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

METEOROLOGICAL APPLICATIONSMeteorol. Appl. (2011)Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/met.292

Optimal stochastic multi-states first-order Markov chainparameters for 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 usedas an important input for mathematical simulation inhydrology, agriculture and water resources models. Inthe case of univariate series daily rainfall models, themissing data could be generated using statistical param-eters that describe the hydrological behaviour withinthe sequence itself. For this purpose, the multiplicativeautoregressive integrated moving average (ARIMA) andThomas–Fiering models have been extensively applied(Delleur and Kavvas, 1978; Sharma, 1985; Vogel, 1988;Mujumdar and Kumar, 1990; Schreider et al., 1997; Tothet al., 1999; Ahmad et al., 2001; Taewichit and Chitta-ladakorn, 2007; Amini et al., 2009). However, the dif-ficulty, complexity and requirement of large statisticalparameters are considered to be their limitations. Onesimplified approach is the use of a model called theStochastic First-Order Markov Chain (SFOMC), which

∗ Correspondence to: P. Soni, Agricultural Systems and Engineering,School of Environment Resources and Development, Asian Institute ofTechnology, Pathumthani 12120, Thailand. E-mail: [email protected]

describes the probability of rainfall occurrences on agiven day using transition probability matrices (TPMs).The SFOMC model is applied to study the occurrences ofdaily rainfall (Gabriel and Neumann, 1962; Moon et al.,2006) and to construct rainfall-runoff synthesizing mod-els (Kottegoda et al., 2000). The main concept behind theSFOMC is the use of conditional probability to describethe occurrences and non-occurrences of rainfall (Gabrieland Neumann, 1962). Initial development and applicationof SFOMC is as a two-state (wet-dry) model (Gabrieland Neumann, 1962; Todorovic and Woolhiser, 1975;Haddada et al., 2000), which is applied to generate wet-dry events. Further development of SFOMC is the useof the two-state model coupled with some probabilitydistribution functions (PDFs) (Tsakeris, 1988) to esti-mate daily rainfall. In addition, modifying the SFOMCmodel by increasing the number of states (NS) in thewet state (Khanal and Hamrick, 1974; Srikanthan andMcMahon, 1984; Hutchinson, 1990; Aksoy, 2003) cou-pled with the uses of PDFs in the last state has beenattempted successfully. This model is popularly knownas the Stochastic Multi-States First-Order Markov Chain

Copyright 2011 Royal Meteorological Society

C. Taewichit et al.

(SMFOMC). However, some of the difficulties noted inusing the SMFOMC (Haan, 1977) were: (1) determiningNS, (2) determining the intervals of the variable understudy to associate with each state, and, (3) assigning anumber to the magnitude of an event once the state isdetermined. Most research studies of SMFOMC still usetrial and error to overcome these limitations and to deter-mine the optimal SMFOMC parameters for which thegenerated daily rainfall sequences are close to those ofhistorical sequences.

To synthesize daily rainfall for a single site (univari-ate model), optimal SMFOMC parameters of the num-ber of states (NS), the preserved proportion in the laststate (PPL), and the state divider (SD) are determined inthis study. Forty-four rainfall stations from five regions(Central, North, North-East, East, and South) in Thai-land (Figure 1) were selected to apply and to evaluate theoptimal parameters of the SMFOMC model. The multi-objective algorithm, multi-objective differential evolution(MODE), was employed in this study. Two conflict-ing objectives of minimizing NS and minimizing totalmonthly absolute total relative error (TMATRE) wereset with statistical parameters of generated daily rain-fall sequences. Performance of three PDFs, the shiftedexponential distribution (SE), the exponential distribu-tion (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 watermanagement for modelling processes (Kottegoda et al.,2000; Aksoy, 2003; Ochola and Kerkides, 2003). In itsfirst order models it employs conditional probability todescribe the process x(t) at the present time t usingonly the outcome at previous time t − 1. A higher-order Markov Chain model, corresponding to the numberof preceding days (Chapman, 1998), could also beformulated (Kulkarni et al., 2002). SMFOMC may thenbe considered as a simple two-state for a dry day (no rain)and a wet day. However, there is no discernible differencereported between the model performance of first andsecond order models in synthesizing daily rainfall (Jimohand Webster, 1996).

SMFOMC is defined by its transition probabilitymatrices (TPMs) and frequency distributions of rainfallamounts (Haan et al., 1976) that can preserve most ofthe daily, monthly and annual characteristics (Srikanthanand McMahon, 2001). The TPMs play a significant rolein estimating the present data j at time t using theprobability pij (t) of moving from state i at time t − 1 tostate j at time t , which is derived from the frequencyof state changes from state i to state j . To obtainthe frequency of daily rainfall for each of the states,a rainfall class limits table (RCLT) is constructed forclassifying rainfall data into successive states of j = 1to j = r . Each state consists of upper bound and lower

bound rainfall amounts. The state interval for each stateis mostly specified through manual trial and error byresearcher’s experience. In the daily data generation, theamount of rainfall in the intermediate state j at time t

(state of j = 2 to state j = r − 1) is estimated by addingthe lower bound of rainfall amounts of state j with theterm of the stochastic uniform random number U ∈ (0, 1)multiplied by the difference of rainfall amounts betweenupper and lower bound of state j (linear interpolation).The U is generated from the most popular generatorscalled linear congruential generators (Salas, 1992; Reddy,1997). In addition, the inverse cumulative probabilitydistribution function is used to estimate rainfall amountin the last state r .

2.2. Probability distributions

Various PDFs have been applied by researchers (Allenand Haan, 1975; Todorovic and Woolhiser, 1975; Suhailaand Jemain, 2007). However, the cumulative probabilitydistribution function (CDF), which is an area under thePDF curve, is popularly used. SE was first proposed byAllen and Haan (1975). The distribution in terms of CDFis given as:

F(x) = 1 − e−(x−Rfc−1)/λ (1)

where F(x) is a CDF, λ is difference of average of allrecorded historical daily rainfall length being greater thanor equal to Rf c−1, Rf c−1 is the rainfall amount of lowerbound in the last state, x is the rainfall amount in thelast state and c is the state of daily rainfall amount.The Maximum-Likelihood method (MLM) is used forestimating parameters of E and G-2. Numerical methods,suggested by Rao and Hamed (2001), are applied tosolve the parameters of both the distributions. In the caseof E, which is a special case of the Gamma family,

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

Copyright 2011 Royal Meteorological Society Meteorol. Appl. (2011)

Synthesizing daily rainfall: multi-objective differential evolution

the distribution can be obtained by setting β = 1 inEquation (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)

is gamma function.From Equation (4), G-2 has extensively been used in

the Markov Chain model (Coe and Stern, 1982; Richard-son and Wright, 1984; Duan et al., 1995; Kottegodaet al., 2000; Aksoy, 2003): its PDF is formulated by elim-inating 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 objec-tives simultaneously. Various objectives are incorporatedfor making decisions to select the desirable solution. Thesolutions of MOPs comprise non-dominated solutions(NDSs). NDSs are often expressed as the Pareto-optimalline (POL). The heuristic stochastic search techniques,evolutionary algorithms (EAs), have been used inten-sively for solving MOPs (Coello et al., 2002) owing totheir population-based nature that allow the generation ofseveral elements of POL in a single run. EAs also provide

a diversification mechanism to obtain a better solution.The context of EAs in MOPs is to find a POL as closeas possible to the true POL and diversify solution onthe POL as much as possible. The POL comprises NDSsthat have been sorted in many front levels using non-dominated sorting algorithm (NDSA). The NDSA, alsoknown as simple modified naıve slow was proposed byDeb (2001) and is applied in the present study. After theset of POL is met, the preferred solution can be chosenusing the compromise programming (CP) technique withweighted importance values of 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 evolu-tion algorithm (DEA) (Storn and Price, 1997) for multi-objective optimization (Sun et al., 2005). The process ofDEA begins with randomly generating the populationof solution vectors size NP of D dimension or ‘targetvectors’. To improve the solution vectors (trial vectors)recombination process is used, which consists of mutationand crossover. Solution values are swapped and changedby chances of probabilities. The probability is launchedby the crossover constant (CR ∈ (0, 1)) when the ran-dom number during trial vector generation is less thanor equal to CR. The trial vector size, NP, is generateddimension-by-dimension by randomly picking three dis-tinct solution vectors and adding the first vector to theproduct of the weighted factor (F ∈ (0, 1)) and the dif-ference of the remaining two vectors. The trial vectorwill replace the temporary target vectors if the objectivevalue of the former is better than the latter.

Table I. 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


C. Taewichit et al.

The DEA is modified to be MODE (Xue, 2003; Reddyand Kumar, 2007; Li and Zhang, 2008), where theinitialization vectors of population is started as with anormal DEA to generate target vectors x with size NP,which is followed by generation of new trial vectorsu with size NP in the recombination process. Thosevectors are combined and sorted to be vectors r using theproposed NDSA to rank NDSs. The selection process issimilar to an elitist non-dominated sorting GA (NSGAII)where the sorted solutions of NP are directly pickedup from the solution fronts, j , which replace the oldset of solutions. The lending mechanisms from NSGAII,called the crowded distance assignment (Deb, 2001), areassigned to each solution in the non-dominated fronts inorder to use these as criteria to select better comparedsolutions when the last required set of solutions islocated in the same front. The larger crowded distanceis preferred to be chosen. Those processes are completedin one generation. Those steps are repeated generation bygeneration until the set of POL does not change further(Table I).

3. Model application

3.1. Study area and locations of selected rainfallstations

For this study, a 38 year (1971–2008) continuous recordof daily rainfall occurrences at 44 stations distributedin 5 regions of Thailand (Figure 1) was used. Therainfall stations were selected based on the data continuityand with length of record for more than 30 years.The daily rainfall data were obtained from the RoyalIrrigation Department and the Meteorological Departmentof Thailand.

3.2. Model formulation for optimization of SMFMOCparameters

Twelve monthly statistical parameters were used tomeasure the adequacy and acceptability of the model:(1) monthly maximum spell length of wet days(MMaxSLWD); (2) monthly maximum spell length ofdry days (MMaxSLDD); (3) monthly maximum dailyrainfall (MMaxDR); (4) monthly minimum daily rainfall(MMinDR); (5) monthly sum of daily rainfall (MSDR);(6) monthly average daily rainfall (MADR); (7) monthlydaily rainfall standard deviation (MDRStd); (8) monthlydaily rainfall skewness (MDRSk); (9) monthly numberof wet days (MNWD); (10) monthly number of drydays (MNDD); (11) monthly average number of wetdays (MANWD), and, (12) monthly average number ofdry days (MANDD). In order to provide the optimalSMFOMC parameters that minimize the TMATRE andreduce large computation of TPMs by minimizing NSfor each rainfall station with respect to the parameterconstrains assigned, the multi-objective was formulated

as:

F irst objective Min. T MAT RE = U∑

u=1

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 themonthly statistical parameters index ssp of historicaldaily rainfall for month m, EP ssp,m is the monthlystatistical parameters index ssp of generated daily rainfallfor month m, ssp is an index of 12 statistical parameters∈ [1,12], m is an index of 12 months ∈ [1, 12] thatstarts from January, i is a state index ∈ [1,10] thatrefers to SD. The three decisions variables are SD, PPL,and NS. PPL is preserved for application of PDFs ∈[0.01, 0.5] to generate rainfall amount in the last state.The maximum of PPL was fixed to be 50%, so thatthe probability of rainfall occurrence is described byprobability of distribution with a maximum of 50%, whilethe remaining probability is described by TPMs. SD is adiscrete 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) andMax(NSi) is the maximum state for which amount ofrainfall in state two after construction of RCLT is notlower than 0.1 mm (dry state) (Srikanthan and McMahon,1982). This value was calculated corresponding to SD,PPL and the maximum daily rainfall of each rainfallstation (Appendix A).

3.3. Model reliability

The 12 monthly statistical parameters of generated andhistorical sequences were calculated on a monthly basisthrough the 38 years. The model reliability for optimalSMFOMC parameters was evaluated based on the repro-duction of the TMATRE of 12 monthly statistical parame-ters (Equation (5)). The TMATRE describes the absoluterelative error between historical and generated data, thelower value of which indicates a satisfactorily generatedsequence.

3.4. Data input

Input data were: (1) the historical daily rainfall datasequence for each rainfall station, (2) U sequences∈ [0,1], and, (3) maximum NS for each SD(Appendix A). Thirty U sequences were generated andcompiled as a single file input to the proposed model.These U sequences were assumed to be several stochas-tic events and used for generating daily rainfall sequences



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.

TPMs

PDFsU∈(0,1)

RCLT

Last state

Intermediate state Generateddaily rainfall

Generation of U∈(0,1)sequences.

Maximum NS.

Figure 2. SMFOMC’s parameters optimization using MODE.

during the optimization step. This assured that theobtained optimal SMFOMC parameters for each rain-fall station would be reliable when the rainfall event ischanged.

3.5. Model application

As illustrated in Figure 2, for each rainfall station, aftercompilation of input data, the modelling process wasstarted with the calculation of required PDF parametersusing MLM. Likewise, the 12 monthly statistical parame-ters of historical daily rainfall data were calculated. Othersteps are described below.

Optimization process:

(a) Thirty U sequences were read from the input file.Sets of NS, SD and PPL were randomly generatedas the vectors of decision variables. The optimalSMFOMC parameters were calculated only in casesof shifted exponential distribution, so as to comparethe performance of the other two PDFs with thisdistribution later in the generation process.

(b) The TPMs of 12 months were created using historicaldata. The RCLT was also constructed with populationvectors of decision variables. The SMFOMC canthen be performed using stochastically generated30 U sequences to obtain the vectors of functionsof two objectives as Equations (5) and (6). Theparameters used in the MODE were pre-tested withthe calculation time and POL observations. Theparameters were updated to observe the optimalPOL. The population vector size of 30, maximumgeneration of 100, weighted factor F of 0.5, andcrossover constant (CR) of 0.95, were found suitablefor the MODE algorithm, since they provide lowcalculation time and stable POL.

(c) After obtaining the POLs of all rainfall stations,the SMFOMC parameters (NS, SD and PPL) wereacquired from compromise programming.

Generation proces:

(d) For each rainfall station optimal parameters fromoptimization under w1s (weighted importance of first

objective) were used to generate three replicates ofsynthesized daily rainfall sequences of each PDF.When the rainfall amount generation of the last statewas enabled, three PDFs were called to generate rain-fall amount with their inverse CDF. Reliability ofthe optimal SMFOMC parameters for each PDF wasmeasured by TMATRE and coefficient of determi-nation (R2). The relative performance of PDFs wasthen measured by a ranking method.

Performance of PDFs

(e) The eigenvalue of each distribution was calculatedby firstly averaging the values of monthly absoluterelative error (MATRE) of 44 rainfall stations foreach statistical parameter using Equation (7). Theaverage MATRE of each statistical parameter forthree PDFs was then normalized by the sum of PDFaverage MATRE of the ith statistical parameter. TheEigenvalue of each PDF was eventually calculatedusing Equation (8) by assigning an equal weightedimportance. This was performed for one w1. In thecase of other w1s, the procedure was repeated untilthe Eigenvalues of all PDFs under w1s were obtained.Moreover, the higher Eigenvalue indicated lowerperformance:

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 an index of 12 months ∈ [1,12]; OP i,st,m and EP i,st,m

are monthly ith statistical parameters of rainfall stationst of month m of historical and generated sequencesrespectively:

Eigenvalued =I∑

i=1

[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


C. Taewichit et al.

of PDF d of ith statistical parameter, wi is a weightedimportance of ith statistical 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 continu-ously performed until 100 runs and the optimal POL val-ues were recorded. The results of discrete optimal POLs(Figure 3) clearly showed that the TMATRE decreasedwith increasing NS. The TMATRE was high with lowNS, and it sharply decreased until the NS reached nearly10. The NS values between 18 and 24 were not foundin the POLs. After NS of 25, the TMATRE decreasedslightly.

4.1.1. Variations in optimal SMFOMC parameters withw1 s

The SMFOMC parameters were controlled by assigningthe amount of rainfall in the upper limit of second rainfallstate (wet) to be larger than 0.1 mm (Figure 4). NS mustbe minimized according to the second objective functionin order to reduce the size of TPMs and time of modelsimulation. Rapidly rising trends of the NS were noticedwhen w1s were higher than 0.5 (Figure 4(a)). The PPL(Figure 4(b)) remained stable when w1s was lower than0.5, after which sharp decreases in PPLs were noted. Vari-ations in SD are illustrated in Figure 4(c). This parameterspecifies the interval of rainfall amounts of the succes-sive intermediate states. A maximum SD of 1.6 was notedupon varying w1s in the range of 0.1–0.6 and a minimumSD of 1.1 was found at values of w1s higher than 0.7.The minimum values of SD visibly corresponded to lowerTMATRE. As shown in Figure 4(d), TMATRE showed

0.0

0.5

1.0

1.5

2.0

2.5

0 10 20 30 40 50 60 70NS

0.0

0.5

1.0

1.5

2.0

2.5

0 10 20 30 40 50 60 70 80

NS

TM

AT

RE

(x1

02 )

0.0

0.5

1.0

1.5

0 10 20 30 40 50 60 70

NS

TM

AT

RE

(x1

02 )

0.0

0.5

1.0

1.5

2.0

2.5

0 10 20 30 40 50 60NS

0.0

0.5

1.0

1.5

2.0

2.5

0 10 20 30 40 50 60 70NS

TM

AT

RE

(x1

02 )

(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.

0

10

20

30

40

50

60

70

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0w1s

Num

ber

of s

tate

s(O

ptim

ized

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0w1s

Pre

serv

ed p

orpo

rtio

n in

the

last

sta

te

(b)(a)

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Stat

e di

vide

r (O

ptim

ized

)

w1s

(c)

w1s

0

50

100

150

200

250

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

TM

AT

RE

(O

ptim

ized

)

(d)

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.



large disparity at w1s lower than 0.50, after which TMA-TRE started gaining stability. It could be summarized thatTMATRE was decreasing as the given w1s were higher.As hypothesized, the conflicting characteristics of TMA-TRE and NS are evidently noticeable in Figure 4(a) and(d). However, for further analysis in the generation pro-cess the compromise set of solutions that obtained byvarying w1s of 0.50 onward was considered, as thosew1s provided low and stable TMATRE.

4.1.2. Daily rainfall generation

In the generation process three replicates of daily rain-fall sequences were generated as historical daily rainfallsequences with an equal length for each rainfall station.Three PDFs were applied to generate the daily rainfallsequences in the last state corresponding to the sametime and the same U. Probability distribution param-eters were obtained from 44 rainfall stations once allRCLT for w1s were constructed (Table II). The gen-erated sequences were summarized monthly, based onthe 12 statistical parameters, and were then averagedfor the three replicates. High TMATRE values (above300) were found in w1s of 0.1–0.4. However, dis-tinctly visible variations were observed in TMATREfor w1s in the range of 0.5–0.7 (Figure 5). In general,the compromise solution is presented with given w1 of0.5 to both objectives. Moreover, the compromise solu-tions could also be taken as desirable solutions underthe variation of w1s. The comparative analysis of w1s

on the variation of ∗TMATRE (normalized sum of allTMATRE of 44 rainfall stations of each PDF under

each w1 for 12 statistical parameters) is presented inFigure 6.

4.1.3. Effect of criteria weights on statisticalparameters of synthetic daily rainfall sequences

Based on the statistical test results, a model was con-sidered to perform satisfactorily if the average of theparameters estimated from the replicates was closeto the historical values. No significant differences in∗TMATRE were found in the statistical parameters forthree PDFs (Figure 6) except in MMaxDR (Figure 6(c)),in MMinDR (Figure 6(l)), in MSDR (Figure 6(k)), inMADR (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 generateddata varied between 2 and 25% and deviated from histori-cal statistical parameters except MMinDR, which showeddistinctly large error in its estimation (Figure 6(l)). Thismay be due to the difficulty in estimating the near zerovalue of daily rainfall generation with the applicationof this model. The daily rainfall generation in the laststate using three PDFs did not show any significantdifference. On the contrary, they provided almost thesame ∗TMATRE for all statistical parameters. Althoughit could not be firmly concluded which PDF performsthe best, to rank the performance of the three PDFsthe Eigenvalues of three PDFs under w1s were latercalculated and ranked. Over half of the 12 statisticalparameters provided low ∗TMATRE in w1s of 0.8–1.0(Figure 6(d–f, k and l)) compared to the others. Hence

Table II. 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.

0

100

200

300

400

500

600

0520

014

033

1447

221

012

2551

125

550

5015

050

160

5017

050

171

5018

057

161

6212

067

220

3812

0138

1301

1011

115

012

3401

243

013

4518

146

013

5822

161

341

0739

116

151

1618

117

081

2011

128

111

4015

163

181

0323

106

121

0916

044

191

4814

166

071

0436

119

351

3102

232

012

5401

256

012

6001

3

Station ID

TM

AT

RE

(G

ener

ated

)

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).


C. Taewichit et al.

b

b baa

a

aaa

aaa

aaa

aaa

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

*TM

AT

RE

(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

(wl).

for this range of w1s (0.8–1.0) the cumulative ratiosof monthly statistical parameters of generated data toyearly historical data versus the cumulative ratios ofmonthly statistical parameters of monthly historical datato yearly historical data for all PDFs were then plot-ted to see correlations (Figure 7). Good correlations(R2 > 0.9) were observed for most parameters exceptin the extreme overestimates of MMinDR. In addition,modest underestimates in MMaxSLWD, MMaxSLDDand MMaxDR also appeared.

4.1.4. Which set of SMFOMC parameters should beselected?

Most of the parameters considered in this study providedlow ∗TMATRE with a range of less than 0.1–0.3 timesthe historical statistical parameters (Figure 6(a–k)) andranges of R2 > 0.9 (Figure 7(a–l)). This indicates thatthe model and optimal parameters could realisticallydescribe the variation of historical daily rainfall data. Anunacceptably high ∗TMATRE was received in the caseof MMinDR (Figure 6(l)) where ∗TMATRE was about3–6 times for w1s of 0.8–1.0, 8–10 times for w1s of0.6–0.7, and about 16 times for w1 of 0.5. This impliesthat if MMinDR is neglected the compromise solution

with w1 of 0.5 would generally be the desired solution.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 withEigenvalues, is presented at different values of w1s

(Table III) as w2s was reduced an importance accordingto higher w1s. The results clearly show consistency forall w1s: the poorest performance was provided by the SE,and the G-2 showed the best performance.

Relative differences of yearly sum of the 12 monthlystatistical parameters through the 38 years between gen-erated and historical sequences of G-2 with w1 of 0.8were interpolated using inverse distance weighing (IDW)and are depicted in GIS maps (Figure 8). Tolerablysmall differences were noted (percentage differenceranged between 1 and 30% with an average of about2%) for all statistical parameters except MminDR,where the yearly sum of MminDR overestimated thegenerated data (17.8 ± 56 mm) compared with that ofthe historical average (3.3 ± 6.4 mm). Statistical param-eters of generated data tended to be underestimatedfor MMaxSLWD, MMaxSLDD, MMaxDR, MDRSk,



0.0

0.4

0.8

1.2

1.6

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Historical data

0.0 0.4 0.8 1.2 1.6 0.0 0.4 0.8 1.2 1.6

Gen

erat

ed d

ata

1:1 1:1 1:1

1:1 1:1 1:1

1:1 1:1 1:1

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.

Table III. 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

MNWD and MANWD (Figure 8(a–c, g–i)) whilst oth-ers were overestimated. The lower middle part of theNortheast region showed a larger deviation of MminDRwhen compared to the other regions (Figure 8(l)). Thisis because of the historical amount of rainfall inthe dry state was lower than 0.1 (e.g. the amountof MminDR 0.067 mm in the case of rainfall sta-tion ID 14033). Nevertheless, the Northeast regionprovided larger error differences of MMaxSLWD andMMaxSLDD, between 5 and 30%, probably due to thefact that this region receives lower and more inconsis-tent rainfall compared to the other regions. This couldbe the probable reasons for larger deviations as the

model loses its accuracy particularly at low rainfalldepths. In the North, overestimation of the generatedMSDR and MNWD was noted (14% max) (Figure 8(d)).In the Central region, the highest error estimation wasfound in MmaxDR (20% max) (Figure 8(c)) with about80 mm year−1 average difference of yearly sum ofMmaxDR. In the Southern region the largest differenceswere observed in some rainfall stations for MDRstd andMDRsk (17% max) (Figure 8(f) and (g)). The modelappears efficiently applied to the East region with verysmall deviation from historical data for all statisticalparameters.


C. Taewichit et al.

(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.

5. Conclusion

With the dataset containing 38 years of daily rainfall from44 rainfall stations located in 5 regions of Thailand, thedaily rainfall was modelled with SMFOMC in terms ofdaily rainfall occurrences and generation. The optimalSMFOMC parameters were appropriately obtained by the

effective use of MODE integration with SMFOMC atdesirable intervals in Markov Chain model parametersspecification. Minimization of two conflicting objectiveson TMATRE and NS was considered with three selecteddecision variables of SD, PPL and NS. The proposedmodel reproduced characteristics of original daily rainfalloccurrences with acceptable seasonality and accuracy.



Table IV. 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 model was described in detail with the sensitivityof weighted importance values of objective one (w1s).Lower w1s adversely affected MMaxDR, MMinDR,MADR, MSDR, MDRStd and MDRSk, whereas higherw1s offered significantly higher accuracy and acceptabil-ity. The model failed to describe MMinDR. This was theparticular case found in the stochastic rainfall simulationin which the period of available historical data wasshort and the extreme rainfall events were rare to model(Regniere and St-Amant, 2007). However, the optimalSMFOMC parameters under w1s of 0.8–1.0 (Table IV)are recommended to be the appropriate solutions thatcould potentially be applied for Thailand. The study alsoverified the performance of PDFs that have been appliedin the research. The shifted exponential, exponential andtwo-parameter gamma distributions were concluded tobe generally adequate for describing the rainfall occur-rences in the last state. Although no significant differencewas found among three PDFs, the performance rankingshowed the higher potential of the two-parameter gammadistribution over the others. Moreover, the usefulness ofTPMs is still considerable for SMFOMC. With daily rain-fall amounts less than 60–70% of the maximum dailyrainfall, TPMs helped the model to be fitted well with

historical data, and the remaining data was mobilized byPDFs (see optimal PPL of w1s in the range of 0.8–1.0in Table IV, where an average PPL is in the range of0.3–0.4).

Abbreviations.

ARIMA = Multiplicative autoregressive integrated mov-ing average

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 errorMADR = Monthly average daily rainfallMANDD = Monthly average number of dry daysMANWD = Monthly average number of wet daysMDRSk = Monthly daily rainfall skewnessMDRStd = Monthly daily rainfall standard deviation


C. Taewichit et al.

MMaxSLDD = Monthly maximum spell length ofdry days

MMaxSLWD = Monthly maximum spell length ofwet days

MMaxDR = 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 1114 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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