# The importance of prewhitening in change point analysis under persistence

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## 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).

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