# Optimization based data enrichment using stochastic dynamical system models

## TL;DR

This paper introduces a new method for estimating system states using continuous time models and noisy measurements, offering a more general approach than traditional Kalman filters.

## Contribution

The novel contribution is a general framework for state estimation without assuming linear mappings or Gaussian noise.

## Key findings

- The optimal solution is interpreted as a continuous time spline influenced by system dynamics and noise distributions.
- The approach provides accurate data estimates at measurement times and continuous estimates in between.
- Monte Carlo simulations show significant performance improvements over existing methods.

## Abstract

We develop a general framework for state estimation in systems modeled with noise-polluted continuous time dynamics and discrete time noisy measurements. Our approach is based on maximum likelihood estimation and employs the calculus of variations to derive optimality conditions for continuous time functions. We make no prior assumptions on the form of the mapping from measurements to state-estimate or on the distributions of the noise terms, making the framework more general than Kalman filtering/smoothing where this mapping is assumed to be linear and the noises Gaussian. The optimal solution that arises is interpreted as a continuous time spline, the structure and temporal dependency of which is determined by the system dynamics and the distributions of the process and measurement noise. Similar to Kalman smoothing, the optimal spline yields increased data accuracy at instants when measurements are taken, in addition to providing continuous time estimates outside the measurement instances. We demonstrate the utility and generality of our approach via illustrative examples that render both linear and nonlinear data filters depending on the particular system. Application of the proposed approach to a Monte Carlo simulation exhibits significant performance improvement in comparison to a common existing method.