Optimal stochastic multi-states first-order Markov chain parameters for synthesizing daily rainfall...

METEOROLOGICAL APPLICATIONSMeteorol. Appl. 20: 20–31 (2013)Published online 13 September 2011 in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/met.292

Optimal stochastic multi-states first-order Markov chain parametersfor synthesizing daily rainfall data using multi-objective

differential evolution in Thailand

Chakkrapong Taewichit,a Peeyush Soni,a* Vilas M. Salokheb and Hemantha P. W. Jayasuriyac

a Agricultural Systems and Engineering, School of Environment Resources and Development, Asian Institute of Technology, Pathumthani 12120,Thailand

b Vice-Chancellor, Kaziranga University, Jorhat, Assam, Indiac Department of Soil, Water and Agricultural Engineering, College of Agricultural and Marine Sciences, Sultan Qaboos University, PC 123

Al-Khod, Oman

ABSTRACT: Stochastic Multi-states First-order Markov Chain (SMFOMC) models have been used to describe occurrenceof daily rainfall. This paper describes optimization of SMFOMC parameters through the generation of synthetic daily rainfallsequences. Three SMFOMC parameters were the number of states (NS), the preserved proportion in the last state (PPL) andthe state divider (SD). The multi-objective differential evolution (MODE) was used to find the Pareto-optimal line (POL) oftwo conflicting objectives; (1) minimization of total monthly absolute total relative error (TMATRE), and, (2) minimizationof NS. Three probability distributions functions (PDFs) for generating daily rainfall amounts in the last Markov Chain statewere compared. They were: (1) the shifted exponential distribution (SE), (2) the exponential distribution (E), and, (3) thetwo-parameter gamma distribution (G-2). The optimal SMFOMC parameters were applied to generate the daily rainfallsequences of 44 rainfall stations located in five regions of Thailand. Reliability of the optimal SMFOMC parameters foreach PDF was measured by TMATRE and coefficient of determination (R2). Performance of PDFs was analysed by aranking method. Results showed that the three PDFs were mostly found to be fitted well with the synthetic daily rainfallsequences. However, highest error was found in case of monthly average minimum daily rainfall values. Out of the threePDFs, the SE demonstrated the lowest performance, while G-2 performed the best. Copyright 2011 Royal MeteorologicalSociety

KEY WORDS daily weather generator; stochastic model; evolutionary algorithms; Markov chain model; optimization

Received 3 January 2011; Revised 31 March 2011; Accepted 4 August 2011

1. Introduction

The synthetic daily rainfall sequences are often used as animportant input for mathematical simulation in hydrology, agri-culture and water resources models. In the case of univariateseries daily rainfall models, the missing data could be gen-erated using statistical parameters that describe the hydrolog-ical behaviour within the sequence itself. For this purpose,the multiplicative autoregressive integrated moving average(ARIMA) and Thomas–Fiering models have been extensivelyapplied (Delleur and Kavvas, 1978; Sharma, 1985; Vogel, 1988;Mujumdar and Kumar, 1990; Schreider et al., 1997; Toth et al.,1999; Ahmad et al., 2001; Taewichit and Chittaladakorn, 2007;Amini et al., 2009). However, the difficulty, complexity andrequirement of large statistical parameters are considered to betheir limitations. One simplified approach is the use of a modelcalled the Stochastic First-Order Markov Chain (SFOMC),which describes the probability of rainfall occurrences on agiven day using transition probability matrices (TPMs). TheSFOMC model is applied to study the occurrences of dailyrainfall (Gabriel and Neumann, 1962; Moon et al., 2006) and toconstruct rainfall-runoff synthesizing models (Kottegoda et al.,

∗ Correspondence to: P. Soni, Agricultural Systems and Engineering,School of Environment Resources and Development, Asian Institute ofTechnology, Pathumthani 12120, Thailand. E-mail: soni@ait.asia

2000). The main concept behind the SFOMC is the use ofconditional probability to describe the occurrences and non-occurrences of rainfall (Gabriel and Neumann, 1962). Initialdevelopment and application of SFOMC is as a two-state (wet-dry) model (Gabriel and Neumann, 1962; Todorovic and Wool-hiser, 1975; Haddada et al., 2000), which is applied to generatewet-dry events. Further development of SFOMC is the use ofthe two-state model coupled with some probability distributionfunctions (PDFs) (Tsakeris, 1988) to estimate daily rainfall. Inaddition, modifying the SFOMC model by increasing the num-ber of states (NS) in the wet state (Khanal and Hamrick, 1974;Srikanthan and McMahon, 1984; Hutchinson, 1990; Aksoy,2003) coupled with the uses of PDFs in the last state has beenattempted successfully. This model is popularly known as theStochastic Multi-States First-Order Markov Chain (SMFOMC).However, some of the difficulties noted in using the SMFOMC(Haan, 1977) were: (1) determining NS, (2) determining theintervals of the variable under study to associate with each state,and, (3) assigning a number to the magnitude of an event oncethe state is determined. Most research studies of SMFOMCstill use trial and error to overcome these limitations and todetermine the optimal SMFOMC parameters for which the gen-erated daily rainfall sequences are close to those of historicalsequences.

To synthesize daily rainfall for a single site (univariatemodel), optimal SMFOMC parameters of the number of states

Synthesizing daily rainfall: multi-objective differential evolution 21

(NS), the preserved proportion in the last state (PPL), and thestate divider (SD) are determined in this study. Forty-four rain-fall stations from five regions (Central, North, North-East, East,and South) in Thailand (Figure 1) were selected to apply and toevaluate the optimal parameters of the SMFOMC model. Themulti-objective algorithm, multi-objective differential evolution(MODE), was employed in this study. Two conflicting objec-tives of minimizing NS and minimizing total monthly absolutetotal relative error (TMATRE) were set with statistical parame-ters of generated daily rainfall sequences. Performance of threePDFs, the shifted exponential distribution (SE), the exponen-tial distribution (E) and the two-parameter gamma distribution(G-2), were evaluated and compared.

2. Theoretical considerations

2.1. Stochastic multi-states first order Markov chain

SMFOMC has been applied in hydrology and water manage-ment for modelling processes (Kottegoda et al., 2000; Aksoy,2003; Ochola and Kerkides, 2003). In its first order modelsit employs conditional probability to describe the process x(t)

at the present time t using only the outcome at previous timet − 1. A higher-order Markov Chain model, corresponding tothe number of preceding days (Chapman, 1998), could alsobe formulated (Kulkarni et al., 2002). SMFOMC may then beconsidered as a simple two-state for a dry day (no rain) and awet day. However, there is no discernible difference reportedbetween the model performance of first and second order mod-els in synthesizing daily rainfall (Jimoh and Webster, 1996).

SMFOMC is defined by its transition probability matrices(TPMs) and frequency distributions of rainfall amounts (Haanet al., 1976) that can preserve most of the daily, monthly andannual characteristics (Srikanthan and McMahon, 2001). TheTPMs play a significant role in estimating the present data j attime t using the probability pij (t) of moving from state i at timet − 1 to state j at time t , which is derived from the frequencyof state changes from state i to state j . To obtain the frequencyof daily rainfall for each of the states, a rainfall class limitstable (RCLT) is constructed for classifying rainfall data intosuccessive states of j = 1 to j = r . Each state consists of upperbound and lower bound rainfall amounts. The state interval foreach state is mostly specified through manual trial and error byresearcher’s experience. In the daily data generation, the amountof rainfall in the intermediate state j at time t (state of j = 2to state j = r − 1) is estimated by adding the lower boundof rainfall amounts of state j with the term of the stochasticuniform random number U ∈ (0, 1) multiplied by the differenceof rainfall amounts between upper and lower bound of state j

(linear interpolation). The U is generated from the most populargenerators called linear congruential generators (Salas, 1992;Reddy, 1997). In addition, the inverse cumulative probabilitydistribution function is used to estimate rainfall amount in thelast state r .

2.2. Probability distributions

Various PDFs have been applied by researchers (Allen andHaan, 1975; Todorovic and Woolhiser, 1975; Suhaila andJemain, 2007). However, the cumulative probability distributionfunction (CDF), which is an area under the PDF curve, ispopularly used. SE was first proposed by Allen and Haan(1975). The distribution in terms of CDF is given as:

F(x) = 1 − e−(x−Rfc−1)/λ (1)

Figure 1. Locations of the selected 44 rainfall stations in 5 regions ofThailand. This figure is available in colour online at wileyonlinelibrary.

com/journal/met

where F(x) is a CDF, λ is difference of average of all recordedhistorical daily rainfall length being greater than or equal toRf c−1, Rf c−1 is the rainfall amount of lower bound in thelast state, x is the rainfall amount in the last state and c isthe state of daily rainfall amount. The Maximum-Likelihoodmethod (MLM) is used for estimating parameters of E and G-2. Numerical methods, suggested by Rao and Hamed (2001),are applied to solve the parameters of both the distributions. Inthe case of E, which is a special case of the Gamma family, thedistribution can be obtained by setting β = 1 in Equation (2)and expressed as Equation (3):

f (x) = 1

αβ.�(β)(x − ε)β−1e−(x−ε)/α (2)

f (x) = 1

αe−(x−ε)/α (3)

where α, β and ε are distribution parameters and �(β) isgamma function.

From Equation (4), G-2 has extensively been used in theMarkov Chain model (Coe and Stern, 1982; Richardson andWright, 1984; Duan et al., 1995; Kottegoda et al., 2000; Aksoy,2003): its PDF is formulated by eliminating ε in Equation (2):

f (x) = 1

αβ.�(β)xβ−1e−(x/α) (4)

2.3. Optimization

2.3.1. Multi-objective optimization problems (MOPs)

The MOPs deal with optimizing various conflict objectivessimultaneously. Various objectives are incorporated for makingdecisions to select the desirable solution. The solutions ofMOPs comprise non-dominated solutions (NDSs). NDSs areoften expressed as the Pareto-optimal line (POL). The heuristicstochastic search techniques, evolutionary algorithms (EAs),have been used intensively for solving MOPs (Coello et al.,2002) owing to their population-based nature that allow thegeneration of several elements of POL in a single run. EAsalso provide a diversification mechanism to obtain a better

22 C. Taewichit et al.

solution. The context of EAs in MOPs is to find a POL asclose as possible to the true POL and diversify solution onthe POL as much as possible. The POL comprises NDSs thathave been sorted in many front levels using non-dominatedsorting algorithm (NDSA). The NDSA, also known as simplemodified naıve slow was proposed by Deb (2001) and isapplied in the present study. After the set of POL is met,the preferred solution can be chosen using the compromiseprogramming (CP) technique with weighted importance valuesof each objective function (Zeleny, 1982; Romero and Rehman,1989).

2.3.2. Multi-objective differential evolution (MODE)

MODE is an advanced version of the differential evolutionalgorithm (DEA) (Storn and Price, 1997) for multi-objectiveoptimization (Sun et al., 2005). The process of DEA beginswith randomly generating the population of solution vectorssize NP of D dimension or ‘target vectors’. To improve thesolution vectors (trial vectors) recombination process is used,which consists of mutation and crossover. Solution valuesare swapped and changed by chances of probabilities. Theprobability is launched by the crossover constant (CR ∈ (0, 1))when the random number during trial vector generation is lessthan or equal to CR. The trial vector size, NP, is generateddimension-by-dimension by randomly picking three distinctsolution vectors and adding the first vector to the product of theweighted factor (F ∈ (0, 1)) and the difference of the remainingtwo vectors. The trial vector will replace the temporary targetvectors if the objective value of the former is better than thelatter.

The DEA is modified to be MODE (Xue, 2003; Reddy andKumar, 2007; Li and Zhang, 2008), where the initializationvectors of population is started as with a normal DEA togenerate target vectors x with size NP, which is followedby generation of new trial vectors u with size NP in therecombination process. Those vectors are combined and sortedto be vectors r using the proposed NDSA to rank NDSs. The

selection process is similar to an elitist non-dominated sortingGA (NSGAII) where the sorted solutions of NP are directlypicked up from the solution fronts, j , which replace the oldset of solutions. The lending mechanisms from NSGAII, calledthe crowded distance assignment (Deb, 2001), are assigned toeach solution in the non-dominated fronts in order to use theseas criteria to select better compared solutions when the lastrequired set of solutions is located in the same front. The largercrowded distance is preferred to be chosen. Those processesare completed in one generation. Those steps are repeatedgeneration by generation until the set of POL does not changefurther (Table 1).

3. Model application

3.1. Study area and locations of selected rainfall stations

For this study, a 38 year (1971–2008) continuous record ofdaily rainfall occurrences at 44 stations distributed in 5 regionsof Thailand (Figure 1) was used. The rainfall stations wereselected based on the data continuity and with length of recordfor more than 30 years. The daily rainfall data were obtainedfrom the Royal Irrigation Department and the MeteorologicalDepartment of Thailand.

3.2. Model formulation for optimization of SMFMOCparameters

Twelve monthly statistical parameters were used to measurethe adequacy and acceptability of the model: (1) monthly max-imum spell length of wet days (MMaxSLWD); (2) monthlymaximum spell length of dry days (MMaxSLDD); (3) monthlymaximum daily rainfall (MMaxDR); (4) monthly minimumdaily rainfall (MMinDR); (5) monthly sum of daily rain-fall (MSDR); (6) monthly average daily rainfall (MADR);(7) monthly daily rainfall standard deviation (MDRStd); (8)

Table 1. Proposed MODE algorithm.

Algorithm : MODE

1 Initialize population vectors to generate target vectors x size NP2 For G = 1 to Max G3 For i = 1 to NP: Randomly select three distinct vectors and randomly select position j ∈ (1, D)4 For k = 1 to D: generate random number rand (k) ∈ (0,1)5 If rand(k) < CR or k = D then generate trial vectors u at position j end if6 J = next position: If j > D then j = 1 end if7 Next k: Next i8 Combine x and u to create new vectors r size NP × 29 Repeat10 Perform non-dominated sorting to vectors r using simple modified naıve slow sorting11 Until all the population size NP × 2 are sorted, store number of front to NF12 Assign crowded distance to vectors x and trial vectors u in r13 Set remaining required solutions (RRSs) = NP14 For j = 1 to NF: if RRSs = 0 then exit for end if15 If RRSs ≥ the number of solutions in front j then16 pull out all solutions from front j to replace in the next generation G17 RRSs = NP- the number of solutions in front j18 Else if RRSs < the number of solutions in front j then19 For k = 1 to RRSs20 compare crowded distance (CD) of solutions of r in front j (the larger CD will be replaced in the21 next generation G)22 Next k23 end if: Next j: Next G

Calculation of twelvemonthly-statistical

parameters of historical data.

SMFOMC using MODERandom generation NS, SD, and PPL using MODE.Construction of RCLT and TPMs of 12 months.Calculation of PDFs' parameters. Repeat until all U∈(0,1) sequences are used: for year=1 to max year:For month=1 to 12: for day=1 to 365 (or 366 for leap year).

Next day: next month: next year.Calculation of 12 monthly statistical parameters and TMATRE.Execute MODE process to generate POL of TMATRE and NS.Continue until POL doesn't change further.

PDFsU∈(0,1)

Last state

Intermediate state Generateddaily rainfall

Generation of U∈(0,1)sequences.

Maximum NS.

Figure 2. SMFOMC’s parameters optimization using MODE.

monthly daily rainfall skewness (MDRSk); (9) monthly num-ber of wet days (MNWD); (10) monthly number of drydays (MNDD); (11) monthly average number of wet days(MANWD), and, (12) monthly average number of dry days(MANDD). In order to provide the optimal SMFOMC parame-ters that minimize the TMATRE and reduce large computationof TPMs by minimizing NS for each rainfall station with respectto the parameter constrains assigned, the multi-objective wasformulated as:

First objective Min. T MAT RE = U∑

M∑m=1

SSP∑ssp=1

|(OPssp,m − EPssp,m)/OPssp,m|]

/u (5)

Second objective Min. NS (6)

Subject to : Min(NSi) ≤ NSi ≤ Max(NSi)

0.01 ≤ PPL ≤ 0.5

1.1 ≤ SD ≤ 2.0

where u is the number of U sequences, OP ssp,m is the monthlystatistical parameters index ssp of historical daily rainfall formonth m, EP ssp,m is the monthly statistical parameters index sspof generated daily rainfall for month m, ssp is an index of 12statistical parameters ∈ [1,12], m is an index of 12 months ∈ [1,12] that starts from January, i is a state index ∈ [1,10] that refersto SD. The three decisions variables are SD, PPL, and NS. PPLis preserved for application of PDFs ∈ [0.01, 0.5] to generaterainfall amount in the last state. The maximum of PPL wasfixed to be 50%, so that the probability of rainfall occurrenceis described by probability of distribution with a maximum of50%, while the remaining probability is described by TPMs. SDis a discrete decimal number of state divider varying as 1.1, 1.2,. . ., 2.0. NS i is the number of states ∈ [Min(NSi), Max(NSi)].Min(NSi) = 3 (two for wet-dry model) and Max(NSi) is themaximum state for which amount of rainfall in state two afterconstruction of RCLT is not lower than 0.1 mm (dry state)(Srikanthan and McMahon, 1982). This value was calculatedcorresponding to SD, PPL and the maximum daily rainfall ofeach rainfall station (Appendix A).

3.3. Model reliability

The 12 monthly statistical parameters of generated and histor-ical sequences were calculated on a monthly basis through the

38 years. The model reliability for optimal SMFOMC parame-ters was evaluated based on the reproduction of the TMATRE of12 monthly statistical parameters (Equation (5)). The TMATREdescribes the absolute relative error between historical and gen-erated data, the lower value of which indicates a satisfactorilygenerated sequence.

3.4. Data input

Input data were: (1) the historical daily rainfall data sequencefor each rainfall station, (2) U sequences ∈ [0,1], and, (3)maximum NS for each SD (Appendix A). Thirty U sequenceswere generated and compiled as a single file input to theproposed model. These U sequences were assumed to beseveral stochastic events and used for generating daily rainfallsequences during the optimization step. This assured that theobtained optimal SMFOMC parameters for each rainfall stationwould be reliable when the rainfall event is changed.

3.5. Model application

As illustrated in Figure 2, for each rainfall station, aftercompilation of input data, the modelling process was startedwith the calculation of required PDF parameters using MLM.Likewise, the 12 monthly statistical parameters of historicaldaily rainfall data were calculated. Other steps are describedbelow.

Optimization process:

(a) Thirty U sequences were read from the input file. Sets ofNS, SD and PPL were randomly generated as the vectors ofdecision variables. The optimal SMFOMC parameters werecalculated only in cases of shifted exponential distribution,so as to compare the performance of the other two PDFswith this distribution later in the generation process.

(b) The TPMs of 12 months were created using historical data.The RCLT was also constructed with population vectors ofdecision variables. The SMFOMC can then be performedusing stochastically generated 30 U sequences to obtain thevectors of functions of two objectives as Equations (5) and(6). The parameters used in the MODE were pre-tested withthe calculation time and POL observations. The parameterswere updated to observe the optimal POL. The populationvector size of 30, maximum generation of 100, weightedfactor F of 0.5, and crossover constant (CR) of 0.95, werefound suitable for the MODE algorithm, since they providelow calculation time and stable POL.

0 10 20 30 40 50 60 70NS

0 10 20 30 40 50 60 70 80

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60NS

0 10 20 30 40 50 60 70NS

(a) (b)

(d) (e)(c)

Figure 3. Pareto-optimal line of 44 selected rainfall stations: (a) Central, (b) East, (c) North, (d) Northeast and (e) South. NS = number of states.

(c) After obtaining the POLs of all rainfall stations, theSMFOMC parameters (NS, SD and PPL) were acquiredfrom compromise programming.

Generation proces:

(d) For each rainfall station optimal parameters from optimiza-tion under w1s (weighted importance of first objective)were used to generate three replicates of synthesized dailyrainfall sequences of each PDF. When the rainfall amountgeneration of the last state was enabled, three PDFs werecalled to generate rainfall amount with their inverse CDF.Reliability of the optimal SMFOMC parameters for eachPDF was measured by TMATRE and coefficient of deter-mination (R2). The relative performance of PDFs was thenmeasured by a ranking method.

Performance of PDFs

(e) The eigenvalue of each distribution was calculated by firstlyaveraging the values of monthly absolute relative error(MATRE) of 44 rainfall stations for each statistical parame-ter using Equation (7). The average MATRE of each statis-tical parameter for three PDFs was then normalized by thesum of PDF average MATRE of the ith statistical parame-ter. The Eigenvalue of each PDF was eventually calculatedusing Equation (8) by assigning an equal weighted impor-tance. This was performed for one w1. In the case of otherw1s, the procedure was repeated until the Eigenvalues ofall PDFs under w1s were obtained. Moreover, the higherEigenvalue indicated lower performance:

Avg.MAT REd,i = Average(

ST∑st=1

M∑m=1

|OPi,st,m

− EPi,st,m|/OPi,st,m) (7)

where Avg.MATRE d,i is an average of MATRE of the ithstatistical parameter of PDF d; st is station ID ∈ [1,44]; m is anindex of 12 months ∈ [1,12]; OP i,st,m and EP i,st,m are monthlyith statistical parameters of rainfall station st of month m ofhistorical and generated sequences respectively:

Eigenvalued =I∑

[Norm(Avg.MAT REd,i) × (wi)] (8)

where Eigenvalued is an Eigenvalue of PDF d; Norm(Avg.MATRE d,i) is a normalized average MATRE of PDF d ofith statistical parameter, wi is a weighted importance of ithstatistical parameter.

4. Results and discussion

4.1. Optimal SMFOMC parameters

During simulation runs, various POLs were generated. Inmost of the events, the POLs did not change further after60 simulation runs. Thus, the simulation was continuouslyperformed until 100 runs and the optimal POL values wererecorded. The results of discrete optimal POLs (Figure 3)clearly showed that the TMATRE decreased with increasingNS. The TMATRE was high with low NS, and it sharplydecreased until the NS reached nearly 10. The NS valuesbetween 18 and 24 were not found in the POLs. After NSof 25, the TMATRE decreased slightly.

4.1.1. Variations in optimal SMFOMC parameters with w1 s

The SMFOMC parameters were controlled by assigning theamount of rainfall in the upper limit of second rainfallstate (wet) to be larger than 0.1 mm (Figure 4). NS must beminimized according to the second objective function in orderto reduce the size of TPMs and time of model simulation.Rapidly rising trends of the NS were noticed when w1s

were higher than 0.5 (Figure 4(a)). The PPL (Figure 4(b))remained stable when w1s was lower than 0.5, after which sharpdecreases in PPLs were noted. Variations in SD are illustratedin Figure 4(c). This parameter specifies the interval of rainfallamounts of the successive intermediate states. A maximum SDof 1.6 was noted upon varying w1s in the range of 0.1–0.6 anda minimum SD of 1.1 was found at values of w1s higher than0.7. The minimum values of SD visibly corresponded to lowerTMATRE. As shown in Figure 4(d), TMATRE showed largedisparity at w1s lower than 0.50, after which TMATRE startedgaining stability. It could be summarized that TMATRE wasdecreasing as the given w1s were higher. As hypothesized, theconflicting characteristics of TMATRE and NS are evidentlynoticeable in Figure 4(a) and (d). However, for further analysisin the generation process the compromise set of solutions that

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0w1s

(b)(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 4. Effects of weighted importance values of objective one (w1s) on: (a) the number of total states, (b) the preserved proportion in thelargest state, (c) the state dividers and (d) TMATRE.

Table 2. Probability distribution parameters.

Statistic value Shifted exponential Exponential Two-parameter gamma

λ α ε α β Csa

Max 3.95 124.5 279.4 14.4 954.3 0.39Min 0.36 2.6 60.8 0.004 26.1 0.01Average ± STD 1.3 ± 0.7 35 ± 23.9 112.5 ± 38.6 2.4 ± 2.3 112.5 ± 132.2 0.23 ± 0.08

a Cs is the skewness coefficient of the observed historical daily rainfall in the last state.

Station ID

NorthNorth-East South East Central

Figure 5. Variations in TMATRE of generated daily rainfall sequences with w1s. , 0.5 (wl); , 0.6 (wl); , 0.7 (wl); , 0.8 (wl); , 0.9 (wl);, 1.0 (wl).

obtained by varying w1s of 0.50 onward was considered, asthose w1s provided low and stable TMATRE.

4.1.2. Daily rainfall generation

In the generation process three replicates of daily rainfallsequences were generated as historical daily rainfall sequenceswith an equal length for each rainfall station. Three PDFswere applied to generate the daily rainfall sequences in the laststate corresponding to the same time and the same U. Prob-ability distribution parameters were obtained from 44 rainfall

stations once all RCLT for w1s were constructed (Table 2).The generated sequences were summarized monthly, basedon the 12 statistical parameters, and were then averaged forthe three replicates. High TMATRE values (above 300) werefound in w1s of 0.1–0.4. However, distinctly visible varia-tions were observed in TMATRE for w1s in the range of0.5–0.7 (Figure 5). In general, the compromise solution ispresented with given w1 of 0.5 to both objectives. More-over, the compromise solutions could also be taken as desir-able solutions under the variation of w1s. The comparativeanalysis of w1s on the variation of ∗TMATRE (normalized

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SE E G-2 SE E G-2 SE E G-2

(b) (c)(a)

(e) (f)(d)

(h) (i)(g)

(k) (l)(j)

Figure 6. ∗TAMTRE variation of 12 statistical parameters of all stations from three PDFs (SE, E, and G-2) with different w1s: (a) MMaxSLWD(days), (b) MMaxSLDD (days), (c) MMaxDR (mm), (d) MADR (mm), (e) MDRStd, (f) MDRSk, (g) MNWD (days), (h) MNDD (days),(i) MANWD (days), (j) MANDD (days), (k) MSDR (mm) and (l) MMinDR (mm). Means within each PDF with the same letter are notsignificantly different (p < 0.05) by Duncan Multiple Range Test. ∗TMATRE = [Normalized sum of all TMATRE of 44 rainfall stations of eachstatistical parameter under each w1/(44 rainfall stations × 12 months)]. , 0.5 (wl); , 0.6 (wl); , 0.7 (wl); , 0.8 (wl); , 0.9 (wl); , 1.0

sum of all TMATRE of 44 rainfall stations of each PDFunder each w1 for 12 statistical parameters) is presented inFigure 6.

4.1.3. Effect of criteria weights on statistical parameters ofsynthetic daily rainfall sequences

Based on the statistical test results, a model was consideredto perform satisfactorily if the average of the parametersestimated from the replicates was close to the historical values.No significant differences in ∗TMATRE were found in thestatistical parameters for three PDFs (Figure 6) except inMMaxDR (Figure 6(c)), in MMinDR (Figure 6(l)), in MSDR(Figure 6(k)), in MADR (Figure 6(d)), in MRStd (Figure 6(e))in MSk (Figure 6(f)). Those differences in ∗TMATRE (p <

0.05) that varied by w1s, indicated that the w1s statisticallyaffected the acceptability of those statistical parameters. The∗TMATRE of all statistical parameters of generated data variedbetween 2 and 25% and deviated from historical statisticalparameters except MMinDR, which showed distinctly largeerror in its estimation (Figure 6(l)). This may be due to thedifficulty in estimating the near zero value of daily rainfallgeneration with the application of this model. The daily rainfallgeneration in the last state using three PDFs did not show anysignificant difference. On the contrary, they provided almost thesame ∗TMATRE for all statistical parameters. Although it couldnot be firmly concluded which PDF performs the best, to rank

the performance of the three PDFs the Eigenvalues of threePDFs under w1s were later calculated and ranked. Over half ofthe 12 statistical parameters provided low ∗TMATRE in w1s of0.8–1.0 (Figure 6(d–f, k and l)) compared to the others. Hencefor this range of w1s (0.8–1.0) the cumulative ratios of monthlystatistical parameters of generated data to yearly historical dataversus the cumulative ratios of monthly statistical parametersof monthly historical data to yearly historical data for allPDFs were then plotted to see correlations (Figure 7). Goodcorrelations (R2 > 0.9) were observed for most parametersexcept in the extreme overestimates of MMinDR. In addition,modest underestimates in MMaxSLWD, MMaxSLDD andMMaxDR also appeared.

4.1.4. Which set of SMFOMC parameters should be selected?

Most of the parameters considered in this study providedlow ∗TMATRE with a range of less than 0.1–0.3 times thehistorical statistical parameters (Figure 6(a–k)) and ranges ofR2 > 0.9 (Figure 7(a–l)). This indicates that the model andoptimal parameters could realistically describe the variation ofhistorical daily rainfall data. An unacceptably high ∗TMATREwas received in the case of MMinDR (Figure 6(l)) where∗TMATRE was about 3–6 times for w1s of 0.8–1.0, 8–10times for w1s of 0.6–0.7, and about 16 times for w1 of 0.5.This implies that if MMinDR is neglected the compromisesolution with w1 of 0.5 would generally be the desired solution.

R2 = 0.986R2 = 0.946R2 = 0.9430.0

R2 = 0.980R2 = 0.990R2 = 0.996

R2 = 0.998R2 = 0.999R2 = 0.998

R2 = 0.2720

0.0 1.0 2.0 3.0 4.0 5.0

R2 = 0.994

Historical data

R2 = 0.999

0.0 0.4 0.8 1.2 1.6 0.0 0.4 0.8 1.2 1.6

1:1 1:1 1:1

(b) (c)(a)

(e) (f)(d)

(h) (i)(g)

(k) (l)(j)

Figure 7. Regression plots for the cumulative ratios of the monthly statistical parameters of generated data to yearly historical dataversus the cumulative ratios of monthly statistical parameters of monthly historical data to yearly historical data for w1s in the range of0.8–1.0 month by month (All plots obtained from three distributions): (a) MMaxSLWD (days; R2 = 0.943), (b) MMaxSLDD (days; R2 = 0.946),(c) MMaxDR (mm; R2 = 0.986), (d) MADR (mm; R2 = 0.996), (e) MDRStd; R2 = 0.990, (f) MDRSk; R2 = 0.980, (g) MNWD (days;R2 = 0.998), (h) MNDD (days; R2 = 0.999), (i) MANWD (days; R2 = 0.998), (j) MANDD (days; R2 = 0.999), (k) MSDR (mm; R2 = 0.994)

and (l) MMinDR (mm; R2 = 0.272). , Regression line. This figure is available in colour online at wileyonlinelibrary.com/journal/met

Table 3. Performance ranking of three distributions under variation of w1s.

PDFs Criteria weights of the first objective (w1s)

1.0 0.9 0.8 0.7 0.6 0.5

Eigenvalues SE 0.3416 0.3357 0.3345 0.3361 0.3360 0.3369E 0.3391 0.3331 0.3334 0.3329 0.3331 0.3325

G-2 0.3192 0.3313 0.3321 0.3310 0.3308 0.3306

Otherwise, the compromise solutions with w1s of 0.8–1.0would be appropriate.

4.2. Performance of PDFs

The performance of the three PDFs, as indicated with Eigen-values, is presented at different values of w1s (Table 3) as w2s

was reduced an importance according to higher w1s. The resultsclearly show consistency for all w1s: the poorest performancewas provided by the SE, and the G-2 showed the best perfor-mance.

Relative differences of yearly sum of the 12 monthlystatistical parameters through the 38 years between generatedand historical sequences of G-2 with w1 of 0.8 were interpolatedusing inverse distance weighing (IDW) and are depicted in

GIS maps (Figure 8). Tolerably small differences were noted(percentage difference ranged between 1 and 30% with anaverage of about 2%) for all statistical parameters exceptMminDR, where the yearly sum of MminDR overestimatedthe generated data (17.8 ± 56 mm) compared with that ofthe historical average (3.3 ± 6.4 mm). Statistical parameters ofgenerated data tended to be underestimated for MMaxSLWD,MMaxSLDD, MMaxDR, MDRSk, MNWD and MANWD(Figure 8(a–c, g–i)) whilst others were overestimated. Thelower middle part of the Northeast region showed a largerdeviation of MminDR when compared to the other regions(Figure 8(l)). This is because of the historical amount ofrainfall in the dry state was lower than 0.1 (e.g. the amountof MminDR 0.067 mm in the case of rainfall station ID14033). Nevertheless, the Northeast region provided larger error

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 8. Maps of relative differences between yearly statistical parameters of generated and historical daily rainfall (All plots resulted byG-2 distribution with w1 of 0.8): (a) MMaxSLWD (days), (b) MMaxSLDD (days), (c) MMaxDR (mm), (d) MSDR (mm), (e) MADR (mm),(f) MDRStd, (g) MDRSk, (h) MNWD (days), (i) MANWD (days), (j) MNDD (days), (k) MANDD (days) and (l) MminDR (mm). Relative

difference derived from difference of generated data and historical data divided by historical data.

differences of MMaxSLWD and MMaxSLDD, between 5 and30%, probably due to the fact that this region receives lowerand more inconsistent rainfall compared to the other regions.This could be the probable reasons for larger deviations as themodel loses its accuracy particularly at low rainfall depths.

In the North, overestimation of the generated MSDR andMNWD was noted (14% max) (Figure 8(d)). In the Centralregion, the highest error estimation was found in MmaxDR(20% max) (Figure 8(c)) with about 80 mm year−1 averagedifference of yearly sum of MmaxDR. In the Southern region

Table 4. Average optimal number of state, preserved proportion in the last state, and state divider for w1s.

Criteria weights ofobjective 1(w1)

SMFOMC’s parameters Minimum Maximum Average ± STD

0.10 3 3 3 ± 0.0000.20 3 4 4 ± 0.3870.30 4 5 4 ± 0.5060.40 6 7 7 ± 0.4770.50 Number of states 9 13 10 ± 0.9910.60 11 18 14 ± 1.7190.70 13 36 20 ± 7.6320.80 15 42 29 ± 8.0800.90 28 49 37 ± 4.9371.00 33 68 49 ± 8.342

0.10 0.4954 0.5000 0.499 ± 0.0010.20 0.4770 0.5000 0.496 ± 0.0050.30 0.4776 0.5000 0.497 ± 0.0050.40 0.4824 0.5000 0.498 ± 0.0030.50 Preserved proportion in the last state 0.4436 0.5000 0.493 ± 0.0100.60 0.4174 0.4999 0.487 ± 0.0200.70 0.3232 0.4999 0.459 ± 0.0430.80 0.2498 0.4994 0.410 ± 0.0660.90 0.1645 0.4956 0.361 ± 0.0791.00 0.1676 0.4890 0.302 ± 0.081

0.10 1.4 1.6 1.5 ± 0.0620.20 1.5 1.6 1.5 ± 0.0390.30 1.5 1.6 1.5 ± 0.0490.40 1.5 1.6 1.5 ± 0.0420.50 State divider 1.5 1.6 1.5 ± 0.0490.60 1.5 1.6 1.5 ± 0.0390.70 1.1 1.6 1.4 ± 0.1700.80 1.1 1.5 1.2 ± 0.1550.90 1.1 1.2 1.1 ± 0.0461.00 1.1 1.2 1.1 ± 0.029

the largest differences were observed in some rainfall stationsfor MDRstd and MDRsk (17% max) (Figure 8(f) and (g)).The model appears efficiently applied to the East region withvery small deviation from historical data for all statisticalparameters.

5. Conclusion

With the dataset containing 38 years of daily rainfall from 44rainfall stations located in 5 regions of Thailand, the daily rain-fall was modelled with SMFOMC in terms of daily rainfalloccurrences and generation. The optimal SMFOMC parame-ters were appropriately obtained by the effective use of MODEintegration with SMFOMC at desirable intervals in MarkovChain model parameters specification. Minimization of twoconflicting objectives on TMATRE and NS was consideredwith three selected decision variables of SD, PPL and NS.The proposed model reproduced characteristics of original dailyrainfall occurrences with acceptable seasonality and accuracy.The model was described in detail with the sensitivity ofweighted importance values of objective one (w1s). Lowerw1s adversely affected MMaxDR, MMinDR, MADR, MSDR,MDRStd and MDRSk, whereas higher w1s offered significantlyhigher accuracy and acceptability. The model failed to describeMMinDR. This was the particular case found in the stochasticrainfall simulation in which the period of available historicaldata was short and the extreme rainfall events were rare tomodel (Regniere and St-Amant, 2007). However, the optimalSMFOMC parameters under w1s of 0.8–1.0 (Table 4) are rec-ommended to be the appropriate solutions that could potentially

be applied for Thailand. The study also verified the performanceof PDFs that have been applied in the research. The shiftedexponential, exponential and two-parameter gamma distribu-tions were concluded to be generally adequate for describingthe rainfall occurrences in the last state. Although no significantdifference was found among three PDFs, the performance rank-ing showed the higher potential of the two-parameter gammadistribution over the others. Moreover, the usefulness of TPMsis still considerable for SMFOMC. With daily rainfall amountsless than 60–70% of the maximum daily rainfall, TPMs helpedthe model to be fitted well with historical data, and the remain-ing data was mobilized by PDFs (see optimal PPL of w1s inthe range of 0.8–1.0 in Table 4, where an average PPL is inthe range of 0.3–0.4).

Abbreviations.

ARIMA = Multiplicative autoregressive integrated movingaverage

CDF = Cumulative probability distribution functionCP = Compromise programmingCR = Crossover constantDEA = Differential evolution algorithmEAs = Evolutionary algorithmsE = Exponential distributionG-2 = Two-parameter gamma distributionIDW = Inverse distance weightingMATRE = Monthly absolute relative error

MADR = Monthly average daily rainfallMANDD = Monthly average number of dry daysMANWD = Monthly average number of wet daysMDRSk = Monthly daily rainfall skewnessMDRStd = Monthly daily rainfall standard deviationMMaxSLDD = Monthly maximum spell length of dry daysMMaxSLWD = Monthly maximum spell length of wet daysMMaxDR = Monthly maximum daily rainfallMMinDR = Monthly minimum daily rainfallMNDD = Monthly number of dry daysMNWD = Monthly number of wet daysMSDR = Monthly sum of daily rainfallMODE = Multi-objective differential evolutionMOPs = Multi-objective optimization problemsMLM = Maximum-Likelihood methodNDSs = Non-dominated solutionsNDSA = Non-dominated sorting algorithmsNS = The number of statesPDFs = Probability distributions functionsPOL = Pareto-optimal linePPL = Preserved proportion in the last stateRCLT = Rainfall class limits tableSD = State dividerSE = Shifted exponential distributionSFOMC = Stochastic First-Order Markov ChainSMFOMC = Stochastic Multi-States First-Order Markov ChainTMATRE = Monthly absolute total relative error∗TMATRE = Normalized monthly absolute total relative error

for each statistical parameterTPMs = Transition probability matrices

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Appendix A. Geographic details of rainfall station and the maximum number of state under varying state dividers

Regions Stationcode

ID Province Latitude Longitude Maximum number of state undervarying state dividers

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Central 04 361 1 Chainat 15°09′57′′ 100°11′32′′ 67 36 25 20 17 15 13 12 11 1019 351 2 Lop Buri 15°20′21′′ 101°22′30′′ 64 34 24 19 16 14 13 12 11 1031 022 3 Nonthaburi 13°54′38′′ 100°30′09′′ 70 37 26 21 18 15 14 13 12 1132 012 4 Pathumthani 14°01′05′′ 100°32′12′′ 69 37 26 21 17 15 14 13 12 1154 012 5 Saraburi 14°31′35′′ 100°54′51′′ 64 34 24 19 16 14 13 12 11 1056 012 6 Singburi 14°53′12′′ 100°24′29′′ 66 35 25 20 17 15 13 12 11 1060 013 7 Suphanburi 14°28′10′′ 100°07′14′′ 68 36 26 20 17 15 13 12 11 11

East 03 231 8 Chachoengsoa 13°28′29′′ 101°37′44′′ 64 34 24 19 16 14 13 12 11 1006 121 9 Chanthaburi 12°47′23′′ 102°15′33′′ 70 37 26 21 18 15 14 13 12 11

9160 10 Chonburi 13°12′04′′ 100°57′59′′ 68 36 26 20 17 15 14 12 11 1144 191 11 Prachinburi 14°10′37′′ 101°47′30′′ 70 37 26 21 18 15 14 13 12 1148 141 12 Rayong 12°55′41′′ 101°19′30′′ 64 34 24 19 16 14 13 12 11 1066 071 13 Trat 12°28′28′′ 102°28′52′′ 79 42 30 23 20 17 15 14 13 12

North 07 391 14 Chaingmai 18°47′21′′ 99°01′01′′ 63 34 24 19 16 14 13 11 11 1016 151 15 Lampang 18°08′09′′ 99°34′53′′ 71 38 27 21 18 16 14 13 12 1116 181 16 Lampang 18°48′12′′ 99°38′45′′ 64 34 24 19 16 14 13 12 11 1017 081 17 Lampang 17°53′16′′ 99°05′20′′ 67 36 25 20 17 15 13 12 11 1120 111 18 Maehongson 19°16′10′′ 97°56′55′′ 66 35 25 20 17 15 13 12 11 1028 111 19 Nan 18°34′05′′ 100°52′28′′ 72 38 27 21 18 16 14 13 12 1140 151 20 Phrae 18°08′44′′ 100°08′42′′ 68 36 26 20 17 15 13 12 11 1163 181 21 Tak 16°45′44′′ 98°45′14′′ 64 34 24 19 16 14 13 12 11 10

North East 05 200 22 Chaiyaphum 15°46′07′′ 101°49′03′′ 65 35 25 20 16 14 13 12 11 1021 012 23 Mahasarakham 16°21′58′′ 103°18′17′′ 68 36 26 20 17 15 14 12 11 1125 511 24 Nakhonratchasima 14°35′20′′ 101°50′30′′ 63 34 24 19 16 14 13 11 11 1025 550 25 Nakhonratchasima 14°50′47′′ 101°42′15′′ 63 34 24 19 16 14 13 11 11 1050 150 26 Sakonnakhon 17°13′43′′ 103°33′08′′ 67 36 25 20 17 15 13 12 11 1150 160 27 Sakonnakhon 17°14′51′′ 103°34′16′′ 68 36 26 20 17 15 13 12 11 1150 170 28 Sakonnakhon 17°13′02′′ 104°02′14′′ 70 37 26 21 18 15 14 13 12 1150 180 29 Sakonnakhon 17°12′57′′ 103°57′24′′ 70 37 26 21 18 15 14 13 12 1157 161 30 Sisaket 14°29′48′′ 104°03′29′′ 66 35 25 20 17 15 13 12 11 1062 120 31 Surin 14°48′48′′ 103°29′50′′ 64 34 24 19 16 14 13 12 11 1067 220 32 Ubonratchathani 15°14′17′′ 104°51′01′′ 67 36 25 20 17 15 13 12 11 11

38 1201 33 Khonkaen 16°26′00′′ 102°50′00′′ 68 36 26 20 17 15 14 12 11 1138 1301 34 Khonkaen 16°20′00′′ 102°49′00′′ 68 36 26 20 17 15 14 12 11 11

14 033 35 Khonkaen 15°48′52′′ 102°36′12′′ 69 37 26 21 17 15 14 13 12 1114 472 36 Khonkaen 15°57′00′′ 102°33′00′′ 67 36 25 20 17 15 13 12 11 11

South 10 111 37 Chumphon 10°37′18′′ 99°03′39′′ 72 38 27 21 18 16 14 13 12 1115 012 38 Krabi 8°03′15′′ 98°55′17′′ 67 36 25 20 17 15 13 12 11 1134 012 39 Phangnga 8°27′35′′ 98°31′54′′ 69 37 26 21 17 15 14 12 12 1143 013 40 Phuket 7°53′18′′ 98°23′14′′ 67 36 25 20 17 15 13 12 11 1145 181 41 Prachuapkirikhan 12°06′55′′ 99°44′20′′ 72 38 27 21 18 16 14 13 12 1146 013 42 Ranong 9°57′55′′ 98°08′12′′ 71 38 27 21 18 16 14 13 12 1158 221 43 Songkhla 6°37′59′′ 100°23′46′′ 67 36 25 20 17 15 13 12 11 1161 341 44 Suratthani 9°25′31′′ 99°09′44′′ 75 40 28 22 19 16 15 13 12 12

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