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On the Whittle-Matérn correlation family Peter Guttorp Tilmann Gneiting NRCSE T e c h n i c a l R e p o r t S e r i e s NRCSE-TRS No. 080 July 11, 2005 The NRCSE was established in 1996 at the University of Washington. Center activities include workshops and other research activities in areas of statistical methodology related to scientific problems in the environmental and ecological sciences. On the Whittle-Matérn correlation family Peter Guttorp and Tilmann Gneiting University of Washington Abstract: Handcock and Stein (1993) introduced the Matérn family of spatial correlations into statistics as a flexible parametric class with one parameter determining the smoothness of the paths of the underlying spatial field. In this note we describe the history of this family, and document its relationship to the Hankel transform. We argue that an appropriate name for this family is the Whittle-Matérn family. Key words: Hankel transform, spatial covariance, Bessel function. d 1. Stationary processes in R The idea of a wide sense stationary stochastic process was introduced by Khintchine (1934), who showed that the correlation function of a continuous such process could be written as a cosine transform ? r(s) = Corr(X (t ),X (t + s)) = ú cos(xs)dF(x ) -? of a cdf F . Cramér (1940) generalized the result to a d-dimensional complex-valued process, assuming continuity at the origin of the correlation function, obtaining