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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:

tex2html_wrap_inline2519 criterion:
the choice of the best function to optimize (minimize or maximize)
tex2html_wrap_inline2519 estimation:
the optimization of the chosen function
tex2html_wrap_inline2519 design:
optimal design to obtain the best parameter estimates
tex2html_wrap_inline2519 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 tex2html_wrap_inline2527 be the (state/parameter) vector containing the parameters to be estimated. The dimension of tex2html_wrap_inline2527 , say m, is the number of parameters to be estimated. Let tex2html_wrap_inline2533 be the (measurement) vector which is the output of the system to be modeled. The system is described by a vector function tex2html_wrap_inline2535 which relates tex2html_wrap_inline2533 to tex2html_wrap_inline2527 such that


In practice, observed measurements tex2html_wrap_inline2541 are only available for the system output tex2html_wrap_inline2533 corrupted with noise tex2html_wrap_inline2545 , i.e.,


We usually make a number of measurements for the system, say tex2html_wrap_inline2547 ( tex2html_wrap_inline2549 ), and we want to estimate tex2html_wrap_inline2527 using tex2html_wrap_inline2547 . As the data are noisy, the function tex2html_wrap_inline2555 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 tex2html_wrap_inline2527 and the function tex2html_wrap_inline2559 has first and second partial derivatives everywhere, necessary conditions for a minimum are




By the last, we mean that the tex2html_wrap_inline2561 -matrix is positive definite.

next up previous contents
Next: Conic Fitting Problem Up: Parameter Estimation Techniques: A Previous: Introduction

Zhengyou Zhang
Thu Feb 8 11:42:20 MET 1996