# The importance of prewhitening in change point analysis under persistence

## Abstract

The presence of serial correlation in hydro-meteorological time series often makes the detection of deterministic gradual or abrupt changes with tests such as Mann–Kendall (MK) and Pettitt problematic. In this study we investigate the adverse impact of serial correlation on change point analyses performed by the Pettitt test. Building on methods developed for the MK test, different prewhitening procedures devised to remove the serial correlation are examined, and the effects of the sample size and strength of serial dependence on their performance are tested by Monte Carlo experiments involving the first-order autoregressive [AR(1)] process, fractional Gaussian noise (fGn), and fractionally integrated autoregressive [ARFIMA(1,*d*,0)] model. Results show that (1) the serial correlation affects the Pettitt test more than tests for slowly varying monotonic trends such as the MK test both for short-range and long-range persistence; (2) the most efficient prewhitening procedure based on AR(1) involves the simultaneous estimation of step change and lag-1 autocorrelation * ρ*, and bias correction of * ρ* estimates; (3) as expected, the effectiveness of the prewhitening procedure strongly depends upon the model selected to remove the serial correlation; (4) prewhitening procedures allow for a better control of the type I error resulting in rejection rates reasonably close to the nominal values. As ancillary results, (5) we show the ineffectiveness of the original formulation of the so-called trend-free prewhitening (TFPW) method and provide analytical results supporting a corrected version called TFPWcu; and (6) we propose an improved two-stage bias correction of * ρ* estimates for AR(1) signals.

### Keywords

Pettitt test Change point analysis Prewhitening Autoregressive process Fractional Gaussian noise Hurst parameter## 1 Introduction

Climate fluctuations and human activities can cause statistical shifts in long-term means of hydro-meteorological variables. Recognition and attribution of these changes is fundamental for infrastructure design, water management strategies, and risk mitigation policies. In this respect, appropriate statistical diagnostics and change detection methods can help understand the nature of historic fluctuations in hydrological time series [e.g., Rougé et al. (2013); Guerreiro et al. (2014) and references therein]. Among many available statistical testing procedures devised for assessing the significance of a change [e.g., Kundzewicz and Robson (2004)], the Pettitt test (Pettitt 1979) is one of the widely used rank-based nonparametric tests to check the presence and timing of abrupt changes in the mean or median of hydro-meteorological variables such as rainfall, runoff, and temperature [e.g., Villarini et al. (2009, 2011); Ferguson and Villarini (2012); Rougé et al. (2013); Tramblay et al. (2013); Guerreiro et al. (2014); Sagarika et al. (2014) among others].

Different aspects of such tests (Pettitt and MK) have been widely studied in the literature. However, the MK test has always received much more attention than the Pettitt test despite their common theoretical background and the potential interest of regime shift detection in hydrological and climate studies compared with monotonic trends. For example, the power of the MK test under different conditions (i.e., sample size, magnitude of deterministic trend, type of the parent distribution) was studied by extensive Monte Carlo simulations about one decade ago (Yue et al. 2002a; Önöz and Bayazit 2003; Yue and Pilon 2004), whereas, to the best of our knowledge, an analogous study was performed only recently for the Pettitt test (Xie et al. 2014; Mallakpour and Villarini 2015).

The same holds for the effect of serial correlation (also referred to as autocorrelation or serial dependence) on the outcome of Pettitt and MK tests. It is well known that a basic assumption for a correct application of tests such as Pettitt and MK is that the data should be randomly ordered (i.e. observations should be serially independent), which is a condition seldom fulfilled by real-world hydro-meteorological data (e.g., Hamed 2009). The effect of the autocorrelation on tests devised for independent data is a general increase of the rejection rate of the null hypothesis (“no change”) of the statistical test, even if no change is present in the data. This over-rejection (compared with the nominal rejection rate) is due to the information redundancy which makes the effective sample size smaller than the observed size, thus implying that the effective variance of the test statistics to be used in the testing procedure under serial dependence is larger than that provided by standard results obtained under the hypothesis of independence (e.g., Bayley and Hammersley 1946; Koutsoyiannis and Montanari 2007). This phenomenon is known as variance inflation. In this respect, there is an extensive literature on the study of the effect of serial correlation on the MK test (see Sect. 2), whereas, to the best of our knowledge, only Busuioc and von Storch (1996) and Rybski and Neumann (2011) (see Sect. 3) tackled the problem for the Pettitt test.

In this study we provide a comprehensive investigation of the effects of serial dependence on the Pettitt test, and propose a set of so-called prewhitening methods (see Sect. 3) in order to make the test procedure suitable for serially correlated data. Such methods involve different autocorrelation structures, and take into account the mutual influence of serial correlation and structural abrupt changes. The capability of controlling the type I error and the sensitivity to model misspecification are tested by extensive Monte Carlo simulations. Since the proposed prewhitening procedures are derived from techniques developed for the MK test, an overview of these methods is given in Sect. 2. Prewhitening approaches for Pettitt are therefore presented in Sect. 3, whilst simulation results are discussed in Sect. 4. Finally, conclusions are drawn in Sect. 5.

## 2 Some aspects of MK analysis of gradual changes under serial correlation

Both procedures (inflated variance correction and prewhitening) require the estimation of the autocorrelation terms at different lags (for nonparametric approaches or ARMA models), \(d\) (for ARFIMA models), or \(H\) (for fGn). However, the presence of deterministic (gradual or abrupt) changes tends to strengthen the autocorrelation among data, resulting in biased estimates of the models’ parameters, and eventually in overestimating the terms of the autocorrelation function. Using such inflated correlation values in computing the variance in Eq. 3 results in an over-inflation of the variance of the test statistic \(S\), thus making the test too liberal (i.e., the rejection rate of the null hypothesis is smaller than expected). Analogously, the effect of inflated correlation on prewhitening is a removal of a portion of the trend (Yue and Wang 2002), thus increasing the chances of not rejecting the null hypothesis when the original MK test is applied to model residuals. The interaction between deterministic trends and autocorrelation structure prompted a rather heated debate about the suitability of the prewhitening procedure and its effect on the test significance level and power (e.g., Bayazit and Önöz 2004; Yue and Wang 2004a, b; Zhang and Zwiers 2004; Hamed 2008a; Bayazit and Önöz 2008).

In this respect, focusing on prewhitening by AR(1) correlation structure, the preliminary removal of the apparent deterministic trend (e.g., Hamed and Rao 1998; Yue et al. 2002b; Yue and Wang 2004c) was shown to reduce the inflation of the lag-1 autocorrelation \(\rho \) used in prewhitening, thus avoiding the problem of overcorrection (also known as over-whitening). However, Hamed (2009) highlighted that the removal of the apparent trend leads to an underestimation of \(\rho \), resulting in an insufficient removal of the autocorrelation, and thus in the persistence of the original problem of over-rejection. He concluded that no prewhitening, prewhitening without trend removal, or prewhitening with trend removal all exhibit a poor performance owing to the presence of the autocorrelation, the overestimation and underestimation of \(\rho \), respectively. To overcome such problems, Hamed (2009) suggested a procedure allowing for the simultaneous estimation of \(\rho \) and the slope \(\beta \) of a possible deterministic linear trend. This approach was shown to balance between under- and over-correction improving the effectiveness of prewhitening and also correcting the bias in the \(\rho \) estimates.

*equivalent*trend (Hamed 2009, p. 148) with

*effective*slope \((1 - \rho ') \beta '\) corresponding to prewhitened observations \(y_t - \rho y_{t-1}\). In order to obtain a prewhitened time series with the same trend slope \(\beta '\) of the observed sequences, Wang and Swail (2001) suggested dividing the prewhitened values by \((1-\rho ')\), obtaining

## 3 Prewhitening methods for the Pettitt test

As mentioned above, unlike the MK test, the Pettitt test has received less attention in the literature. Dealing with the impact of serial correlation, Busuioc and von Storch (1996) showed the adverse effect of the autocorrelation (namely, AR(1) correlation structure) and the presence of possible gradual (linear) trends on the rejection rate. Busuioc and von Storch (1996) recommend prewhitening before performing the test, and highlight the detrimental effects of the presence of linear trends. Indeed, the preliminary removal of a linear trend corrects for the over-rejection of the Pettitt test if only a linear trend is present. However, when both linear trend and one or more abrupt changes are present, spurious trends can results from the presence of abrupt changes, and trend removal reduces the power of the test making it sometimes useless. Thus they “recommend using the Pettitt test as a mere exploratory tool and calculating Pettitt’s statistic and dealing with change points as unproven hypotheses, which plausibility should be supported by physical arguments”. Similarly, Rybski and Neumann (2011) discussed the over-rejection introduced by a long-range power-law decaying correlation structure, thus confirming the results of Busuioc and von Storch (1996) and suggesting the modification of the expression of the distribution of \(K_T\) under the null hypothesis accounting for short-range and long-range correlation. However, they do not discuss such procedures. Dealing with a sequential regime shift detection method (Rodionov 2004), which is different to the Pettitt test but is similarly affected by serial correlation, Rodionov (2006) investigated the effect of prewhitening, highlighting the importance of performing a bias correction of the ordinary least squares (OLS) or maximum likelihood estimates of \(\rho \).

Based on these remarks and the results reported in the previous section concerning the MK test, in this study, we investigate the effect of the autocorrelation on the rejection rate of the Pettitt test and the effectiveness of prewhitening, bearing in mind the concealing effects of the interaction between serial correlation and “true” abrupt changes, and the bias affecting the parameters’ estimates.

### 3.1 TFPWcu adapted for the Pettitt test

- Step 1:
The Pettitt test is applied to the original data. If the value of the test statistic \(K_T\) is not significant, it can be concluded that there is no evidence to reject the null hypothesis (“no change”).

- Step 2:If \(K_T\) is significant, the position \(\tau \) of the possible change point is used to split the time series in two sub-series (before and after \(\tau \)), the difference of the medians or means, \(\hat{\mu }_{\text {b}}\) and \(\hat{\mu }_{\text {a}}\), of the two sub-series is computed as \(\hat{\mathrm{\Delta }}'= \hat{\mu }_{\text {b}} - \hat{\mu }_{\text {a}}\) and used to remove the step change as follows:$$ x_t = y_t - \hat{\mathrm{\Delta }}' \cdot {\mathbf{1}}_{\left\{ t > \tau \right\} } . $$(12)
- Step 3:The value of the lag-1 autocorrelation \(\rho \) of \(x_t\) is estimated by the OLS estimator and corrected for bias using the two-stage bias correction described in the Appendix; then the AR(1) structure is removed bywhere \(\hat{\rho }^*\) is the bias corrected estimate of \(\rho \) and \(\varepsilon _t'\) should be an uncorrelated series.$$ \varepsilon _t' = x_t - \hat{\rho }^* x_{t-1}, $$(13)
- Step 4:The step change and the residuals \(\varepsilon _t'\) are combined byand the Pettitt test is applied to these prewhitened series to assess the significance of the abrupt change.$$ \hat{\mathrm{\Delta }}' \cdot {\mathbf{1}}_{\left\{ t > \tau \right\} } + \dfrac{\varepsilon _t' }{1-\hat{\rho }^*} , $$(14)

### 3.2 Hamed’s methods adapted for the Pettitt test

#### 3.2.1 AR(1) prewhitening

**z**is a \((T-1)\times 3\) design matrix containing observations from \(y_1\) to \(y_{T-1}\) in the first column, a vector of \((T-1)\) ones in the second column, and a sequence of integers from 2 to

*T*in the third column; \({\mathbf{y}} \) is the vector of observation from \(y_2\) to \(y_{T}\). The simultaneous estimation allows for the correction of the bias in \(\rho \) related to the estimation of nuisance parameters, i.e. the coefficients of the linear (or polynomial) mean function. In particular, for both OLS and maximum likelihood estimators, and a linear trend, Kang et al. (2003) and van Giersbergen (2005) showed that \({\text {E}}[\hat{\rho }- \rho ] = -(2+4\rho )/T\), yielding the bias-corrected value

*effective*magnitude of the step change. Thus, the testing procedure consists of applying the original Pettitt test to the prewhitened signal

#### 3.2.2 Prewhitening with models different from AR(1)

- Step 1:
The Pettitt test is applied to the original data. If the value of the test statistic \(K_T\) is not significant, it can be concluded that there is no evidence to reject the null hypothesis (“no change”).

- Step 2:
If \(K_T\) is significant, the abrupt change is removed as for Step 2 of the TFPWcu approach (Sect. 3.1), and the parameters of the selected model are calculated on this detrended time series.

- Step 3:
The original data are prewhitened by the model calibrated in the previous step and the Pettitt test is applied. If the value of the test statistic \(K_T\) is not significant, it can be concluded that there is no evidence to reject the null hypothesis (“no change”), otherwise the null hypothesis can be rejected at a given significance level.

*model*-UPW and

*model*-CPW, where

*model*refers to the model used to prewhiten (e.g., AR(1)).

## 4 Monte Carlo results

To test the effectiveness of the procedures described in Sect. 3, we used a set of models accounting for both short-range and long-range serial correlation, namely, AR(1), fGn, and ARFIMA(1,*d*,0). The analyses are based on Monte Carlo simulations of samples from AR(1) with \(\rho \) ranging from 0 to 0.9 by 0.1, fGn with Hurst parameter ranging from 0.5 to 0.95 by 0.05, and ARFIMA(1,\(d\),0) with ten combinations of the parameters \(\rho \) and \(d\) (detailed below), and sample size \(T\in \left\{ 20,40,60, 80, 100, 150, 200, 250 \right\} \). For each configuration, 1000 time series were simulated.

Figure 2 also shows the effect of model misspecification. In particular, fGn-based methods do no provide a sufficient prewhitening (which is known as under-whitening) for small sample sizes owing to the difficulty of reliably estimating the Hurst parameter in these cases (e.g., Tyralis and Koutsoyiannis 2011). On the other hand, fGn-CPW and fGn-UPW yield over-whitening, and so under-rejection, as the sample size increases and the removed fGn depedence structure is stronger than the actual AR(1). ARFIMA(1,\(d\),0)-CPW and ARFIMA(1,\(d\),0)-UPW provide results similar to fGn-UPW and fGn-CPW for small sample sizes, whereas their short-range correlation component prevents over-whitening for larger sample sizes. Finally, there is no significant difference between conditional and unconditional prewhitening. A map of the rejection rate as a function of \(\rho \) and sample size \(T\) is also provided for the “best” performing method to highlight the dependence of the rejection rates on the pairs \((\rho ,T)\).

## 5 Conclusions

- 1.
A preliminary analysis of prewhitening techniques developed for MK showed that the well-known TFPW method as introduced by Yue et al. (2002b) can provide an effective prewhitening of the series only if the trend residuals are multiplied by a magnification factor equal to \(1/(1-\rho )\). As this correction was introduced for instance in software such as zyp (Bronaugh and Werner 2013) based only on empirical results, we provide a theoretical justification showing that it is not an option but a must to guarantee the actual prewhitening of the series and the fulfillment of the basic hypotheses required for a correct application of the MK test.

- 2.
Focusing on AR(1) signals and Pettitt test, we found that the simultaneous estimation of the model parameters (\(\rho \) and \({\mathrm {\Delta }}\)) provides the best results, thus confirming the suitability of this method not only for the MK test but also for the Pettitt test. On the other hand, model misspecification yields systematic over- or under-whitening, and thus under- and over-rejection, respectively. In this respect, it should be noted that we considered a range of sample sizes corresponding with hydro-meteorological series at annual or seasonal time scales, which often makes the estimation of the parameters of long-range dependence components difficult.

- 3.
As far as fGn signals are concerned, the long-range dependence further increases the actual rejection rate confirming the difficulty of distinguishing between deterministic change points and long-range persistence (see e.g., Beran et al. 2013, pp.700–701, and references therein). However, also in this case, prewhitening provides significant reduction of the over-rejection, even though the correction is not as effective as in the case of AR(1). For fGn, model misspecification yields only under-whitening as the alternative models exhibit autocorrelation structures weaker than fGn.

- 4.
When short-range and long-range serial dependence structures are mixed via ARFIMA(1,\(d\),0), the performance of the Pettitt test depends on the combination of the model parameters. However, the overall result is that AR(1)-based prewhitening generally yields better results than the correct model specification. Indeed, the small sample size prevents the reliable estimation of the model parameters, especially of the long-range component, which is not easy to recognize in short time series. This partly explains the performance of AR(1)-based methods for ARFIMA(1,\(d\),0) time series.

Finally, it should be mentioned for the sake of completeness that the methods described in this study represent simple approaches (adapted for the Pettitt test) similar to those commonly applied in MK trend analyses of hydro-meteorological data. However, there is quite an extensive literature concerning other tests, especially the so-called CUSUM test, and providing asymptotic results in terms of inflation factors to be used in presence of short-range and long-range serial correlation (see e.g. Basseville and Nikiforov 1993; Beran et al. 2013 (Chap. 7.9), and references therein for an overview].

## Notes

### Acknowledgments

This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) grant EP/K013513/1 “Flood MEMORY: Multi–Event Modelling Of Risk & recoverY”, and Willis Research Network. The comments of two anonymous reviewers are gratefully acknowledged. The analyses were performed in R (R Development Core Team 2014) by using the contributed packages fArma (Wuertz et al. 2013), FGN (McLeod and Veenstra 2012), fracdiff (Fraley 2012), fractal (Constantine and Percival 2014).

### References

- Basseville M, Nikiforov IV (1993) Detection of abrupt changes: theory and application. Prentice Hall, New JerseyGoogle Scholar
- Bayazit M, Önöz B (2004) Comment on “Applicability of prewhitening to eliminate the influence of serial correlation on the Mann–Kendall test” by Sheng Yue and Chun Yuan Wang. Water Resour Res 40(8):W08801CrossRefGoogle Scholar
- Bayazit M, Önöz B (2007) To prewhiten or not to prewhiten in trend analysis? Hydrol Sci J 52(4):611–624CrossRefGoogle Scholar
- Bayazit M, Önöz B (2008) Reply to discussion of “To prewhiten or not to prewhiten in trend analysis?”. Hydrol Sci J 53(3):669–669CrossRefGoogle Scholar
- Bayley GV, Hammersley JM (1946) The “effective” number of independent observations in an autocorrelated time series. Suppl J R Stat Soc 8(2):184–197CrossRefGoogle Scholar
- Beran J, Feng Y, Ghosh S, Kulik R (2013) Long-memory processes: probabilistic properties and statistical methods. Springer, New YorkCrossRefGoogle Scholar
- Bronaugh D, Werner A (2013) zyp: Zhang + Yue-Pilon trends package. http://www.CRAN.R-project.org/package=zyp,rpackageversion0.10-1
- Busuioc A, von Storch H (1996) Changes in the winter precipitation in Romania and its relation to the large-scale circulation. Tellus A 48(4):538–552CrossRefGoogle Scholar
- Cochrane D, Orcutt GH (1949) Application of least squares regression to relationships containing auto-correlated error terms. J Am Stat Assoc 44(245):32–61Google Scholar
- Constantine W, Percival D (2014) Fractal: fractal time series modeling and analysis. http://www.CRAN.R-project.org/package=fractal,rpackageversion2.0-0
- Ferguson CR, Villarini G (2012) Detecting inhomogeneities in the Twentieth Century Reanalysis over the central United States. J Geophys Res Atmos 117(D5):D05–123CrossRefGoogle Scholar
- Fraley C (2012) fracdiff: fractionally differenced ARIMA aka ARFIMA(p, d, q) models. URL http://www.CRAN.R-project.org/package=fracdiff,rpackageversion1.4--2 (S original by Chris Fraley and U. Washington and Seattle. R port by Fritz Leisch at TU Wien; since 2003–2012: Martin Maechler; fdGPH and fdSperio and etc by Valderio Reisen and Artur Lemonte.)
- Guerreiro SB, Kilsby CG, Serinaldi F (2014) Analysis of time variation of rainfall in transnational basins in Iberia: abrupt changes or trends? Int J Climatol 34(1):114–133CrossRefGoogle Scholar
- Gurevich G (2009) Asymptotic distribution of Mann–Whitney type statistics for nonparametric change point problems. Comput Model New Technol 13(3):18–26Google Scholar
- Gurevich G, Raz B (2010) Monte Carlo analysis of change point estimators. J Appl Quant Methods 5(4):659–669Google Scholar
- Hamed KH (2008a) To prewhiten or not to prewhiten in trend analysis? Hydrol Sci J 53(3):667–668CrossRefGoogle Scholar
- Hamed KH (2008b) Trend detection in hydrologic data: the Mann–Kendall trend test under the scaling hypothesis. J Hydrol 349(3–4):350–363CrossRefGoogle Scholar
- Hamed KH (2009) Enhancing the effectiveness of prewhitening in trend analysis of hydrologic data. J Hydrol 368(1–4):143–155CrossRefGoogle Scholar
- Hamed KH, Rao AR (1998) A modified Mann–Kendall trend test for autocorrelated data. J Hydrol 204(1–4):182–196CrossRefGoogle Scholar
- Hawkins DM (1977) Testing a sequence of observations for a shift in location. J Am Stat Assoc 72(357):180–186CrossRefGoogle Scholar
- Kang W, Shin DW, Lee Y (2003) Biases of the restricted maximum likelihood estimators for ARMA processes with polynomial time trend. J Stat Plan Inference 116(1):163–176CrossRefGoogle Scholar
- Katz RW (1988) Statistical procedures for making inferences about climate variability. J Clim 1(11):1057–1064CrossRefGoogle Scholar
- Khaliq MN, Ouarda TBMJ, Gachon P, Sushama L, St-Hilaire A (2009) Identification of hydrological trends in the presence of serial and cross correlations: a review of selected methods and their application to annual flow regimes of Canadian rivers. J Hydrol 368(1–4):117–130CrossRefGoogle Scholar
- Koutsoyiannis D (2003) Climate change, the Hurst phenomenon, and hydrological statistics. Hydrol Sci J 48(1):3–24CrossRefGoogle Scholar
- Koutsoyiannis D, Montanari A (2007) Statistical analysis of hydroclimatic time series: uncertainty and insights. Water Res Res 43(5):W05–429CrossRefGoogle Scholar
- Kulkarni A, von Storch H (1995) Monte Carlo experiments on the effect of serial correlation on the Mann–Kendall test of trend. Meteorol Z 4(2):82–85Google Scholar
- Kundzewicz ZW, Robson AJ (2004) Change detection in hydrological records-a review of the methodology. Hydrol Sci J 49(1):7–19CrossRefGoogle Scholar
- Lenton RL, Schaake JC (1973) Comments on ‘Small sample estimation of \(\rho _1\)′ by James R. Wallis and P. Enda O’Connell. Water Resour Res 9(2):503–504CrossRefGoogle Scholar
- Mallakpour I, Villarini G (2015) A simulation study to examine the sensitivity of the Pettitt test to detect abrupt changes in mean. Hydrol Sci J. doi:10.1080/02626667.2015.1008482
- Marriott FHC, Pope JA (1954) Bias in the estimation of autocorrelations. Biometrika 41(3–4):390–402CrossRefGoogle Scholar
- Matalas NC, Sankarasubramanian A (2003) Effect of persistence on trend detection via regression. Water Resour Res 39(12):1342CrossRefGoogle Scholar
- McLeod AI, Hipel KW (1978) Preservation of the rescaled adjusted range: 1. A reassessment of the Hurst Phenomenon. Water Resour Res 14(3):491–508CrossRefGoogle Scholar
- McLeod AI, Veenstra J (2012) FGN: fractional Gaussian noise, estimation and simulation. http://www.CRAN.R-project.org/package=FGN,rpackageversion2.0
- McLeod AI, Yu H, Krougly ZL (2007) Algorithms for linear time series analysis: with R Package. J Stat Softw 23(5):1–26CrossRefGoogle Scholar
- Mudelsee M (2001) Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes. Stat Pap 42(4):517–527CrossRefGoogle Scholar
- Önöz B, Bayazit M (2003) The power of statistical tests for trend detection. Turk J Eng and Environ Sci 27(4):247–251Google Scholar
- Orcutt GH (1948) A study of the autoregressive nature of the time series used for Tinbergen’s model of the economic system of the United States, 1919–1932. J R Stat Soc Ser B 10(1):1–53Google Scholar
- Pettitt AN (1979) A non-parametric approach to the change-point problem. J R Stat Soc Ser C 28(2):126–135Google Scholar
- R Development Core Team (2014) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. http://www.R-project.org/, ISBN3-900051-07-0
- Rodionov SN (2004) A sequential algorithm for testing climate regime shifts. Geophys Res Lett 31(9):L09204CrossRefGoogle Scholar
- Rodionov SN (2006) Use of prewhitening in climate regime shift detection. Geophys Res Lett 33(12):L12707CrossRefGoogle Scholar
- Rougé C, Ge Y, Cai X (2013) Detecting gradual and abrupt changes in hydrological records. Adv Water Resour 53:33–44CrossRefGoogle Scholar
- Rybski D, Neumann J (2011) A review on the Pettitt test. In: Kropp J, Schellnhuber HJ (eds) In extremis. Springer, Dordrecht, pp 202–213CrossRefGoogle Scholar
- Sagarika S, Kalra A, Ahmad S (2014) Evaluating the effect of persistence on long-term trends and analyzing step changes in streamflows of the continental United States. J Hydrol 517:36–53CrossRefGoogle Scholar
- Sen PK (1968) Estimates of the regression coefficient based on Kendall’s tau. J Am Stat Assoc 63(324):1379–1389CrossRefGoogle Scholar
- Serinaldi F (2010) Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series. Phys A Stat Mech Appl 389(14):2770–2781CrossRefGoogle Scholar
- Shumway RH, Stoffer DS (2011) Time series analysis and its applications: with R examples, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
- Tramblay Y, El Adlouni S, Servat E (2013) Trends and variability in extreme precipitation indices over Maghreb countries. Nat Hazards Earth Syst Sci 13(12):3235–3248CrossRefGoogle Scholar
- Tyralis H, Koutsoyiannis D (2011) Simultaneous estimation of the parameters of the Hurst–Kolmogorov stochastic process. Stoch Environ Res Risk Assess 25(1):21–33CrossRefGoogle Scholar
- van Giersbergen NPA (2005) On the effect of deterministic terms on the bias in stable AR models. Econ Lett 89(1):75–82CrossRefGoogle Scholar
- Villarini G, Serinaldi F, Smith JA, Krajewski WF (2009) On the stationarity of annual flood peaks in the continental United States during the 20th century. Water Resour Res 45(8):W08417CrossRefGoogle Scholar
- Villarini G, Smith JA, Serinaldi F, Ntelekos AA (2011) Analyses of seasonal and annual maximum daily discharge records for central Europe. J Hydrol 399(3–4):299–312CrossRefGoogle Scholar
- von Storch H (1999) Misuses of statistical analysis in climate research. In: von Storch H, Navarra A (eds) Analysis of climate variability. Springer, Dordrecht, pp 11–26CrossRefGoogle Scholar
- Wallis JR, O’Connell PE (1972) Small sample estimation of \(\rho _1\). Water Resour Res 8(3):707–712CrossRefGoogle Scholar
- Wang XL, Swail VR (2001) Changes of extreme wave heights in northern hemisphere oceans and related atmospheric circulation regimes. J Clim 14(10):2204–2221CrossRefGoogle Scholar
- Wuertz D, many others, see the SOURCE file (2013) fArma: ARMA Time Series Modelling. http://www.CRAN.R-project.org/package=fArma, rpackageversion3010.79
- Xie H, Li D, Xiong L (2014) Exploring the ability of the Pettitt method for detecting change point by Monte Carlo simulation. Stoch Environ Res Risk Assess 28(7):1643–1655CrossRefGoogle Scholar
- Yue S, Pilon P (2004) A comparison of the power of the t test, Mann–Kendall and bootstrap tests for trend detection. Hydrol Sci J 49(1):21–37CrossRefGoogle Scholar
- Yue S, Wang C (2002) Applicability of prewhitening to eliminate the influence of serial correlation on the Mann–Kendall test. Water Resour Res 38(6):41–47CrossRefGoogle Scholar
- Yue S, Wang C (2004a) Reply to comment by Mehmetcik Bayazit and Bihrat Önöz on “Applicability of prewhitening to eliminate the influence of serial correlation on the Mann–Kendall test”. Water Resour Res 40(8):W08802CrossRefGoogle Scholar
- Yue S (2004b) Reply to comment by Xuebin Zhang and Francis W. Zwiers on “Applicability of prewhitening to eliminate the influence of serial correlation on the Mann–Kendall test”. Water Resour Res 40(3):W03806CrossRefGoogle Scholar
- Yue S, Wang CY (2004c) The Mann–Kendall test modified by effective sample size to detect trend in serially correlated hydrological series. Water Resour Manag 18(3):201–218CrossRefGoogle Scholar
- Yue S, Pilon P, Cavadias G (2002a) Power of the Mann–Kendall and Spearman’s rho tests for detecting monotonic trends in hydrological series. J Hydrol 259(1–4):254–271CrossRefGoogle Scholar
- Yue S, Pilon P, Phinney B, Cavadias G (2002b) The influence of autocorrelation on the ability to detect trend in hydrological series. Hydrol Process 16(9):1807–1829CrossRefGoogle Scholar
- Zhang X, Zwiers FW (2004) Comment on “Applicability of prewhitening to eliminate the influence of serial correlation on the Mann–Kendall test” by Sheng Yue and Chun Yuan Wang. Water Resour Res 40(3):W03805Google Scholar
- Zhang X, Vincent LA, Hogg WD, Niitsoo A (2000) Temperature and precipitation trends in Canada during the 20th century. Atmos Ocean 38(3):395–429CrossRefGoogle Scholar

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