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

• Wigley T
• Briffa K
• Jones P
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#### Abstract

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 r̄ as SSS ≡ (R̄n,N)2 ≈ n(1 + (N − 1)r̄)/ N(1 + (N − 1)r̄), EPS ≡ R̄N)2 ≈ Nr̄/1 + (N − 1)r̄.Similar formulas are given relating these mean correlations to the fractional common variance which arises as a parameter in a...

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