VII. On a method of investigating periodicities disturbed series, with special reference to Wolfer's sunspot numbers

Abstract

If we take a curve representing a simple harmonic function of the time, and superpose on the ordinates small random errors, the only effect is to make the graph somewhat irregular, leaving the suggestion of periodicity still quite clear to the eye. Fig. 1 (a) shows such a curve, the random errors having been determined by the throws of dice. If the errors are increased in magnitude, as in fig. 1 (b), the graph becomes more irregular, the suggestion of periodicity more obscure, and we have only sufficiently to increase the “errors” to mask completely any appearance of periodicity. But, however large the errors, periodogram analysis is applicable to such a curve, and, given a sufficient number of periods, should yield a close approximation to the period and amplitude of the underlying harmonic function. When periodogram analysis is applied to data respecting any physical phenomenon in the expectation of eliciting one or more true periodicities, there is usually, as it seems to me, a tendency to start from the initial hypothesis that the periodicity or periodicities are masked solely by such more or less random superposed fluctuations— fluctuations which do not in any way disturb the steady course of the underlying periodic function or functions. It is true that the periodogram itself will indicate the truth or otherwise of the hypothesis made, but there seems no reason for assuming it to be the hypothesis most likely a priori.