Identification of trends in Malaysian monthly runoff under the scaling hypothesis

Abstract

Statistical tests have been widely used for several decades to identify and test the significance of trends in runoff and other hydrological data. The Mann-Kendall (M-K) trend test is commonly used in trend analysis. The M-K test was originally proposed for random data. Several variations of the M-K test, as well as pre-processing of data for use with it, have been developed and used. The M-K test under the scaling hypothesis has been developed recently. The basic objective of the research presented in this paper is to investigate the trends in Malaysian monthly runoff data. Identification of trends in runoff data is useful for planning water resources projects. Existence of statistically significant trends would also lead to identification of possible effects of climate change. Monthly runoff data for Malaysian rivers from the past three decades are analysed, in both five-year segments and entire data sequences. The five-year segments are analysed to investigate the variability in trends from one segment to another in three steps: (1) the M-K tests are conducted under random and correlation assumptions; (2) the Hurst scaling parameter is estimated and tested for significance; and (3) the M-K test under the scaling hypothesis is conducted. Thus the tests cover both correlation and scaling. The results show that the number of significant segments in Malaysian runoff data would be the same as those found under the assumption that the river flow sequences are random. The results are also the same for entire sequences. Thus, monthly Malaysian runoff data do not have statistically significant trends. Hence there are no indications of climate change in Malaysian runoff data.

Citation Rao, A. R., Azli, M. & Pae, L. J. (2011) Identification of trends in Malaysian monthly runoff under the scaling hypothesis. Hydrol. Sci. J. 56(6), 917–929.

Résumé

Les tests statistiques ont été largement utilisés depuis plusieurs décennies en vue d'identifier et de tester la signification des tendances des débits et d'autres données hydrologiques. Le test de tendance de Mann-Kendall (M-K) est couramment utilisé pour ces analyses. Le test de M-K a été initialement proposé pour des données aléatoires. Plusieurs variantes du test M-K, ainsi que des prétraitements des données pour son utilisation, ont été développés et utilisés. Le test de M-K sous l'hypothèse de scalance a été développé récemment. L'objectif fondamental de la recherche présentée dans cet article est d'étudier les tendances dans les données mensuelles des débits de Malaisie. L'identification des tendances des débits est utile pour la planification des projets de ressources en eau. L'existence de tendances statistiquement significatives permettrait également de mettre en évidence de possibles conséquences du changement climatique. Les données mensuelles de débit des rivières malaises des trois dernières décennies ont été analysées, par segments de cinq ans et pour la séquence complète. Les segments de cinq ans ont été analysés pour étudier la variabilité dans l'évolution d'un segment à l'autre selon trois étapes: (1) les tests de M-K sont appliqués sous les hypothèses aléatoires et de corrélation, (2) le paramètre d'échelle de Hurst est estimé et testé pour la signification, et (3) le test de M-K sous l'hypothèse de scalance est appliqué. Ainsi, les essais couvrent à la fois la corrélation et la mise à l'échelle. Les résultats montrent que le nombre de segments significatifs pour les données de débits malaises serait le même que celui trouvé dans l'hypothèse où les séquences de débits seraient aléatoires. Les résultats sont identiques pour les séquences complètes. Ainsi, les données mensuelles de débits de Malaisie n'ont pas de tendances statistiquement significatives. Par conséquent il n'y a aucune indication du changement climatique dans les données de débits de Malaisie.

Key words:

• climate change
• monthly rainfall
• runoff data
• trend analysis
• Malaysia

Mots clefs:

• changement climatique
• précipitations mensuelles
• données de ruissellement
• analyse des tendances
• Malaisie

INTRODUCTION

It is important and sometimes even essential to study trends in runoff, because changes in runoff affect water and food supply. Knowledge of trends in runoff would be of considerable use in planning and management of water supply. Climate change would affect runoff, and hence water supply. Consequently, trend analysis of runoff is of considerable importance.

In recognition of this importance, trends in runoff data have been widely investigated. Trend tests and other methods have been used to investigate impacts of climate change and variability in runoff time series in different parts of the world. In the United States, Lettenmaier et al. (1994), Lins and Slack (1999), and Douglas et al. (2000) studied trends in river flows. Over the years a number of studies have been conducted in Canada. Westmacott and Burn (1997), Yulianti and Burn (1998), Ouarda et al. (1999), Whitfield and Cannon (2000), Zhang et al. (2001), Burn and Hag Elnur (2002), Yue et al. (2003), Rood et al. (2005), Dery and Wood (2005) and Abdul Aziz and Burn (2006) have investigated trends in Canadian streamflows. In Mexico, Molnar and Ramirez (2001) have investigated streamflows. In the UK, Robson (2002), Hannaford and Marsh (2006) and Dixon et al. (2006) have studied trends in streamflows. In Australia, Chiew and McMahon (1993); in Europe, Hisdal et al. (2001), Wang et al. (2005), Birsan et al. (2005); in Turkey, Kahya and Kalayaci (2004); and in China, Xiong and Shenglian (2004) and Zhang et al. (2006) have investigated different characteristics of streamflows. Kundzewicz et al. (2005) and Svensson et al. (2005) have analysed river flow data from around the world. This list is not comprehensive, but is indicative of the extensive activity in trend analysis of runoff data. Most of these studies have been conducted in Europe and North America. Relatively few studies of trend analysis of runoff appear to have been conducted in Asia, although recently there has been some activity. As far as we can find there are no trend analyses of Malaysian runoff.

Several methods of trend analysis have been in use. The Mann-Kendall (M-K) trend test (Mann 1945, Kendall 1975) has been widely used in trend studies. The Spearman rank correlation test, which is based on the Spearman rank correlation coefficient (Dahmen and Hall 1990), has been used as an alternative to the M-K test. Although Sen's (1968) slope has been used to assess the magnitude of the slope of trends, it does not provide a statistical test. The linear least squares test (Haan 1977) is used to test the statistical significance of the slope of the fitted regression line.

Cox and Stuart (1955) discussed the important role which serial correlation plays in trend tests. The technique of modifying trend tests to account for the effect of serial correlation in the data has been used by several investigators. Lettenmaier (1976), Hirsch and Slack (1984) were early investigators who have used this approach. Hamed and Rao (1998) introduced a correction factor to the variance used in the M-K trend test. This correction factor is based on the correlations of ranks of the data. Yue and Wang (2004) proposed a correction factor for the variance in the M-K test, which is based on the correlation coefficients of the data. Matalas and Sankarasubramanian (2003) investigated the effect of persistence on trend detection via regression. They also gave expressions for the variance inflation factor. Cohn and Lins (2005) have given a modified likelihood ratio test for testing data in the presence of long-term persistence. Because monthly runoff data are usually correlated, using trend tests which consider correlation in data is especially important in the analysis of monthly runoff data.

The importance of inhomogeneities in data and its effects are discussed in detail by Peterson et al. (1998). The runoff data used in this study has been checked by the Department of Irrigation and Drainage (DID), Malaysia. In rainfall data, inhomogeneities are introduced by changing rain gauge locations (Peterson et al., 1998). Such changes are much less frequent with runoff data because runoff gauging stations are relocated much less frequently. Usually the original data are not released to others who are not associated with the governmental agency collecting the data. Consequently, the data were inspected by us. No obvious inhomogeneities were found in the data.

Recently, Hamed (2008) modified the M-K test to consider the effect of scaling. Scaling and modelling of hydrological data are discussed in Koutsoyiannis (2006). Scaling is a natural way of considering the variability in hydrological time series. Hamed (2008) demonstrated that considering the scaling effect would explain inconsistencies reported in trend studies. Contradictory results, such as highly significant increasing and decreasing trends in the same time series, are explained by the scaling effect. These contradictory results are shown to be a regular feature of time series if scaling assumption is considered in the analysis (Koutsoyiannis 2003, 2006, Koutsoyiannis et al. 2008). There are only a few cases where trends are tested under the scaling hypothesis (Hamed 2008, Kumar et al. 2009). In view of these findings, testing trends under the scaling hypothesis is quite important.

The basic objective of research discussed in this paper is to analyse trends in Malaysian runoff data under the scaling hypothesis. As using the trend test under the scaling hypothesis is a relatively new procedure, the results are of interest. The significant trends found by this study would be useful for water resources planning in Malaysia. These results would also be useful in understanding the effects of climate changes on runoff in Malaysia.

DATA USED IN THE STUDY

Monthly data from 13 rivers in Malaysia, acquired from the Department of Irrigation and Drainage (DID), Malaysia, were used in this study. The names and locations of the rivers, and the watershed areas are given in Table 1. Six of the rivers are in Peninsular Malaysia and seven in East Malaysia. The larger rivers are in East Malaysia. The locations of the gauging stations are shown in Fig. 1.

Table 1  Runoff data from rivers in Malaysia

Station no.	River	Watershed area (km^2)	Duration	N
1105401	Sadong River, Sarawak	3543	1978–1997	20
1737451	Johor River, Johor	1130	1978–2007	30
1932408	Balleh River, Sarawak	12416	1978–2007	30
2519421	Linggi River, Negri Sembilan	523	1978–2003	26
3152408	Baram River, Sarawak	22209	1978–2005	28
3519426	Bentong River, Pahang	241	1978–2007	30
4310401	Kinta River, Perak	1700	1978–2002	25
4959401	Padas River, Sabah	3185	1983–2007	25
5074401	Kuamut River, Sabah	2950	1978–2007	30
5721442	Kelantan River, Kelantan	11900	1978–2007	30
5760401	Papar River, Sabah	365	1978–2007	30
5806414	Muda River, Kedah	1710	1978–2007	30
6264401	Kadamaian River, Sabah	308	1978–2007	30

Fig. 1 Locations of gauging stations in Malaysia.

The data are selected so that there is at least one river represented in each of the states in Malaysia. Most of these rivers are in pristine condition without any dams or other major diversionary structures on them.

The data duration ranges from 20 to 30 years, starting from 1978 and ending at different years, as shown in Table 1. An attempt was made to obtain the longest series possible for each river. Some of these rivers do not have 30 years of data, so the longest available series was used. Significant trends in runoff could shed some light on the effects of global warming on Malaysian runoff. Two plots of river flows are shown in Fig. 2 as examples: the first is of Baram River in Sarawak (3152408) and the second of Papar River in Sabah (5760401). The seasonality and trends in Malaysian river flows are seen in Fig. 2.

Fig. 2 River flow data for Baram River (3152408) for 1978–2005 and Papar River (5760401) for 1978–2007.

Data from five-year segments for a river as well as the entire sequence of data are investigated. The five-year segments are analysed to investigate the variation in trends from one five-year period to the next. The entire data sequences are analysed to investigate the trends in long-period data.

TREND TESTS USED IN THE STUDY

Procedures proposed by Hamed (2008) are used in this study, with one exception: Hamed (2008) did not use the modified M-K test for correlated data, whereas it is used in the present study. Because monthly runoff data are usually correlated, the modified M-K test (Hamed and Rao, 1998) which accounts for correlation in the data is used with the procedure proposed by Hamed (2008).

The trend analysis is conducted in three steps. In the first step, the M-K (Kendall 1975) and the modified M-K (Hamed and Rao 1998) tests are used to test the statistical significance of trends. If the data have significant trends, they are further tested in the second and third steps. In the second step, Hurst's H is estimated and tested for statistical significance. If H is significant and if the data have significant trends according to M-K and modified M-K tests, the third step is carried out to test the trends under the scaling hypothesis. The conclusions at each of the steps lead to inferences about trends in the data. At each stage, the statistical significance is tested by using three significance levels (10%, 5% and 2.5%) to examine the robustness of the inferences. This procedure is necessary because both the correlation in the data and the scaling affect inferences about trends. The details of the tests are found in Hamed (2008) and Hamed and Rao (1998). Therefore, only the test procedure is briefly given in the following discussion.

TEST PROCEDURE

Mann-Kendall test

Consider a time series X = [x ₁, x ₂, …, x_n ]. The test statistic S is computed by:

(1)

where

(2)

In equation (2)(2) R_i and R_j are the ranks of observations x_i and x_j of the time series, respectively. Assuming that the data are independent and identically distributed, Kendall (1975) showed that the expected value and the variance of S read, respectively:

(3)

The significance of trends is tested by comparing the standardised variable u ₁ in equation (4)(4) with the standard normal variate at significance α (Kendall, 1975):

(4)

If u ₁ is not significant, the observed trend is not significant. If u ₁ is significant, the observed trend is significant.

When data are correlated, positive serial correlation results in an increase in the rate of false identification of trends, even when no trends exist in the data. This is due to underestimation of V ₀(S) when autocorrelation is ignored. Therefore, when u ₁ is significant, the data are tested again by using the modified M-K test (Hamed and Rao 1998), which considers correlation in the data.

Modified Mann-Kendall test

The variance V ₀(S) in equation (3)(3) is recalculated as V*(S) by:

(5)

where (n/n_s*) represents a correction to V ₀(S) due to autocorrelations in the data.

The approximation for (n/n_s*) used is the empirical expression:

(6)

where ρ _s (i) are the autocorrelation coefficients of the ranks of the data. The expression for ρ _s (i) is given by:

(7)

where N is the number of ranks of observations.

As the ranks of observations are used in equation (6)(6), V*(S) is computed without using either the data or their autocorrelation functions. In the present study, significant correlation coefficients up to N/10 of N ranks are used. The modified statistic u ₂ is used in the significance test:

(8)

If the decision is that u ₂ is insignificant, then it is concluded that there is no trend in the data. If u ₂ is significant, then the data are tested under the scaling hypothesis. The trend test under the scaling hypothesis starts with computation of H.

Computation of H

The data are detrended by using Sen's (1968) non-parametric trend estimator S ₀, which is given by equation (9)(9). There may be periodicities in monthly data. In this study, the periodicities in monthly runoff data are removed by subtracting the monthly mean from the data and dividing the resulting value by the monthly standard deviation.

(9)

The equivalent normal variates, z_t , computed by using the transformation in equation (10)(10), are used to form the vector z:

(10)

The scaling coefficient H is obtained by maximising the log-likelihood function (McLeod and Hipel 1978):

(11)

where γ₀ is the variance of z _t , and C _n(H) is the correlation matrix of H given by:

(12)

(13)

The estimate of H computed by equation (11)(11) is approximately normally distributed for the uncorrelated case when true H is 0.5, with the mean and variance given by (Hamed 2008):

(14)

The significance of H is tested by using μ _H and σ _H in equation (14)(14). If H is not significant, then the decision of the M-K or the modified M-K test is accepted. If H is significant, the trend test is conducted under the scaling hypothesis.

Mann-Kendall test under the scaling hypothesis

The modified variance of the test statistic V(S) is computed by:

(15)

where:

(16)

The variance V(S) in equation (15)(15) is corrected for bias by multiplying it with the bias correction factor B:

(17)

The coefficients a ₀, a _1, ..., a ₄ in equation (17)(17) are functions of the sample size n. These are found in Hamed (2008). The modified test statistic u ₃ is computed by using the modified variance and equation (4)(4). If u ₃ is significant, then the trend is significant; otherwise, it is not. Thus, both correlation and the scaling effects are considered in these tests.

RESULTS AND DISCUSSION

Analysis of five-year segments of data

Results from five-year segments of data are presented first; 72 five-year segments are analysed to examine how the results vary from one segment to another in the same river. Results from two runoff sequences are presented, as the discussion of results from all 72 segments would be repetitive. The entire data are also analysed and the results are presented next.

Mann-Kendall test

The statistics S, V ₀(S), and u ₁ of the M-K test for two stations are presented in Table 2. The statistic u ₁ changes sign often, indicating that the trends change from increasing to decreasing from one segment to another. Even when there are sustained trends, such as in Kadamaian River in Sabah from 1993 to 2007, all the trends are not statistically significant. The significance of trends is also given in Table 2. The reason for the variability of the statistic u ₁ is clear if the variability in the statistic S is examined. Examples of variability of S along with 95% confidence intervals are shown in Fig. 3. Frequently, the statistic S changes from increasing to decreasing states, even crossing the confidence interval, as seen in Fig. 3. Hence the trend also changes.

Fig. 3 Variation of S with time of an entire sequence (Papar River, 5760401) and of a 5-year segment (Kadamaian River, 6264401) along with 95% confidence limits.

Table 2  Results from the original Mann-Kendall test for stations 3519426 and 6264401 for five-year segments (V ₀(S) = 24 583)

Station no.	Period	S	u 1	Significance
3519426	1978–1982	55	0.344
	1983–1987	30	0.185
	1988–1992	−621	−3.954*	10%, 5%, 2.5%
	1993–1997	468	2.978*	10%, 5%, 2.5%
	1998–2002	−842	−5.364*	10%, 5%, 2.5%
	2003–2007	−58	−0.364
6264401	1978–1982	64	0.402
	1983–1987	57	0.357
	1988–1992	−268	−1.703*	10%, 5%
	1993–1997	208	1.320*	10%
	1998–2002	27	0.166
	2003–2007	426	2.711*	10%, 5%, 2.5%
*Values marked with an asterisk are significant at levels indicated in the last column.

The total numbers of significant trends given by this test for the 72 segments at 10%, 5% and 2.5% significance levels are 32, 28 and 20, respectively. The expected number and confidence interval of the number of segments with significant trends only due to chance in 72 segments at a specified significance level may be calculated by using the binomial distribution. At 10% level, the expected number of significant segments is seven. The upper and lower confidence limits for the expected number of 7 are 3 and 12, respectively. The actual number of significant trends for these data at the 10% level is 32, which is almost three times the number due to chance. For independent segments, the maximum numbers of expected trends at the above stated significance levels are 12, 8 and 5, respectively. The actual number of trends is much higher than the number due to chance. But these trends are affected by correlation and scaling effects. Therefore, the trends are re-estimated by the modified M-K test in which the effect of correlation is considered.

Modified Mann-Kendall test

The statistics V*(S), V*(S)/V ₀(S), and u ₂ from the modified M-K test are given in Table 3. The strong effect of the correlation in the data is brought about by the differences in the results in Tables 2 and 3. In the modified M-K test, the total numbers of significant trends for the 72 segments at 10%, 5% and 2.5% significance levels are 27, 15 and 12, respectively. These correspond to reductions of 15.6%, 46.4% and 40%, respectively. The reduction in the number of significant trends is solely due to the correlation in the data. Examples of correlograms are shown in Fig. 4. In the correlogram of Bentong River in Pahang (3519426), because of a large baseflow component in the streamflow, the correlation coefficients remain high for large lags. However, the actual numbers of significant trends are still higher than the numbers expected due to chance which are 12, 8, and 5.

Fig. 4 Correlograms of a five-year segment (Bentong River, 3519426) and of an entire sequence (Papar River, 5760401) with two standard deviation limits.

Table 3  Results from the modified Mann-Kendall test for stations 3519426 and 6264401 for five-year segments

Station no.	Period	V*(S)	V*(S)/V 0(S)	u 2	Significance
3519426	1978–1982	27 332	1.112	0.333
	1983–1987	72 733	2.959	0.111
	1988–1992	48 400	1.969	−2.823*	10%, 5%, 2.5%
	1993–1997	180 374	7.337	1.102
	1998–2002	139 907	5.691	−2.251*	10%, 5%, 2.5%
	2003–2007	48 495	1.973	−0.263
6264401	1978–1982	44 093	1.794	0.305
	1983–1987	63 454	2.581	0.226
	1988–1992	46 980	1.911	−1.236
	1993–1997	60 239	2.450	0.847
	1998–2002	40 091	1.631	0.135
	2003–2007	38 665	1.573	2.166*	10%, 5%, 2.5%
*Values marked with an asterisk are significant at levels indicated in the last column.

All u ₂ values are smaller than u ₁ values, reflecting the effect of correlation on the statistic u ₂. These segments with significant values of u ₂ are tested next under the scaling hypothesis.

Estimation and testing the scaling parameter H

The Hurst H is estimated by the maximum likelihood method. The mean μ _H and standard deviation σ _H , computed by equation (14)(14), are given in Table 4. The significance of H is tested and the results are also given in Table 4. With the exception of the segment corresponding to years 2003–2007 in the Kadamaian River, all other segments have significant H values. The results of the modified M-K test are accepted for the 2003–2007 segment of the Kadamaian River. Those segments which gave significant results in the trend test and which have significant H value are tested under the scaling hypothesis. At 10%, 5% and 2.5% significance levels, 66, 65 and 62 segments out of 72 had significant H values. Therefore, a high percentage of the five-year segments indicate that they must be tested under the scaling hypothesis.

Table 4  Results from the estimation of the Hurst parameter for stations 3519426 and 6264401 for five-year segments (μ _H  = 0.430; σ _H  = 0.094)

Station no.	Period	H 0	H	Significance
3519426	1978–1982	0.70	0.70*	10%, 5%, 2.5%
	1983–1987	0.97	0.98*	10%, 5%, 2.5%
	1988–1992	0.80	0.71*	10%, 5%, 2.5%
	1993–1997	0.98	0.98*	10%, 5%, 2.5%
	1998–2002	0.97	0.95*	10%, 5%, 2.5%
	2003–2007	0.84	0.87*	10%, 5%, 2.5%
6264401	1978–1982	0.82	0.82*	10%, 5%, 2.5%
	1983–1987	0.90	0.89*	10%, 5%, 2.5%
	1988–1992	0.82	0.80*	10%, 5%, 2.5%
	1993–1997	0.73	0.69*	10%, 5%, 2.5%
	1998–2002	0.80	0.80*	10%, 5%, 2.5%
	2003–2007	0.68	0.54
*Values marked with an asterisk are significant at levels indicated in the last column.

The estimates H and H ₀ are close to each other. Removal of the trend has little effect on the estimate of H. In other words, the trends in the data are weak. Also, quite a few of the H values are high. These high H values would strongly affect the statistic u ₃.

Trend test under the scaling hypothesis

The variance V(S) which is computed by using equation (15)(15), the variance inflation factor V(S)/V ₀(S) (Matalas and Sankarasubramanian, 2003), the bias correction factor B, and the statistic u ₃ are listed in Table 5. The variance inflation factor ranges from 0.378 to 9.054. The great majority (61 of 72) of the variance inflation factors are greater than unity and several are quite large. There is a high bias in the variance, as indicated by values of B which range from 1.771 to 3.531. These factors affect the statistic u ₃ so strongly that all but one of the u ₃ values are statistically insignificant. The single significant u ₃ value is from the 2003–2007 segment of the Kadamaian River. The flow for this segment, along with a trendline, is shown in Fig. 5. However, the H value for this segment is statistically insignificant and hence the result from the trend test under the scaling hypothesis is not valid. The previous trend test result of significant increasing flows is accepted for this segment.

Table 5  Results from the Mann-Kendall test under scaling hypothesis for stations 3519426 and 6264401 for five-year segments

Station no.	Period	V(S)	V(S)/V 0(S)	B	u 3	Significance
3519426	1978–1982	9 300	0.378	2.046	0.391
	1983–1987	38 530	1.567	3.531	0.079
	1988–1992	153 510	6.244	2.068	−1.101
	1993–1997	222 570	9.054	3.531	0.527
	1998–2002	175 160	7.125	3.240	−1.116
	2003–2007	15 820	0.644	2.659	−0.278
6264401	1978–1982	36 560	1.487	2.411	0.212
	1983–1987	25 180	1.024	2.781	0.212
	1988–1992	66 070	2.688	2.330	−0.680
	1993–1997	46 380	1.887	2.025	0.675
	1998–2002	44 210	1.798	2.330	0.081
	2003–2007	54 330	2.210	1.786	1.364*	10%

Fig. 5 Five-year 2003–2007 Kadamaian River flows (6264401) with a linear regression trend line.

Summary of results of the five-year segments

The summary of trend test results of the five-year segments is presented in Table 6. Ranges of different statistics are also presented in Table 6. When the ranges of u ₁ values are considered, they are negative for three rivers in Sarawak: Sadong River (1105401), Balleh River (1932408) and Baram River (3152408). The trend in Baram River is decreasing and is significant even when it is tested under scaling hypothesis. The only other river with a significant (positive) trend is the Papar River in Sabah (5760401). None of the rivers in Peninsular Malaysia have significant trends. Except for the three rivers in Sarawak, the five-year trends oscillate from increasing to decreasing and vice versa. Therefore the trends in river flows in East and Peninsular Malaysia are different. This aspect may be considered in future studies.

Table 6  Summary of test results for five-year segments

Station no.	Period	u 1	u 2	H 0	H	V(S)/V 0(S)	B	u 3
1105401	1978–1997	−1.346*	−15.448	0.86	0.83	0.685	2.455	−0.708
		−0.880*	−0.669	0.98	0.98	1.776	3.531	−0.409
1737451	1978–2007	−3.387	−2.040	0.76	0.77	0.443	2.227	−1.001
		3.010	1.604	0.94	0.92	5.221	2.992	0.856
1932408	1978–2007	−2.851	−2.291	0.53	0.53	0.446	1.771	−1.206
		−0.026	−0.028	0.80	0.81	3.103	2.369	−0.022
2519421	1978–2003	−1.869	−1.357	0.70	0.55	1.133	1.800	−0.846
		2.959	2.146	0.93	0.93	3.110	3.070	1.251
3152408	1978–2005	−3.686	−1.831	0.57	0.55	1.100	1.800	−1.023
		−0.395	−0.402	0.74	0.67	6.601	1.987	−0.279
3519426	1978–2007	−5.364	−2.823	0.70	0.70	0.378	2.046	−1.116
		2.978	1.102	0.98	0.98	9.054	3.531	0.527
4310401	1978–2002	−1.881	−1.411	0.81	0.78	0.859	2.259	−0.958
		2.870	2.010	0.93	0.90	2.530	2.847	1.159
4959401	1983–2007	−3.757	−1.988	0.69	0.60	2.086	1.873	−1.070
		2.704	2.172	0.83	0.83	5.761	2.455	1.014
5074401	1978–2007	−2.290	−1.703	0.56	0.55	0.470	1.800	−0.864
		1.231	1.237	0.81	0.81	3.395	2.369	0.869
5721442	1978–2007	−1.952	−1.696	0.91	0.91	0.785	2.917	−0.696
		1.971	1.465	0.99	0.98	2.454	3.531	0.703
5760401	1978–2007	−0.389	−0.420	0.71	0.62	0.946	1.903	−0.253
		3.023	2.211	0.91	0.91	3.404	2.917	1.152
5806414	1978–2007	−1.001	−0.738	0.78	0.79	0.550	2.294	−0.412
		2.436	1.504	0.98	0.97	2.374	3.429	0.955
*The maximum and minimum values within the five-year segments are shown one below the other (e.g. the range is between –1.346 to –0.880).

The values of H range from 0.53 to 0.98. Most of these values are higher than 0.60 and are significant. Hence, trend test under the scaling hypothesis is necessary for these data. Most of the u ₃ values are small and statistically insignificant.

Analysis of entire sequences of data

Results from the analysis of entire sequences of data are summarised in Table 7. When u ₁ is considered, there are 11 which are significant at 10%, and nine each at 5% and 2.5% levels. The maximum values in the confidence intervals for independent sequences calculated by binomial distribution are three, two and two, respectively. Therefore the number of trends detected by the trend test is much larger than those that are obtained by chance. The M-K test gives the result that almost all the river flows have significant trends. But these trends vary. Some of them are increasing and others are decreasing, even in the same region.

Table 7  Summary of test results for entire sequences

Station no.	Period	u 1	u 2	H 0	H	V(S)/V 0(S)	B	u 3
1105401	1978–1997	−6.773*	−2.115*	0.98	0.96*	17.045	2.954	−0.954
1737451	1978–2007	−3.792*	−1.608*	0.87	0.89*	8.016	2.162	−0.911
1932408	1978–2007	−0.661	−0.406	0.71	0.71*	1.205	1.321	−0.524
2519421	1978–2003	6.842*	2.241*	0.89	0.85*	16.590	1.882	1.224
3152408	1978–2005	−8.691*	−2.489*	0.75	0.68*	20.423	1.285	−1.696*
3519426	1978–2007	−6.489*	−1.699*	0.93	0.92*	15.756	2.446	−1.045
4310401	1978–2002	2.758*	1.240	0.91	0.91*	9.518	2.367	0.581
4959401	1983–2007	1.632*	0.912	0.79	0.79*	2.636	1.578	0.800
5074401	1978–2007	−0.155	−0.153	0.72	0.72*	0.549	1.341	−0.180
5721442	1978–2007	−1.570*	−0.710	0.91	0.92*	3.498	2.446	−0.537
5760401	1978–2007	4.500*	2.182*	0.87	0.84*	5.066	1.799	1.490*
5806414	1978–2007	3.361*	1.482*	0.98	0.97*	5.615	3.051	0.812
6264401	1978–2007	2.481*	1.367*	0.82	0.80*	4.181	1.591	0.962
*Values marked with an asterisk are significant at 10% level.

The statistic u ₂ gives a different result. The number of streamflows with significant trends come down to eight, five and four at 10%, 5% and 2.5% significance levels. These values are still higher than the number of significant trends for independent sequences, but are lower than the corresponding numbers from the M-K test. These trends also vary irregularly.

The H and H ₀ values are very close to each other, which indicates that trends in the data are weak. The H values range from 0.68 to 0.97, whereas H ₀ values range from 0.71 to 0.98. All the H values are significant and, hence, trend tests need to be conducted under the scaling hypothesis.

The variance inflation factors are high and range from 0.549 to 20.423. All but one inflation factor are greater than unity. The bias correction factor B ranges from 1.285 to 3.051, while u ₃ values range from –1.696 to 1.490. All but two of these values are insignificant. The maximum number of trends under the assumption of independence is three at 10% and two at 5% levels. The number of significant trends determined by trend test under the scaling hypothesis, two, is within this limit. The results of trend tests of segments and of entire sequences agree with each other.

SUMMARY AND CONCLUSIONS

Change detection in Malaysian monthly runoff data of the past three decades is carried out. The M-K test is used first for five-year segments of data. It identified quite a few trends as statistically significant. The number of trends identified by the modified M-K test is lower than the number identified by the M-K test, because the modified test alters the test statistic to account for the correlation in the data. The Hurst H values are significant in the large majority of the five-year segments indicating the need to test the trends under the scaling hypothesis. High H values also justify trend testing under the scaling hypothesis. The trend test under the scaling hypothesis, in contrast to the results from the modified M-K test, show that very few of the trends are statistically significant. These results demonstrate the importance of trend testing under the scaling hypothesis.

The numbers of significant trends identified in the five-year segments by the test under the scaling hypothesis are within the limits expected from independent series. Therefore, the number of identified trends is within the natural stochastic variability of the data. Consequently there are no changes in Malaysian runoff data that may be attributed to the effects of climate change. In a previous study of Malaysian monthly rainfall (Rao et al. 2011) similar results were obtained. Results from the present investigation of trends in monthly runoff and previous study of trends in monthly rainfall thus support each other in the conclusion that the changes in Malaysian rainfall and runoff are within the normal climatic variability.

The numbers of significant trends found in the analysis of entire sequences of data are also within the normal stochastic variability. As these results of analyses of five-year segments and entire sequences support each other, it may be concluded that there are no changes in Malaysian runoff that may be attributed to climate change.

Acknowledgements

We would like to extend our gratitude to Dr Khaled H. Hamed of Cairo University for clarifying several points in his paper and to Professor Datuk Dr Ghauth Jasmon, Vice-Chancellor of the University of Malaya, for his support. The Department of Irrigation and Drainage, Ministry of Natural Resources and Environment, Malaysia, provided the data used in this study. Their help is acknowledged. We would also like to thank the University of Malaya for partially supporting this work.