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. Jhametric tests require 5 to 35% more data than parametrictests (Bethea and Rhinehart, 1991).Very few studies are reported wherein t-test hasbeen used to examine the stationarity of hydrologictime series (e.g., Jayawardena and Lai, 1989). TheMann-Whitney Test for detecting a shift in themean or median of hydrological time series hasbeen applied by McCuen and James (1972), Lazaro(1976), Lettenmaier (1976), Helsel and Hirsch(1988), Kiely (1999), Kiely et al. (1998), Yue andWang (2002). Also, stationary stochastic modelssuch as AR (Auto Regressive), MA (Moving Average),or ARMA (Auto Regressive Moving Average)models are frequently used to characterize thestandardized time series (Hipel and McLeod, 1994).However, the standardization procedure does notensure stationarity in the transformed series (Salas,1993). Moreover, some researchers (Appel andBrandt, 1983; Lovell and Boashash, 1987; Imbergerand Ivey, 1991; Chen and Rao, 2002) have developedsegmentation algorithms to determine stationarysegments and to estimate the parameterscharacterizing each segment in order to establishpiecewise stationary time series models.2.3 TrendThere can be a variety of situations that wouldresult in the measured value of some climatologicalvariables changing over time, which in turn cause alinear or nonlinear trend in the time series of theclimatological variable. The trend in a time seriescan be expressed by a suitable linear or nonlinearmodel; the linear model is widely used in hydrology(Shahin et al., 1993). The simplest of lineartrend detection models is Student’s t-test (Hameedet al., 1997), which requires that the series undertesting should be normally distributed. Thus,whether or not the sample data follow a normaldistribution has to be examined prior in order toapplying the Student’s t-test to assess the statisticalsignificance of these two types of trends (Hoel,1954). Unfortunately, some researchers ignore thisimportant check (e.g., Fanta et al., 2001). If normalityis violated, the nonparametric test such asthe Mann-Kendall Test (Mann, 1945; Kendall,1975) is commonly applied to assess the statisticalsignificance of trends. This test detects a monotonictrend in the mean or median of a time series. Asmentioned earlier, the nonparametric tests are moresuitable for non-normal data and censored datacompared to the parametric t-test (Helsel andHirsch, 1988; Hirsch and Slack, 1984). The applicationof the nonparametric Mann-Kendall Test fordetecting monotonic trends in hydrological timeseries is reported by Hirsch et al. (1982), Hirschand Slack (1984), Burn (1994), Burn and Elnur(2002), Lettenmaier et al. (1994), Gan (1992,1998), Lins and Slack (1999), Douglas et al. (2000),Zhang et al. (2001), Yue et al. (2003), and others.Another important trend test is the Spearman RankOrder Correlation Test, which has been applied byKhan (2001) and Adeloye and Montaseri (2002).However, in some hydrologic studies, the Kendall’sRank Correlation Test has been preferred (Jayawardenaand Lai, 1989; Zipper et al., 1998;Kumar, 2003).Most of the tests (Turning Point, Kendall’sPhase, Wald-Wolfowitz Total Number of Runs,Sum of Square Lengths, Adjacency, DifferenceSign, Run Test on Successive Differences, Wilcoxon-Mann-Whitney,and Inversions tests) describedin Shahin et al. (1993) have not attractedthe attention of hydrologists, may be because of theavailability of some sound statistical trend detectiontests. Esterby (1996) and Hess et al. (2001) presentan excellent overview of the statistical methods fortrend detection and estimation in environmentaltime series (e.g., water quality and atmosphericdeposition monitoring data). Hess et al. (2001)evaluated six methods for trend detection usingreal-life data and provided recommendations basedon a simulation study. The t-test adjusted for theseasonality and the Seasonal Kendall tests are reportedto be more powerful than the remaining fourtests viz., the Spearman Partial Rank CorrelationTest, Ordinary Least Square Regression, GeneralizedLeast Square Regression, and the Kolmogorov-ZurbenkoTest. However, Hess et al. (2001)did not consider all the available trend detectiontests, which are sound and widely employed in thehydrologic time series analysis. Some more statisticaltests can be found in Mahé et al. (2001).2.4 PeriodicityPeriodicity in the hydrologic time series can bedetected if the time series are defined at time intervalsless than a year. In most cases, a periodicity of6 and 12 months is very common. The Fourier serieshas been primarily used for the detection ofperiodic components in the hydrologic time series(e.g., Maidment and Parzen, 1984; Kite, 1989; Jayawardenaand Lai, 1989; Fernando and Jayawardena,1994; Pugacheva et al., 2003). However,some researchers (e.g., Hurd and Gerr, 1991; Vec-240
Time series analysis of hydrologic data for water resources planning and management: a reviewchia and Ballerini, 1991) have suggested differentmethods for testing the periodically correlated timeseries.2.5 Persistence/non-randomnessIn few hydrologic time series studies, no distinctionis made between persistence and randomness.Therefore, the tests to examine the randomness of ahydrologic time series are used for detecting bothtrend and persistence. Generally, randomness ornon-persistence is defined as the independenceamong data in a hydrological time series. On thecontrary, the series is called persistent if the data inthe series are dependent on each other. Practically,persistence is a tendency of the successive values ofa climatological series to ‘remember’ their antecedentvalues, and to be influenced by them (Gilesand Flocas, 1984). Mathematically, persistence isdefined as the correlational dependency of order kbetween each i th element and the (i–k) th element ofthe series (Kendall, 1973), and is measured byautocorrelation (i.e., a correlation between the twoterms of the same time series). Here, ‘k’ is usuallycalled time lag. The detection of persistence can bemade by autocorrelation technique (time domain)and/or spectral technique (frequency domain).However, the autocorrelation technique has beenapplied in several studies such as Mirza et al.(1998), Maidment and Parzen (1984), Schwankl etal. (2000), etc. Here it is worth mentioning thatsome researchers (e.g., Jayawardena and Lai,1989) have used the autocorrelation technique fortesting the periodicity in time series. Such a misconceptionis quite common in the hydrologic timeseries analysis.3. Salient merits and demeritsof the time series tests(i) Cumulative Deviations Test is superior to theclassical von Neuman Test for a model with onlyone change in the mean (Buishand, 1982).(ii) The major limitation with all the multiple comparisontests of homogeneity (i.e., Tukey, Link-Wallace, Dunnett, Bartlett and Hartley tests) is therequirement that populations should be normallydistributed with equal variances, which makes thetests parametric in nature. Though the Link-Wallace Test, the Dunnett’s Test and the Hartley’sTest can be employed for the same purpose as theTukey’s Test, the former two tests have the limitationthat the sample size of all populations must beequal, while the Hartley’s Test is applicable to approximatelyequal sample size.(iii) Of the three stationarity tests, both the t-testsare parametric and the Mann-Whitney Test is nonparametricin nature.(iv) Among the trend tests, although the linearmodel (i.e., Regression Test) is most commonlyused, it has a demerit that it does not distinguishbetween trend and persistence. The test can also bemisleading if seasonal cycles are present, the dataare not normally distributed, and/or the data areserially correlated.(v) The Spearman Rank Order Correlation Testovercomes the problem associated with the linearmodel. Another advantage of this test is its nearlyuniform power for detecting linear as well as nonlineartrends (WMO, 1966; Dahmen and Hall,1990).(vi) The Turning Point Test is easy to apply, especiallywhen the time series is plotted graphically. Itis an effective test for randomness against systematicoscillation. However, if the turning points tendto bunch together, the Kendall’s Phase Test is morerelevant (Shahin et al., 1993). Here, the difficulty isthat a comparison of observed and theoretical numbersof phases by the usual chi-square test is invalidateddue to the fact that the lengths of phases arenot independent. Also, the distribution of phaselengths does not tend to be normal for large lengthsof a series, but the number of phases follows anormal distribution (Kendall, 1973).(vii) Among the trend tests, the superiority of oneover other is mainly associated with the extent ofadaptability of a chosen test to the structure of thetime series to be tested. The Turning Points andNumber of Phases tests are practically out-dateddue to the availability of much more powerful tests(Shahin et al., 1993).(viii) The Wald-Wolfowitz Test does not take intoaccount the length of runs and considerable informationis ignored. Hence, this test is not very powerfulnor efficient, but can be used to determinewhether observations of a random variable are independent(if they are, and there is no trend). TheSum of Squared Lengths Test is a more powerfultest (Himmelblau, 1969).(ix) The limitation of the Adjacency Test is thebasic assumptions that the observations are obtainedindependently and under similar conditions(Kanji, 2001).(x) The Difference Sign Test is applied with theassumptions that the number of observations is241
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D. Machiwal, M. K. Jhametric tests require 5 to 35% more data than parametrictests (Bethea and Rhinehart, 1991).Very few studies are reported wherein t-test hasbeen used to examine the stationarity of hydrologictime series (e.g., Jayawardena and Lai, 1989). TheMann-Whitney Test for detecting a shift in themean or median of hydrological time series hasbeen applied by McCuen and James (1972), Lazaro(1976), Lettenmaier (1976), Helsel and Hirsch(1988), Kiely (1999), Kiely et al. (1998), Yue andWang (2002). Also, stationary stochastic modelssuch as AR (Auto Regressive), MA (Moving Average),or ARMA (Auto Regressive Moving Average)models are frequently used to characterize thestandardized time series (Hipel and McLeod, 1994).However, the standardization procedure does notensure stationarity in the transformed series (Salas,1993). Moreover, some researchers (Appel andBrandt, 1983; Lovell and Boashash, 1987; Imbergerand Ivey, 1991; Chen and Rao, 2002) have developedsegmentation algorithms to determine stationarysegments and to estimate the parameterscharacterizing each segment in order to establishpiecewise stationary time series models.2.3 TrendThere can be a variety of situations that wouldresult in the measured value of some climatologicalvariables changing over time, which in turn cause alinear or nonlinear trend in the time series of theclimatological variable. The trend in a time seriescan be expressed by a suitable linear or nonlinearmodel; the linear model is widely used in hydrology(Shahin et al., 1993). The simplest of lineartrend detection models is Student’s t-test (Hameedet al., 1997), which requires that the series undertesting should be normally distributed. Thus,whether or not the sample data follow a normaldistribution has to be examined prior in order toapplying the Student’s t-test to assess the statisticalsignificance of these two types of trends (Hoel,1954). Unfortunately, some researchers ignore thisimportant check (e.g., Fanta et al., 2001). If normalityis violated, the nonparametric test such asthe Mann-Kendall Test (Mann, 1945; Kendall,1975) is commonly applied to assess the statisticalsignificance of trends. This test detects a monotonictrend in the mean or median of a time series. Asmentioned earlier, the nonparametric tests are moresuitable for non-normal data and censored datacompared to the parametric t-test (Helsel andHirsch, 1988; Hirsch and Slack, 1984). The applicationof the nonparametric Mann-Kendall Test fordetecting monotonic trends in hydrological timeseries is reported by Hirsch et al. (1982), Hirschand Slack (1984), Burn (1994), Burn and Elnur(2002), Lettenmaier et al. (1994), Gan (1992,1998), Lins and Slack (1999), Douglas et al. (2000),Zhang et al. (2001), Yue et al. (2003), and others.Another important trend test is the Spearman RankOrder Correlation Test, which has been applied byKhan (2001) and Adeloye and Montaseri (2002).However, in some hydrologic studies, the Kendall’sRank Correlation Test has been preferred (Jayawardenaand Lai, 1989; Zipper et al., 1998;Kumar, 2003).Most of the tests (Turning Point, Kendall’sPhase, Wald-Wolfowitz Total Number of Runs,Sum of Square Lengths, Adjacency, DifferenceSign, Run Test on Successive Differences, Wilcoxon-Mann-Whitney,and Inversions tests) describedin Shahin et al. (1993) have not attractedthe attention of hydrologists, may be because of theavailability of some sound statistical trend detectiontests. Esterby (1996) and Hess et al. (2001) presentan excellent overview of the statistical methods fortrend detection and estimation in environmentaltime series (e.g., water quality and atmosphericdeposition monitoring data). Hess et al. (2001)evaluated six methods for trend detection usingreal-life data and provided recommendations basedon a simulation study. The t-test adjusted for theseasonality and the Seasonal Kendall tests are reportedto be more powerful than the remaining fourtests viz., the Spearman Partial Rank CorrelationTest, Ordinary Least Square Regression, GeneralizedLeast Square Regression, and the Kolmogorov-ZurbenkoTest. However, Hess et al. (2001)did not consider all the available trend detectiontests, which are sound and widely employed in thehydrologic time series analysis. Some more statisticaltests can be found in Mahé et al. (2001).2.4 PeriodicityPeriodicity in the hydrologic time series can bedetected if the time series are defined at time intervalsless than a year. In most cases, a periodicity of6 and 12 months is very common. The Fourier serieshas been primarily used for the detection ofperiodic components in the hydrologic time series(e.g., Maidment and Parzen, 1984; Kite, 1989; Jayawardenaand Lai, 1989; Fernando and Jayawardena,1994; Pugacheva et al., 2003). However,some researchers (e.g., Hurd and Gerr, 1991; Vec-240
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