TIME SERIES ANALYSIS OF HYDROLOGIC DATA FOR WATER ...
TIME SERIES ANALYSIS OF HYDROLOGIC DATA FOR WATER ... TIME SERIES ANALYSIS OF HYDROLOGIC DATA FOR WATER ...
D. Machiwal, M. K. Jhalarge and that they have been obtained under similarconditions.(xi) The necessary condition for applying the RunTest on Successive Differences is that the observationsin the sample should be obtained under similarconditions.(xii) The Wilcoxon-Mann-Whitney Test is a nonparametrictest (i.e., distribution-free) and is applicableonly when the observations are random andindependent.(xiii) The Kendall’s Rank Correlation Test is themost popular test for trend detection in the hydrologictime series.(xiv) The Mann-Kendall Test is a nonparametrictest for trend detection in a time series withoutspecifying whether the trend is linear or nonlinear.Existence of serial correlation in a time series willaffect the ability of the Mann-Kendall Test to assessthe site significance of a trend, and the presenceof cross correlation among sites in a networkwill influence the ability of the test to evaluate thefield significance of trends over the network (Yue etal., 2003).(xv) In general, the parametric methods to assesssignificance of trend employ pre-specified modelsand associated tests, whereas the nonparametricmethods generally apply rank tests to the data. Neitherapproach is suitable for exploratory analysis(Ramesh and Davison, 2000).(xvi) The assumptions of the classical parametrictests viz., normality, linearity, and independenceare usually not met by the hydrological time seriesdata, especially in case of surface water qualitydata. Therefore, recently some nonparametric testshave been proposed to determine the trend in surfacewater quality time series (Kalayci and Kahya,1998). At the same time, the statistical tests fortrend detection in surface water quality time seriesare normally confounded by one or more of thefollowing problems: missing values, censored data,flow relatedness, and seasonality.4. Theoretical research on time seriesanalysis techniquesHamed and Rao (1998) studied the effects ofautocorrelation on the variance of the Mann-Kendall trend test-statistic. A theoretical relationshipwas derived to calculate the variance of theMann- Kendall test statistic for autocorrelated data.The special cases of AR(1) and MA(1) dependencewere discussed as examples. Based on the modifiedvalue of the variance of the Mann-Kendall trendtest statistic, a modified nonparametric trend testsuitable for the autocorrelated data is proposed. Themodified test was applied to rainfall and streamflowdata to demonstrate its performance compared tothe original Mann-Kendall Trend Test. The accuracyof the modified test was found to be superiorto that of the original Mann-Kendall Trend Testwithout any loss of power.Tsakalias and Koutsoyiannis (1999) developed anew approach for the computer-aided explorationand analysis of hydrologic time series with a focuson identification of multiple stage-discharge relationshipsin a river section, analyses for homogeneityand temporal consistency, detection of outliers,shifts and trends. To demonstrate the developedmethodology, initially a mathematical representationwas proposed based on the set theory. It wasdemonstrated that an exhaustive search of all candidatesolutions is intractable. Therefore, a heuristicalgorithm is proposed, which emulates the exploratorydata analysis of the human expert. This algorithmencodes a number of search strategies in apattern directed computer program, and results inan automatic determination of a satisfactory solution.Yue et al. (2002a) investigated the interaction betweena linear trend and a lag-one autoregressive[AR(1)] model using Monte Carlo simulation.Simulation analysis indicated that the existence ofserial correlation alters the variance of the Mann-Kendall (MK) statistic estimate, and the presence ofa trend alters the magnitude of serial correlation.Furthermore, it was found that the commonly usedpre-whitening procedure for eliminating the effectof serial correlation on the MK Test leads to inaccurateassessments of the significance of a trend.Therefore, it was suggested that firstly trend shouldbe removed prior to ascertaining the magnitude ofserial correlation. Both the suggested approach andthe existing approach were employed to assess thesignificance of a trend in the serially correlatedannual mean and annual minimum streamflow dataof some pristine river basins in Ontario, Canada. Itwas concluded that the researchers might have incorrectlyidentified the possibility of significanttrends by using the already existing approach.Yue et al. (2002b) studied the efficacy of the twononparametric rank-based statistical tests (theMann-Kendall Test and Spearman's Rho Test) byMonte Carlo simulation. These two tests were usedto assess the significance of trends in annual maximumstreamflow data of 20 pristine basins in Ontario,Canada. The results indicated that their effec-242
Time series analysis of hydrologic data for water resources planning and management: a reviewtiveness depends on the pre-assigned significancelevel, magnitude of trend, sample size, and theamount of variation within a time series. Thus, thebigger the absolute magnitude of trend or larger thesample size, the more powerful are the tests; but asthe amount of variation in a time series increases,the power of the tests decreases. When a trend ispresent, the power is also dependent on the distributiontype and skewness of the time series. It wasalso found that these two tests have practicallysimilar power in detecting a trend.Clarke (2002) described a model in which theGumbel distribution has a (possibly) time-variantmean. The time-trend in mean value was determinedby a single parameter β estimated by MaximumLikelihood (ML). The large-sample varianceof the ML estimate was compared with the varianceof the trend calculated by linear regression; thelatter was found to be 64% greater. The simulatedsamples from a standard Gumbel distribution weregiven superimposed linear trends of different magnitudes,and the efficacy of three trend-testingmethods viz., Maximum Likelihood, Linear Regression,and the nonparametric Mann-Kendall Testwas compared. The ML Test was found alwaysmore powerful than the Linear Regression orMann-Kendall Test regardless of the value (positive)of the trend β; the MK Test was found leastpowerful for all the values of β.Ducré-Robitaille et al. (2003) evaluated eighthomogenization techniques for the detection ofdiscontinuities in the temperature series using simulateddatasets reproducing a vast range of possiblesituations. The simulated data represented homogeneousseries and the series having one or moresteps. Although the majority of the techniques consideredin this study performed very well, twomethods are reported to work slightly better thanthe others: the standard normal homogeneity testwithout trend, and the multiple linear regressiontechnique.Yue and Wang (2004) proposed effective samplesize (ESS) to modify the MK statistic for eliminatingthe effect of serial correlation on the MK Test.The Monte Carlo simulation indicated that when notrend exists within time series, ESS can effectivelylimit the effect of serial correlation on the MK Test.When trend exists within time series, the existenceof trend will contaminate the estimate of the magnitudeof sample serial correlation, and the ESS computedfrom the contaminated serial correlation cannotproperly eliminate the effect of serial correlationon the MK Test. However, if the ESS is computedfrom the sample serial correlation that is estimatedfrom detrended series, the ESS can stilleffectively reduce the influence of serial correlationon the MK Test.5. Application of time series analysisin climatology5.1 Precipitation/precipitation with other dataFortuniak (1995) used the daily precipitation totalsand mean daily temperature for the period 1956– 1990 from 10 Polish meteorological stations(Gdansk, Szczecin, Suwalki, Poznan, Lódz, Warszawa,Wroclaw, Kraków, Przemysl, Zakopane) totest their periodicity. The annual course of temperaturewas removed by subtracting the 35th Fourierharmonic. The classical Blackman and Tukey Testwas used to detect the cyclic behaviour of the analysedseries. The power spectrum of temperaturefor each station exhibited two significant peaks:around 7.4 years and 193 days. For the precipitation,the power spectra were found different foreach station and it was impossible to find a characteristiccycle for the entire region.Nieplová (1995) applied five statistical homogeneitytests (Student's, Bartlett's, Kruskal-Wallis's,Abbe Criterion, Spearman Rank Correlation Test)and the Craddock's Relative Homogeneity Test tothe annual and monthly air temperature means,precipitation totals and relative air humidity meansof 40 years and longer series. It was found thatmost of inhomogeneities were caused by changedobservation terms and by the relocation of measuringstations. These results were used for selectingstations for long-term monitoring of climate changein Slovakia.Walanus-Gliwice (1995) analyzed the periodicityby using the Fast Fourier Transform (FFT). Thewater stages of Vistula River at Szczucin, dischargesof Warta (Poland) and Tisa (Hungary) Rivers,Dnieper River (Ukraina), precipitation fromWarsaw, Cracow, Wroclaw and other towns, airtemperature, dendroclimatological curves and thethickness of yearly strata (warws) from GosciazLake (Central Poland) were analyzed. The 3.5-year(3.5±0.15 yr) periodicity of unknown origin in theriver discharge and the precipitation was confirmed.For rivers, the 3.5-yr signal was found much less inSzczucin, but it was visible. For the precipitation,the signal was still less, especially in comparison tothe more dominant seasonal periodicity. The perio-243
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D. Machiwal, M. K. Jhalarge and that they have been obtained under similarconditions.(xi) The necessary condition for applying the RunTest on Successive Differences is that the observationsin the sample should be obtained under similarconditions.(xii) The Wilcoxon-Mann-Whitney Test is a nonparametrictest (i.e., distribution-free) and is applicableonly when the observations are random andindependent.(xiii) The Kendall’s Rank Correlation Test is themost popular test for trend detection in the hydrologictime series.(xiv) The Mann-Kendall Test is a nonparametrictest for trend detection in a time series withoutspecifying whether the trend is linear or nonlinear.Existence of serial correlation in a time series willaffect the ability of the Mann-Kendall Test to assessthe site significance of a trend, and the presenceof cross correlation among sites in a networkwill influence the ability of the test to evaluate thefield significance of trends over the network (Yue etal., 2003).(xv) In general, the parametric methods to assesssignificance of trend employ pre-specified modelsand associated tests, whereas the nonparametricmethods generally apply rank tests to the data. Neitherapproach is suitable for exploratory analysis(Ramesh and Davison, 2000).(xvi) The assumptions of the classical parametrictests viz., normality, linearity, and independenceare usually not met by the hydrological time seriesdata, especially in case of surface water qualitydata. Therefore, recently some nonparametric testshave been proposed to determine the trend in surfacewater quality time series (Kalayci and Kahya,1998). At the same time, the statistical tests fortrend detection in surface water quality time seriesare normally confounded by one or more of thefollowing problems: missing values, censored data,flow relatedness, and seasonality.4. Theoretical research on time seriesanalysis techniquesHamed and Rao (1998) studied the effects ofautocorrelation on the variance of the Mann-Kendall trend test-statistic. A theoretical relationshipwas derived to calculate the variance of theMann- Kendall test statistic for autocorrelated data.The special cases of AR(1) and MA(1) dependencewere discussed as examples. Based on the modifiedvalue of the variance of the Mann-Kendall trendtest statistic, a modified nonparametric trend testsuitable for the autocorrelated data is proposed. Themodified test was applied to rainfall and streamflowdata to demonstrate its performance compared tothe original Mann-Kendall Trend Test. The accuracyof the modified test was found to be superiorto that of the original Mann-Kendall Trend Testwithout any loss of power.Tsakalias and Koutsoyiannis (1999) developed anew approach for the computer-aided explorationand analysis of hydrologic time series with a focuson identification of multiple stage-discharge relationshipsin a river section, analyses for homogeneityand temporal consistency, detection of outliers,shifts and trends. To demonstrate the developedmethodology, initially a mathematical representationwas proposed based on the set theory. It wasdemonstrated that an exhaustive search of all candidatesolutions is intractable. Therefore, a heuristicalgorithm is proposed, which emulates the exploratorydata analysis of the human expert. This algorithmencodes a number of search strategies in apattern directed computer program, and results inan automatic determination of a satisfactory solution.Yue et al. (2002a) investigated the interaction betweena linear trend and a lag-one autoregressive[AR(1)] model using Monte Carlo simulation.Simulation analysis indicated that the existence ofserial correlation alters the variance of the Mann-Kendall (MK) statistic estimate, and the presence ofa trend alters the magnitude of serial correlation.Furthermore, it was found that the commonly usedpre-whitening procedure for eliminating the effectof serial correlation on the MK Test leads to inaccurateassessments of the significance of a trend.Therefore, it was suggested that firstly trend shouldbe removed prior to ascertaining the magnitude ofserial correlation. Both the suggested approach andthe existing approach were employed to assess thesignificance of a trend in the serially correlatedannual mean and annual minimum streamflow dataof some pristine river basins in Ontario, Canada. Itwas concluded that the researchers might have incorrectlyidentified the possibility of significanttrends by using the already existing approach.Yue et al. (2002b) studied the efficacy of the twononparametric rank-based statistical tests (theMann-Kendall Test and Spearman's Rho Test) byMonte Carlo simulation. These two tests were usedto assess the significance of trends in annual maximumstreamflow data of 20 pristine basins in Ontario,Canada. The results indicated that their effec-242
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