32 



NATURE 



[September 9, 1915 



LETTERS TO THE EDITOR. 



[The Editor does not hold himself responsible for 

 opinions expressed by his correspondents. Neither 

 can he undertake to return, or to correspond with 

 the writers of, rejected manuscripts intended for 

 this or any other part of Nature. No notice is 

 taken of anonymous communications.] 



The Probable Error of the Amplitudes in a Fourier Series 

 obtained from a Given Set of Observations. 



Mr. Dines's question (Nature, August 12, 

 p. 644) as to the degree of significance to 

 be attached to the amplitudes of waves of given 

 period found in short series of observations and the 

 Ukelihood of their permanence as the series is extended 

 brings home again the reflection that whilst probable 

 errors are easily found where the experience is large 

 and they are least wanted, their determination from a 

 limited experience where they would be most useful 

 is largely hypothetical. 



One of the principal sources of error springs from 

 the process of selection. For example, we may find 

 by counting that twelve persons out of twenty in an 

 omnibus, picking up people at random, are males ; 

 but to predict from this the probable constitutions in 

 like respect of other omnibuses, or to say that the 

 percentage of males in London has been determined 

 as sixty, with a probable error of 7-5, is clearly a mis- 

 use of statistical theory. The result is fallacious, 

 because of the false premises. Omnibuses do not pick 

 up people at random, or even -travelling people at 

 random. Persons go about in small groups, and there 

 are larger groups going to the shopping centres and 

 going citywards, and still larger groups travelling at 

 the extreme hours and travelling at the mid hours 

 of the day, and so on. The constitutions of the groups 

 are different, and the probable error of the percentage 

 is much in excess of that calculated on the basis of 

 random sampling, this being due to a process which, 

 applied to phenomena in general, may be called a 

 tendency to cluster. 



In all cases where a space or time content is taken 

 as the sample it is necessary, owing to unknown 

 clustering, to repeat the samples many times over, 

 before a just estimate of the statistical constants and 

 of their probable errors can be obtained. 



Applied to meteorological and such-like phenomena, 

 in which some value fluctuates in time, the successive 

 observations are seldom at such a distance as to be 

 haphazard, and the clustering, in this instance, is of 

 the nature of waves similar to the waves which are 

 the subject of observation. In these circumstances 

 it does not seem possible, from observations of a 

 single period, to make any estimate of the significance 

 of an observed amplitude. This can only be done by 

 repeating the period and noting the fluctuations in 

 phase and amplitude found in such repetitions. If ob- 

 servations for many periods are to hand, then it is 

 only following a well-understood practice to divide the 

 material up into groups and calculate the constants 

 of each group. From the fluctuations found in the 

 several groups an empirical gauge of error may be 

 constructed, upon which may be based a measure of 

 error, suitable for application to the constant found 

 for the whole. 



If the phenomenon were one fully observed, and a 

 frequency distribution of wave period and amplitude 

 had been calculated by one of the several methods 

 that have been proposed, then it would not be difficult 

 to obtain the probability of error of any sustained 

 wave found in the observations during a limited 

 period of time. But such- a frequency distribution is 

 seldom a subject of research, although it is the first 



NO. 2303, VOL. 96] 



step in a description of the phenomenon, and one that 

 cannot fail, by giving the salient periods, to mark out 

 regions of investigation relative to the causes of un- 

 dulation. 



The more limited statement of the problem, as 

 enunciated by Mr. Dines, renders it amenable to 

 algebraical expression in the following manner. 



A variable quantity has a periodic wave of amplitude 

 a, and is subject, in addition, to casual fluctuations 

 the mean of which is zero, and standard deviation o-. 

 It is easily shown that if p and q are the amplitudes 

 of the waves, phasal with, and in quadrature with, 

 the above wave, calculated from the casual variations 

 alone, the mean values of * and q are zero, and their 

 standard deviations are crva/v^n. Supposing that the 

 distributions of p and q are normal, what will be the 

 errors of the calculated amplitudes, and phase angles, 

 due to these fluctuations? 



Draw OA = a, AC = p, CA' at right angles = 5. 



Then, clearly, a' = OA' is the calculated amplitude 

 due to p and q, superimposed upon a, and 6 = AOA' is 

 the phase displacement. 



Now A' has varying positions, due to the variations 

 of p and q, and Its frequency upon the element of area, 

 dp, dq, about A', is that of a normal distribution, 

 viz. : — 



V{2n)s ^-^ 



k-A^ dg 



2TrS^ 



where ^ is written ior (t\^2J i/n, the s.d. of p or q. 



Putting />^ + g"^ = AA'2 = a'^ - 2a'a cos 6 + a'\ and 

 dpdq = a'da'de, the variables are changed from p, q to 

 a', 6, and the frequency distribution of a', 9 -'' "' ^'""" 



therefore 



// 



J_^-i<«'2-,z'« COS e+a2)/.2 ^'^^,'^^_ 

 2irs^ , 



When this is integrated with respect to 6 the result 

 will be the frequency distribution of amplitudes a'. 

 It appears to be a problem of the nature of random 

 migration, and the solution of the above integral will 

 probably be found (I am unable to consult my refer- 

 enced in Prof. Karl Pearson's memoir upon this sub- 

 ject. 



If there is no initial wave, so that the calculated 

 amplitudes are all due .to casual fluctuations, then 

 a = o and the frequency distribution of a' becomes :— 



/^.-r'"-'.w. 



This is a well-known distribution, being, for_ instance, 

 the distribution of arithmetical velocity of wind when 

 the NS and EW components vary independently, in 

 normal manner, with s.d. equal to s. 



The mean is -/(•|ff).5 = 1-2535, the standard deviation 

 is \/(2 — |n-). 5 = 0-6555, and the number per cent, ex- 

 ceeding xs is Iooxe-i•^^ 



Applying these formulae to Mr. Dines's experimental 

 data, the comparison of experiment with theory comes 

 out as follows : — 



