Impacts of Data Transformations on Least-Squares Solutions and Their Significance in Data Analysis and Evaluation

Abstract

In order to illustrate a potential problem which lies implicitly in the least-squares method, an anomaly known as Peelle's Pertinent Puzzle is introduced which appears in obtaining a least-squares average of two strongly-correlated data. This anomaly is then generalized as a change of least-squares solution when data and associated covariance matrix are transformed by non-linear functions. Reason of the change of the least-squares solution with respect to such data transformations is explained by the inconsistency in transforming the data covariance and sensitivity matrices. General criteria which can resolve this anomaly are derived. It is shown that if either one of these criteria is satisfied, the least-squares method gives the correct answer even if the data are discrepant, strongly correlated and the number of data points is small. Effects of data truncations are illustrated by a numerical example, which give an explanation on another aspect of Peelle's Puzzle. An approximate method is also proposed which should be applied when the correct method derived in this paper is not feasible.

KEYWORDS:

• least-squares method
• least square fit
• anomaly
• Peelle's Pertinent Puzzle
• nonlinear data transformation
• invariant solution
• covariance matrix
• sensitivity matrix
• truncation
• approximation