By Terence C. Mills (auth.)

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**Extra resources for A Very British Affair: Six Britons and the Development of Time Series Analysis During the 20th Century**

**Example text**

3, . . 4], and quite slightly to decrease the maximum correlation attainable; it is not until the sampleinterval becomes as large as half a period, or thereabouts, that the contours of the curve round off and the maximum undergoes a rather sudden drop. , page 8) Yule then used these curves to construct the frequency distribution of the correlation coefficient for a given value of 2h/n. 6 and led Yule to conclude that the3 answer to our question, how the distribution of isolated frequencies at +1 and −1 closes up to the distribution of an isolated clump of frequency at zero, is then that the distribution first of all becomes a U-shaped distribution, with limits not far from +1 and −1, and that these limits, at first gradually and then more rapidly, close in on zero; but the distribution always remains U-shaped, and values of the correlation as far as possible removed from the true value (zero) always remain the most frequent.

Again, the serial correlations differ considerably from series to series. 12 Serial correlations up to r(10) for three experimental series (of 100 terms) with positively correlated (conjunct) differences Yule: Nonsense Correlations 39 Finally, Yule briefly considered the case when the second differences of a series were random, so that the series is the ‘second sum’ of a random series, that is, t t st = uj , j=1 xt = sj = tut + (t − 1)ut−1 + · · · + u1 , t = 1, . . , T j=1 In this case the first differences of xt are the sum of a random series and therefore the serial correlations of xt are given by 2 1 rx (k + 1) = 0, or 1 rx (k) = k1 rx (1) − (k − 1) = 1 − k(1 − 1 rx (1)) = 1 − αk with α = 1 − 1 rx (1).

This attention led to yet another seminal paper (Yule, 1927), in which Yule’s starting point was to take a simple harmonic function of time and to superpose upon it a sequence of random errors. 20(a). , page 267). , page 268). But Yule did not see this setup as being the most likely hypothesis in most physical situations, leading him to suggest a delightful thought experiment, based on the following set-up of a pendulum. 20]. But by improvement of apparatus and automatic methods of recording, let us say, errors of observation are practically eliminated.