Orthogonal and Spectral Decomposition Algorithms for the Optimal Estimation of Parameters from Indirect Measurements

Abstract

The optimal estimation of the parameters of a physical system from a set of measurements is a task commonly performed in numerical analysis of data in engineering and all applied quantitative fields. In a statistical sense, experimental measurements are usually considered as a mixture of both useful signals and unwanted noise and so, often, the aim of measurement data processing is to separate these components of a measurement in an optimal way. In solving these problems, researchers often adopt the method of direct formation and inversion of normal equations for the solution of a least squares problem. This paper however presents the method of orthogonal decomposition as an easier, simpler and equally valid alternative approach to direct formation and inversion of normal equations. The basic principles of L2-norm (a vector norm defined for a complex vector) optimization and its geometric structure are discussed and implemented through the techniques of orthogonal and spectral decompositions. A simple regression problem was used to demonstrate these algorithms from which conclusions are made that: the combined use of these decomposition algorithms produces both the parameter values and their variances simultaneously, making it possible to solve an optimization problem without an express formation of normal equations.