To prewhiten or not to prewhiten in trend analysis?

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1 Hydrological Sciences Journal ISSN: (Print) (Online) Journal homepage: To prewhiten or not to prewhiten in trend analysis? M Bayazit & B Önöz To cite this article: M Bayazit & B Önöz (2007) To prewhiten or not to prewhiten in trend analysis?, Hydrological Sciences Journal, 52:4, , DOI: /hysj To link to this article: Published online: 15 Dec Submit your article to this journal Article views: 5373 View related articles Citing articles: 97 View citing articles Full Terms & Conditions of access and use can be found at

2 Hydrological Sciences Journal des Sciences Hydrologiques, 52(4) August To prewhiten or not to prewhiten in trend analysis? M. BAYAZIT & B. ÖNÖZ Division of Hydraulics, Civil Engineering Faculty, Istanbul Technical University, Istanbul, Turkey Abstract The Mann-Kendall test, used to detect a trend in a time series, yields an incorrect (too large) rejection rate when applied to an autocorrelated series with no trend. Prewhitening corrects this situation, but reduces the power of the test when a trend exists. A simulation study is performed to determine when prewhitening can be applied with no real loss of power. It is found that, in general, prewhitening should be avoided when the test has a high power, i.e. when the coefficient of variation is very low, the slope of trend is high, and the sample size is large. In other cases, prewhitening will prevent the false detection of a non-existing trend, without a significant power loss in identifying a trend that exists. Key words trend analysis; Mann-Kendall test; prewhitening; power of a test; serial correlation Pré-blanchir ou ne pas pré-blanchir dans l analyse de tendance? Résumé Le test de Mann-Kendall, employé pour détecter une tendance dans une série temporelle, présente un taux de rejet incorrect (trop grand) lorsqu il est appliqué à une série auto-corrélée sans tendance. Le pré-blanchissement corrige cette situation, mais réduit la puissance du test lorsqu une tendance existe. Une étude de simulation est réalisée pour déterminer quand le pré-blanchissement peut être appliqué sans véritable perte de puissance. Il apparaît que, en général, le pré-blanchissement devrait être évité quand le test a une puissance élevée, i.e. quand le coefficient de variation est très bas, la pente de la tendance est haute et la dimension de l'échantillon est grande. Dans les autres cas, le préblanchissement empêchera la fausse détection d'une tendance inexistante, sans perte significative de puissance pour l identification d une tendance qui existe. Mots clefs analyse de tendance; test de Mann-Kendall; pré-blanchissement; puissance d un test; corrélation sérielle 1 INTRODUCTION Trend analysis of hydrometeorological time series has gained importance in recent years, as evidenced by the large number of articles published (e.g. Lettenmaier, 1976; Burn, 1994; Lins & Slack, 1999; Douglas et al., 2000; Zhang et al., 2001; Yue et al., 2003). The reason for this is the effect on precipitation and streamflow, of the global warming caused by the green-house effect, due to the increased concentrations of gases such as carbon dioxide in the atmosphere. Although no consensus has been achieved on the sign and magnitude of the changes in the hydrometeorological variables in various parts of the Earth, attempts are being made to investigate whether or not significant trends exist in recent precipitation and streamflow series. There are various statistical methods that can be used to test the hypothesis that there is no trend in a time series. Kundzewicz & Robson (2004) reviewed the methodology for trend detection in hydrological records. The rank-based Mann- Kendall (MK) test (Helsel & Hirsch, 1992) is the most frequently used nonparametric test for this purpose. Yue et al. (2002a) investigated the power of this test as a function of the slope of trend, sample size, assigned significance level, coefficient of variation, and type of probability distribution. The MK test was found to have the same power for detecting the trend as the other commonly used nonparametric test, Spearman s rho test. It is well known that the MK test, devised for independent data, rejects the null hypothesis of no trend more often than specified by the significance level when the data are serially correlated (Von Storch, 1995; Yue & Wang, 2002) unless its variance Open for discussion until 1 February 2008

3 612 M. Bayazit & B. Önöz is modified accordingly. In other words, positive serial correlation increases the Type I error when the time series has no trend, independent of the sample size. This is because the variance of the MK test statistic increases with the magnitude of serial correlation (Yue et al., 2002b). Matalas & Sankarasubramanian (2003) showed that the variance of the estimate of the slope of trend increases with the autocorrelation. Because the existence of positive serial dependence increases the probability of rejecting the no-trend hypothesis, it may cause trends to be detected that would not be found significant if the series were independent. Von Storch (1995) proposed a procedure called prewhitening (PW) to eliminate this undesired effect. The essence of the PW procedure is to remove the serial correlation assuming a lag-one autoregressive (AR(1)) model, and then apply the MK test to the serially independent residuals. It has been shown (Von Storch, 1995; Yue et al., 2002b) that PW effectively decreases the probability of rejecting the null hypothesis in the MK test. Matalas & Sankarasubramanian (2003) showed that PW greatly reduces the inflation factor of the variance of the estimate of the trend slope. Douglas et al. (2000) and Zhang et al. (2001), among others, applied the PW procedure to reduce the influence of serial correlation on the MK test. Hamed & Rao (1998) applied a different method to eliminate the effect of serial dependence, by modifying the variance of the MK test statistic to compensate for this effect. Yue et al. (2002b) showed that, although this procedure decreases the rejection rate of the null hypothesis, it is still much higher than the assigned significance level. Zhang & Zwiers (2004) discussed other methods to estimate the magnitude and statistical significance of trends in the presence of serial correlation. This paper starts with a description of the problems encountered when the PW procedure is applied to the data before performing the MK test. The cases of trend and no trend are discussed, and the advantages and disadvantages of PW are evaluated. The modified PW (MPW) procedure proposed by Yue et al. (2002b, 2003) is assessed for its effects on the magnitude of the test errors. This is followed by a discussion of the loss of power caused by the PW procedure. A simulation study is carried out to investigate the effects of the sample size, slope of trend, lag-one autocorrelation coefficient and coefficient of variation on the loss of power. The results of this study are used to determine when it is advantageous to perform the PW procedure so that no real loss of power is caused by it in comparison with the case of serial independence. The parameters corresponding to the coefficient of variation, slope of trend and lag-one autocorrelation coefficient must be estimated from the available sample. The sampling errors of these statistics are investigated by a simulation study, described in the next section of the paper. A case study is presented in which the effects of prewhitening on trend detection are investigated using the annual flow data of Turkish rivers. The paper ends with a discussion of the advantages and disadvantages of prewhitening in the light of the results of the simulation study. 2 PROBLEMS OF PREWHITENING The MK test is used for checking the null hypothesis of no trend (no change in the mean of the series) versus the alternative hypothesis of the existence of trend

4 To prewhiten or not to prewhiten in trend analysis? 613 (increasing or decreasing of the mean over time). The result of the test can have two types of error, as in all statistical tests: (a) Type I error is the incorrect rejection of the null hypothesis when there is actually no trend. The probability of this error is equal to the assigned significance level. (b) Type II error is the incorrect acceptance of the null hypothesis when a trend exists. The power of the test is the probability of not making this type of error. The MK test is designed for serially independent series. Therefore, the probability of Type I error is correct only when the data are independent. As described above, serial correlation increases the variance of the test statistic and, hence, the rejection rate of the null hypothesis is increased. This has the following results: When there is no trend, the MK test has a Type I error that is incorrect (too large) Trends will then be detected that are not in fact significant. When a trend exists, Type II error will be decreased and the power of the test will be increased. This causes no problem. Prewhitening is applied to remove (or rather to reduce) the problem of too frequent trend detection. The PW procedure decreases the inflation of the variance of the test statistic due to serial correlation, and thus reduces the rejection rate below the rate before PW. This has the following results: When there is no trend, the probability of Type I error is significantly reduced, almost to the theoretically correct value. When a trend exists, however, the power of the test is decreased as compared with the power before PW, so that sometimes it may cause a significant trend not to be detected. It is seen that the PW procedure has an advantage and a disadvantage. It should be determined when it is advantageous to apply it. Yue & Wang (2002) stated that prewhitening is not suitable for eliminating the effect of serial correlation on the MK test when trend exists in a time series, because prewhitening will remove a portion (equal to the lag-one autocorrelation coefficient) of trend and hence reduces the probability of rejecting the null hypothesis when it is false. They argued that the effect of the serial correlation on the rejection rate of the null hypothesis is not significant for large sample sizes (larger than about 70) and large magnitudes of trend (larger than about 0.005), in which case it is better to use the MK test on the original data rather than after prewhitening. Yue et al. (2002b, 2003) proposed a modified PW (MPW) procedure. In this procedure, the slope of trend is first estimated and the record is detrended. Then the lag-1 serial correlation coefficient of the detrended series is estimated, and the series is prewhitened using this estimate. Finally, the identified trend is added to the prewhitened series. The MK test is applied to this series to assess the significance of the trend. They argued that the removal of the trend as a first step may allow for a more accurate estimate of the population s lag-1 autocorrelation coefficient, and a subsequent better estimation of the significance of the trend. The arguments summarized above seem to be against the use of the PW procedure, at least for large samples and large magnitudes of trend, even when significant autocorrelation exists in the time series, in order not to reduce the probability of detecting the trend. Bayazit & Önöz (2004) and Zhang & Zwiers (2004) are not in

5 614 M. Bayazit & B. Önöz agreement with the above reasoning. Bayazit & Önöz (2004) pointed out the risk of too frequent detection of a trend when it is not actually present if PW is not applied in the presence of serial dependence. They argued that the prevention of an error of false detection of trend when it does not exist may be preferred to an error of not being able to detect a weak trend. Zhang & Zwiers (2004) commented that power is a secondary consideration that only comes to play after we are satisfied that a chosen technique will not produce an unexpectedly large number of false positives, and concluded that a suitable method of eliminating the serial dependence should be used before applying the MK test. Yue & Wang (2004b), in response to the above comments, remarked that our consideration is that it is not worth getting a reduced rejection rate of the null hypothesis by reducing the magnitude of a real trend, which is caused by prewhitening. and that Prewhitening a time series using spurious or contaminated serial correlation is fundamentally wrong because the existence of a trend in a time series will produce a spurious serial correlation when there is no serial correlation, and the presence of trend will increase the estimate of positive serial correlation when the serial correlation exists (Yue & Wang, 2004c). Yue & Wang (2004a) investigated the use of the effective sample size (ESS) employed by Lettenmaier (1976) to estimate the effect of serial correlation on the MK test. They found that the ESS is an appropriate approach when there is no trend in a time series. However, when there is a trend, the ESS overcorrects the effect of the true serial correlation and reduces the power of the MK test. This is because the trend (upward or downward) always has a positive contribution to serial correlation. They proposed that the existing trend component should be removed from a time series first, and then the lag-one serial correlation coefficient may be computed from the residuals, so that it is no longer affected by the trend. 3 WHEN SHOULD WE PREWHITEN? The PW procedure will reduce the rejection rate of the null hypothesis. This is not desired when the time series has a trend. However, it should be remembered that the MK test in a series with a serial dependence yields a much larger rejection rate than in an independent series. A real loss of power is caused by the PW only when the power of the test is reduced to a value below the power for an independent series for a given sample size, magnitude of trend and coefficient of variation. Otherwise, the reduction of the rejection rate is simply a reduction of the inflated power. It is important that the rejection rate of the test be brought down to the nominal rate in order to avoid spurious trend detection, which can be achieved by prewhitening. However, when PW brings the power of the MK test below the power for a corresponding record with no autocorrelation, its application may cause a significant trend to be missed. In spite of this, it may be preferred to prewhiten in such cases, in order to preserve the nominal rejection rate of the null hypothesis. A simulation study is performed to determine the effect of the PW procedure on the power of the MK test. Twenty thousand normally-distributed time series are generated by the AR(1) model (lag-one autocorrelation coefficient r = 0, 0.2, 0.4, 0.6, 0.8) for each sample size n = 25, 50, 75, 100 with the mean 1 and the coefficient of

6 To prewhiten or not to prewhiten in trend analysis? 615 variation C v = 0.1, , 0.7, 0.9. Linear trends with slopes b = 0.002, 0.004, 0.006, 0.008, 0.01, 0.015, 0.020, are superimposed on each of the generated series. The MK test is applied on the series, and the power of the test is computed as the rejection rate of the null hypothesis of no trend for the generated series in each case. Then, each time series is prewhitened in the usual way, and the MK test is applied to these series to compute the power after prewhitening. The power ratio (PR) of the PW procedure is defined as the ratio of the power of a correlated series after prewhitening for a certain case to the power without PW for the same case when there is no serial dependence (r = 0). When the power ratio is equal to or larger than one, PW causes no real power loss, because the power of the test after PW is still higher than that without PW of a similar series with r = 0. A power ratio above one shows that PW did not completely remove the inflation of the test statistic due to serial correlation. However, a power ratio less than one implies that PW reduced the power of the test below its uninflated value for r = 0, in which case it causes an existing trend to be detected with less success (real loss of power). The results of the simulation study are presented in Figs 1 3 for C v = 0.1, 0.3 and 0.5, respectively, where the power ratio is plotted as a function of b, n and r (results for C v = 0.7 and 0.9 are similar to those for C v = 0.5). These results are summarized and discussed below. The loss of power due to prewhitening is related to the power of the MK test. It is known that the power increases with the slope of trend and the sample size, and decreases as the coefficient of variation increases (Yue et al., 2002a). Figures 1 3 show that, in general, the power ratio is inversely related to the power of the test. It is usually higher for high values of C v and low values of b and n. The lag-one autocorrelation coefficient r has a complicated effect on PR. In most cases, the lowest values of the PR procedure are for r = It is important to know in which cases the power ratio is less than one. For C v = 0.1, there is almost always power loss (PR < 1). For higher values of C v, PR < 1 when n is larger than a certain value which usually decreases with the slope of trend. Thus, for C v = 0.3, PR < 1 when b 0.002, n = 100; b 0.004, n 75; and b 0.006, n 50. For C v = 0.5, PR < 1 when b 0.004, n = 100; and b 0.006, n 75. For C v = 0.7, PR < 1 when b 0.006, n = 100; and b = 0.008, n 75. For C v = 0.9, PR < 1 only when b 0.004, n = 100. Power loss is largest (PR < 0.5) when C v = 0.1, b 0.006, n = 25. In conclusion, it may be stated that prewhitening should be avoided when the coefficient of variation is very low (C v = 0.1) for all sample sizes; when C v is low (C v = 0.3) and b is high (b 0.006) for moderate sample sizes (n = 50 75); when C v is low (C v = 0.3) and b is low (b 0.004) for large samples (n 75); when C v is moderate (C v = ) and b is high (b 0.006) for large samples (n 75); and when C v is high (C v = 0.9) and b for very large samples (n = 100). When C v is very low, it is better not to prewhiten to prevent significant loss of power. Considering the findings of Yue & Wang (2002), prewhitening is not needed in any case for n > 70 and b > 0.005, autocorrelation not having much effect on the rejection rate in these conditions.

7 616 M. Bayazit & B. Önöz Fig. 1 Power ratio PR as function of the slope of trend b, sample size n and lag-one autocorrelation coefficient r (coefficient of variation C v = 0.1). 4 SAMPLING ERRORS OF PARAMETERS OF COMBINED SERIES The above discussion on when the prewhitening should be avoided in performing the MK test assumed that the population values of the parameters C v, b and r are known. In fact, these parameters have to be estimated from the available sample with unavoidable sampling errors. The combined series with trend and autocorrelation generated in the simulation study described above are used to estimate the relative bias and relative root mean square error (rmse) of the statistics corresponding to C v, b and r. Yue et al. (2002b)

8 To prewhiten or not to prewhiten in trend analysis? 617 Fig. 2 Power ratio PR as function of the slope of trend b, sample size n and lag-one autocorrelation coefficient r (coefficient of variation C v = 0.3). performed a similar study where the bias and probability density function of b and r were found only for C v = 0.5 and n = 100. The results of the present study are described below and compared with those of the earlier study.

9 618 M. Bayazit & B. Önöz Fig. 3 Power ratio PR as function of the slope of trend b, sample size n and lag-one autocorrelation coefficient r (coefficient of variation C v = 0.5).

10 To prewhiten or not to prewhiten in trend analysis? 619 As an example, Figs 4 6 show the relative bias and rmse of b, r and C v, respectively, for C v = 0.1 and b = Slope of trend, b The relative bias and rmse of b are largest for small samples (n = 25). They increase with the increase of C v and the decrease of b and r. The bias decreases as the sample size increases, the rmse is the largest for n = 25 and the smallest for n = 75. Yue et al. (2002b) estimated comparable biases, but much smaller values of rmse. In their study, the bias of the slope of trend was largest for r = 0.2, and its variance was largest for r = 0.8. These results are somewhat different from those of the present study. High values of the relative bias and especially those of the relative rmse of the slope of trend make it very difficult to estimate trend from the samples with reasonably low sampling errors, except for very small values of C v and large samples. However, for larger slopes (b 0.008) and n 50, sampling errors are relatively low (Fig. 4) bias BIAS of b: of Cv = b 0.1; b Cv=0.1 = b= n=25 n=50 n=75 n=100 r rmse RMSE of b: of Cv b = 0.1; Cv=0.1 b = b=0.008 n=25 n=50 n=75 n= r Fig. 4 Relative bias and rmse of the slope of trend b as function of the lag-one autocorrelation coefficient r and sample size n (b = 0.008; coefficient of variation C v = 0.1). Lag-one autocorrelation coefficient, r This parameter also has rather high relative bias and rmse, especially for low values of C v and r, and high values of b and n. Yue et al. (2002b) found that the trend component increases the autocorrelation coefficient, especially for small C v and large b and n. This is substantiated by the results of the present study, where the largest (positive) bias and rmse of r is found at

11 620 M. Bayazit & B. Önöz Fig. 5 Relative bias and rmse of the lag-one autocorrelation coefficient r as function of r and sample size n (slope of trend b = 0.008,;coefficient of variation C v = 0.1) bias BIAS of b: of Cv Cv = 0.1; CV=0.1 b = b=0.008 n=25 n=50 n=75 n= r rmse RMSE of b: of Cv Cv = 0.1; b Cv=0.1 = b=0.008 n=25 n=50 n=75 n= Fig. 6 Relative bias and rmse of the coefficient of variation C v as function of the lagone autocorrelation coefficient r and slope size n (C v = 0.1; slope of trend b = 0.008). r

12 To prewhiten or not to prewhiten in trend analysis? 621 C v = 0.1, r = 0.2, b = and n = 100 (Fig. 5). The comparable simulation results of Yue et al. (2002 b) for C v = 0.5 and n = 100 agree with the results of the present study. It is seen that the parameter r cannot be estimated from a combined series sample with a low sampling error when C v and r are low and b and n are large. However, the effect of r on the power ratio is small, and, therefore, the sampling error of r is not important. Coefficient of variation, C v The relative bias and rmse of C v of the combined series are small for low values of C v (Fig. 6), but assume high values for C v 0.7 when b 0.004, n = 25 and r 0.6. Therefore, the parameter C v has large sampling errors when estimated from a combined sample with high C v and r, and low b and n. The above discussion shows that the parameters b, r and C v all have significant sampling errors when they are estimated from a combined sample with both trend and autocorrelation, for most values of n, b, r and C v considered in the simulation study, which covers the ranges of these parameters in practical applications. These errors are smaller when the slope of trend is high (b 0.008) and the sample size is not too small (n 25). In the light of the above results, the proposal by Yue et al. (2003) and Yue & Wang (2004a) to remove the trend from the series first, before computing the autocorrelation coefficient is of not much use because estimated trend will have large sampling errors, unless the sample size is large and coefficient of variation is small, or the slope of trend is high, conditions which may be unrealistic in practice. 5 CASE STUDY The effects of prewhitening on the detection of trend by the MK test are investigated in a case study using the PR results summarized in Section 3. Annual mean flows of 107 sites from 24 river basins all over Turkey are tested for the possible existence of trend by the MK test. The series are chosen as those that are known to be reasonably free of human influence and sufficiently long for trend analysis (sample size varies in the range of years). Detailed information about the streamflow series can be found in Önöz & Bayazit (2003). The MK test rejected the null hypothesis of no trend in 31 cases at the significance level α = The characteristics of these streamflow series are given in Table 1. All the detected trends are negative. Dimensionless value of trend, b, varies in the range of to After prewhitening, the number of rejections (sites with trend) dropped to 27. In 27 cases out of 31, PW did not change the verdict of the MK test, implying that the trend was strong enough not to be missed, even after prewhitening. In the remaining four cases (Site nos 809, 1717, 1721 and 2505), the question whether a significant trend could not be detected because of the power loss due to PW, or the apparent trend before PW was due to the inflation by serial correlation, can be answered using the PR results of this study. For Site 2505 (n = 29, C v = 0.24, r = 0.28, b = 0.008) PR is higher than 1 (Fig. 2), implying that prewhitening cancelled the inflatory effect of serial correlation without a real loss of power, and that therefore the PW procedure can be safely applied. For the other three sites, however, PR values are slightly below 1. For Site 809 (n = 45,

13 622 M. Bayazit & B. Önöz Table 1 Characteristics of annual streamflow series with detected trend. Site no. River basin Catchment n (years) C v r b area (km 2 ) 105 Ergene n M. Kemalpaşa ç Orhaneli ç Emet ç Karamenderes n Medar ç Küçükmenderes n Çine ç Esen ç Dalaman ç Esen ç Boz ç Sivrikaya d Porsuk ç Seydi s Sakarya n Aladağ ç Ova ç Kirmir ç Çarşamba s Peçeneközü d Göksu n Lamas ç Ermenek ç Ermenek ç Anamur ç Göksu n Çağçağ s Bey d Tohma s Bendimahi C v = 0.53, r = 0.42, b = 0.014), the power ratio is about 0.8 (Fig. 3). For Site 1717 (n = 35, C v = 0.42, r = 0.63, b = 0.026) PR = 0.9 (Fig. 3), and for Site 1721 (n = 33, C v = 0.31, r = 0.46, b = ) PR = 0.9. Therefore, prewhitening causes a real power loss at these sites, although not more than 10 20%, which could account to some extent for the acceptance of the no-trend hypothesis after prewhitening. In these cases it could be better not to prewhiten, because the rejection rate would be only slightly increased by autocorrelation, slopes of trend being rather high ( b 0.01). The above analysis assumes that the C v, r and b values of flow series estimated from the samples are equal to their population values. Sampling variability may cause significant differences between the population values and their sample estimates. But because the errors are small for b 0.008, which is the case for all the sites of this study where the PW reverses the outcome of the MK test, it may be considered that the analysis of the PR is relatively free of the sampling effects.

14 To prewhiten or not to prewhiten in trend analysis? CONCLUSIONS Prewhitening is the most commonly used procedure to eliminate the effect of serial correlation in trend analysis. It efficiently removes the possibility of finding a significant trend in the Mann-Kendall test when actually there is no trend. But prewhitening has the disadvantage of accepting the hypothesis of no trend with a high probability when a trend exists. The results of a simulation study have shown that prewhitening causes a real loss of power only when the coefficient of variation is very low, or the slope of trend and sample size exceed certain values when the coefficient of variation is moderate or high. Otherwise, although there is a reduction of power in the MK test due to prewhitening, it will not bring the power below the value the test has when there is no autocorrelation. The results depend on the assumption that autocorrelation is Markovian, because the AR(1) model is used in simulations and for prewhitening. The simulation study has shown that the statistics corresponding to the slope of trend, lag-one autocorrelation coefficient and coefficient of variation usually have large bias and rmse when they are estimated from samples with both trend and autocorrelation. The idea of removing the trend before applying the prewhitening to prevent its overcorrecting effect on the serial correlation is not useful in practice because the trend can usually not be estimated from a record with reasonably low sampling error. The prewhitening procedure is not needed for large samples (n 50) and high slopes of trend (b 0.01), in which cases it would cause significant power loss if applied, because serial correlation has negligible effect on the rejection rate of the test in these cases. It should be applied, however, in other cases to prevent the detection of a non-existant trend. Prewhitening will not cause a significant loss of power in these cases, with the possible exception of very low values of the coefficient of variation, which can be estimated from a sample with small sampling error when its population value is small. It can be concluded that prewhitening should be applied except when the sample size and the magnitude of slope are large, or when the coefficient of variation is very small. Prewhitening will prevent the false detection of a trend when it does not exist. When there is a trend, prewhitening will not reduce the power of the test significantly in cases where its use is recommended (coefficient of variation not too low, sample size and slope of trend not large). Because the power of the MK test is high in the circumstances when it is recommended to be used, a small loss of power will, in general, not prevent a strong trend being detected. It should be remembered that prewhitening helps to keep the nominal rejection rate of the null hypothesis of no trend, which is a characteristic of a test that must not be compromised. Acknowledgement The authors would like to thank two anonymous reviewers for their constructive comments. REFERENCES Bayazit, M. & Önöz, B. (2004) Comment on Applicability of prewhitening to eliminate the influence of serial correlation on the Mann-Kendall test by S. Yue & C. V. Wang. Water Resour. Res. 40(8), W Burn, D. H. (1994) Hydrologic effects of climate change in West Central Canada. J. Hydrol. 160, Douglas, E. B., Vogel, R. M. & Knoll, C. N. (2000) Trends in floods and low flows in the United States: impact of serial correlation. J. Hydrol. 240,

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