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1 殣殣檺檺檺檺檺檺檺檺檺檺檺檺檺檺檺檺殣殣 MathematicalModeligadItsApplicatios Vol.3No.1 Feb , 2 1, (1. 南伊利诺伊州立大学数学系, 伊利诺伊州卡本代尔 62901, 美国 ;2. 北京工业大学应用数理学院, 北京 ) : 系统综述了自 19 世纪开始至今常用的统计相关性的方法, 例如 Pearso 和 Spearma 相关系数,CorGc 和 CovGc 相关性及距离相关性方法重点介绍了 2011 年提出的 MIC 方法以及由此引发的毁誉参半的大量评述, 旨在揭示这一热点领域的研究面貌该领域不仅受到统计学家的关注, 而且受到了分析大样本和异质数据的应用研究领域的学者们的追捧, 例如基因组生物学家和网络信息研究者这些研究者期望在众多已有方法的理解和剖析中更恰当地付诸应用, 并提出新的应用问题来推动新的分析方法的创造 : 相关分析 ;Pearso 相关系数 ;Spearma 相关系数 ;Kedal 相关系数 ; 互信息 ; 距离相关 ; 最大信息系数 :O212 :A : (2014) ,,,,,,, 2 :,,,, X, Y,,,,,, Spearma (Spearmarakcorrelatiocoeficiet) (mutualiformatioestimators) [1-3] (maximumcorrelatiocoeficiet) [4-5] (priciplecurve-basedmethods) [6-10] (distacecorrelatio) [10] (maximaliformatiocoeficiet,mic) [11-12] 1959,Reyi [4], 2, 7,,, δ(x,y) X Y, : 1)δ(X,Y) X Y,X Y 1 ; : :, rogfa@siu.edu 1

2 )δ(X,Y)=δ(Y,X); 3)0 δ(x,y) 1; 4)δ(X,Y)=0 X Y ; 5) X Y, X =g(y) Y =f(x),,g( ) f( ) Borel, δ(x,y)=1; 6) Borel g( ) f( ), δ(f(x),g(y))=δ(x,y); 7) X Y, δ(x,y)= R(X,Y),,R(X,Y) X Y Pearso,Bel [13],Schweizer Wolf [14],Grager [15] Nelse [16] [17] 7, Shao,, Bjerve Doksum [18] (correlatio curve), 2011 Sciece MIC, Pearso [19] , ;, ;,,, 2 10, KarlPearso [20] Pearso :r= 槡 i=1 (X i - 珡 X)(Y i - Y) 珚 i=1 (X i - X) 珡 2 i=1 槡 (Y i - Y) 珚 2 r, Rodgers Nicewader [21] Pearso r r 2, -1~1 r 1, 2, ; -1, ; 0, Spearma Spearma, 2, 2, 1 Spearma ρ 2 X = (X 1,,X ) Y = (Y 1,Y 2,,Y ) Pearso : ρ= i=1 槡 (r i -r - )(s i - s) 珋 i=1 (r i -r - 2 ) i=1 槡 (s i - s) 珋 2,r i s i x i yi,i=1,2,, ( ), ρ [-1,1], ρ = 1,, ρ =-1 Spearma, 2,, Spearma 1 [22] 2

3 3 1 Vol.3No.1 Feb.2014 Pearso Spearma 2 X Y Pearso 0.88, 0.88;Spearma 1,, Kedal Kedal (Kedalcoeficietofcocordace) [23], 2 X,Y, X i,y i X Y i (X i,y i ) (X i, Y i ) (X j,y j ), X i > X j Y i >Y j, X i < X j Y i <Y j, ; X i > X j Y i <Y j, X i < X j Y i >Y j, ; X i = X j Y i =Y j, 2 2 X Y, Kedal τ : 2P τ= -1= 4P 1 2 (-1) (-1) -1,P Kedal τ -1~1, τ=1, 2 ; τ=-1, 2 ; τ=0, 1.4 X 1 X 2 (X,A,P), (X 1,B 1 ) (X 2,B 2 ) X i : (X,A) (X i,b i ) A A i = Xi -1 (B i ),i=1,2 P i P A i,i= 1,2 φ E φ = φ dp ( φ 1, φ 2)=E( φ 1, φ 2),L 2 =L 2 (P) A φ Hilbert, L 2 i =L 2 i(p) A i φ Hilbert, i=1,2 Hirschfeld [24] Gebelei [25] 2 X 1 R(X 1 )=sup{r( φ (X 1 ), ψ (X 2 ))},r(x 1 ) X 1 X 2 Pearso, φ (X 1 ) L 2 1, ψ (x 2 ) L 2 2, φ, ψ r ( φ, ψ ),,, 2, Pearso ; X 1 X 2, R(X 1 )=0,, 2 L 2 1 L ,Reyi [4],, φ (X 1 ), ψ (X 2 ) R(X 1 )=r( φ (X 1 ), ψ (X 2 ))=ρ, Eφ ( X 1 )= Eψ ( X 2 )=0,Eφ ( X 1 ) 2 = Eψ ( X 2 ) 2 =1, 3

4 2014 2,E( φ (X 1 ) X 2 )=ρψ ( X 2 ),E( ψ (X 2 ) X 1 )=ρφ ( X 1 ),1985,Breima Fridema [26] φ ψ, r ( φ, ψ ) ( X 1 ), φ, ψ (X 1 ),Pearso r, r, R(X 1 )= r Sethurama [27] X 1 X 2, Dembo [28] Pearso Czaki Fisher [29],,R(X 1 ) 2 L 2 1 L 2 2, R(X 1 )=cos(l 2 1,L 2 2) 1.5 Shao 1948 (mutualiformatio) [17], 2 X,Y : I(X,Y)= p (x,y)log( p (x,y) p(x)p(y) ) :p(x,y) X Y ;p(x) p(y) X Y, Reyi 7 ;, Kulback-Leibler,,,,Moo [1] (kereldesityestimatio,kde) ;Kraskov [3] k- (k-earest eighbor distaces,knn) Walters-Wilias [30] 2008,, k N,KDE KNN,,KDE KNN 1.6 CorGc CovGc Delicado [31],, Covariacealog a Geeratig Curve(CovGc) CorrelatioalogaGeeratigCurve(CorGc) R 2 CovGc CorGc, X Y,, X Y : Σ = σ2 X σxy [ σyx σ ] = [ cosα -siα 2 Y siα cos α ] T λ1 烄 0烌 cosα -siα 烆 0 λ [ 2 烎 siα cos α ] :λ1 λ2 Σ ;α λ1 x X,Y α : Var(X)=σ 2 X =λ1cos 2 α+λ2si 2 α, Var(Y)=σ 2 Y =λ1si 2 α+λ2cos 2 α, Cov(X,Y)=σXY = (λ1 -λ2)cosαsiα, σxy ρxy = = σxσy (λ1 -λ2)cosαsiα (λ1cos 2 α+λ2si 2 α) 1 2 (λ1si 2 α+λ2cos 2 α) 1 2 (X,Y) c(i) 3 (X,Y) (X,Y)=χ c (S,T):A R 2 :c(i) ;(S,T) ; χ c:i R R 2 χ (s,t)=c(s)+tv(s), c c(s):i R R 2 s I c (s) =1;v(s), 4

5 3 1 Vol.3No.1 Feb.2014 s I c (s) T v(s)=0 (X,Y) c(i) I,c(s) c (s) (pricipalcurves), (middle), [32-34] Hastie Stuetzle Delicado 3 R Matlab, (X,Y) c(i), (X,Y) c(i),x Y,, c(i) c(s) (X,Y) c(s),, c(s) (X s,y s ), (X,Y) 2 [33] 2 c(s) c(t) (a) c(s) (b) c(t) 2 (x,y) c(s) c(t) 4 (X,Y)=χ (S,T) c(i) s I, α(s) c (s) x c, X Y c(s) c(s) c(s) : LV X (S)=V(S)cos 2 α(s)+v(t S =s)si 2 α(s), LV Y (S)=V(S)si 2 α(s)+v(t S =s)cos 2 α(s), LCov(X,Y)(S)= (V(S)-V(T S =s))cosα(s)siα(s), LCor(X,Y)(S)=LCov(X,Y)(S)/(LV X (S)LV Y (S)) ,X Y c(i) ( ) : CovGc(X,Y)= { E S [(LCov(X,Y)(S)) 2] } 1 2, c(i) ( ) : CorGc(X,Y)= { E S [(LCor(X,Y)(S)) 2] } 1 2 Reyi 7,, X Y,CorGc ; X Y 5

6 2014 2, X Y,CorGc Pearso, R, 1.7 (distacecorrelatio,dcor) Szkely [35] 2007,, 0, X p,y q,x Y, X Y (theoreticaldistacecovariace) : V(X,Y)= 槡 c p c q 瓗 p+q 槡 V 2 (X,Y )= 槡 fx,y(t,s)-fx(t)fy(s) 2 = 1 fx,y(t,s)-fx(t)fy(s) 2 dtd s t 1+p p s 1+q q π 1+d/2 :fx(t) fy(s) X Y ;fx,y(t,s) X Y ;c d = Γ((1+d)/2), d =p q (distacevariace) : V(X)= 槡 V(X,X )= 槡 fx,y(t,s)-fx(t)fy(s) (distacecorrelatio) : 烄 V 2 (X,Y), (X)V R(X,Y)= 槡 R 2 (X,Y )= 槡 V 2 (X)V 2 (Y ) V2 2 (Y)>0 烅烆 0, V 2 (X)V 2 (Y)= 0 R(X,Y) :10 R(X,Y) 1 ;2R(X,Y)=0 X Y Pearso, X Y, R(X,Y) r (X,Y), r(x,y)=±1 X Y (X,Y)= {(X k,y k ),k= 1,2,,},, : a kl = X k -X, l p 珔 a k = 1 a kl, 珔 a l = 1 l=1 k=1 a kl, 2 珔 a = 1 a 2 kl,a kl =a kl - 珔 a k - 珔 a l + 珔 a,k,l=1,2,, k,l=1 b kl = Y k -Y, l q B kl =b kl - 珔 b k - 珔 b l + 珔 b,k,l=1,2,, (empiricaldistacecovariace,dcov) : V (X,Y)= V (X) 槡槡 V 2 (X,Y )= 1 2 A klb kl k,l=1 V (X)= 槡 V 2 (X )= 槡 V 2 (X,Y )= 1 2 (empiricaldistacecorrelatio)r (X,Y) : 槡 A 2 kl k=1 烄 V(X,Y) 2 R (X,Y)= 槡 R 2 (X,Y )= 槡 V 2 (X)V 2 (Y ), V 2 (X)V (Y) 2 >0 烅烆 0, V 2 (X)V (Y)= 2 0 R (X,Y) :10 R (X,Y) 1;2R (X,Y)=0 X Y R (X,Y)= 1, b C, Y =a+bxc dcov,, dcov 6

7 3 1 Vol.3No.1 Feb.2014 dcov [11] 2 MIC Reshef [11] Sciece (maximaliformatiocoeficiet,mic) : 2, 2, MIC Pearso, MIC,, MIC, MIC, MIC (computer-itesive) [36] 2.1 MIC 2 :1,, x y ;2, x a, a x,, : I(D,X,Y)= x X, p(x,y)log( p (x,y) y Y p(x)p(y) ) :D ;X,Y ;p(x,y), ;p(x),p(y) (k,k+1) (l,l+1), 0 k x-1,0 l y-1,,, I(D, x,y), [0,1]: M(D) x, y = I (D,x,y) log(mi{x,y}) 6 2 D, B() (MIC) MIC(D)= max xy<b() {M(D) x, y } MIC 3 (A) (x,y) (B) m xy A x y (C) M = [m xy ],MIC (B), (C) MIC 2,,MIC 2,MIC R 2 ( Pearso ),MIC Pearso r MIC 0~1 MIC 0, 2 ; MIC 1,, 7

8 [11] MIC 2.2 MIC MIC 2 :,MIC,,, ( ),MIC (R 2 ) 2 4 8

9 3 1 Vol.3No.1 Feb [11] MIC 4,(a) MIC ;(b) 4 2 R 2,MIC 4,MIC R 2, 2.3 MIC MIC 2,MIC, MIC 1,MIC, 2,MIC Pearso 1; 2,MIC Spearma 1,MIC 1 MIC 9

10 MIC 7 (maximalasymmetryscore,mas) 2, MAS(D)= max xy<b M(D) x, y -M(D) y,x 8 (maximumedgevalue,mev),, MEV(D)= max xy<b {M(D) x, y : x =2 y =2} 9 (miimumcelumber,mcn) MIC, MCN(D,ε)= max xy<b {log(xy):m(d) x, y (1-ε)MIC(D)} maximaliformatio-basedo-parametricexploratio(mine) 2 MIC [11] 3 MIC Reshef MIC, MIC Gorfie [37], Simo [38] Kiey [39] Gorfie MIC 2 dcor HHG 2 :1 ( ), MINE MIC HHG, dcor ;2 (30~100),HHG dcor MIC;3Reshef dcor, 30~100, MIC (>100), dcor HHG,,dCor HHG, MIC ;4 Reshef MIC,M. Gorgie : 2 (di- amodrelatio),,mic,, 2 :1, 1;2, Simo Tibshirai MIC,, : 2,MIC, 2 Kiey Atwal [42],MIC Reshef, MIC Reshef,, MIC,, Bjerve Doksum [18],, MIC,,,,, 10

11 3 1 Vol.3No.1 Feb.2014, [1]MooYI,RajagopalaB,LalU.Estimatioofmutualiformatiousigkereldesityestimators[J].PhysicalReview E, 1995,52(3): [2]DarbelayG,VajdaI.EstimatiooftheIformatiobyaadaptivepartitioigoftheobservatiospace[J].IEEE Trasac- tioiformatio,1999,45(4): [3]KraskovA,St gbauerh,grassbergerp.estimatig mutualiformatio[j/ol].( )[ ].physicalre- view E,2004.htp://jourals.aps.org/pre/abstract/ /PhysRevE [4]RéyiA.Omeasuresofdepedece[J].Acta MathematicaAcademiaeScietiarum Hugarica,1959,10(3/4): [5]BreimaL,FriedmaJH.Estimatigoptimaltrasformatiosformultipleregressioadcorrelatio[J].JouraloftheA- mericastatisticalassociatio,1985,80(391): [6]HastieT,Stuetzle W.Pricipalcurves[J].JouraloftheAmericaStatisticalAssociatio,1989,84(406): [7]TibshiraiR.Pricipalcurvesrevisited[J].StatisticsadComputig,1992,2(4): [8]KéglB,KrzyzakA,LiderT,etal.Apolygoalliealgorithmforcostructigpricipalcurves[C] AdvacesiNeuralI- formatioprocessigsystems.cambridge:mitpress,1999(11): [9]DelicadoP,SmrekarM.Measurigo-lieardepedecefortworadomvariablesdistributedalogacurve[J].Statistics adcomputig,2009(19): [10]DelicadoP.Aotherlookatpricipalcurvesadsurfaces[J].JouralofMultivariateAalysis,2001(77): [11]ReshefD N,ReshefY A,FiucaeH K,etal.Detectigovelassociatiosilargedatasets[J].Sciece,2011(334): [12]SpeedT.Acorrelatioforthe21(upst)cetury[J].Sciece,2011(334): [13]BelC.Mutualiformatioadmaximalcorrelatioasmeasuresofdepedece[J].TheAalsofMathematicalStatistics, 1962(33): [14]SchweizerB,WolfEF.Ooparametricmeasuresofdepedeceforradomvariables[J].TheAalsofMathematical Statistics,1981,9(4): [15]GragerC W,MassoumiE,RacieJ.A depedecemetricforpossiblyoliearprocesses[j].jouraloftheamerica StatisticalAssociatio,1989(84): [16]NelseRB.Aitroductiotocopulas,2(upd)ed,sprigseriesistatistics[M].New York:Spriger,2006. [17]ShaoCE,WeaverW.Themathematicaltheoryofcommuicatio[M].Champaig:UiversityofIlioisPress,1949. [18]BjerveS,Doksum K.Correlatiocurves:measuresofassociatioasfuctiosofcovariatevalue[J].TheAalsofMathe- maticalstatistics,1993(21): [19]GaltoF.Regressiotowardsmediocrityihereditarystature[J].JouraloftheAthropologicalIstitute,1885(15): [20]PearsoK.Notesothehistoryofcorrelatio[J].Biometrika,1920(13): [21]RodgersJL,NicewaderW A.Thirteewaystolookatthecorrelatiocoeficiet[J].TheAmericaStatisticia,1988,42 (1): [22]Wikimedia.File:Spearmafig1.svg[EB/OL].[ ].htp://commos.wikimedia.org/wiki/File:Spearma_fig1. svg?uselag=zh-c. [23]KedalM G.Aew measureofrakcorrelatio[j].biometrika,1938(30): [24]HirschfeldH O.Acoectiobetweecorrelatioadcotigecy[J].ProceedigsoftheCambridgePhilosophicalSocie- ty,1935,31(4): [25]Gebelei H.Dasstatistischeproblemderkorrelatioalsvariatios-udeigewertproblem udseizusammehagmitder Ausgleichsrechug[J].JouralofApplied Mathematicsad Mechaics,1941,21(6): [26]BreimaL,FriedmaJ.Estimatigoptimaltrasformatiosformultipleregressioadcorrelatio(withdiscussio)[J]. JouraloftheAmericaStatisticalAssociatio,1985(80): [27]SethuramaJ.Theasymptoticdistributioofthereyimaximalcorrelatio[J].CommuicatiosiStatistics,Theory Meth- od,1990,19(11): [28]DemboA,KagaA,SheppL A.Remarksothemaximumcorrelatiocoeficiet[J].Berouli,2001,7(2):

12 [29]CzakiP,FisherJ.Othegeeralotioofmaximumcorrelatio[J].Publ.Math.Ist.Hug.Acad.Sci,1963(8): [30]Walters-WiliamsJ.Estimatioofmutualiformatio:Asurvey[J].LectureNotesiComputerSciece,2009(5589): [31]DelicadoP,SmrekarM.Measurigo-lieatdepedecefortworadomvariablesdistributedalogacurve[J].Statistics adcomputig,2009(19): [32]HastieT,Stuetzle W.Pricipalcurves[J].JouraloftheAmericaStatisticalAssociatio,1989(84): [33]KeglB.Learigaddesigofpricipalcurves[J].IEEE Tras,PaterAalysisad MachieIteligece,2000(22): [34]DelicadoP.Aotherlookatpricipalcurvesadsurfaces[J].JouralofMultivariateAalysis,2001(77): [35]SzekelyG,Rizzo M.Measurigadtestigidepedecebycorrelatiodistaces[J].TheAalsofStatistics,2007(35): [36]DiacoisP,EfroB.Computer-itesivemethodsistatistics[J].ScietificAmerica,1983(248): [37]GorfieM,HelerR,HelerY.Commeto DetectigNovelAssociatiosiLargeDataSets[EB/OL].[ ].ht- tp:// [38]SimoN,TibshiraiR.Commeto Detectigovelassociatiosilargedatasets byreshefet,al,sciece,dec16,2011 [EB/OL].[ ].htp://statweb.staford.edu/~tibs/reshef/commet.pdf. [39]KieyJB,AtwalGS.Equitability,mutualiformatioadthemaximaliformatiocoeficiet[J].ProceedigsoftheNa- tioalacademyofscieces,2014,111(9): [40]HelerR,HelereY,GorfieM.Acosistetmultivariatetestofassociatiobasedoraksofdistaces[EB/OL].[ ].htp://xxx.tau.ac.il/pdf/ v3.pdf. SurveyofResearchProcessoStatisticalCorrelatioAalysis FaRog 1,MegDazhi 2,XuDashu 1 (1.DepartmetofMathematics,SoutherIlioisUiversity,Carbodale,IL62901,USA; 2.DepartmetofApplied Mathematics,BeijigPoletychicUiversity,Beijig100022,Chia) Abstract:Correlatioaalysisisamajorresearchtopiciboththeoreticalstatisticalstudyadpracticalapplicatios.Ithasbee paidmoreadmoreatetioastheamoutofdataisicreasigsigificatly.thisarticlereviewsseveralmethodsthatarecom- molyused,icludigthepearsocorrelatioadspearmacorrelatiodevelopedi19(upth)ceturyadcorgcadcovgci- troducedi21(upst)ceturyetc.iparticular,weiclude MICthatwasproposedi2011aditspositiveadegativecom- mets,aimigatsketchigthewholeresearchtopic.methodsofcorrelatioaalysisthemselvesplayakeyroleistatistics,es- pecialyiaalyziglargeheterogeeitydatasets,suchascomplexiformatioetworksadgeome-proteomedatasets.this surveytriestoprovidesomeuderstadigofexistigmethodsadtheirapplicatios.wehopetoecouragesomeewapplica- tios,whichiturmaypromotesomeew methodsdevelopig. Keywords:correlatioaalysis;Pearsocorrelatiocoeficiet;Spearmacorrelatiocoeficiet;Kedalcorrelatiocoeficiet; mutualiformatio;distacecorrelatio;mic 樊嶸 (1979-), 女, 博士, 主要从事概率统计模型及其应用徐大舜 (1971-), 男, 副教授, 博士, 主要从事微分方程理论及其应用 12

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殣 殣 檺檺檺檺檺檺檺檺檺檺檺檺檺檺檺檺殣 殣 MathematicalModelingandItsApplications Vol.3No.1 Feb , 2 1, (1. 南伊利诺伊州立大学数学系, 伊利诺伊州卡本代尔 62901, 美国 ;2. 北京工业大学应

殣殣檺檺檺檺檺檺檺檺檺檺檺檺檺檺檺檺殣殣 MathematicalModelingandItsApplications Vol.3No.1 Feb , 2 1, (1. 南伊利诺伊州立大学数学系, 伊利诺伊州卡本代尔 62901, 美国 ;2. 北京工业大学应