A Glance over Parameter Estimation in General

Parameter estimation is a discipline that provides tools for the efficient use of data for aiding in mathematically modeling of phenomena and the estimation of constants appearing in these models [2]. It can thus be visualized as a study of inverse problems. Much of parameter estimation can be related to four optimization problems:

criterion:

the choice of the best function to optimize (minimize or maximize)

estimation:

the optimization of the chosen function

design:

optimal design to obtain the best parameter estimates

modeling:

the determination of the mathematical model which best describes the system from which data are measured.

In this paper we are mainly concerned with the first three problems, and we assume the model is known (a conic in the examples).

Let be the (state/parameter) vector containing the parameters to be estimated. The dimension of , say m, is the number of parameters to be estimated. Let be the (measurement) vector which is the output of the system to be modeled. The system is described by a vector function which relates to such that

In practice, observed measurements are only available for the system output corrupted with noise , i.e.,

We usually make a number of measurements for the system, say ( ), and we want to estimate using . As the data are noisy, the function is not valid anymore. In this case, we write down a function

which is to be optimized (without loss of generality, we will minimize the function). This function is usually called the cost function or the objective function.

If there are no constraints on and the function has first and second partial derivatives everywhere, necessary conditions for a minimum are

and

By the last, we mean that the -matrix is positive definite.