A Data-Driven Approach to State Construction in Markov Models

We propose a robust feature-weighted jump model for time-dependent clustering. A penalty is used to encourage smoothness of transitions over time, while robustness is achieved through the use of a Tukey's biweight loss function. An additional parameter controls the variability of feature weights across states, allowing the model to assign state-specific relevance to each feature. We illustrate in simulation how the method accurately recovers the true cluster sequence and reliably identifies relevant features, outperforming competing approaches, particularly in the presence of outliers. We conclude with two empirical applications, one on the number of conflict-related homicides in Kosovo in the period 1998-2000, and another on macroeconomic performance of twelve European countries in the period 1949-2024.

Learning latent representations from temporal, multimodal, and partially observed data requires specifying what information a latent state should retain, discard, and organize. Existing approaches encode these requirements through heterogeneous objectives, making methods difficult to compare and learned representations difficult to interpret. We propose Constrained Latent State Modeling (CLSM), a conceptual framework that characterizes latent states through six complementary properties: predictive sufficiency, minimality, temporal coherence, observation compatibility, invariance to nuisance factors, and structural constraints. CLSM separates these properties from the surrogate objectives used to induce them and from the diagnostics used to evaluate them, and clarifies how combinations of constraints can improve identifiability by restricting the space of admissible representations. We reinterpret major representation-learning families through this common design space and illustrate the framework with a controlled synthetic benchmark. The experiments show how objectives produce distinct latent organizations and empirical trade-offs depending on the prediction target, surrogate formulation, parameterization, and optimization. Companion repository containing the reference implementation, reproducible experiments, documentation, and model cards: https://github.com/gwenole-quellec/clsm

In many domains, practitioners seek models that produce accurate forecasts while faithfully capturing latent system dynamics. Existing approaches typically sacrifice one of these goals: deep state space models often assume Gaussian latent transitions, limiting fit and forecasting, while diffusion models are highly expressive but lack principled inference for the underlying dynamics. To combine the strengths of both, we introduce the Diffusion-Driven State Space Model (DDSSM), which replaces the conventional Gaussian transition distribution with a diffusion model. Our DDSSM resolves the open problem of how to jointly train an autoencoder and a diffusion model on sequential data, thereby extending the literature on latent diffusion models for time series. Moreover, we find that the DDSSM empirically outperforms a state-of-the-art deep SSM at fitting and forecasting a simulated time series with multimodal transitions.