Abstract

There are many practical applications based on the Least Square Error (LSE) approximation. It is based on a square error minimization “on a vertical” axis. The LSE method is simple and easy also for analytical purposes. However, if data span is large over several magnitudes or non-linear LSE is used, severe numerical instability can be expected. The presented contribution describes a simple method for large span of data LSE computation. It is especially convenient if large span of data are to be processed, when the “standard” pseudoinverse matrix is ill conditioned. It is actually based on a LSE solution using orthogonal basis vectors instead of orthonormal basis vectors. The presented approach has been used for a linear regression as well as for approximation using radial basis functions.

Highlights

  • It should be noted that the Total Least Square Error (TLSE) was originally derived by Pearson [16](1901)

  • MATLAB, Fig.2, if, = + + is used for the Least Square Error (LSE). In this case of the LSE defined by Eq(1) the conditionality improvement is even higher, as

  • In the case of the least square approximation, we want to minimize using a polynomial of degree

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Summary

INTRODUCTION

This algorithm is quite complex and solution can be found in [18] It should be noted, that all methods above do have one significant drawback as values are taken in a squared value. In the vast majority the Least Square Error (LSE) methods measuring vertical distances are used. This approach is acceptable in the case of explicit functional dependences , = h, resp. The LSE methods are sensitive to a rotation as they measure vertical distances. It should be noted, that rotational and translation invariances are fundamental requirements especially in geometrically oriented applications. The overdetermined system can be difficult to solve

NUMERICAL STABILITY
PROJECTIVE NOTATION AND GEOMETRY ALGEBRA
MATRIX CONDITIONALITY
LEAST SQUARE METHOD WITH POLYNOMIALS
HILBERT’S MATRIX CONDITIONALITY
CONCLUSIONS
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