Overview

In Chapter 4, the maximum likelihood estimator was developed for the Fisher model, formulated as

where ? is a vector of unknown constant parameters and v is (0, R).

It was shown that for a linear measurement equation z= H ?+ v, the maximum likelihood estimator reduces to a least-squares estimator with weighting equal to the inverse of the noise covariance matrix. This chapter starts with the development of the maximum likelihood estimator for a stochastic dynamic system described by differential equations with process noise. In this case, the measurements are a nonlinear function of the parameters, as shown in Chapter 4. The general form of the relevant measurement equation is therefore z= h( ?)+ v.

In general, model parameter estimates are found by maximizing a likelihood function, which involves minimizing the weighted least-squares difference between measured outputs and model outputs. The solution combines a state estimator represented by a Kalman filter and a nonlinear parameter estimator. The state estimator is necessary because the presence of process noise in the dynamic equations means that the states are random variables. A nonlinear parameter estimator is required because of the nonlinear connection between model parameters and model outputs mentioned above. Deterministic inputs are still assumed to be measured without error, as in Chapter 5. Because the maximum likelihood estimator includes a Kalman filter to estimate the states, and the outputs are computed from the resulting state estimates, this algorithm for parameter...