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International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of ...

International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.

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  • Obednego Dominggus Nara, Muhammad Bisri, Aniek Masrevaniah3, Agus Suharyanto /International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622www.ijera.com Vol. 3, Issue 3, May-Jun 2013, pp.1170-11761170 | P a g eExtreme And Distribution Data Of Drainage System In The CityOf AmbonObednego Dominggus Nara1, Muhammad Bisri2, Aniek Masrevaniah3,Agus Suharyanto 31Student of Doctoral Program in Civil Engineering, Faculty of Engineering, Brawijaya University Malang2Professor of Civil Engineering, Faculty of Engineering, Brawijaya University Malang3Lecturer in the Department of Civil Engineering, Faculty of Engineering, Brawijaya University Malang(Indonesia)ABSTRACTClimate change has been a significantenvironmental issue during this century. Variousstudies have been conducted in order to identifythe causes of climate change. To avoid theharmful effects of extreme rainfall events andclimate, it is necessary to give special attention tothe extreme values, given the inability of humanbeings to avoid or escape from disaster that iscaused by phenomena of precipitation andclimate (temperature). One theory thatspecifically addresses extreme events is EVT(Extreme Value Theory). The purpose of thisstudy is to provide an overview of extreme,distribution and modeling values as well asdiagnostics model which is modeled with GEVdistribution. The family of GEV distribution hasthree sub-families namely Gumbel, Fréchet andWeibull. This distribution is characterized bythree parameters: the location, scale, and shapeparameter. From the data distribution test witherror rate (α) of 0.05, it gives a result that thetested distribution fits with GEV and Normaldistribution. For the suitability distribution, inaddition to specific inferential test, it can also betested using the P-P plot and Q-Q plot. Whileempirical CDF graph is used to evaluate the fit ofthe data compared to the data distribution of thecumulative distribution of samples in which bothprobability values gives a similar profile at bothlevels axis. Based on return level of 100 yearsfrom the analysis of the maximum temperature,it increases only 10.66% or the xk value = 29.367°C which means that in the period (return period)of 100 years, temperatures in the study area ofAmbon will surpass 29.367 °C. Therefore, it iscertain that the temperature changes in the studyarea are not too significant and have extremephenomenon compared to the one with themaximum rainfall.Keywords - Climate change, GEV distribution,return levelI. INTRODUCTIONClimate change has been a significantenvironmental issue during this century. Variousstudies have been conducted in order to identify thecauses of climate change. Various measurablechanges and consequences are needed to responseand adapt to climate change, particularly anadaptation that can be done in urban areas. By 2030,an estimated 60% of the world population will livein urban areas, this is a tough challenge. IPCC(Intergovernmental Panel on Climate Change)found that, over the last 100 years (1906-2005) theEarths surface temperature has risen to an averageof about 0.74 °C, with greater warming mainlyoccurred at the land rather than at the sea. The late1990s and the early 21st century were the warmestyears since the existence of modern data archive.Warming increase by 0.2 °C is projected to occurfor each decade and of the next two decades [8]. Theissue of global climate change impacts is deeply feltin several regions in Indonesia, which contributes tothe occurrence of some extreme phenomena, andone of the affected areas is the city of Ambon(Moluccas-Indonesia). To avoid the harmful effectsof extreme rainfall events and climate, it isimportant to give special attention against extremevalues, given the inability of human beings to avoidor escape from disaster that is caused by rainfall andclimate phenomena. One theory that specificallyaddresses extreme events is EVT (Extreme ValueTheory) [2,13]. EVT focuses on the information ofextreme events based on the extreme valuesobtained to form the distribution function from theextreme values of the anomalous precipitation andclimate (temperature).II. METHODSome types of extreme value, thedistribution, analysis and diagnostics of the modelwill be a method in the concept of the maximumvalue of the random variable modeling. Based onthe theory, asymptotically, the extreme values ofrainfall and temperature distribution will convergefollowing the GEV (Generalized Extreme Value)[1,5,9,10,12] which is an extreme value in a givenperiod. Suppose we have independent randomvariables x1, x2, ... xn, each variable xi has the samedistribution function F(x), which later on will be
  • Obednego Dominggus Nara, Muhammad Bisri, Aniek Masrevaniah3, Agus Suharyanto /International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622www.ijera.com Vol. 3, Issue 3, May-Jun 2013, pp.1170-11761171 | P a g econsidered that the maximum Mn = max (X1, X2, Xn)as written as the following equation;The ξ parameter determines the characteristics of thetip of the distribution: if ξ <0 then the probabilityfunction has a finite right tip point and if ξ ≥ 0 theprobability function will have an infinite right endpoint of. Extreme value distribution, introduced byJenkinson (1955), is a combination of three types ofextreme limited value distributions to a single formas derived by Fisher and Tippet (Hosking et al.1985) [4]. Those three types are Gumbel, Fréchetand Weibull distributions whose equation are asfollowThe ξ parameter determines the characteristics of thetip of the distribution: if ξ <0 then the probabilityfunction has a finite right tip point and if ξ ≥ 0 theprobability function will have an infinite right endpoint of. Extreme value distribution, introduced byJenkinson (1955), is a combination of three types ofextreme limited value distributions to a single formas derived by Fisher and Tippet (Hosking et al.1985) [4]. Those three types are Gumbel, Fréchetand Weibull distributions whose equation are asfollows:μ is location parameter, σ> 0 is scale parameter ξ> 0is shape parameter (Stephenson, 2003)[9]2.2 Parameter EstimationEstimation method that is frequently usedis the maximum likelihood estimation (MLE) [3,12].Suppose y1, ..., yn are n independent observations ofrandom variables Y1, ..., Yn which are GEVdistributed respectively with indeterminateparameter θ = (μ, σ, ξ). In addition, the data isassumed not dependent one another, then:F(yi,...,yn|θ)=f(y1|θ)...f(yn|θ). This is called thelikelihood function, often referred to l(θ|y). Log ofthe likelihood function: nitiKyL 1/11exp)(11111 yini2.3 Model DiagnosisA data modeling usually begins withassumptions, the data is assumed that it adheres to aparticular distribution. Subsequently, with the dataand the parameters of the distribution to beestimated. Consequently, the initial assumption mustbe tested, that is the similarity of data with a modelthat assumed. Two diagnostic test techniques basedon the graph will be explained as follows.2.4 Probability and QQ Plots GraphsSuppose y1, ..., yn is a series ofobservations of extreme values (block maxima)which are independent with function of Gdistribution (unknown). If yi is assumed as GEVdistributed, and from MLE, we get the estimation ofthe G distribution. The accuracy in estimating G canbe tested by comparing the estimation model to theempirical distribution Suppose y(1), y(2), .., y(n) is adata sequence such a way that y(1) ≤ y(2) ≤ ..... ≤y(n), then the empirical function is easily obtained,as follows,1)(~( )()(niyGyYP iiThis empirical distribution can be used to estimatethe real data distribution (G). If1~, 1)(iiGy i  )1log(1ˆ11~ 1ninGyi ≈ GEV assumption is true, then G ≈ G for each.As a result, the graph of the pair of points(G(y(i)),G(y(i))) for i = 1,...,n, will lie along thediagonal line (gradient 1). This is known as theprobability graph (probability plot) or P-PPlot. While the chart quantile (Q-Q plot) in theother part is a plot of the pair of points. Similar tothe probability graph, if the model assumption iscorrect, then the points in the QQ plot will belocated along the diagonal line.  00,expexp,1exp)(1xxxF
  • Obednego Dominggus Nara, Muhammad Bisri, Aniek Masrevaniah3, Agus Suharyanto /International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622www.ijera.com Vol. 3, Issue 3, May-Jun 2013, pp.1170-11761172 | P a g e2.5 The extreme values determination of rainfalland temperatureExtreme values determination can be done in twoways:1.By taking the maximum values for a certainperiod, such as weekly or monthly, the observationof these values is considered as extreme values.2.By taking the values that exceed a threshold value,all values that exceed the threshold are consideredas extreme values.2.6 Return LevelIn the practice, the concernedamount/quantity is not only focused on theestimation of the parameters themselves but also onthe quantile which is also referred to as return levelof the GEV estimators [6,7,14]. Assumed returnvalue of the maximum rainfall and temperature willbe used for the validation of the rainfall andtemperature data. If F is the distribution of themaximum value for the same period of observation,then; kFxk 111rainfall and temperature will reach a maximumvalue once (Gilli and Kellezi, 2003) after assumedparameters μ, σ and ξ can be substituted using aboveequation.0;p1-1ln-ln-0;p1-1ln--1-^^^^^kxIII. RESULTTime series is an observation sequenceused to measure the scale of time and space. Thedata, which is tested periodically, is the maximumrainfall data which consists of monthly rainfall datafor 29 years (1984-2012) and temperature data for16 years (1995-2011) from taken from PattimuraAmbon weather station as in Figure.1 below.It appears in the Boxplot on annual data,that the average value which is indicated by the linein the middle of the box is always in the downposition. The pattern is read that most of the data areon the left side, while others have high extremevalue with the advent of the ‘*’. Characteristics ofthe data are not symmetrical with the proportion ofmore data on the left side; it is assumed that therainfall data will follow a generalized extreme valuedistribution while the temperature data follows thenormal distribution. of the work or suggestapplications and extensions.From the table above, the results of the datadistribution test with with error rate (α) of 0.05 willgive a result that the tested distribution equivalentwith the data distribution, if the calculatedprobability value is greater than 0.05. Like themodel fit test results conducted by the three modelsabove, the acceptable distribution test is theKolmogorov-Smirnov, the GEV distribution have aprobability = 0.781 and has the highest valuecompared to other distributions. So that the testresults have given sufficient proof that the rainfalldata follows the GEV distribution with mean (µ) =218.32, standard deviation (σ) = 170.78 and shape(k) = 0.0718. While temperature follows Normaldistribution; probability = 0.455 and has the highestvalue compared to other distributions. So that thetest results have given sufficient proof that thetemperature follows a normal distribution with mean= 26.537, standard deviation = 1.201.Several diagnostic plots are done inferential test toassess the accuracy of the model as shown in Figure4 and 6. Points on both the Q-Q and P-P plot layalong the diagonal line, an indication that the GEVmodel is suitable for rainfall and Normal fortemperature. If this image can establish a pattern ofstraight lines, it can be concluded that thedistribution matches the distribution of researchdata. Suitability distribution based on the results ofthis test will be illustrated through "fit data" towardshistograms and cumulative distribution. While theempirical CDF graphs are used to evaluate the fitdata from distribution data compared to the sampledistribution cumulatively in which both probabilityvalues gives a similar shape on both axes.3.1 Return levelThe return level analysis of rainfall andtemperature in the city of Ambon gives an idea ofhow big a maximum value expected to be surpassedon average once in certain period. Return levelvalues of rainfall and temperature are obtained andused as a benchmark for forecasting the occurrenceof rainfall and maximum temperature with certainProbabilities. The estimated value of the GEVparameters for the rainfall and temperature results inthe return level[11] for 100 years as the tablebelow:IV CONCLUSION1.Frequently, from several analyzed extreme datatypes can meet certain conditions, which aremodeled with the GEV distribution, GEVdistribution family has three subfamilies, namelyGumbel, Fréchet and Weibull. This distribution ischaracterized by three parameters: the location,scale, and shape parameter.2.From the result of distribution analysis andestimation, the histogram of rainfall data in thisstudy has a generalized extreme value distributionpattern while the temperature data in this study has anormal distribution pattern.3.Suitability distribution based on the test results isdescribed through "fit data" to histograms andcumulative distribution. The empirical cdf graphs
  • Obednego Dominggus Nara, Muhammad Bisri, Aniek Masrevaniah3, Agus Suharyanto /International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622www.ijera.com Vol. 3, Issue 3, May-Jun 2013, pp.1170-11761173 | P a g eare used to evaluate the data fit from datadistribution which compared from sampledistribution cumulatively in which the value of bothprobabilities gives a similar shape on both axes.4.Based on the return level on 100 years from theanalysis of the maximum temperature, it increasedonly for 0.66% or the xkvalue = 29.367°C, whichmeans that in return period of 100 years, thetemperatures in the study area of Ambon city willpass 29.367°C. Therefore, it is certain that thetemperature changes in the study area are not toosignificant and have extreme phenomenon comparedto the one with the maximum rainfall.AcknowledgementsMy grateful thanks go to Prof. Dr. Ir. Mohammad Bisri, MS, Dr.Ir. Aniek Masrevaniah, Dipl.HE. and Ir. AgusSuharyanto, M.Eng, Ph.D. for proofreading the manuscript.Figure 1 The maximum series data of rainfall and temperatureFigure 2 Boxplot graph of rainfall and temperatureFigure 3 Histogram of rainfall suitability distribution
  • Obednego Dominggus Nara, Muhammad Bisri, Aniek Masrevaniah3, Agus Suharyanto /International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622www.ijera.com Vol. 3, Issue 3, May-Jun 2013, pp.1170-11761174 | P a g eFigure 4 Graph of rainfall diagnosticsFigure 5 Histogram of temperature suitable distribution
  • Obednego Dominggus Nara, Muhammad Bisri, Aniek Masrevaniah3, Agus Suharyanto /International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622www.ijera.com Vol. 3, Issue 3, May-Jun 2013, pp.1170-11761175 | P a g eFigure 6 Graph for temperature diagnosticsTable 1 Compatibility Test of rainfall distributionDistributionKolmogorov-Smirnov Anderson Darling Chi-SquaredSampleStatisticsProbabilityrankingSampleStatisticsranking(°)FreeStatisticsProbabilityrankingGama 384 0.0408 0.592 2 384 1.399 2 8 10.054 0.261 2GEV 348 0.0347 0.781 1 348 0.401 1 8 9.847 0.276 1Normal 348 0.0981 0.0026 3 348 6.813 3 8 24.309 0.002 3Weibull 348 0.1487 3.4E-07 4 348 14.999 4 8 95.586 0 4Table 2 Compatibility Test of temperature distributionDistributionKolmogorov-Smirnov Anderson Darling Chi-SquaredSampleStatisticsProbabilityrankingSampleStatisticsranking(°)FreeStatisticsProbabilityrankingGama 192 0.066 0.347 2 192 0.67 1 7 15.84 0.0265 4GEV 192 0.073 0.245 3 192 0.687 2 7 15.834 0.0266 3Normal 192 0.0609 0.455 1 192 0.702 3 7 15.839 0.026 2Weibull 192 0.098 0.043 4 192 3.807 4 7 12.911 0.074 1Table 3 MLE distribution estimation for rainfallDistributionParameterα β σ μ ξGama 1.8617 177.196GEV 170.7843 218.3223 0.0718Normal 241.78 329.9043Weibull 0.9111 418.7987
  • Obednego Dominggus Nara, Muhammad Bisri, Aniek Masrevaniah3, Agus Suharyanto /International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622www.ijera.com Vol. 3, Issue 3, May-Jun 2013, pp.1170-11761176 | P a g eTable 4 Estimation of temperature distribution for the MLEDistributionParameterα β σ μ ξGama 489.606 0.0542GEV 1.1704 26.13 -0.2437Normal 1.201 26.537Weibull 27.113 27.089Table 5 The estimated value of the parameters of GEV and Return level resultsREFERENCES1. Ramachandra Rao, Shih-Chieh Kao.Statistical Analysis of Rainfall DataIndiana, Joint Transportation ResearchProgram, Purdue University. 20062. Ci Yang and David Hill. Modeling streamextremes under non-stationery-timeconditions, XIX International Conferenceon Water Resources CMWR 2012University of Illinois at Urbana-champaign June 17-22, 2012.3. Dana Draghicescu, Rosaria Ignaccolo.Modeling threshold exceedanceprobabilities of spatially correlated timeseries, Electronic Journal of Statistics Vol.3 (2009) 149-164 ISSN: 1935-7524 DOI:10.1214/08-EJS252.2009.4. Hosking, JRM, and Wallis, JR, andquantile Parameter Estimation for theGeneralized Pareto Distribution,Technometrics, 29, 1987, pp. 339-349.5. Indriati Bisono. Knowing Extreme Dataand Distribution, Industrial EngineeringJournal, Vol. 13, No.. 2, December 2011,81-86. 20116. James A. Smith and Mary Lynn Baeck,Julia E. Morrison. The RegionalHydrology of Extreme Floods in anUrbanizing Drainage Basin, AMS OnlineJournal Volume 3, Issue 3 (June 2002).7. Julia E. Morrison, James A. Smith.Stochastic modeling of flood peaks usingthe generalized extreme value distribution,Water Resources Research Volume 38,Issue 12, pages 41-1-41-12, December2002.8. P. Willems, K. Arnbjerg-Nielsen, J.Olsson, VTV Nguyen. Climate changeimpact assessment on urban rainfallextremes and urban drainage: Methodsand shortcomings, Atmos-02 405; No. ofPages 13.20119. Richard W. Katz, Marc B. Parlange,Philippe Naveau. Statistics of extremes inhydrology, Advances in Water Resources25 (2002) 1287-1304.200210. K. Riipunjaii. Shuklla, M.. Triivedii,Manoj Kumar. Proffiiciientt On tthe useoff GEV disttriibuttiion, anale. SeriaInformatica. Vol. VIII fasc. I - 2010.11. Slobodan P. Simonovic, Angela Peck.Updated rainfall intensity durationfrequency curves for the City of Londonunder the changing climate, Department ofCivil and Environmental Engineering TheUniversity of Western Ontario London,Ontario, Canada.201012. Song Feng, Saralees Nadarajah, Qi Hu.Modeling annual extreme precipitation inchina using the generalized extreme valuedistribution, Journal of the MeteorologicalSociety of Japan, Vol.85.no.5, pp.599-613.2007.13. Stevan Hadživukovic, RA Fisher andModern Statistics, International StatisticalInstitute, 55th Session 2005: StevanHadzivukovic, Emilija Nikolic-Doric. ,2005.14. Xin Liu, Data Analysis in Extreme ValueTheory, Department of Statistics andOperations Research University of NorthCarolina at Chapel Hill. April 30, 2009.VariableParameter Tr(Year)σ μ ξ 2 5 10 20 50 100Max rainfall 170.784 218.322 0.072 280.917 474.489 602.65 725.585 884.712 1003.96Max temperature 1.1704 26.13 -0.2437 26.5404 27.6004 28.1574 28.6039 29.0769 29.367728