On the Average Value of Correlated Time Series, with Applications in Dendroclimatology and Hydrometeorology

Abstract

In a number of areas of applied climatology, time series are either averaged to enhance a common underlying signal or combined to produce area averages. How well, then, does the average of a finite number (N) of time series represent the population average, and how well will a subset of series represent the N-series average? We have answered these questions by deriving formulas for 1) the correlation coefficient between the average of N time series and the average of n such series (where n is an arbitrary subset of N) and 2) the correlation between the N-series average and the population. We refer to these mean correlations as the subsample signal strength (SSS) and the expressed population signal (EPS). They may be expressed in terms of the mean inter-series correlation coefficient r? as SSS ≡ (R?n,N)2 ≈ n(1 + (N ? 1)r?)/ N(1 + (N ? 1)r?), EPS ≡ R?N)2 ≈ Nr?/1 + (N ? 1)r?.Similar formulas are given relating these mean correlations to the fractional common variance which arises as a parameter in analysis of variance. These results are applied to determine the increased uncertainty in a tree-ring chronology which results when the number of cores used to produce the chronology is reduced. Such uncertainty will accrue to any climate reconstruction equation that is calibrated using the most recent part of the chronology. The method presented can be used to define the useful length of tree-ring chronologies for climate reconstruction work. A second application considers the accuracy of area-average precipitation estimates derived from a limited network of raingage sites. The uncertainty is given in absolute terms as the standard error of estimate of the area-average expressed as a function of the number of gage sites and the mean inter-site correlation.