Discrete-Time & Piecewise-Constant Models

• Discrete-Time and Piecewise-Constant Models are defined by time partitioning with constant parameters, bridging continuous and discrete frameworks in survival analysis, control, and neural modeling.
• They simplify analytical solutions by enabling closed-form updates and likelihood computations, thus optimizing tasks like system identification and hazard rate estimation.
• These models integrate efficiently with neural architectures, reducing computational load via interval-wise approximations and supporting scalable deep learning applications.

Discrete-time and piecewise-constant models form a foundational conceptual bridge between purely continuous- and purely discrete-time approaches in stochastic processes, dynamical systems, neural modeling, and control. These models extract computational and statistical advantages by leveraging time partitioning and simple segment-wise parameterizations, preserving analytical tractability while enabling flexibility and efficiency in both inference and learning. Under various mathematical and engineering frameworks, such as survival analysis, optimal control, neural time series, and the theory of hybrid dynamical systems, discrete-time and piecewise-constant models appear with distinct but structurally related definitions.

1. Mathematical Foundations of Discrete-Time and Piecewise-Constant Models

At their core, piecewise-constant models partition time into intervals—either uniform or non-uniform—on each of which a relevant parameter (e.g., hazard rate, system control, state velocity) is held constant. The general form in a continuous-time context is: for time grid $0 = t_0 < t_1 < \cdots < t_K$, a function $h(t)$ or $f(t)$ is defined by

$$h(t) = \sum_{k=1}^K h_k \mathbf{1}_{[t_{k-1}, t_k)}(t)$$

where each $h_k$ is either constant or a simple function (e.g., linear in "piecewise-linear" extensions).

In discrete-time models, the process is observed or defined only at the grid points (e.g., $T \in \{[t_{k-1}, t_k)\}_{k=1}^K$), and probabilities, hazards, or updates are assigned per interval.

The piecewise-constant parameterization simplifies the solution of model equations—e.g., ODE integration, Kolmogorov equations, or likelihood computation—leading to closed-form or efficiently computable expressions for key quantities such as survival probabilities, transition kernels, or system responses (Holmer et al., 2024, Greydanus et al., 2021).

2. Survival Analysis: Discrete-Time and Piecewise-Constant Hazard Models

In survival analysis, the modeling of event times uses either discrete-time or continuous-time representations. Discrete-time survival models treat time-to-event $T$ as confined to intervals; the discrete hazard (failure probability conditional on survival up to an interval) is

$$p_k(x) = P(T \in [t_{k-1}, t_k) | T \geq t_{k-1}, x)$$

A neural parametrization of $p_k(x)$ often relies on sigmoid or softmax outputs for flexibility and identifiability.

The piecewise-constant hazard model (a piecewise exponential model) extends this by modeling the hazard rate as constant within each time interval but potentially varying between intervals: $$h(t|x) = \sum_{k=1}^K \lambda_k(x) \mathbf{1}_{[t_{k-1}, t_k)}(t)$$ Survival and density functions obtain closed-form representations. With neural parametrization (e.g., $h(t)$0), the approach generalizes both discrete-time and continuous-time models and, as the grid refines, the discrete-time likelihood converges to the piecewise-constant continuous likelihood.

Moreover, parameterizing either the hazard or the density function, and allowing for either constant or linear segments, yields a comprehensive family of four models on the chosen time grid—all of which support efficient likelihood-based training, even under censoring, and offer significant computational advantages over nonparametric ODE-based survival models (Holmer et al., 2024).

3. Control and System Identification: Piecewise-Constant Controls and Discrete-Time Approximations

In optimal control, especially for uncertain or multi-model systems, piecewise-constant (zero-order hold) controls are standard, enabling discretization of continuous-time plants and straightforward application of discrete-time Riccati-based optimization.

Given a plant

$h(t)$1

where $h(t)$2 is restricted to be constant on time intervals $h(t)$3, each interval's dynamics can be mapped to a discrete-time update via matrix exponentiation and convolution: $h(t)$4 This structure underpins min-max LQ (linear–quadratic) control for multi-model uncertainty, with the optimal policy arising through solving discrete Riccati equations and optimizing over the simplex of plant indices (Miranda et al., 2014).

In recursive system identification, piecewise-constant (zero-order-hold) models provide fast, accurate simulations under band-limited assumptions, with closed-form error bounds and efficient recursive implementations (Relan et al., 2018).

4. Piecewise-Constant Neural ODEs and Discrete-Time Deep Learning

Neural ODEs—models in which the dynamics of a latent state are parametrized by neural networks—benefit from piecewise-constant derivative approximations, which allow for exact integration via Euler steps: $h(t)$5 With an adaptive partition ($h(t)$6 selected to maintain error tolerance), such models allow step sizes to vary, reducing the number of required function evaluations per trajectory by 3–20× compared to RNNs or ODE-RNN architectures, while matching their predictive accuracy on sequential data tasks (Greydanus et al., 2021).

This approach ensures computational invariance under constant-derivative intervals, eliminating numerical integration error, and permits efficient training and inference with modern deep learning frameworks.

5. Hybrid and Piecewise-Constant Argument Differential Equations

Hybrid systems involving both continuous evolution and discrete jumps are naturally modeled using ODEs with piecewise-constant arguments (EPCAG/PCAD). On every interval $h(t)$7, the right-hand side is fixed (possibly depending on previous discrete-time states), and the solution patches together continuous and discrete-time updates: $h(t)$8 The solution is continuous over $h(t)$9, but inherits a discrete stepping map at $f(t)$0. Existence, uniqueness, and qualitative properties of such solutions reduce to properties of the induced discrete map (e.g., exponential dichotomy, almost-automorphicity). This ties the qualitative theory of ODEs to that of difference equations (Chavez et al., 2013, Fen et al., 11 Mar 2025).

Quasilinear systems with generalized piecewise-constant arguments further admit rigorous construction of homoclinic and heteroclinic solutions via Banach's fixed point theorem, with discrete-time maps driving the overall dynamics and lifting chaotic structures between discrete and continuous settings.

6. Stochastic Processes and Phase-Type Approximations with Discrete-Time Segmentation

Phase-type distributions—dense in the cone of positive measures on $f(t)$1—are classically realized through continuous-time Markov chains (CTMCs). For distributions with regions of zero density, or where efficiency and state-space size are a concern, a hybrid CTMC model with discrete-time deterministic jumps (d-CTMC) can construct piecewise segmentations: deterministic jumps partition the support, and within each interval, standard phase-type fitting applies.

This interval PH approach allows the approximation of arbitrary nonnegative densities using a moderate number of states and leverages discrete-timed transitions to optimize transient analysis, supporting dense modeling for non-Markovian distributions (Korenčiak et al., 2014).

7. Computational and Modeling Advantages

Piecewise-constant and discrete-time models provide significant computational and analytical benefits:

• Efficiency: Closed-form updates and likelihoods replace black-box ODE solvers or large matrix exponentiations.
• Flexibility: Piecewise-linear extensions capture non-constant rates/densities accurately without requiring very fine grids.
• Hybridization: These models form the backbone of hybrid systems theory, merging discrete and continuous modeling.
• Statistical Tractability: Likelihoods under right-censoring, complex hazard structures, and multi-plant uncertainty are straightforward to express and optimize.
• Scalability: Neural parametrization, as in neural survival or neural ODE models, integrates seamlessly with deep learning frameworks, supports backpropagation, and enables large-scale data-driven applications (Holmer et al., 2024, Greydanus et al., 2021).

The common structural paradigm of time segmentation and interval-wise simple parameterization unifies a wide spectrum of approaches—from classical recurrence-relation-driven models, through modern deep learning temporal models, to rigorous hybrid systems analysis.