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# 5 Spurious periodicities

The period with the smallest is not always "real'', because "spurious'' periods are introduced by the interplay between distribution and real periodicity. The phases of folded with an arbitrary period are circular data. This distribution can be studied with numerous methods for analysing circular data (e.g. Batshelet 1981; Jetsu & Pelt 1996). We apply the Deeming (1975) window function

 (15)

where the n complex vectors are more parallel for a significant window period . If varies with P1, then P0 induces an important set of spurious periods

 (16)

where , and . This relation (Tanner 1948) states that ti separated by k2P0 have an integer phase difference k1 with the periods P1 and P'. These do not significantly alter the sequence in . However, produce a "mirror image'' by reversing the sequence.

But Eq. (16) cannot separate P1 from . The following modification of another method by Tanner (1948) may reveal, if a window P0 induces a spurious period P1: (1) Derive . Transform to . (2) Model the with P1, and measure the "phase residuals'' ( ), i.e. the minimum distance for each yi from the model . These are unique, when the distance between data and model is defined as , where and . Transform to . The d1 and d2 are comparable, because d2 is independent of the yi units. The d minimum determines the point closest to uniquely, which yields the phase residuals . Transform to . (3) The linear correlation coefficient ( ) between and is higher for "spurious'' than "real'' periods. Note that we subtract the means from and (Figs. 1q-u, 3q-u, 5q-u and 7q-u), since this does not influence . Our "correlation hypothesis'' is

HR: The and are uncorrelated random samples.
Under HR, the value of is the probability (i.e. the critical level) that reaches , or an even larger, value.

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