analysis (Zaman et al., 2020). The problem decouples row-wise, each node update uses recursively maintained sufficient statistics, and the proximal map is soft-thresholding on off-diagonal edge coefficients while leaving the exogenous coefficient unchanged (Zaman et al., 2020). The principal theoretical object is dynamic regret relative to the time-varying clairvoyant minimizer. The resulting bound depends on the path length

$i$7

so tracking quality is controlled by how rapidly the optimal topology moves (Zaman et al., 2020). When the path length is sublinear in $i$8, the average excess cost vanishes asymptotically.

A related but distinct structural-discovery line models time-dependent data through

$i$9

where $t$0 is a contemporaneous structural matrix and $t$1 are lag matrices (Lin et al., 2023). The contemporaneous part is constrained to be a DAG for identification, and prior partial ordering information is incorporated via hard-zero constraints on forbidden entries of $t$2 (Lin et al., 2023). The estimation problem uses $t$3-penalization and acyclicity constraints within a multi-block ADMM scheme, with convergence to a stationary point established through sufficient descent, subgradient bounds, and the Kurdyka–Łojasiewicz property (Lin et al., 2023). This line is DSEM-like in separating within-time structure from lagged structure, but its inferential focus is high-dimensional structural discovery rather than multilevel latent-process estimation.

5. Distributional, functional, and latent-state extensions

One important extension concerns outcome families. Earlier samplers for categorical intensive longitudinal data were limited to Bernoulli outcomes with a probit link, whereas recent work makes binomial/logit DSEM practical through the combination of Pólya–Gamma augmentation, state-space marginalization, and hybrid NUTS-Gibbs sampling (Sørensen et al., 31 Mar 2026). This establishes that DSEM is not intrinsically tied to Gaussian or probit-only observation models.

Another extension adds functional predictors and multi-resolution covariates. The partially functional DSEM (PFDSEM) for environmental data combines scalar predictors, functional covariates, latent variables, temporal dependence, and province-level heterogeneity in one Bayesian model (Liu et al., 6 Jul 2026). Its measurement component is

$t$4

with basis expansion used to obtain a finite-dimensional regression and Bayesian P-spline smoothing applied to the functional coefficient functions (Liu et al., 6 Jul 2026). Temporal dependence is encoded by a CAR(1)-type structure, while cross-variable covariance is handled by a Linear Model of Coregionalization, yielding a separable decomposition of adjacent-time covariance and between-variable covariance (Liu et al., 6 Jul 2026).

A further extension adds discrete regime change. Dynamic Latent Class Structural Equation Modeling (DLCSEM) is presented as DSEM plus latent dynamic states that can change over time, typically via a Hidden Markov Model (Faleh et al., 18 Aug 2025). In the tutorial’s formulation, DSEM handles continuous dynamic variation, whereas DLCSEM is intended for categorical shifts or state transitions, such as sudden gains in therapy, response-pattern changes, or switching between responder and non-responder states (Faleh et al., 18 Aug 2025).

Taken together, these developments indicate that current DSEM research is expanding simultaneously along three axes: richer observation models, richer predictor structures, and richer latent-state dynamics. A plausible implication is that the traditional boundary between DSEM, state-space SEM, and dynamic latent-variable regression is becoming increasingly methodological rather than conceptual.

6. Applications, interpretation, and recurrent assumptions

DSEM has been applied across clinical psychology, environmental science, macroeconomics, gene-network analysis, and social-media diffusion. In a clinical tutorial example based on the IMPLEMENT trial, a one-factor latent anxiety DSEM for $t$5 patients over $t$6 sessions reported an intercept of approximately $t$7 (SD $t$8) and an autoregressive coefficient of approximately $t$9 (SD $$\mathbf y_{it}=\mathbf y_{1,it}+\mathbf y_{2,i}+\mathbf y_{3,t}.$$0); the model also showed moderate person-specific random intercept variance, moderate person-specific random slope variance, and time-specific random slope variance of approximately $$\mathbf y_{it}=\mathbf y_{1,it}+\mathbf y_{2,i}+\mathbf y_{3,t}.$$1, interpreted as session-level changes in persistence (Faleh et al., 18 Aug 2025).

In ecological momentary assessment, the 2026 hybrid sampler was illustrated on data from 43 patients over 402 days, with a daily binary panic-attack outcome predicted by daily mean heart rate, daily heart rate variance, and step count (Sørensen et al., 31 Mar 2026). The model combined Bernoulli/logit and Gaussian measurement components within a multivariate within-person VAR structure. Missingness was handled naturally by prediction-only Kalman updates, and posterior intervals for the cross-lagged effects from heart-rate and activity variables to panic-attack probability included zero (Sørensen et al., 31 Mar 2026).

In environmental panel data, the PFDSEM was applied to 30 Chinese provinces from 2015 to 2020 using 10 pollutant or emission indicators and 10 socio-environmental factor categories (Liu et al., 6 Jul 2026). The paper reported strong temporal and spatial clustering, significant positive spatial autocorrelation, province-level heterogeneity in structural effects, and pronounced seasonal patterns in the functional effects of sea-level pressure and $$\mathbf y_{it}=\mathbf y_{1,it}+\mathbf y_{2,i}+\mathbf y_{3,t}.$$2 m temperature. Estimated temporal dependence parameters were $$\mathbf y_{it}=\mathbf y_{1,it}+\mathbf y_{2,i}+\mathbf y_{3,t}.$$3 and $$\mathbf y_{it}=\mathbf y_{1,it}+\mathbf y_{2,i}+\mathbf y_{3,t}.$$4, with credible intervals excluding zero (Liu et al., 6 Jul 2026).

In dynamic network inference, the 2013 social-media case study analyzed 360 websites and 466 cascades related to “Kim Jong-un” over 45 weeks and found that the inferred network became progressively denser around major political events (Baingana et al., 2013). In high-dimensional structural discovery, partial ordering on DREAM4 gene-expression data improved AUPRC and AUROC by about $$\mathbf y_{it}=\mathbf y_{1,it}+\mathbf y_{2,i}+\mathbf y_{3,t}.$$5, and the macroeconomic application with 78 quarterly variables produced a contemporaneous network and lag matrices interpreted as economically plausible (Lin et al., 2023).

Several assumptions recur across these literatures. In multilevel latent DSEM, measurement invariance across people and time, homoskedasticity across time, and equal observation spacing are explicit simplifying assumptions in tutorial implementations (Faleh et al., 18 Aug 2025). In contemporaneous-structure discovery, acyclicity of $$\mathbf y_{it}=\mathbf y_{1,it}+\mathbf y_{2,i}+\mathbf y_{3,t}.$$6 and sometimes partial ordering are used for identification (Lin et al., 2023). In Bayesian state-space estimation, missing observations are marginalized naturally rather than treated as additional latent parameters in the main sampling loop (Sørensen, 4 Mar 2026). The literature also emphasizes that estimation behavior depends strongly on how well latent states are observed: when indicators are noisy, Gibbs-style samplers mix poorly and Kalman-marginalized HMC is especially advantageous (Sørensen, 4 Mar 2026).

The cumulative picture is that DSEM is neither a single algorithm nor a single discipline-specific model. It is a broad methodological class centered on structurally specified temporal dependence. In one branch, it unifies latent measurement, autoregression, and hierarchical variation for intensive longitudinal data; in another, it provides sparse dynamic equations for topology tracking and structural discovery. Recent work has made this class markedly more computationally tractable and materially broader, extending it to binomial/logit outcomes, partial ordering constraints, functional covariates, and latent regime switching (Sørensen et al., 31 Mar 2026, Lin et al., 2023, Liu et al., 6 Jul 2026, Faleh et al., 18 Aug 2025).