A geometric approach to the linear modelling

Authors:
B. Yagoubi
Affiliations:
Labo. Electromagnétisme et Optique Guidée, Département d'Électronique, Faculté des Sciences et Sciences de l'Ingénieur, Université de Mostaganem, Algeria
Venue:
CSS'11 Proceedings of the 5th WSEAS international conference on Circuits, systems and signals
Year:
2011

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MMES'11/DEEE'11/COMATIA'11 Proceedings of the 2nd international conference on Mathematical Models for Engineering Science, and proceedings of the 2nd international conference on Development, Energy, Environment, Economics, and proceedings of the 2nd international conference on Communication and Management in Technological Innovation and Academic Globalization

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Abstract

Since the real model output is a physical reaction to any excitation (input), it should not be modified by the way of measuring or observing it. This is an intrinsic behaviour which allows us to consider the inputs as a base of a relative geometric space of observation. Any physical component that is not observed, such as the measurement or the modelling error, is orthogonal to the observation space, and any completely observed component, is entirely within this space. In this geometric approach, we consider a linear modelling as breaking the real model output in the input linear geometric space. To determine the model output contra variant components which represent conventional linear model coefficients, we use the co and contra variant components transformation relation by means of the input space metric tensor. We will apply this, instead of the least mean square (LMS) and the Yule-Walker equations, to estimate the autoregressive model coefficients, by breaking its output (the predicted vector) in the input past values space. Furthermore, using the intrinsic property mentioned above, we have broken the predicted vector in a space with one more orthogonal dimension to the input space in order to be able to estimate the autoregressive prediction error variance.