Sequential Maximal Updated Density Parameter Estimation for Dynamical Systems with Parameter Drift

TL;DR

The paper addresses sequential parameter estimation for dynamical systems subject to parameter drift by embedding the problem in a data-consistent (DC) framework and using Maximal Updated Density (MUD) estimates to quantify epistemic uncertainty. It couples push-forward/pullback measure theory with data-derived QoI maps learned via PCA, estimating the predicted density $\pi_{pred}$ with kernel-density methods and updating to $\pi_{update}$ through $\pi_{update}(\lambda) = \pi_{init}(\lambda)\frac{\pi_{obs}(Q(\lambda))}{\pi_{pred}(Q(\lambda))}$, thereby achieving selective regularization in data-informed directions. The sequential scheme relies on three diagnostics—solution validity, weight collapse, and change-point identification (CPI)—to ensure reliability and detect drift, and it is demonstrated on three problems: wind-drag parameter estimation in ADCIRC storm surge modeling, a KL-expanded thermal diffusivity field in a heat-conduction problem, and change-point detection in SEIRS epidemiological dynamics. The results show that sequential updates can match or improve non-sequential analyses while reducing uncertainty as more data are assimilated, and they enable drift detection in near-real-time settings with data arriving in packets. The work provides practical, open-source tooling and a framework with clear pathways to theory, optimal experimental design connections, and comparisons with data assimilation approaches in complex, time-varying systems.

Abstract

We present a novel method for generating sequential parameter estimates and quantifying epistemic uncertainty in dynamical systems within a data-consistent (DC) framework. The DC framework differs from traditional Bayesian approaches due to the incorporation of the push-forward of an initial density, which performs selective regularization in parameter directions not informed by the data in the resulting updated density. This extends a previous study that included the linear Gaussian theory within the DC framework and introduced the maximal updated density (MUD) estimate as an alternative to both least squares and maximum a posterior (MAP) estimates. In this work, we introduce algorithms for operational settings of MUD estimation in real or near-real time where spatio-temporal datasets arrive in packets to provide updated estimates of parameters and identify potential parameter drift. Computational diagnostics within the DC framework prove critical for evaluating (1) the quality of the DC update and MUD estimate and (2) the detection of parameter value drift. The algorithms are applied to estimate (1) wind drag parameters in a high-fidelity storm surge model, (2) thermal diffusivity field for a heat conductivity problem, and (3) changing infection and incubation rates of an epidemiological model.

Figures (9)

• Figure 1: Time Evolution of Water Elevation (Left Axis): True state (black line), Observed state (blue dots), and a sample of 100 Predicted States (red). Vertical lines mark the 12-hour intervals used for transmission times, while the parameter with the highest sensitivity, average wind speed, is located at the top banner. Notably, three time windows exhibit high wind speeds, with the second window featuring lower speeds. This aligns with the most favorable update in the $\lambda_1$ (wind-drag slope) direction due to reduced cut-off parameter dynamics.
• Figure 2: Parameter sample scatter plots, colored by the learned $Q_{\mathrm{PCA}}$ over the 1st (left) and last (right) iterations of data. Note the difference in implied contour structures across iterations, which illustrates how the sequential estimation allows updating of parameter estimates to occur in distinct directions informed by each time window of data.
• Figure 3: ADCIRC Wind Drag Parameter Estimation Results: Initial (black, non-filled in) and updated density plots (purple/green filled in densities) along with true parameter values $\lambda^\dagger$ (vertical black dotted-dash lines) and $\lambda^\mathrm{MUD}$ estimates (vertical orange lines) for the $\lambda_1$ slope parameter (left) and $\lambda_2$ cut-off parameter (right) using three different approaches: (top) Non-iterative, 2 principle component QoI map, (middle) non-iterative 1 principle component QoI map, and (bottom) 12-hour iterative 1 principle component QoI map. Note how the iterative, 1 component map performs similarly well as the non-iterative 2-component map.
• Figure 4: True thermal diffusivity field using 10 KL modes (top) along with estimate fields according to the $\lambda^\mathrm{MUD}$ parameter estimate (left) and the error between the true and approximate fields (right) on iterations 1, 3, and 6 ($t=0.5, 1.5, 3.0$).
• Figure 5: The SEIRS model with demography. Rates are $\lambda_{1}$ (contact), $\lambda_{2}$ (latency), $\lambda_{3}$ (recovery), $\lambda_{4}$ (loss of immunity). The relationship between $R_0 = \lambda_1 / \lambda_3,$ known as the basic reproduction number determines the periodicity of the model, with periodic behavior when $R_0 > 1.0$

Theorems & Definitions (1)

• Definition 1: Predictability Assumption