October 14, 1922] 



NA TURE 



5ii 



Letters to the Editor. 



I The Editor does not hold himself responsible for 

 opinions expressed by his correspondents. Neither 

 can lie 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^ 



Periodicities. 



The recent paper by Sir William Beveridge on 

 " Wheat Prices and Rainfall" (Journal of the Royal 

 Statistical Society, vol. 85, pp. 412-478, 1922) raises 

 a rather important question of principle which is 

 involved not only in discussions over the existence 

 of periodicities, but also over relationships between 

 different variables. 



Before Schuster's papers on the periodogram it was 

 customary for a period to be accepted as real provided 

 that it had an amplitude comparable with that of the 

 original figures under analysis ; and he revolutionised 

 the treatment of the subject by showing that if the 

 squares of the intensities of the various periodic terms 

 are plotted in a periodogram, and if the data are those 

 of an entirely chance distribution, then the average 

 value of an ordinate being a, the probability that a 

 particular ordinate will equal or exceed ka is e' k . Sir 

 William Beveridge is accordingly perfectly justified 

 in taking Schuster's sunspot period of 11-125 years, 

 or Bruckner's 34-8 year period, and deciding that 

 these periods probably occur in his wheat prices if 

 the corresponding intensities are three or four times 

 the average. But he, like many other investigators, 

 goes a stage further, and after picking out the largest 

 from a large number of intensities he applies the same 

 criterion as if no selection had occurred. It is, how- 

 ever, clear that if we have a hundred intensities the 

 average of which, a, is derived from a number of random 

 figures, then the probable value of the largest of these 

 chance intensities will not be a but will be considerably 

 greater, and it is only when the largest amplitude 

 actually derived materially exceeds the theoretical 

 chance value thus obtained that reality can be in- 

 ferred. 



Taking the periodicities of wheat prices on pp. 

 457-459 between 5 years and 40 years, 1 I estimate 

 that the "width of a line" ranges from o-i year for 

 a 5 years' period, through 0-5 at 12 years to 4 years at 

 33 vears ; and accordingly that the number of inde- 

 pendent periods between 5 years and 40 is in this 

 case about 51. The value of a, the average intensity, 

 being 5-898, it is easily seen that the chanqe of 

 all the 51 random intensities being less than 3a is 

 (1 -e -3 ) 51 , or 0-074, so tnat the- chance of at least one 

 intensity greater than 3a is 0-926, not e' z or 0-050, as 

 is habitually assumed. Instead of the chance of an 

 occurrence of 3a " making a prima facie case for 

 enquiry " (p. 424), the odds are 12 to 1 in favour of 

 its production by mere chance. The chance of at 

 least two intensities above 30 is 0-728, of three it is 

 0-470, of four 0-248, of five 0-109, of six 0-0403, of 

 seven 0-0127, of nine 0-00085, an( i °f eleven 0-00003. 



Thus it is not until six intensities over 3a are found 

 that the chance of production by pure luck is less 

 than 1 in 20. It is also easily found that if the chance 

 of all the 51 intensities being less than na is to be 

 19/20, 11 is 6-9 ; i.e. the greatest intensity for wheat 

 price fluctuations must be 41, not 18, before the 

 probability of its being due to luck is reduced to 1 '20 ; 



1 Sir William Beveridge points out on pp. 423-424 that amplitudes for 

 pel i ids of less than 5 years are inevitably diminished, while those above 31 

 are diminished by the process employed for eliminating secular trend : I 

 calculate that the intensity at 35 years should be multiplied by (0-87)-= or 1 -3, 

 and that at 54 by 3-8. 



NO. 2763, VOL. I IO] 



andif the likelihood is to be i/ioo we must have n =8-5, 

 the corresponding wheat-price intensity being 50. 

 Of intensities greater than 4 1 Sir William Beveridge 

 found four, and greater than 50 only two. 



At first sight it might seem that the agreement 

 between Sir William Beveridge's forecasted synthesis 

 rainfall curve and the actual rainfall was too great 

 to be explained by a few harmonic terms ; but the 

 correlation co-efficient of 0-38 (see p. 475) indicates 

 that while 0-38 of the rainfall variations are accounted 

 for, only (o^S) 2 , or about a seventh, of the independ- 

 ent factors which control these variations have been 

 ascertained. 



As pointed out in a paper " On the Criterion for 

 the Reality of Relationships or Periodicities," in the 

 Indian Meteorological Memoirs (vol. 21, No. 9, 1914), 

 the same principle is valid when discussing relation- 

 ships. If we are examining the effect of rainfall on 

 temperature and ascertain that the correlation co- 

 efficient between the rainfall and temperature of the 

 same month in a particular English county is four 

 times the probable error, we may infer that the effect 

 is highly probable. But if we work out the co- 

 efficients of that temperature with a hundred factors 

 taken at random, e.g. with the monthly rainfall of 

 Tashkend 5-8 years previously, and pick out the 

 largest co-efficient, it would be wrong to compare it 

 with the average co-efficient produced by mere 

 chance ; as shown .in the paper referred to, the 

 probable value of the largest of 100 co-efficients is 

 4-01 times as great as the probable value of one taken 

 at random. Gilbert T. Walker. 



Meteorological Office, Simla, August 24. 



Dr. Walker's note contains, I think, a valid and 

 valuable criticism of the procedure commonly 

 adopted hitherto in comparing individual intensities 

 with the average intensity in harmonic analysis. It 

 would lead me now to modify in several ways my 

 general discussion of the " test of intensity " (pp. 

 422-424 of my paper in the Journal of the Royal 

 Statistical Society. I was particularly careful, how- 

 ever, in that paper to avoid laying stress on intensity 

 as such. The net result of Dr. Walker's calculations 

 is not to weaken but to confirm my main thesis : 

 that a number of real periodicities exist in European 

 wheat prices from 1550 to 1850. 



According to these calculations, the chance of my 

 getting by pure luck between five and fortv years 

 one intensity as great as 3a is 0-926, but the chance 

 of my getting seven such intensities is 0-0127, and 

 that of getting eleven is 0-00003. Actually I have, 

 between five and forty years, fifteen intensities above 

 30 (=17-69); the odds are therefore 80 to 1 that at 

 least nine of these intensities, and 33,000 to 1 that at 

 least five of them, are not due to luck. Obviously 

 every such intensity does, in the circumstances, 

 present a prima facie case for further inquiry, the 

 object of the inquiry being to determine which of 

 the 15 intensities have the strongest probabilities of 

 being due to real periods. 



In that inquiry the actual height of the intensity 

 in any case (the " test of intensity ") is only one and 

 not necessarily the most important point for con- 

 sideration. As Dr. Walker shows, an intensity in my 

 periodogram of nearly seven times the average might 

 well be due to pure luck (the odds being only 20 to 

 1 against it). On the other hand, a much lower 

 intensity might represent a true and perfectly 

 regular but weak periodicity, just as a quite small 

 correlation co-efficient may prove a real though weak 

 connexion, if the number of cases compared is very 

 large. Indication of the same period in each half of 

 a sequence when analysed separately (the " test of 



