Fitting straight lines to poor quality ( χ, y) data

Abstract

Estimating model parameters from poor quality data is an unfortunately common problem in practice. Point estimation procedures like least squares will yield numbers, but their utility as parameter estimates may be highly questionable. If worst-case error limits can be obtained, estimated, or crudely guessed for the available data, set-theoretic (“unknown-but-bounded”) parameter estimation strategies can be used to estimate explicit model parameter bounds. To illustrate the quality of inferences that may be drawn from very poor data (e.g., uncertainties of 100% or more), explicit geometric solutions are developed for the set-theoretic estimation of slope and intercept parameters for linear models of ( χ, y) data with arbitrarily large uncertainties in either χ or y or both. This problem is first solved for a single pair of uncertain data points to obtain bounding sets for the slope and intercept parameters of all lines passing through these two regions. These results may be extended to an arbitrary collection of N data points by intersecting the two-point bounding sets obtained for all possible point pairs. In practice, the O( N 2) complexity of this approach is frequently impractical, so intersections over smaller sets of pairings must be used. This paper addresses these issues, presenting the details necessary to obtain worst-case parameter estimates from arbitrarily uncertain data for the problem of fitting ( χ, y) measurements to straight lines. One of the interesting results is that parameter estimate sets obtained from highly uncertain data may not be connected, extending the observation of Gay that the solution sets for interval linear equations may be nonconvex.