Modeling and Prediction of Time-Varying Environmental Data Using Advanced Bayesian Methods

The problem of state/parameter estimation represents a key issue in crop models which are nonlinear, non-Gaussian and include a large number of parameters. The prediction errors are often important due to uncertainties in the equations, the input variables, and the parameters. The measurements needed to run the model (input data), to perform calibration and validation are sometimes not numerous or known with some uncertainty.

In these cases, estimating the state variables and/or parameters from easily obtained measurements can be extremely useful. In this work, we address the problem of modeling and prediction of leaf area index and soil moisture (LSM) using state estimation. The performances of various conventional and state-of-the-art state estimation techniques are compared when they are utilized to achieve this objective. These techniques include the extended Kalman filter (EKF), unscented Kalman filter (UKF), particle filter (PF), and the more recently developed technique variational Bayesian filter (VF).
The objective of this work is to extend the state and parameter estimation techniques (i.e., EKF, UKF, PF and VF) to better handle nonlinear and non-Gaussian processes without a priori state information, by utilizing a time-varying assumption of statistical parameters. In this case, the state vector to be estimated at any instant is assumed to follow a Gaussian model, where the expectation and the covariance matrix are both random. The randomness of the expectation and the covariance of the state/parameter vector are assumed here to further capture the uncertainty of the state distribution. One practical choice of these distributions can be a Gaussian distribution for the expectation and a multi-dimensional Wishart distribution for the covariance matrix. The assumption of random mean and random covariance of the state leads to a probability distribution covering a wide range of tail behaviors, which allows discrete jumps in the state variables.