By Terence C. Mills (auth.)
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Additional resources for A Very British Affair: Six Britons and the Development of Time Series Analysis During the 20th Century
8 is that meaningless correlations between time series could arise because the series are in some way analogous to harmonic functions, leading Yule to ask (w)hat characteristics must two empirical series possess in order that small random samples, taken from them in the same way that we took Yule: Nonsense Correlations 23 the small samples from the sine-curves, may tend to give a U-shaped frequency-distribution for the resultant correlations? , page 14) The phenomenon is clearly related to the fact that a small segment of a sine curve, when taken at random, will usually be either rising or falling and so will tend to be highly correlated (of either sign) with other segments taken at random.
UT , then xt = u1 + u2 + · · · + ut is a series with random differences (in the simulations T is set at 100). 10. 697 for A1 , B1 and C1 , respectively. , page 60). 8 20 30 40 50 60 70 80 90 100 Three random series of k and will cease to be valid for large values. , page 60; italics in original) Yule next considered the case when the differences are correlated such that 1 rx (k) is a linear function of k. This can be expressed as 1 rx (k) = 1 − αk since 1 rx (0) = 1. 9) j=1 To utilize this result, Yule constructed a series with correlated differences by taking the random series ut and cumulating 11-period Yule: Nonsense Correlations 37 moving sums, that is, by calculating t t st = uj , sj = ut + 2ut−1 + · · · +2ut−10 + ut−11 , xt = j=t−10 t = 11, .
The average of first or last deviations, respectively, called +1000 Term Expectation Experimental results a and b Term Expectation Experimental results c and d (1) (2) (3) (4) (5) (6) 1 2 3 4 5 6 7 8 9 10 +1000 +870 +689 +467 +215 −59 −347 −644 −945 −1247 +1000 +868 +691 +489 −258 −10 −300 −637 −1002 −1357 10 9 8 7 6 5 4 3 2 1 +1000 +733 +473 +226 −1 −203 −377 −520 −629 −702 +1000 +726 +459 +206 −10 −205 −367 −502 −615 −692 are really only concerned with the serial correlations for the differences of our given series, and not with the serial correlations of those series themselves.
A Very British Affair: Six Britons and the Development of Time Series Analysis During the 20th Century by Terence C. Mills (auth.)