Comment on the paper of Willems, P.: Multidecadal oscillatory behaviour of rainfall extremes in Europe. Published in: Climatic Change 120 (4), p. 931–944 Svenja Fischer & Andreas Schumann
Climatic Change An Interdisciplinary, International Journal Devoted to the Description, Causes and Implications of Climatic Change ISSN 0165-0009 Volume 130 Number 2 Climatic Change (2015) 130:77-81 DOI 10.1007/s10584-015-1361-y
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Author's personal copy Climatic Change (2015) 130:77–81 DOI 10.1007/s10584-015-1361-y COMMENTARY
Comment on the paper of Willems, P.: Multidecadal oscillatory behaviour of rainfall extremes in Europe. Published in: Climatic Change 120 (4), p. 931–944 Svenja Fischer & Andreas Schumann
Received: 29 April 2014 / Accepted: 26 June 2014 / Published online: 5 March 2015 # Springer Science+Business Media Dordrecht 2015
Abstract In his article Willems (Clim Chang 120(4):931–944, 2013) proposed a methodology to analyse extremes in rainfall series. When applying it to artificially generated, non-cyclic random variables we were able to detect cyclic behavior. Therefor we had a closer look on the methodology. Here we discuss our considerations, why this method generates cycles, depending on chosen subperiods and their coherence between detected cycle lengths. To verify these relationships some examples based on random data samples are given. P. Willems proposed a methodology to analyse extremes in rainfall series. His approach is described in the following (cited from his paper in BClimate Change^): BThe ranked extremes represent quantiles x(L/i) with empirical recurrence intervals L/i, where i is the rank number (1 for the highest). The full series thereafter is divided in subperiods with fixed length Lb, moving from the first to the last subperiod with a moving step of 1 year. When the quantiles present in each subperiod are denoted xb(Lb/ ib), where ib is the rank number (1 for the highest in the subperiod), the anomaly percentage of each of these quantiles is calculated by 100 * (xb(Lb/ib) / x(Lb/ib)−1). When Lb/ib does not match one of the L/i recurrence intervals, the closest L/i is considered. The anomaly percentages for all empirical recurrence intervals above the considered threshold are then averaged to obtain the mean anomaly per subperiod. It is plotted at the central time moment of the subperiod. Results are hereafter shown for subperiods sliding with a 1 year step, hence representing the temporal variability in extreme quantile anomalies.^
An author's reply to this comment is available at http://dx.doi.org/10.1007/s10584-015-1364-8. This comment refers to the article available at: http://dx.doi.org/10.1007/s10584-013-0837-x. S. Fischer (*) : A. Schumann Institute for Hydrology, Ruhr- University Bochum, Bochum, Germany e-mail:
[email protected] A. Schumann e-mail:
[email protected]
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Applying this methodology on stochastic simulated non-cyclic time series, we found cyclic behavior. This result initiated this closer look on the proposed methodology. The results of our considerations are described below. Let us consider a series X1,…,XL of quasi-equidistant extremes and the related order statistic X(1) ≥…≥X(L). Using the method of Willems we take subsets of extremes in subperiods ðX 1 ; …; X Lb Þ; ðX 2 ; …; X Lb þ1 Þ; …; ðX L‐Lb þ1 ; …; X L Þ of length Lb. Now let us regard the subsample, which contains the maximum of all Xi, that is X(1), for the first time. For simplicity of notation let us assume that X ð1Þ ∈ðX 1 ; …; X Lb Þ and X ð1Þ ¼ X Lb therefore. Naturally X Lb ≥X j for all j=1,…,Lb −1. We want to calculate the recurrence interval of X Lb in our subperiod to finally gain the anomaly. RI b1 ðX Lb Þ ¼ iLbb ¼ Lb , where b1 means that we have ranks concerning the first subperiod. 1
This is compared with the corresponding quantiles derived from the full series: RI ðX Lb Þ ¼ Li ¼ L. As one can see, the recurrence interval of X(1) is much smaller considering it is based on the subsample instead of the whole series. That leads automatically to a comparison 0 of X(1) with a1much B X ð1 Þ C 1A will smaller value, indeed with X ðL=Lb Þ , and therefore the anomaly aib1 ¼ 100⋅@ X ðL=Lb Þ |fflfflffl{zfflfflffl}>1 be large. Since Willems 2013 uses the mean of all anomalies aav above a certain threshold T, this single value will result in a large positive anomaly as the mean is known as being very nonrobust. In general, single small values of a data sample in a subsample will not affect the mean of all anomalies very much. The reason is, that we have a large quantity of small and mid-level values, which do not affect the mean-anomaly as much as large values (s. below). Thus, the threshold is negligible or even intensifies the henceforth-described phenomenon. Now let us have a look at the second subperiode ðX 2 ; …; X Lb þ1 Þ. The only change in the subsample is that X1 is replaced by X Lb þ1, leading to two different cases. Case 1: X 1 ≥X Lb þ1 If X1 is a small value of the whole series, there is no significant effect on the mean-anomaly (s. above). If X1 is large and X Lb þ1 is much smaller, the mean-anomaly will be smaller than before. Case 2: X 1 ≤X Lb þ1 Since X Lb is, as long as it is contained, the greatest value in the subperiod and compared with X ðL=Lb Þ , the second greatest value is compared with X ð2L=Lb Þ . Therefore, if the second greatest value equals X(i) with i=2,…,2L/Lb −1 it delivers again a positive anomaly. This proceeds for every further great value, where the interval of possible Bover-compared^ values increases by L/Lb −1. Therefore we have a large increase of the mean-anomaly, if the difference between X1 and X Lb þ1 is large, and a small increase otherwise. This behavior remains valid at least as long X Lb is part of the subperiods, that is Lb times. Thus in general, as long as we have a very large value in the subsample, all other (large) values are very likely compared with too small values of the whole sample and the meananomaly increases or stays nearly constant. This holds on for mostly Lb steps (only if the large value lies at the very beginning or end of the whole sample, we have less steps).
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Fig. 1 Mean anomaly and auto correlation function of the mean anomaly (with white noise confident intervals) for 100 Pareto(100,3) random variables with Lb =5 (top), Lb =10 (middle) and Lb =15 (bottom), T=0
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Fig. 2 Mean anomaly and auto correlation function of the mean anomaly (with white noise confident intervals) for the sample in Fig. 1 with Lb =20 (top), Lb =40 (middle) and Lb =60 (bottom), T=0
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At some time point t a local maximum value aav;max is achieved, when there are many large Lb þt values in the subsample ∑ X k is large : aav is saturated. From this point on, these large k¼1þt Lb þt
values vanish one by one out of the subsample and ∑ X k and therefore aav decreases. k¼1þt
So by having this in mind we are able to explain why we can very often detect a periodic behavior of the mean-anomaly. If we have a minimum value of the mean-anomaly, we only have very small values in our subsample. At some point, a relatively large value occurs. This value causes the abovementioned behavior, as long as it is in the subsample or another even larger value occurs, which leads to the same behavior. So even a uniform increasing and decreasing of the mean-anomaly will cause finally a periodicity. The amplitude of the periodicity is determined by the variance of the data set. If Lb converges towards L the periodicity vanishes. If there is a trend in the data, the trend can also be found in the mean-anomaly. If the data set belongs to a heavy tailed distribution, which is very likely for the data considered in Willems (2013), the effect of periodicity should be enlarged. In this case, we have some kind of lower threshold and therefore no lower outlier in the data and additionally very large values occurring more frequently than in medium-tailed distributions. In the following we generated two samples of L=100 i.i.d. random variables which are α 1½x ≥ μ P a r e t o - d i s t r i b u t e d h a v i n g d i s t r i b u t i o n f u n c t i o n F ðxÞ ¼ 1 μx 1½x ≥ μ being the indicator function with parameter μ=100 and shape parameter α=3, that is heavy tailed random variables, and Normal-distributed with mean 100 and standard deviation 10, that is a symmetric distribution, respectively. We took Lb =15 and calculated the mean-anomaly above the threshold T=x0.6 as well as T= 0, the 60 %-quantile and no threshold at all respectively. This procedure was repeated 50 times. The results were (for T=0): 23 times a smooth periodicity, 12 no periodicity at all and 15 times a periodicity with changing period-length and –height for the Pareto-distributed sample. For the Normal distribution the results were nearly the same. In a second step we investigated the influence of the length of the subperiod Lb. Therefore we again generated samples like the ones above and increased Lb. Some generic results can be found in Figs. 1 and 2. We can see periodic behavior of the mean-anomaly which is mirrored also in the autocorrelation function. The length of the period increases and finally vanishes for Lb→L Nevertheless we cannot find a concrete, e.g., linear, connection between the length of the detected period and the chosen length of the subperiod. We only show the results for the Pareto-distributed sample since it seems to be more relevant in practice. Nevertheless the results for Normal-distributed samples showed in parts even more periodic behavior, a greater dependence of the length of the subperiod and a smoother period. Therefore the phenomenon of generated periodicity does not depend on the heaviness of the tail. We come to the conclusion that the method to calculate the mean-anomaly of Willems 2013 generates cycles since it amplifies the influence of very high values and diminishes the influence of middle and lower values. The length of the cycle depends much on the length of the chosen subperiod.
Reference Willems P (2013) Multidecadal oscillatory behaviour of rainfall extremes in Europe. Clim Chang 120(4):931–944
