The best probability models for dry and wet spells in Peninsular Malaysia during monsoon seasons

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 0: 9 0 (00) Published online June 009 in Wiley InterScience ( DOI: 0.00/joc.9 The best probability models for dry and wet spells in Peninsular Malaysia during monsoon seasons Sayang Mohd Deni, a * Abdul Aziz Jemain b and Kamarulzaman Ibrahim b a Center for Statistical Studies, Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA (UiTM), 00 Shah Alam, Selangor, Malaysia b School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia (UKM), 00, Bangi, Selangor, Malaysia ABSTRACT: The identification of the most appropriate models to represent the rainfall distribution is very important as it can be utilised in various applications such as those in water resource management, agricultural planning department and hydrological sectors. Various types of probability models have been introduced and applied to the distribution of dry (wet) spells by previous researchers in the field. With the application of these probability models, this study will investigate the best probability models to describe the distribution of dry and wet spells at each of the 8 rainfall stations in Peninsular Malaysia for the period of The analysis will be conducted on annual basis and also through monsoon seasons. Thirteen types of probability models with one up to three parameters including the family of geometric and log series distributions will be considered. In selecting the best models to represent the distribution of the dry (wet) spells, the Akaike s information criteria (AIC) and the Kolmogorov-Smirnov goodness-of-fit (GOF) test are used. The results revealed that the modified log series () and the compound geometric () distributions were found to be the best models for the dry and wet spells respectively during the monsoon seasons. Meanwhile, the success of the mixed log series with the truncated Poisson distribution () was also proven in the study since this model was the most frequent best model selected in representing the observed distribution of both the annual dry and wet spells in most of the stations over the Peninsula. Copyright 009 Royal Meteorological Society KEY WORDS dry and wet spells; best probability models; log series distribution; geometric distribution; Akaike s Information Criteria Received March 008; Revised April 009; Accepted May 009. Introduction The analysis of the daily rainfall data particularly in identifying the best mathematical models in representing the observed data is becoming important not only in Malaysia, but everywhere in the world. The outcome of the analysis will provide some useful information in various applications including the hydrological and water related sectors. The best probability model obtained from the analysis could also be used for data generation and prediction purposes. The analysis of the spatial distribution of dry and wet spells in the daily rainfall events has always been of interest to the researchers in the field (Chapman, 99; Wilks, 999; Dobi-Wantuch et al., 000; Anagnostopoulou et al., 00; Di Giuseppe et al., 00; Tolika and Maheras, 00; Deni and Jemain, 009a, 009b, 008a, 008b; Deni et al., 009a, 009b, 008a, 008b). Besides, modeling extreme dry spells has been another important aspect in the rainfall occurrence event that has received attention and has been * Correspondence to: Sayang Mohd Deni, Center for Statistical Studies, Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA (UiTM), 00 Shah Alam, Selangor Malaysia. sayan99@salam.uitm.edu.my discussed in the literature (Vicente-Serrano and Begueria- Portugues, 00; Lana et al., 00a, 00b). In addition, there have been a number of studies in Malaysia focusing on determining the best fitting of other aspects of rainfall distribution such as the amount as well as the annual maximum amount (Zalina et al., 00; Wan Zin et al., 009; Suhaila and Jemain, 009, 00a, 00b). Modeling the distribution of dry and wet spells is one of the important aspects in analyzing the daily rainfall data which could provide useful information for data generation and other applications. Various types of probability models for the distribution of dry (wet) spells have been explored and developed by previous scientists since the early part of the 0th century. A thorough literature review on the application of several probability models to the distribution of dry (wet) spells such as the geometric distribution (GD), log series distribution (), modified log series distribution (), truncated negative binomial distribution (), Polya distribution (), compound geometric distribution (), mixed two geometric distributions () and mixed geometric Poisson distribution (MGPD) had been discussed by Deni et al. (008a) and Deni and Jemain (009a). By considering the seven types of probability models which Copyright 009 Royal Meteorological Society

2 THE BEST PROBABILITY MODELS FOR DRY AND WET SPELLS IN PENINSULAR MALAYSIA 9 had been introduced by previous researchers, Deni et al. (008a) conducted a study on the spatial distribution of wet and dry spells at 0 selected rainfall stations in Peninsular Malaysia. They found that the and the were more appropriate for the dry and wet spells respectively. In addition, by including the other two distributions i.e. and MGPD, Deni et al. (008b) and Deni and Jemain (008b) reported that the was found to best fit the distribution of wet and dry spells respectively, in Peninsular Malaysia during two different periods, 90 9 and The development of the probability models in representing the underlying observed distribution of dry (wet) spells has continuously been explored by Deni et al. (009b) who proposed five types of new mixed probability models, i.e. mixed two log series distribution (M), mixed log series geometric distribution (), mixed log series Poisson distribution (), mixed log series truncated Poisson distribution () and mixed geometric truncated Poisson distribution (MGTPD). The findings indicated that was the best probability model to represent the distribution of dry spells at each of the rainfall stations over Peninsular Malaysia. In order to evaluate the performance of the three of the new mixed probability models i.e. M, and, with the other nine existing models, Deni and Jemain (009a) reanalysed data sets which had been published by several authors from the various locations around the world together with additional six new data sets from Peninsular Malaysia in the study. Their findings demonstrated that the newly proposed models, and, showed a better fit as more than half of the data sets successfully fitted the distribution of wet and dry spells. The success of in representing the distribution of dry spells at rainfall stations in Peninsular Malaysia has also been reported by Deni and Jemain (009b). Moreover, the performance of the new model MGTPD in comparison with the other five existing models, including the MGPD, was tested at each of the five selected rainfall stations in Peninsular Malaysia (Deni and Jemain, 008a). The findings indicated that the new MGTPD produced a better fit than the existing model MGPD in describing the distribution of wet spells at the five selected stations. Although quite a number of studies had been done in assessing the performance of the various probability models in representing the distribution of dry (wet) spells, those studies were conducted based on the overall basis where the seasonality in the data series was ignored. It would be interesting to explore the type of the best model in describing the distribution of rainfall particularly on the spells by taking into account the seasons in the country. Therefore, by considering the two types of monsoons and the inter monsoon periods, i.e. the first inter monsoon with the southwest monsoon (FISW) and the second inter monsoon with the northeast monsoon (SINE), this study will identify the best models among all the existing models with the newly mixed probability models proposed by Deni et al. (009b). The analysis will also be conducted on the annual (ANNL) basis. In selecting the best probability models to describe the observed distribution of dry and wet spells at each of the 8 rainfall stations in Peninsular Malaysia, the Akaike s Information Criteria (AIC) decision criteria and the Kolmogorov-Smirnov goodness-of-fit (GOF) test will be considered.. Data and Methodology.. The study area and data Peninsular Malaysia lies entirely in the equatorial zone, situated in the northern latitude between and N andin the eastern longitude from 00 to 0 E. The two types of monsoons which influence the climate of the country are the southwest monsoon and the northeast monsoon which occur in the months of May to August and November to February respectively. Meanwhile, the first and second inter monsoon periods occur in the months of March to April and September to October respectively. As mentioned earlier in the Introduction section, this study will be conducted on the basis of the monsoon seasons (FISW and SINE) and also on the annual (ANNL) basis. The geographical coordinates and the locations of the 8 selected rainfall stations are provided in Table I and Figure respectively. Here, a wet day is defined as a day with a rainfall amount of at least mm. Otherwise, the day is considered as a dry day. A wet spell is a period of consecutive days of exactly, say x wet days, immediately preceded and followed by a dry day. The minimum length of a wet spells is taken as one day which means a single wet day. The homogeneity of the data series was checked using the four types of homogeneity tests as recommended by Wijngaard et al. (00); the standard normal homogeneity test, the Pettit test, the Buishand range test and the Von Neumann ratio test. The results indicate that the annual series of wet days were found homogenous for each of the 8 rainfall stations. The detailed explanation on the results of the homogeneity test on the data series could be referred in Suhaila et al. (008b). The data used in this present study can be considered good quality data with all of the stations obtaining less than 0% missing values throughout the 0-year period, except at Station. The missing values in the data series for the period of 9 00 were estimated using the various types of weighting methods such as the inverse distance, the normal ratio and the correlation between the target and the neighbouring stations (Eischeid et al., 000; Sullivan and Unwin, 00; Teegavarapu and Chandramouli, 00; Suhaila et al., 008a)... The probability models and the parameter estimation The probability models are developed to provide a statistical description of the observations for the various purposes such as for data generation and modeling climatic events that can be used to describe the physical explanation of the rainfall occurrence. In this study, types Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

3 9 S. M. DENI ET AL. Table I. Geographical coordinates and the percentage of missing data for each of the 8 rainfall stations in Peninsular Malaysia. No. Station Name Latitude Longitude % Missing data No. Station Name Latitude Longitude % Missing data Johor Bahru Kemaman. 0.. Senai Ladang BOH Kota Tinggi Ipoh Batu Pahat Dungun Kluang Gua Musang Melaka Kg. Menerong Mersing Bt. Kurau Sikamat Selama Sg. Lui Halt Bayan Lepas Subang Kuala Terengganu. 0.. Ampang Bukit Berapit Bkt. Ibam Air Itam Genting Kelang Bukit Bendera Gombak Bumbong Lima Pekan. 0.. Kota Bharu Bagan Terap Alor Setar Kuantan Ampang Pedu Paya Kangsar Kangar Sitiawan Kg. Bahru E 0 E 0 E 0 E 0 E km N 8 0 S T R A I T S O F M E L A K A S O U T H C H I N A S E A N N N LEGEND 8 Below 0 meter Between 0 to 000 meter Above 000 meter Rainfall Station N Figure. The physical map showing the locations of the 8 rainfall stations in Peninsular Malaysia. of probability models introduced by previous researchers as mentioned earlier in the Introduction section will be applied to the distribution of dry and wet spells at each of the 8 rainfall stations in Peninsular Malaysia. The probability models with their probability function as well as the parameter(s) are displayed in Table II. The parameters of the probability models are estimated using the maximum likelihood method. The maximum Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

4 THE BEST PROBABILITY MODELS FOR DRY AND WET SPELLS IN PENINSULAR MALAYSIA 9 Table II. The probability models, their probability functions and the parameter(s) used for fitting the distribution of dry and wet spells. For each of the following probability functions, x =,,... Probability model Probability function Parameter(s) Geometric Distribution (GD) P(X = x) = p( p) x p Log Series Distribution P(X = x) = p x /x log( p) p () Modified Log Series P(X = x) = cp x /(x + h) 0 h<, c = ( p x /(x + h)) p, h Distribution () (normalising constant) Truncated Negative Binomial P(X = x) = r+x C x ( p) x p r /( p r ), r p, r Distribution () Polya Distribution () P(X = ) = ( + d) m/d m, d P(X = x) = m + (x )d P(X = x ), x =,,... (x )( + d) Compound Geometric P(X = 0) = b(a ) a, b Distribution () P(X = x) = a + a + b + x x P(X = x ) Mixed Two Geometric P(X = x) = Wp ( p ) x + ( W)p ( p ) x W, p, p Distribution () Mixed Geometric Poisson Distribution (MGPD) Mixed Geometric Truncated Poisson Distribution (MGTPD) Mixed Two Log Series Distribution (M) Mixed Log Series Geometric Distribution () Mixed Log Series Poisson Distribution () Mixed Log Series Truncated Poisson Distribution () λ x P(X = x) = Wp( p) x + ( W) (x )! e λ W, p, λ P(X = x) = WP( p) x + ( W) λ x e λ x!( e λ ) p x P(X = x) = W x log( p ) ( W) p x x log( p ) p x W, p, λ W, p, p P(X = x) = W x log( p ) + ( W)p ( p ) x W, p, p p x P(X = x) = W x log( p) + ( e λ W)λx (x )! p x x e λ P(X = x) = W + ( W) λ x log( p) x!( e λ ) W, p, λ W, p, λ likelihood estimates are found by implementing the R or S-Plus function for optimisation as suggested by Venables and Ripley (00) under the quasi-newton procedure. When minimising the negative log likelihood during the estimation of the parameters, the initial values for the parameters are required. The initial values are obtained using the observed data which are on the basis of the method of moment or the method of factorial moment. The estimators of the one- and two-parameter models can be referred in Deni et al. (008a). For the new mixed probability models proposed, i.e. MGTPD, M,, and, the choice of the initial values depends on the definition with respect to each parameter for each model as defined in Deni et al. (009b). For the other existing two three-parameter models i.e. and MGPD, the initial parameters could be obtained in the same approach of the new mixed probability models... Model selection In this study, the performance of the probability models in representing the distribution of dry (wet) spells will be evaluated in two stages. Firstly, the selection is on the basis of the minimum value of the AIC and secondly the selection of the best models is obtained through the results of the Kolmogorov-Smirnov GOF test. If a particular data set shows the minimum value of the AIC but does not show a significant fit of the Kolmogorov- Smirnov GOF test, the next best model with the minimum AIC value which fulfills the second condition will be chosen. The performance of the probability models is measured by using the AIC (Akaike, 9) which is given by AIC = (log likelihood P(X = x)) + k where P(X = x) represents the probability functions of interest and k denotes the number of parameters in these functions. The selection of the best probability model for each of the rainfall station is on the basis of the minimum value of AIC. In addition, the significant fit of the expected and observed frequency distribution of dry (wet) spells is evaluated by using the Kolmogorov-Smirnov GOF test. The test will compare the observed and expected frequency distributions of the dry (wet) spells obtained from the probability models. The maximum absolute difference between the two cumulative values of the observed and expected frequencies, D max, is computed Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

5 98 S. M. DENI ET AL. and if this value is found to be less than or equal to the critical value, D 0.0 at % level of significance, a particular probability model is considered as significantly fit to the observed distribution of the dry (wet) spells. The probability model which corresponds to the minimum value of D max represents the best fit model in representing the distribution of dry (wet) spells.. Results and discussion.. Spatial distribution of the mean and maximum length of dry (wet) spells Figures and display the spatial distribution of the mean and maximum length of the dry and wet spells respectively at each of the 8 rainfall stations over Peninsular Malaysia. It is observed in Figure a-c that the mean of the dry spells ranged from to days over the Peninsula for all seasons. The findings in Figure a indicated that one area which is located in the center of the Peninsula between the latitudes. and N was slightly drier during the FISW season than the other areas. Meanwhile, the northwestern area was found to be the driest area during the SINE season and this was supported by some of the previous results (Dale, 90; Camerlengo, 999; Deni et al., 008a, 009a). Generally, Figure c showed that the mean length of more than days was observed in the northwestern, eastern and central regions of the Peninsula which indicated the driest areas throughout one year during the 0-year period. Furthermore, the results of the maximum length of dry spells as shown in Figure d indicated that the eastern and central areas were observed as having an average of more than weeks without any rainy days. As mentioned earlier, the northwestern area is the driest area in the Peninsula during the SINE season and the ANNL. Further analysis on the maximum length of dry spells in Figure e and f also indicated that this area showed an average of more than 0 days of dry spells. For the analysis of the mean length of wet spells as shown in Figure a-c, the findings indicated that the upper western area was found to be slightly wetter than the other regions. Generally, the averaged mean length of the wet spells ranged between to days over Peninsular Malaysia for all seasons. Similarly, for the maximum length of wet spells, the results in Figure d-f indicated that an average of the maximum length of wet spells of more than weeks was observed in the areas between the latitude and N... The selection of the best mixed probability models for wet spells The selected probability models as shown in Table III (in bold face) were on the basis of the minimum value of the AIC. If these models showed a significant fit of the Kolmogorov-Smirnov GOF test as shown in Table IV, the selected probability models were the best models (as located in the final column of Table IV) in describing the distribution of the dry or wet spells according to a particular probability model at the % level of significance. However, not all of the selected probability models with the minimum AIC values showed a significantly fit of the Kolmogorov-Smirnov GOF test. The results in Table IV indicated that Station failed to fulfill the second criteria. It is shown that the minimum AIC values at Station indicated that was found to be the best model selected; however, the GOF test revealed that the expected frequencies obtained from this model failed to fit the observed distribution of dry spells during the SINE season. In this situation, the next model which satisfied both the criteria would be chosen as the best model. The results in Table IV indicated that there were three models i.e., and that significantly fitted the observed data sets at Station. With a minimum value of AIC and a significant fit of the data set at % level, is chosen as the best model in describing the observed distribution of dry spells at Station during the SINE season. In this study, both the criteria of selecting the best probability models were needed since by applying the minimum AIC alone, the degree of significance of the expected frequency distribution produced could not be identified. In this situation, the GOF test was applied in order to further investigate whether the expected frequency produced by the best probability models well fitted the observed distribution of dry (wet) spells. Furthermore, the application of the Kolmogorov-Smirnov GOF test alone in selecting the best model was not a good solution, since the best models obtained from this GOF test would produce more than one best model for a particular data set. Therefore, further investigation on this model was carried out to determine whether it also produced a minimum AIC value in selecting the best probability models for representing the distribution of dry (wet) spells in Peninsular Malaysia. For example, Table IV showed that the D max values produced by the Kolmogorov-Smirnov GOF test in Station indicated that out of the probability models tested best fitted the distribution of the dry spells during the SINE season. Although these four threeparameter probability models, i.e., M, and could be used to describe the observed distribution of the dry spells at Station, further analysis on determining the minimum AIC value revealed that the was the best probability model for the particular data set. Figure displays the observed and the expected distributions of both types of spells during both the monsoon seasons and on annual basis at Station obtained from the best probability models selected on the basis of both the criteria. It could be seen that the expected frequencies of dry and wet spells which were obtained from the best Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

6 THE BEST PROBABILITY MODELS FOR DRY AND WET SPELLS IN PENINSULAR MALAYSIA 99 (a) FISW_mean dry spells (b) SINE_mean dry spells (c) ANNL_mean dry spells (d) FISW_maximum dry spells (e) SINE_ maximum dry spells (f) ANNL_ maximum dry spells Figure. The spatial distribution of (a c) the mean and (d f) the maximum length of dry spells at each of the 8 rainfall stations in Peninsular Malaysia. Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

7 00 S. M. DENI ET AL. (a) FISW_mean wet spells (b) SINE_mean wet spells (c) ANNL_mean wet spells (d) FISW_maximum wet spells (e) SINE_ maximum wet spells (f) ANNL_ maximum wet spells Figure. The spatial distribution of (a c) the mean and (d f) the maximum length of wet spells at each of the 8 rainfall stations in Peninsular Malaysia. Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

8 THE BEST PROBABILITY MODELS FOR DRY AND WET SPELLS IN PENINSULAR MALAYSIA 0 Table III. The Akaike s Information Criteria (AIC) values of the probability models for the distribution of dry spells in Peninsular Malaysia during the SINE season. The bold face indicates the minimum AIC values which represent the best model selected. Station One-parameter Two-parameter Three-parameter GD MGPD MGTPD M models selected actually described the observed distributions. With the application of each probability model to the data sets, further analysis will identify the best models selected at each of the rainfall stations in Peninsular Malaysia... The spatial distribution of the best models for dry (wet) spells The best probability models selected in describing the distribution of the dry and wet spells at each of 8 rainfall stations over the Peninsular during both the FISW and SINE seasons and the ANNL are displayed in Figure. It is observed that out of the 8 data sets such as at Stations, 8, 0,, 8, and showed the same types of the best probability models selected for the dry spells during all seasons considered. Meanwhile, Stations, and showed the similar types of the best models selected for the observed distribution of wet spells. It can be seen the and were found to be the best probability models selected at Station for the distribution of wet and dry spells respectively. For the other stations, the best models selected for dry (wet) spells were found to be varied at each season. The variability in the types of the best models selected at each station may be due to the characteristics of the observed distribution during a particular season. For example, it is noticed that none of the stations located in the northwestern areas showed the same type of the best models for dry (wet) spells during all seasons. This may be due to the longer period of dry spells and shorter period of wet spells that always occurred in this area than in the other areas over the Peninsula during the SINE Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

9 0 S. M. DENI ET AL. Table IV. The D max and critical values, D 0.0, of the Kolmogorov-Smirnov GOF test for the probability models at each of the dry spells data set during the SINE season in Peninsular Malaysia. Station Oneparameter Two-parameter Three-parameter D 0.0 Best model GD MGPD MGTPD M MGPD season. According to Chapman (99) who also found different types of the best models for the dry and wet spells at locations in Australia, the variability may have been due to the relatively short periods of record and also to the location of the stations. The performance of each of the probability model in describing the distribution of dry (wet) spells for each of the data sets could be assessed by its ranking. The mean rank of each probability model was indicated as to for the best to the lowest performing model for each data set. In order to identify the overall performance of each of the probability model, the mean ranks of each model was averaged for all the 8 data sets in each season as displayed in Table V. The least value of the averaged mean ranks indicated the most frequent model selected to be the best probability model in representing the whole data sets of dry (wet) spells from the 8 rainfall stations during a particular season. It is shown in Table V that the one-parameter model obtained better results than the GD. This is also supported in the findings of Deni et al. (008a, 008b) and Deni and Jemain (009a, 009b, 008a, 008b) who reported that the GD failed to fit the distribution of dry and wet spells in Peninsular Malaysia. Deni and Jemain (008a) reported that the MGTPD was found well fitted for the distribution of wet spells at each of the five selected rainfall stations in Peninsular Malaysia. The success of the new model MGTPD was also proven in this study since this model outperformed the existing MGPD in describing the distribution of the dry (wet) spells over Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

10 THE BEST PROBABILITY MODELS FOR DRY AND WET SPELLS IN PENINSULAR MALAYSIA 0 (a) FISW Length of dry spells Frequency (e) FISW Length of wet spells Frequency (b) SINE Length of dry spells (f) SINE Length of wet spells (c) ANNL Length of dry spells (g) ANNL Length of wet spells Figure. The observed (dots) and expected (strip) frequencies of the best probability models fitted for the distribution of (a c) dry and (d f) wet spells at Station during FISW, SINE and ANNL. the Peninsula. Deni et al. (008a) reported that the and were found to be the most frequent selected probability models in representing the distribution of the wet and dry spells in Peninsular Malaysia. It is shown in Table V that the and were the most frequent best models selected for the distribution of dry spells during the FISW and SINE seasons respectively. Also, for the distribution of wet spells, it is observed that these models, and, were found to be the best models selected in most of stations during the SINE and FISW seasons respectively. Meanwhile, the analysis on the annual basis revealed that the was found to be the most frequent best model selected for both the dry and wet spells. Although the M was not found to be among the most frequent best model selected, this model showed better rankings than the one-parameter model.. Conclusions The analysis of the rainfall behaviour of the daily rainfall event particularly on its occurrence is necessary for providing useful information in the various applications which could be used as an input to the climate monitoring system to obtain a better prediction for future climatic events. In determining the best probability models for the distribution of dry (wet) spells, the models with less number of parameters estimated are preferred. But, not all of the cases of the one parameter, two parameter or even three parameter models were able to fit the observed data (Deni and Jemain, 009b; Deni et al., 009b). The investigation on the performance of the existing and new mixed probability models for distribution of dry (wet) spells, which were proposed by Deni et al. (009b), is extended in this study for the Malaysian dry (wet) spells data sets by considering the monsoon seasons and spells on annual basis. The performance of the probability models such as and, is also proven not only for the published data sets from the various locations around the world as reported by Deni and Jemain (009a), but also for most of the data sets from the 8 rainfall stations over Peninsular Malaysia during both the monsoon seasons and on annual basis. Not only these three models, but also the analysis on annual basis indicated that the was the most frequent best probability models selected for both the dry and wet spells in most of the stations over the Peninsula for the period of The variability in the types of the best probability models selected in describing the distribution of dry (wet) spells in most of the stations on annual basis and during the monsoon seasons may be due Table V. The averaged mean ranks obtained by each of the probability models based on the best models selected for dry and wet spells over Peninsular Malaysia. The bold face indicates the lowest averaged mean rank which represents the most frequent best model selected. One-parameter Two-parameter Three-parameter GD MGPD MGTPD M Dry FISW SINE ANNL Wet FISW SINE ANNL Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

11 0 S. M. DENI ET AL. (a) FISW_dry spells 00 E S T R A I T S O F M E L A K A MGTPD 0 00km S O U T H C H I N A S E A MGTPD N (b) SINE_dry spells 00 E MGPD N (c) ANNL_dry spells 00 E M M M M (d) FISW_wet spells 00 E (b) SINE_wet spells 00 E (c) ANNL_wet spells 00 E GD N GD MGPD N M GD MGTPD MGPD MGPD M MGPD GD MGPD M M M Figure. The best probability models for (a c) dry and (d f) wet spells at each of the 8 rainfall stations in Peninsular Malaysia. N N Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)

12 THE BEST PROBABILITY MODELS FOR DRY AND WET SPELLS IN PENINSULAR MALAYSIA 0 to several reasons such as the length of records, as well as the location. The best probability models for each of the data sets will be very useful in the various applications as these models could provide a better explanation of the rainfall occurrence process. Further analysis will be carried out in applying these models with some other techniques such as copula modeling and general mixed-data model in order to give a more comprehensive analysis of the daily rainfall for the Malaysian data sets, as well as the various locations around the world, by taking into account the seasonal and regional aspects. Acknowledgements The authors are indebted to the staff of the Drainage and Irrigation Department and the Malaysian Meteorological Department for providing the daily rainfall data for this study. This research would not have been possible without the sponsorship from Universiti Teknologi MARA (UiTM) and Universiti Kebangsaan Malaysia (UKM). They also acknowledge their sincere appreciation to Prof Glenn McGregor and the reviewer for their valuable suggestions and remarks which improved the manuscript. This research was funded by Universiti Kebangsaan Malaysia (UKM-GUP-PI-08--8). References Akaike H. 9. A new look at the statistical model identification. IEEE Transactions on Automatic Control 9:. Anagnostopoulou CHR, Maheras P, Karacostas T, Vafiadis M. 00. Spatial and temporal analysis of dry spells in Greece. Theoretical and Applied Climatology : 9. Camerlengo AL Monthly frequency distributions of both dry spells and the number of days with precipitation greater than mm over Peninsular Malaysia. GEOACTA : 8. Chapman TG. 99. Stochastic models for daily rainfall in the Western Pacific. Mathematics and Computers in Simulation : 8. Dale WL. 90. The rainfall of Malaya, Part II. Journal of Tropical Geography : 8. Deni SM, Jemain AA. 008a. Mixed geometric truncated Poisson model for sequences of wet days. Journal of Applied Sciences 8: Deni SM, Jemain AA. 008b. Probability models for dry spells in Peninsular Malaysia. Asia-Pacific Journal of Atmospheric Sciences :. Deni SM, Jemain AA. 009a. Fitting the distribution of dry and wet spells with alternative probability models. Meteorology and Atmospheric Physics 0:. Deni SM, Jemain AA. 009b. Mixed log series geometric distribution for sequences of dry days. Atmospheric Research 9:. Deni SM, Jemain AA, Ibrahim K. 008a. The spatial distribution of wet and dry spells over Peninsular Malaysia. Theoretical and Applied Climatology 9:. Deni SM, Jemain AA, Ibrahim K. 009a. Fitting optimum order of Markov chain models for daily rainfall occurrences in Peninsular Malaysia. Theoretical and Applied Climatology 9: 09. Deni SM, Suhaila J, Jemain AA. 008b. Probability models for wet spells in Peninsular Malaysia during the periods of 90s 9 & Journal of Quality Measurement and Analysis : 8. Deni SM, Jemain AA, Ibrahim K. 009b. Mixed probability models for dry and wet spells. Statistical Methodology : Di Giuseppe E, Vento D, Epifani C, Esposito S. 00. Analysis of dry and wet spells from 80 to 000 in four Italian sites. Geophysical Research Abstracts : 0. Dobi-Wantuch I, Mika J, Szeidl L Modeling wet and dry spells with mixture distributions. Meteorology and Atmospheric Physics :. Eischeid JK, Pasteris PA, Diaz HF, Plantico MS, Lott NJ Creating a serially complete, national daily timeseries of temperature and precipitation for the Western United States. Journal of Applied Meteorology 9: Lana X, Martinez MD, Burgueno A, Serra C, Martin-Vide J, Gomez L. 00a. Distributions of long dry spells in the Iberian Peninsula, years International Journal of Climatology : Lana X, Martinez MD, Burgueno A, Serra C. 00b. Statistical distributions and sampling strategies for the analysis of extreme dry spells in Catalonia (NE Spain). Journal of Hydrology : 9. Suhaila J, Jemain AA. 00a. Fitting daily rainfall amount in Malaysia using the Normal transform distribution. Journal of Applied Sciences : Suhaila J, Jemain AA. 00b. Fitting daily rainfall amount in Peninsular Malaysia using several types of Exponential distributions. Journal of Applied Sciences Research : 0 0. Suhaila J, Jemain AA Investigating the impacts of adjoining wet days on the distribution of daily rainfall amounts in Peninsular Malaysia. Journal of Hydrology 8:. Suhaila J, Deni SM, Jemain AA. 008a. Revised Spatial Weighting Methods for Estimation of Missing Rainfall Data. Asia-Pacific Journal of Atmospheric Sciences : 9 0. Suhaila J, Deni SM, Jemain AA. 008b. Detecting inhomogeneity in Peninsular Malaysian rainfall series. Asia-Pacific Journal of Atmospheric Sciences : Sullivan DO, Unwin DJ. 00. Geographic Information Analysis. Wiley: Hoboken. Teegavarapu RSV, Chandramouli V. 00. Improved weighting methods, deterministic and stochastic data-driven models for estimation of missing precipitation records. Journal of Hydrology : 9 0. Tolika K, Maheras P. 00. Spatial and temporal characteristics of wet spells in Greece. Theoretical and Applied Climatology 8: 8. Venables WN, Ripley BD. 00. Modern Applied Statistics with S. Statistics and Computing, th edn. Springer-Verlag: New York. Vicente-Serrano SM, Begueria-Portugues S. 00. Estimating extreme dry-spell risk in the middle Ebro Valley (Northeastern Spain): a comparative analysis on spatial duration series with a general Pareto distribution and annual maxima with a Gumbel distribution. International Journal of Climatology : 0 8. Wan Zin WZ, Jemain AA, Ibrahim K The best fitting distribution of annual maximum rainfall in Peninsular Malaysia based on methods of L-moment and LQ-moment. Theoretical and Applied Climatology 9:. Wijngaard JB, Klien Tank AMG, Konnen GP. 00. Homogeneity of 0 th century European daily temperature and precipitation series. International Journal of Climatology : 9 9. Wilks DS Interannual variability and extreme-value characteristics of several stochastic daily precipitation models. Agricultural and Forest Meteorology 9: 9. Zalina MD, Desa MN, Nguyen VTV, Hashim MK. 00. Selecting a probability distribution for extreme rainfall series in Malaysia. Water Science and Technology : 8. Copyright 009 Royal Meteorological Society Int. J. Climatol. 0: 9 0 (00)