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Use of cross correlation between hydrological time series to improve estimates of lag one autoregressive parameters
Author(s) -
Frost Jean,
Clarke R. T.
Publication year - 1973
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr009i004p00906
Subject(s) - series (stratigraphy) , autoregressive model , mathematics , statistics , correlation , lag , autocorrelation , time series , converse , geometry , paleontology , computer network , computer science , biology
The problem considered requires estimation of parameters characterizing a serially correlated hydrologic time series { y t }, where data are also available from a longer time series { x t }, itself serially correlated and cross correlated with { y t }. On the assumption that both series are lag one autoregressions (when { y t } is characterized by three parameters μ y , β, and σ ϵ 2 ), large‐sample variances for the autoregressive parameter β are derived from the short series { y t } alone or from both series { x t } and { y t }, and the relative information is investigated numerically. It is concluded that, when the two serial correlations are approximately equal and the length of { x t } is twice that of { y t }, the gain in precision of the estimate β is about 4% when the cross correlation is 0.2, about 15% when the cross correlation is 0.4, about 31% when the cross correlation is 0.6, and about 49% when the cross correlation is 0.8. The equations giving maximum likelihood (ML) estimates are examined, and a relatively simple numerical technique is developed for one particular case of some practical importance. Small‐sample properties of both ML estimates and an estimate derived from work by Matalas are investigated, and tentative conclusions are that (1) the modified Matalas estimates have smaller mean square error than ML estimates, and (2) ML estimates of μ y and β derived from both series have smaller mean square error than ML estimates of μ y and β derived from the single series only, the converse being true for σ ϵ 2 .