A bootstrap calculation of confidence regions for proportions of sediment contributed by different source areas in a ‘fingerprinting’ model

Abstract

AbstractWhen ‘fingerprinting’ is used to identify what proportions Ps (s = 1, …, g) of suspended sediment come from g different source areas, measures of the uncertainties in estimates of the Ps are also required. These uncertainties are influenced by two kinds of correlation whose effects are rarely recognized in the literature. These are (i) correlation between the estimated Ps because they must add to 1 and (ii) correlation between the geochemical tracers measured in sediment samples. This paper uses bootstrap procedures to identify joint confidence regions for the estimated proportions [responding to correlation of type (i)] and to explore alternatives to the ‘standard’ least‐squares criterion used to estimate the proportions when tracer measurements are correlated [correlation of type (ii)].Using a limited dataset with three sediment source areas for illustration (g = 3), results were obtained from 5000 bootstrap samples, using two criteria (standard and generalized least squares, GLS) with two inequality constraints: (a) 0 ≤ Ps ≤ 1, where Ps is the fraction of suspended sediment contributed by the s‐th source area (s = 1, 2, 3) and (b) 0 < Ps < 1, which, the paper argues, better represents reality. Approximate 95% confidence regions for the Ps, given by the two criteria and two inequality constraints, were compared. Using inequality constraint (a), the confidence region given by the GLS criterion was slightly smaller than that given by the standard; using constraint (b), the two confidence regions' boundaries were almost identical, suggesting that the effects of correlations between tracers were not large for the dataset used. For both criteria, the scatter amongst estimated proportions Ps obtained by bootstrapping was large, raising issues concerning the efficiency of sampling and the allocation of sampling effort, both in source areas and in transported suspended sediment. The results suggest that apparently small differences in the constraints applied to the proportions Ps can give quite different numerical results. Copyright © 2014 John Wiley & Sons, Ltd.