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
Tables XV and XVI). 19. , page 49). 15 Yule concluded his address with the following summary which, since it encapsulates what are arguably the most important concepts so far developed for the foundations of time series analysis, is again quoted in detail. Starting from a question that may have seemed to some silly and unnecessary, we were led to investigate the correlations between samples of two simple mathematical functions of time. It appeared that small samples … of such functions tended to give us correlations departing as far as possible from the truth, the correlations tending to approach ±1 if the time for which we had experience was very small compared with the time necessary to give the true correlation.
Conjunct series: Series for which all serial correlations are positive. With finite series, r(k) may well decrease with k and become negative at some point, in which case the series is said to be ‘conjunct up to r(k)’. Disjunct series: Series for which the serial correlations are all negative. , pages 62–3) provided a setup that would generate such a series, the conditions under which this might occur are extremely stringent. 4). Oscillatory series: Series for which the serial correlations change sign, alternating between runs of positive and negative values.
XT is a zero mean series with standard deviation σx for which the serial correlations are rx (1), rx (2), . . 9. Then, if T is 32 A Very British Affair assumed to be large, T −1 T −1 (xt+1 − xt )2 = t=1 T −1 x2t+1 + t=1 T −1 x2t − 2 t=1 T −1 ≈2 T −1 x2t − 2 t=1 xt+1 xt t=1 T −1 =2 xt+1 xt t=1 T −1 t=1 xt+1 xt T −1 2 t=1 xt x2t 1 − t=1 or (cf. 7) Suppose that the differences are random, so that all the 1 rx (k) are zero and 2 rx (k + 1) = 0 for all k, implying that rx (k) = 2rx (k − 1) − rx (k − 2) Yule: Nonsense Correlations 33 Successive serial correlations are then generated by the arithmetical progression rx (2) = 2rx (1) − rx (0) = 2rx (1) − 1 rx (3) = 2rx (2) − rx (1) = 3rx (1) − 2 ..