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摘要
We consider the %26apos;multifrequency%26apos; periodogram, in which the putative signal is modelled as a sum of two or more sinusoidal harmonics with independent frequencies. It is useful in cases when the data may contain several periodic components, especially when their interaction with each other and with the data sampling patterns might produce misleading results. %26lt;br%26gt;Although the multifrequency statistic itself was constructed earlier, for example by G. Foster in his CLEANest algorithm, its probabilistic properties (the detection significance levels) are still poorly known and much of what is deemed known is not rigorous. These detection levels are nonetheless important for data analysis. We argue that to prove the simultaneous existence of all n components revealed in a multiperiodic variation, it is mandatory to apply at least 2(n) - 1 significance tests, among which most involve various multifrequency statistics, and only n tests are single-frequency ones. %26lt;br%26gt;The main result of this paper is an analytic estimation of the statistical significance of the frequency tuples that the multifrequency periodogram can reveal. Using the theory of extreme values of random fields (the generalized Rice method), we find a useful approximation to the relevant false alarm probability. For the double-frequency periodogram, this approximation is given by the elementary formula (pi/16)W(2)e(-z)z(2), where W denotes the normalized width of the settled frequency range, and z is the observed periodogram maximum. We carried out intensive Monte Carlo simulations to show that the practical quality of this approximation is satisfactory. A similar analytic expression for the general multifrequency periodogram is also given, although with less numerical verification.
- 出版日期2013-11
