🤖 AI Summary
This work addresses the challenge of jointly inferring multi-level functional states—such as curves, derivatives, and integrals—in functional data modeling, where existing methods struggle to account for derivative uncertainty, cross-level covariance, and identifiability of integration constants. The authors propose an anchored Gaussian process differential ensemble framework that explicitly models integration constants by embedding anchor points together with their mean-square derivatives and repeated integrals into a joint Gaussian state, enabling efficient computation via transformed Hilbert spaces. A key innovation is the separation of anchor-induced covariance from boundary uncertainty, revealing that integration constants cannot be uniquely identified from anchors alone. To enhance derivative recovery accuracy, the method introduces the TARTARE calibration strategy. Theoretical analysis combines Laplace–Dirichlet basis functions, finite-rank approximations, and operator-level approximation bounds. Experiments demonstrate substantially improved posterior derivative estimation in second-order simulations while preserving accuracy in anchors and integrals, and a motorcycle crash case study confirms coherent inference of coupled kinematic states and functional turning points.
📝 Abstract
Functional data are often modeled through one likelihood-linked curve, while the scientific target is a larger state containing rates, accumulated quantities, boundary values, or nonlinear functionals of several linked levels. These targets require more than smoothing the observed curve: derivative uncertainty, cross-level covariance, and integration constants must be handled jointly. We introduce anchored Gaussian process differential ensembles, embedding an anchor \(f_0\) in a joint Gaussian state with its mean-square derivatives and repeated integrals. Integral levels add explicit Gaussian integration constants. This separates the anchor-induced covariance from finite-dimensional boundary uncertainty and clarifies why anchor-only observations do not identify independent integration constants. For stationary one-dimensional kernels, we compute the ensemble with a transformed Hilbert space Gaussian process approximation that applies derivative and integral operators to Laplacian--Dirichlet basis functions while retaining the integration-constant covariance exactly. We establish operator-level approximation bounds and conditional finite-grid posterior convergence. We introduce TARTARE, a target-aware calibration procedure for finite-rank differential ensemble approximations, to address derivative under-resolution by anchor-calibrated bases. In second-order simulations, derivative-aware calibration improves derivative posterior recovery relative to anchor-only calibration while preserving anchor and integral summaries. A motorcycle crash analysis illustrates coherent posterior inference on a coupled kinematic state and short-horizon turning-point functionals.