Pooling of low flow regimes using cluster and PrinciPal ... · Števková, et al. (2010), and...

A. Števková, M. SAbo, S. kohnová

Pooling of low flow regimes using cluster and PrinciPal comPonent analysis

Key words

• Low flow,• Cluster analysis,• Principal component analysis.

aBstract

This article deals with the regionalization of low flow regimes lower than Q95 in Slovakia. For the regionalization of 219 small and medium-sized catchments, we used a catchment area running from 4 to 500 km2 and observation periods longer than 20 years. The relative frequency of low flows lower than Q95 was calculated. For the regionalization, the non-hierarchical clustering K-means method was applied. The Silhouette coefficient was used to determine the right number of clusters. The principal components were found from the pooling variables on the principal components. The K-means clustering method was applied. Next, we compared the differences between the two methods of pooling data into regional types. The results were compared using an association coefficient.

Andrea ŠTEVKOVÁDepartmentofLandandWaterResourcesManagementemail:[email protected]:seasonalityandregionalizationoflowflows

Miroslav SABODepartmentofMathematicsandConstructiveGeometryemail:[email protected]:Multivariatestatistics,machinelearning,datamining,programmingg

Silvia KOHNOVÁDepartmentofLandandWaterResourcesManagementemail:[email protected] field:Design discharge estimation, regional-ization,rainfallrunoffmodeling

Address:FacultyofCivilEngineering,SlovakUniversityofTechnologyinBratislava,Radlinského11,81368Bratislava.

Vol. XX, 2012, No. 2, 19 – 27

1. introduction

Low flows, their regimes, and their influence on the biologicalfunctionsofariveranditsstabilityisanimportantdiscussiontopic.Lowflowisinfluencedbyanumberofaspects,mainlygeological,hydrologicalandclimaticfactors(airtemperatureandprecipitation).The occurrence of low flows also has agrowing importanceeconomically, because of the increased use of surface watersourcesforirrigation,freshwaterproductionandpowergeneration.Knowledgeoflowflowsisalsoimportantforthebiologicalbalanceofariverandtheprotectionofanecosystemanditsbiodiversity,astheyhaveanimpactonwatertemperature,lighting,speed,oxygenandchangesinsedimentation.Alimitingfactorfororganismsisalsothedurationoflowflowperiods,asashortdurationperioddoesnotcauseexplicitdamagetoorganisms.Oneofthepossiblemethodsforassessingalowflowregimeistousea seasonality analysis. Themost widely used seasonality analysiswas presented by Burn (1997). This method was developed for

testingseasonalityoccurrencesoffloodsoncatchmentsinCanada.Also,thismethodhasbeenusedfortheoccurrenceofseasonalityforlowflowsinAustria(Laaha,2006a,b).Theauthorsusedaseasonalitycluster analysis to represent themonthlyoccurrenceof low flows.TheyclassifiedAustriancatchmentsintotwoclustersofwinterandsummerlowflowtypes.Young(2000)appliedalow-flowseasonalityindexfortheestimationoflowflowseasonalityintheUK.Schreiber(1997)analysedtheseasonalityofmeanannual10-dayminimumsoftotaldischargesmeasuredin169catchmentsinsouthwestGermany.The results indicated the typical occurrence of low flows fromSeptembertoOctoberforlargepartsoftheareastudied,apartfromthePre-Alps,whicharedominatedbylowwinterflows.Clusteranalysishasbeenused inmanystudies.RaoandSrinivas(2006) used cluster analysis on the attributes and flow recordsfrom245gauging stations in Indiana.Theyusedhierarchical andnon-hierarchical analyses.Of the hierarchical clusteringmethods,they used single and complete linkage and Ward clusteringmethods,andforthenon-hierarchicalclusteringanalysis,theyused

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K-meansclustering.Theclusteringalgorithmyieldedsixacceptablyhomogeneoushydrologicregionsthathavebeentestedtoberobust.Kahay (2007) used hierarchical clustering (single, complete andWardmethods) to create regional types ofmonthly low flows inTurkey.TheyfoundthattheWardmethodwithEuclideandistancewas more effective in the production of homogenous clusterscomparedtosinglelinkageandcompletelinkagemethods.InNathanandMacMahon(1990),theauthorsidentifiedhomogeneoussubregions that can be considered to behave in ahydrologicallysimilar fashion. The relative performance of several techniquesis evaluated using the prediction of low-flow characteristics inaheterogeneous group of 184 catchments located in southeasternAustralia.Theauthorsofferedapproachesbasedonacombinationofclusteranalysis,multipleregression,principalcomponentanalysis,andthegraphicrepresentationofmulti-dimensionaldata,whenusedfor the identificationof poolinggroups.The techniques presentedallowfortheassessmentofthesuitabilityofapplyingtheregionalequationsderivedtoanungaugedcatchment.Kohnová, et al. (2008) used the Burn vector and histograms ofoccurrences for abetter understanding of low-flow seasonality inSlovakia.ThisarticlefollowsthestudiesofKohnová,etal.(2009),Števková,etal.(2010),andŠtevková,etal.(2011)wheretheaimofthestudieswastoexamineseasonalityindicesandtheirpotentialintheregionalisationoflowflowsinSlovakia.WecomparedheretheresultsoftheK-meansclusteringmethodandacombinationof thePCAandclusteringmethodsbyselectingthepoolingvariables.Forreducingthehigh-dimensionaldata,Principalcomponent analysis (PCA) was used and for regionalization,the non-hierarchical K-means clustering method was used. Fordeterminingtherightnumberofclusters,theSilhouettecoefficientwasused.Wecompared thedifferencesbetween the resultsof thepoolinggroups.

2. cluster analysis

Theinformationclusteranalysisgroupsarebasedondatadescribingtheobjectsor their relationships. Thegoal is tomake theobjectsinagroupsimilar(orrelated)tooneanotheranddifferentfromtheobjectsinothergroups.Thegreaterthesimilarity(orheterogeneity)withinagroupandthegreater thedifferencesbetweenthegroups,thebetterormoredistincttheclustering(Tan,etal.,2006).Data clustering (or just clustering), which is also called clusteranalysis, is amethod of creating groups of objects or clusters insuchawaythatobjectsinoneclusterareverysimilarandobjectsindifferentclustersarequitedistinct.Dataclusteringisoftenconfusedwith classification, by which objects are assigned to predefinedclasses(Gan,etal.,2007).

2.1 non-hierarchical cluster analysis – K – means clustering

Thealgorithmisverysimple.DenoteDasthesetofallobjects,i.e.D={x1,x2,...,xn}.Eachobjectxihaspfeatures,i.e.,Xi=(Xi1,Xi2,...,Xip).Then,theK-meansalgorithmcanbedescribedasfollows:

1.Theusersetsthenumberofclustersk.DenotetheseclustersX1,X2,...,Xk.

2.Thealgorithmrandomlyinitializesthekclustercenters{µ1,µ2,...,µk}asrandompointsinap-dimensionalspace.

3.Thenitrepeatsthosetwostepsuntiltheconvergencecriterionissatisfied(forexample,untiltheclusterschange)

–∀i∈{1,2,...,n}assignxitoclusterXjif

(1),

whered(.,.)issomemeasureofdistance(mostlyEuclidean)

–findnewclustercenterssuchas

(2),

where l ∈ {1, 2,..., k} and |.| denotes the cardinality of the set(McQueen,1967).Asilhouettecoefficientwasusedtodeterminethecorrectnumberofclusters(Rousseeuw,1987):Letaidenotetheaveragedistancebetweenthei-thobjectandalltheotherobjectsinthesamecluster,i.e.:

(3),

Letbidenotetheaveragedistancebetweenthei-thobjectandalltheotherobjectsfromthenearestcluster,i.e.:

(4),

where C(i) is acluster, where the i-th object has been assigned;dist(i, j) is somemeasure of the distance of the i-th and the j-thobject;and|.|denotesthecardinalityofthesetasbefore.Thenwecandefineforthei-thobjectthewidthofitssilhouetteas:

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21Pooling of low flow regimes using cluster and PrinciPal comPonent...

(5),

This measure takes the values from <-1, 1>; values close to 1indicatethattheobjectiswellassigned,whilenegativevaluescloseto-1indicateapoorassignmentofthei-thobjectintoitscluster.Ifwecomputethewidthoftheaveragesilhouetteforalltheassumedobjects,wewillagainobtainameasurewithvaluesfrom<-1,1>,andagain,valuescloseto1meanabetterpartition.Accordingto(Meloun,2005),thefollowingruleofthumbisusedtodeterminethecorrectnumberofclusters:0.71–1.00 wecansaythatthei numberofclustersisappropriate,0.51–0.70 thestructurehasanaveragestructure;thenumberof

clustersisappropriate,0.26–0.50 the structure is breakable, and we have to use

anothermethodtodeterminethecorrectnumberofclusters,

≤0.25 thestructureisnotgood.

For determining the right number of clusters,we can define it asanaveragesilhouettevalue.Weusedthehighestaveragesilhouettevalueforthecorrectnumberofclusters.

DissimilaritiesinthemeasuresbetweentheclustersweredeterminedwiththeSquaredEuclideandistance.Squared Euclidean distance

Let x = (x1, x2,..., xp) and y = (y1, y2,..., yp) are two points inp-dimensional space Rp (in our case, two observations of thep-dimensional randomvectorX = (X1,X2,...,Xp)).TheEuclideandistancebetweenthemisdefinedas(Meloun,2005):

(6),

2.2 Principal component analysis (Pca)

Principalcomponentanalysis(“PCA“,alsoknowastheKarhunen-Loeve transformation or Hotelling transformation or properorthogonal decomposition) was originally proposed by KarlPearsonin1901(Pearson,1901)andlatergeneralizedbyHotelling(Hotelling,1935).Thissectionisprocessedaccordingto(Harman,2011). Since the PCA is derived from matrix algebra, we firstmentionatheoremthatiscrucialtotheideaofPCA.Theorem 1:IfSisapositivelysemidefinitep×pmatrix,thenthereexistsasystemu1,u2,...upoforthogonaleigenvectorswithanormof1.MatrixS canbeexpressedintheform

(7),

where li is the eigenvalue of matrix S, which corresponds toeigenvectorui∀i∈{1,2,...,p};U=u1,u2,...up isanorthogonalmatrixandΛ=diag(l1,l2,...lp).Ifl1>l2>...>lp,thenvectorsu1,u2,...upareuniquelydefined.

Nowconsiderap-dimensionalrandomvectorXwithameanµandcovariancematrixS,which is a positive semidefinite.Theorem1implies that a system of orthonormal eigenvectorsu1,u2,...up ofmatrixSandl1,l2,...lpascorrespondingeigenvalueshavetoexist.Asbefore,wewilldenoteasUtheorthogonalmatrix(u1,u2,...,up).Withoutalossingeneralitywecanassumethatl1≥l2≥ ...≥lp,since we can easily reorder the eigenvalues with a suitablerenumberingofthecorrespondingeigenvectors.

Theorem 2:RandomvectorY= UT (X–µ)hasazeromeanandcovariancematrixD(Y)=diag(l1,l2,...lp) , i.e., thecomponentsofthisvectorareuncorrelatedwiththevariancesl1≥l2≥...≥lp.

EigenvectorY=(Y1,Y2,...,Yp)Tfromtheprevioustheoremiscalled

the vector of the principal components of the random vector X.∀i∈{1,2,...,p},randomvariable iscalledthei-thprincipalcomponentoftherandomvectorX.∀k∈{1,2,...,p}

iscalledthetotalvarianceproportion,whichis

explained by the first k principal components. In addition, thefollowingtheoremholds:

Theorem 3:ThefirstprincipalcomponentY1oftherandomvectorXhasamaximalvariancefromallthenormalizedlinearcombinationsofthecomponentsofvectorX,i.e.,D(Y1)=D(bTX)∀b∈Rp ||b||=1.Inaddition,fork≥2,thek-thprincipalcomponentYkoftherandomvectorX has the largest variance from all such normalized linearcombinations of the components of vectorX that are uncorrelatedwith theprincipalcomponentsY1,Y2,...,Yk–1, i.e.D(Yk)=D(bTX)∀b ∈ Rp ||b|| = 1, b ⊥ u1,..., b ⊥ uk – 1 (⊥ is the symbol fororthogonality).

According to the previous calculations,we can perform the PCAanalysis in the followingway.We can decompose the covariancematrix to obtain the eigenvalues and corresponding eigenvectors.Wechosesuchanumberofeigenvaluesthattheproportionofthetotalvariancewillbeat least90%.If thisnumberisk, theneachobjectwillbedescribedbythekdimensionalvector(insteadoftheoriginalvectorthatwasp-dimensional).


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Associationcoefficientswereusedtocomparethevariousclusteringtechniques.Acomparison of the similarities between the Jaccard,Sokal-Sneath(1),DiceandRusselRaoclusterswereused,andthesimilarities between clustering techniques of theRand coefficientwereused(Nánásiová,etal.2010;Jackson,etal.1989).

Jaccardcoefficient (8),

Sokal-Sneath(1) (9),

RusselRaocoefficient (10),

Dicecoefficient (11),

Randcoefficient (12)

The detailed code from the R language for determining thesimilaritiesbetweentheclustersusingsimilaritycoefficientscanbefoundintheappendix.

3. data

Theseasonalityanalysis in thisstudywasbasedon thedata from219 small andmedium-sized catchments selected from the entireterritory of Slovakia, with catchment areas ranging from 4 to

500km2andwithobservationperiodslongerthan20years.Figure1 illustrates the spatial distribution of the selected catchments inSlovakia.The number of low-flow days with adischarge lower than theQ95 was calculated in each catchment. Subsequently, the relativefrequencyof low-flowoccurrences in separatemonths during theyear was estimated inall the observation stations. The relativefrequency was estimated and used as apooling variable in theclusteringmethods.

4. results

It is possible to see in Tab. 1 – the correlation matrix, that thestrongest correlations aremostly between two succeedingmonths(i.e., between the diagonals). We had to reduce the correlations

Fig. 1 Spatial distribution of the analyzed catchments in Slovakia.

Tab. 1 Correlation matrix of the relative frequency of low flows lower than Q95.I II III IV V VI VII VIII IX X XI XII

I 1II 0.795 1III 0.603 0.808 1IV 0.175 0.319 0.538 1V -0.209 -0.207 -0.162 0.184 1VI -0.520 -0.504 -0.428 -0.061 0.501 1VII -0.717 -0.672 -0.553 -0.235 0.183 0.606 1VIII -0.821 -0.790 -0.668 -0.317 0.102 0.430 0.796 1IX -0.823 -0.835 -0.726 -0.333 0.121 0.395 0.573 0.757 1X -0.415 -0.467 -0.475 -0.292 -0.028 0.0468 -0.038 0.1093 0.4463 1XI 0.170 -0.004 -0.124 -0.148 -0.089 -0.256 -0.435 -0.366 -0.195 0.412 1XII 0.695 0.405 0.194 0.017 -0.151 -0.380 -0.591 -0.642 -0.593 -0.148 0.440 1

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between the inputdata,sowereduced thevalueshigher than0.7.Thereduceddataforpoolingintothepoolinggroupswasused.Forthepoolingofthecatchmentsintopoolinggroups,weusedthenon-hierarchical K-means clustermethod.Asilhouette coefficientwas used to determine the right number of pooling groups. Thehighestaveragesilhouetteof0.383occurredintwopoolinggroups.Theanalyzedcatchmentswerepooledintotwopoolinggroups,forwhich the average silhouette was the highest (0.383). These twogroupsrepresenttworegimes.Theoccurrenceoflowflowsinthefirstpoolinggroupisduringthewinter(November–April),andthesecond pooling group has an occurrence of low flows during thesummer–autumn(July–October).Subsequently, we used the principal component analysis for theselection of the derived variables. Principal component analysisreducesthedimensionalityofthedataandcreatesnewvariables–principalcomponents.Thereasonwhytheprincipalcomponentsanalysiswasusedisthestrongcorrelationbetweentheinputdata.Inthecorrelationmatrix(Tab. 1), you can see that the strongest correlations are mostlybetweentwosucceedingmonths(i.e.betweenthediagonals).The percentageof the total variance is set at ahorizon of 90%(Stankovičová, et al. 2007;Meloun, et al. 2005) for the selectionoftheprincipalcomponent;fortheanalyzeddatathehorizonwassetat91.54%.Thefirstprincipalcomponentmakesup66.78%ofthe total variability of the data; the second principal componentmakesup12.62%; the thirdprincipalcomponentmakesup6.9%;and the fourthmakesup5.22%.For thenext steps,weused fourprincipalcomponents.Thefirstcomponentexplainsalmost66.78%of the total variability of the data, andwe can conclude that this

componentisthemostimportant.ThefirstPCwasmostdominant(-0.53); thesecondPCwasmostdominantinOctober(-0.54); thethirdPCwasmostdominant in July (-0.65); and the lastPCwasmostdominantinAugust(-0.47).Thesefourprincipalcomponentswereusedfortheclusteranalysis,andthenon-hierarchicalK-meansclusteringmethodwasused.Thehighestaveragesilhouetteof0.703occurredintwopoolinggroups.Theanalyzedcatchmentswith thehighest silhouette (0.703)werepooledintotwopoolinggroups.Thefirstpoolinggrouprepresents

Tab. 2 Table of principal components. The four principal components (PC) are bolded. 1 PC 2 PC 3 PC 4 PC 5 PC 6 PC 7 PC 8 PC 9 PC 10 PC 11 PC 12 PCI 0.39 -0.03 -0.10 -0.46 0.33 -0.06 0.45 -0.46 0.13 0.00 -0.04 0.29II 0.45 0.24 0.28 0.10 0.03 -0.64 -0.35 0.13 -0.11 0.00 -0.03 0.29III 0.29 0.28 0.35 0.36 -0.18 0.55 0.21 -0.10 -0.23 0.25 0.05 0.29IV 0.03 0.05 0.05 0.08 -0.03 0.22 -0.02 0.12 0.17 -0.81 -0.39 0.29V -0.01 0.01 -0.02 0.02 0.02 0.08 -0.07 0.13 0.41 -0.17 0.83 0.29VI -0.06 0.05 -0.10 0.06 0.06 0.07 -0.08 0.26 0.66 0.49 -0.38 0.29VII -0.27 0.51 -0.65 0.25 0.10 -0.12 0.04 -0.12 -0.24 -0.01 0.02 0.29VIII -0.53 0.26 0.37 -0.47 -0.39 -0.16 0.16 -0.02 -0.06 0.03 -0.02 0.29IX -0.39 -0.23 0.32 0.09 0.67 0.10 -0.28 -0.16 -0.20 0.05 -0.02 0.29X -0.11 -0.54 -0.02 0.41 -0.15 -0.34 0.53 0.14 -0.08 0.02 0.00 0.29XI 0.05 -0.37 -0.21 -0.02 -0.47 0.07 -0.48 -0.52 -0.01 0.06 -0.04 0.29XII 0.16 -0.21 -0.27 -0.42 0.02 0.23 -0.10 0.58 -0.43 0.10 0.01 0.29

Tab. 3 Percent explained of the principal components for finding the principal components.

percent explained66.786412.62353 91.54085%6.9025765.2283472.6593512.0048811.3612641.0698590.8639660.3201550.1796731.16E-29


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awinterregimewith theoccurrenceof lowflowsfromDecemberuntilMarch,and thesecondpoolinggrouprepresentsasummer–autumn regime,with theoccurrenceof low flows fromJulyuntilNovember.Subsequently, similarities between the PCA (K-means) and theK-meansclusteringmethodsusing theRussel-Rao,Dice, Jaccard,Sokal-Sneath’s(1) and Rand coefficients were determined. TheRand coefficient reached avalue of 0.9539, which proved thatthese two clusteringmethods are comparable.Table 4 shows thatthecomparedmethodsarevery similar; thedifferencesare in tencatchments.Infig.2thecatchmentsarepooledintoregionaltypes,wherecatchmentsaremarkedwhen therearedifferencesbetween

thepoolingused.Hereitispossibletoseethedifferencesareinthecatchmentsbetweentheregionaltypes.

5. conclusion

This article dealswith comparing the classical clusteringmethodandacombinationofPCAandclusteringbyselectingthepoolingvariables.Inthefirststep,therelativefrequencyoftheoccurrenceof low flows lower than Q95 was used as the pooling variablefor the clustering and created pooling groups using the non-hierarchical,K-meansmethod.Inthesecondstep,wecalculatedthe

Fig. 2 Comparison of the low-flow regimes in the catchment following the derived pooling group’s winter regime, and the summer – autumn regime.

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principalcomponentsandcreatedpoolinggroupsfromtheprincipalcomponents.Also,thenon-hierarchicalK-meansclusteringmethodwasused.The K-means method pooled the analyzed catchments into twopoolinggroupswithasummer-autumnregimeandawinterregimeof occurrences of low flows lower than Q95. The coefficientassociation which was used in this study also shows very smalldissimilarities in the analyzed regional types. Adifference wasfoundintencatchmentswhicharesituatedontheboundaryoftheclusters.Ifthereisastrongcorrelationbetweendata,itisbettertousePCA,becausetherewerequitebetterresultsinourcase.

The derived pooling groups can be subsequently used for theindirect estimation of design minimum discharges on ungaugedcatchments.

acknowledgement

ThisworkwassupportedbytheSlovakResearchandDevelopmentAgencyunderContractNo.APVV-0015-10andtheSlovakVEGAGrantAgencyunderProjectNo.1/0908/11.Theauthorsaregratefulto RNDr. Beate Demeterovej, PhD., from SHMÚKošice for hercooperationandforprocessingtheinputdata.

Tab. 4 Comparison of the pooling catchments into pooling groups, PCA (K-means) and K-means.K-means PCA (K-means) Russell and Rao Dice Jaccard Sokal and Sneath's 1

1 1 0.374 0.943 0.891 0.8041 2 0.014 0.028 0.014 0.0072 1 0.032 0.063 0.032 0.0162 2 0.58 0.962 0.927 0.864

Fig. 3 Pooled catchments into regional types, with marked catchments where there are differences.


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APENDIX 1 koeficienty<-function(x,y){

n<-length(x)

r<-(sort(x))[n]

s<-(sort(y))[n]

a1<-NULL

for(iin1:r){

for(jin1:s){

u<-0

v<-0

z<-0

w<-0

for(kin1:n){if(x[k]==i&y[k]==j)u<-u+1

if(x[k]==i&y[k]!=j)v<-v+1

if(x[k]!=i&y[k]==j)z<-z+1

if(x[k]!=i&y[k]!=j)w<-w+1

}

a<-c(i,j,round(u/n,3),round(((2*(u/n))/(2*(u/n+v/n+z/n)-v/n-

z/n)),3),round((1-

(v/n+z/n)/(u/n+v/n+z/n)),3),round((((u/n))/((u/n+v/n+z/n)+v/n+z/n)),3),round(((1-v/n-

z/n)/(1+v/n+z/n)),3),round(((2*(1-v/n-z/n))/(2-v/n-z/n)),3))

a1<-rbind(a1,a)

}}

colnames(a1)<-c("Cislo zhluku 1","Cislo zhluku 2","Russel and

Rao","G1:Dice","G1:Jaccard","G2:Sokal-Sneath`s 1","G2:Rogerr-Tanimoto","G2:Sokal-

Sneath`s2")

a1

}

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