500
Table1.(Continued).Site#343344345346347348349350351352353354355356
Figure#44444444444444
Longitude
(?)355.48354.28353.96353.95352.02351.97349.92349.91347.1346.57346.15345.95343.72343.78
Latitude(?)?52.63?51.45?51.16?51.11?49.69?49.57?48.34?48.2?47.12?47.06?46.14?45.97?42.8?42.75
SurendraPVermaetal
Country/Province
MidAtlanticRidge(Fracturezones;52?S)
MidAtlanticRidge(Fracturezones;51?S)
MAR,southernregion(50.3–52.46?S)
MAR,southernregion(50.3–54.4?S)
MAR,southernregion(49.3–50.0?S)
MAR,southernregion(49.3–50.0?S)
MAR,southernregion(47.4–49.2?S)
MAR,southernregion(47.4–49.2?S)
MAR,centralregion(46.9–47.45?S)
MAR,centralregion(46.9–47.45?S)
MAR,northernregion(43.4–46.6?S)
MAR,northernregion(43.4–46.6?S)
MAR,northernregion(40.6–44.8?S)
MAR,northernregion(40.6–44.8?S)
No.ofsamples119166639614716141133418
Reference
LeRoexetal(1987)LeRoexetal(1987)LeRouxetal(2002b)LeRouxetal(2002a)LeRouxetal(2002a)LeRouxetal(2002b)LeRouxetal(2002b)LeRouxetal(2002a)LeRouxetal(2002b)LeRouxetal(2002a)LeRouxetal(2002b)LeRouxetal(2002a)LeRouxetal(2002a)LeRouxetal(2002b)
Table2.Samplesizesinthetrainingandtestingsetsandcomparisonwithourearlierwork.
Trainingset
GroupNameIABCRBOIBMORBTotal
Number(1)(2)(3)(4)
Thiswork3636714728262332
Agrawaletal
(2004)
2242092273991059
Testingset
Thiswork100100100100400
Agrawaletal
(2004)
25252525100
Total
Thiswork4637715729262732
Agrawaletal
(2004)
2492342524241159
IAB=islandarcbasicrock;CRB=continentalriftbasicrock;OIB=oceanislandbasicrockandMORB=mid-oceanridgebasicrock.Thenumbersinparenthesesaregroupnumbersdiscussedinthetext.
turn,dependsontheresultingrocktype.BecausemostpersonshavenotusedSINCLAS,weonceagainemphasizethatitisnoteasytorelyonahandcalculatororsomesimplecomputerprogrambecausetheproperFe-adjustmentrequiresaniter-ativeprocess.Toremindtheuseraboutthisbasicneed,wehavecalledthenewmajor-elementarraywiththesubscriptadj,suchas(SiO2)adjforSiO2,andsoon.
3.Dataprocessing3.1Theneedfortransformationof
thecompositionaldata
AspointedoutintheIntroductionsection,weshouldthinkintermsofratiosandabandonthetraditionalwayoflookingatthecompositionsintermsofpercentages(major-elements),orppm,
Fivegeochemicaldiagramsforbasicandultrabasicrocks
Table3.Meansandstandarddeviationsoflog-transformedmajorelementratiosforthefourgroupsetsIAB–CRB–OIB–MORB.Loge-transformedratiovariableln(TiO2/SiO2)adjln(Al2O3/SiO2)adjln(Fe2O3/SiO2)adjln(FeO/SiO2)adjln(MnO/SiO2)adjln(MgO/SiO2)adjln(CaO/SiO2)adjln(Na2O/SiO2)adjln(K2O/SiO2)adjln(P2O5/SiO2)adj
IAB(1)(n=463)mean?4.03?1.06?3.38?1.82?5.61?2.02?1.50?3.02?4.4?5.7
s0.270.120.150.140.200.390.140.240.80.7
CRB(2)(n=771)mean?3.05?1.17?3.08?1.64?5.58?1.81?1.61?2.67?3.6?4.5
s0.300.100.200.190.200.360.200.250.60.5
OIB(3)(n=572)mean?2.90?1.24?3.15?1.58?5.56?1.8?1.51?2.91?4.4?4.9
s0.430.160.210.170.160.50.180.331.00.7
MORB(4)(n=926)mean?3.56?1.19?3.40?1.80?5.69?1.88?1.48?2.95?5.7?5.83
s0.260.080.140.140.180.150.090.150.90.43
501
Subscriptadjreferstotheconcentrationsrecalculatedtoananhydrous100-justedbasisusingcomputerprogramSINCLASemployingtheMiddlemost(1989)recommendationforiron-oxidationratioadjustment(Vermaetal2002).Thiswasdonepriortothelogtransformationofthedata.Roundingofmeanvalueswasdoneaccordingtothecorrespondingstandarddeviation(s)data(Verma2005).Tectonicsettings,groupnumbers,andthenumberofcompiledsamples(n)arethesameasintable2.
orotherconcentrationunits(traceelements).Thepercentagedataoccupyarestrictedregionofsam-plespace;theratiosallowthemtospreadmorefreelythroughthisspace.Infact,thecalculationoflog-ratioshastheconsequenceoffreeingspacevaluesfromarestrictedrangetovarybetween?∞and+∞,whichisahighlydesirableprop-ertyformoststatisticalinferencesincludingthediscriminantanalysis.Aitchison(1986)proposedthatcompositionaldatashouldbeexpressedasthecovariancesoflog-ratiosofthevariablesratherthantherawpercentages.Thus,wedecidedtoratioeveryvariableagainstacommondivisor.AccordingtoAitchison(1986)thechoiceofavari-ableasthedivisorisimmaterialbecauseonlythestructureoftheratiomatrixisimportant.Forourmajor-elementdata,wedecidedtousethemostabundantpartSiO2asthedivisorandthentotransformtheratiodatausingloge(naturallog-arithmbasee,orln).Infact,weexperimentedwithseveralothervariablesasthedivisorandobtainedsimilarresults.Wedidnotexperimentwithothertypesoflogarithmssuchaslog10,buttheyshouldyieldsimilar?nalresults.Thus,our?nalvariablesforthediscriminantanaly-siswere(onelessthantheoriginalvariables):ln(TiO2/SiO2)adj,ln(Al2O3/SiO2)adj,ln(Fe2O3/
ln(FeO/SiO2)adj,ln(MnO/SiO2)adj,SiO2)adj,
ln(Na2O/ln(MgO/SiO2)adj,ln(CaO/SiO2)adj
SiO2)adj,ln(K2O/SiO2)adj,andln(P2O5/SiO2)adj.Althoughtheadjustmentfactor(adj)willcanceloutfromtheratiosofconcentrationdata,thisistoremindthereaderthatthecalculationofFe2O3andFeOusingtheMiddlemostoption(Verma
etal2002)isstillarecommended,probablyamandatorystepduringtheapplicationprocedure.
3.2Comparisonoflog-transformedmajor-elementratiosofbasicrocksfrom
di?erenttectonicsettingsInsteadofreportingtheaveragemajor-elementcompositionsandcorrespondingstandarddevia-tionvalues,wepresentthebasicstatisticalinfor-mationofloge-transformedmajor-elementratiosofthefourtectonictypesofbasicrocks(table3).Noteallmeanvaluesarenegativebecausethedivi-sor(SiO2)usedwasthemostabundantelementrenderingallratiostobe<1,andconsequentlytheirlogarithmstobenegative.TheStudentttest(forANOVAorFvaluetest,seebelow)revealedthatformostratios,therewerestatisticallysig-ni?cantdi?erencesbetweenthemeanvaluesfromdi?erenttectonicsettings.However,thesmallestdi?erenceswereobservedbetweentheCRBandOIB,ortheOIBandMORBgroups,makinganysimplebivariateorternaryplotsoftheseloge-transformedratiosunsuccessfulindiscriminatingbetweenthesefourtectonicsettings(plotsnotshown).Therefore,multivariatetechniquessuchaslineardiscriminantanalysis,wereactuallyrequiredtoobtainmeaningfuldiscriminationofthesesettings.
Anotherwayoflookingatthesedataistodeter-mineifthegeochemicalvariationwithinthegroupislowcomparedtothatbetweengroups.Table4showsWilk’slambda,alsocalledU-statistic,andrepresentstheratioofthewithin-groupsumof
502SurendraPVermaetal
Table4.Testforequalityofgroupmeans.Loge-transformedratiovariableln(TiO2/SiO2)adjln(Al2O3/SiO2)adjln(Fe2O3/SiO2)adjln(FeO/SiO2)adjln(MnO/SiO2)adjln(MgO/SiO2)adjln(CaO/SiO2)adjln(Na2O/SiO2)adjln(K2O/SiO2)adjln(P2O5/SiO2)adj
Wilks’Lambda
0.39910.81000.60050.73200.91460.93920.89100.76270.47770.4979
F-ratio1168182516284735095241848783
Signi?cance0.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
BecauseSiO2wasusedforlogtransformations,thisrowisabsent.Wilks’Lambda(U-statistic)andunivariateF-ratiowithdegreesoffreedom,DF1=g?1=4?1=3andDF2=n?g=2332?4=2328,wheregisthenumberofgroupsandnisthetotalnumberofsamples.
squarestothetotalsumofsquares(therefore,thisstatisticcanhavevaluesbetween0and1).Alambdaof1isobservedwhenallgroupmeansareequal,whereassmallvaluesclosetozerosignifythatwithin-groupvariabilityissmallcomparedtothetotalvariability.Inotherwords,largevaluesoflambdaindicatethatgroupmeansarenotstatis-ticallydi?erent,whereassmallvaluesindicatetheopposite.TheWilk’slambdavaluesformostloge-transformedmajor-elementratiosaremuchlowerthan1,indicatingthattheirgroupmeansaredif-ferent.
Yetanotherstatistic(table4)isthesigni?-cancetestfortheequalityofgroupmeansforeachratio.TheFvalueisobtainedbydividingthebetween-groupvariationbywithin-groupvariation.Theobservedsigni?cancelevel(0.0000;table4)foreveryloge-transformedratioislessthan0.01(even<0.0001),thehypothesis(H0)thatallgroupmeansareequalisrejectedineachcaseatthecon-?dencelevelof99%(evenatthehighestcon?dencelevelof99.99%).Thismeansthat,intheory,eachratiocanbeconsideredasapossiblecandidateforpredictorvariableinthediscriminantanalysis.
3.3Selectionofpredictorvariables,numberofgroupsconsideredsimultaneously,discriminantscore,andclassi?cationruleTherestofourprocedurewassimilartotheonedescribedindetailbyAgrawaletal(2004).Thesuccessofdiscriminantanalysisdependsonselec-tionofappropriatepredictorvariables.Thisselec-tionwasbasedonthetestofWilk’slambda,andthesevariableswereselectedbyastepwisemethod,withminimizationofWilk’slambdaasthecrite-rionforvariableselection.Ateachstepthevariable
thatresultedinthesmallestlambdaforthedis-criminantfunctionwasselected.
Theuseofdiscriminantfunctionsisadvanta-geousbecauseeachdiscriminantfunctioncanrep-resentseveralofthecompositionalvariables.Thediscriminantanalysisisoneofthestatisticalmul-tivariatetechniquesmostcommonlyusedtodis-tinguishbetweenmembersofdi?erent,pre-de?ned,groupsorclasses,onthebasisofobservationsregardingthem.Theactualaiminmathematicaltermsistosummarizep-dimensionalobservationsfromgclassesonaone-dimensionallinearfunctionthatdiscriminatesbetweentheseclassesbysomemeasureofmaximumseparation,andlaterservesthepurposeofclassifyingsamplesofunknownclasses.An‘unknown’classshouldbeoneoftheclassesthatwereactuallyusedforthediscriminantanalysistobeginwith.Althoughtheconceptofdis-criminantanalysisissimple,moredetailsonlineardiscriminantanalysis,beinghighlymathematicalandcomplex,areprobablybeyondthescopeofthepresentpaper;thesecanbeconsultedinstandardbooksonmultivariatetechniquessuchasMorrison(1990)andReymentandSavazzi(1997).Finally,asinAgrawaletal(2004)weusedtheSPSS/PC+statisticalpackageforperformingthediscriminantanalysis.
Weperformedthediscriminantanalysisforallcombinationsofthefourgroupsbytakingonlyfourandthreegroupsatatime.Thediscriminantscore(D)oftheindividualcaseineachgroupcanbeobtainedas:
D=B1X1+B2X2+···+BpXp+B0,
(1)
whereXiisthevalueoftheloge-transformedithmajor-elementratio(e.g.,the?rstratioisln(TiO2/SiO2)adj),Biistheithdiscriminant
Fivegeochemicaldiagramsforbasicandultrabasicrocks
503
functioncoe?cient,andB0isaconstant.Thesepcoe?cientsandtheconstantcanhavepositiveornegativevalues.Thediscriminantanalysismakesuseofallorsomeoftheavailablelogmajor-elementratios(Xand?xese-transformedi??s)thebestval-uesofBiforeachchosenXitoobtainthemaxi-mumseparationbetweenallgroups.
g?If1gdiscriminantgroupsareinvolvedfunctionsinarediscriminantobtained.Theanalysis,num-berofdiscriminantfunctionsinthree-andfour-groupdiscriminantanalyses,therefore,are2and3respectively.ThediscriminantscoresD1,D2,andD3werecalculatedusingequation(1).Fromthediscriminantscores,themeandiscriminantscoresforthegroups,knownasgroupcentroids,werecal-culated.
TheruleforclassifyingagivenbasicrocksampleintooneofthetectonicgroupswasobtainedfromitsdiscriminantscoreD.TheprobabilitythatabasicrockwithascoreDbelongstogroupiisestimatedby:
P(gi/D)=P(D/gi)P(gi)
(2)
whereP(D/gP(gi/D)istheposteriorprobability,i)istheconditionalprobability,andP(gisthepriorprobability.
i)Thepriorprobability–anestimateofthelike-lihoodthatacasebelongstoaparticulargroupwhennoinformationisavailable–wasassumedtobeequalforallgroups.Theconditionalprob-abilityP(D/gfromthei)ofDforagivengroupiscalcu-latedprobabilityofobtainingaparticulardiscriminantscorevalueDifthecaseisamem-berofthatgroup,undertheassumptionthatthediscriminantscoresofthecasesusedintheanalysisarenormallydistributedandthepara-metersofthedistributioncanbeestimatedfromthemeandiscriminantscores.Thecaseisassumedtobelongtoaparticulargroupandtheprob-abilityoftheobservedscoreforthatparticularmembershipisestimated.Fromthesepriorandconditional/D)probabilities,isthenestimatedtheposteriorusingBayes’probabil-ityP(girule.FromitsdiscriminantscoreD,acaseisclassi?edinagroupforwhichtheposteriorprobabilityislargest.
Finally,concerning?eldboundariesingeo-chemicaldiscriminationdiagrams,Agrawal(1999)arguedatlengththatthepracticeofde?ningassignmentruleswiththeleastpreciseandsubjec-tivemethodofeye-?ttedlinesmustbeabondoned.Heillustratedtheproposedstatisticalmethodusingasimplecaseoftwovariablesfromtwoclasses.Wehaveadoptedthisapproachtodrawprobabilitybasedboundariesinourdiagrams.
4.ResultsofthediscriminantanalysisTable5showstheresultsofthediscriminantanalysisofthepresentdatabaseandtheircom-parisonwithourearlierwork(Agrawaletal2004).Inthe?rstcolumn,therowsmarkedAareforAgrawaletal(2004).TheresultsoftherowsBwereobtainedinexactlythesameman-nerasinAgrawaletal(2004),i.e.,usingmajor-elementconcentrations,butfromtheextendeddatabase.Finally,therowsCcorrespondtotheresultsusingloge-transformationofthemajor-elementratiosfromexactlythesamedatabaseasinrowsB.
Thediscriminantanalysiswasperformed?vetimesusingthetrainingset:onceforallfourtec-tonicgroups;andfourtimesforallpossiblecom-binationsofthreegroupstakenatatime.Allratioswerefoundtobeusefulaspredictorvari-ablesunderthecriterionofreductionofWilk’slambdaforthediscriminantfunction.Then,theassignmentrulesfromthediscriminantanalysisofthetrainingsetwereusedtoclassifysamplesofthetestingset,treatingthemas‘unknown’cases.
Theratesofcorrectclassi?cationofthetrain-ingaswellastestingsetshavebeencalculated,andarecomparedwithourearlierwork(table5).Theoverall%successrate(rateofcorrectclas-si?cation)fordiscriminantfunctiondiagramspro-posedbyAgrawaletal(2004)variedfrom～83to93%(casesAintable5).Thesameprocedureforthelargerdatasetresultedinsimilar%suc-cessratesof～82to94%(casesBintable5).Theloge-transformationprocedure,however,pro-videdhigheroverall%successratesof～83to97%(inallcasesCascomparedtotherespec-tivecasesBintable5),suggestingthatsuchatransformationisastatistically-correctprocedureand,therefore,shouldbeusedinfutureforcom-positionaldatahandling.However,smalldecreasesin%successratesforsomeindividualtectonicgroups(6outofatotalof32cases;table5)occur,thereasonforwhichisnotclearatpresent,butmightberelatedtothestatisticalnatureofthemultivariatetechniques.Inthiscontext,wemustemphasizethatthesediscriminantfunctiondiagramsshouldbeusedonthisverystatisticalbasis,i.e.,bytakingintoaccountthatthebound-ariesarestatistically-basedandagiven?eldatbestenclosesonlyacertainpercentageofandnotallrocksamplesfromagivensetting(e.g.,forournewdiagrams83to97%ofsamplesusedforthetrainingortestingsets).Thiswillbebetterillustratedintheapplicationsectionofthispaper.
504SurendraPVermaetal
Table5.Assessmentofcorrectclassi?cation(%)betweentheIAB(1),CRB(2),OIB(3),andMORB(4)groups.DatasetABCABCABCABCABCABCABCABCABCABC
Group1–2–3–41–2–3–41–2–3–41–2–3–41–2–3–41–2–3–41–2–31–2–31–2–31–2–31–2–31–2–31–2–41–2–41–2–41–2–41–2–41–2–41–3–41–3–41–3–41–3–41–3–41–3–42–3–42–3–42–3–42–3–42–3–42–3–4
SetTrainingTrainingTrainingTestingTestingTestingTrainingTrainingTrainingTestingTestingTestingTrainingTrainingTrainingTestingTestingTestingTrainingTrainingTrainingTestingTestingTestingTrainingTrainingTrainingTestingTestingTesting
n1059233223321004004006601506150675300300832233223327540040085016611661753003008351969196975300300
IAB(1)80.881.890.484.091.094.084.092.691.292.094.094.080.884.891.288.092.094.084.886.895.692.093.099.0
71.876.681.176.079.083.0CRB(2)70.370.977.980.078.084.079.477.980.088.078.083.084.784.291.496.094.097.0
95.685.287.396.082.084.082.478.076.988.072.071.0OIB(3)80.677.376.788.073.071.079.779.985.084.079.083.0
93.594.197.692.095.099.094.596.296.792.098.0100.097.598.496.096.098.099.0MORB(4)
93.093.894.892.094.097.0
Overall83.382.085.686.084.086.585.082.184.385.083.786.787.988.794.192.093.796.792.291.093.893.391.094.386.986.186.386.783.084.7
A=resultsfromAgrawaletal(2004);B=newresultswithlargerdatasetusingrawdata,andC=newresultswithlargerdatasetusinglogetransformationofratios.
4.1DiscriminationbetweenIAB–CRB–OIB–MORBgroups
AsinAgrawaletal(2004),wedeveloped?venewdiscriminantfunctiondiagramsforthefourtec-tonictypesofbasic(andultrabasic)rocks(IAB,CRB,OIB,andMORB).Figure5includesallfourtectonictypesofbasic(andultrabasic)rocks,whereaseachofthefourdiagramsin?gure6(a–d)presentssamplesfromthreegroups.Thus,eachtec-tonicgroupappearsinfourofthese?vediagrams.Eachdiagramisaplotoftwodiscriminantfunc-tions,DF1andDF2,respectivelyinx-andy-axes.Thediscriminantfunctionsreducethedimension-alityoftheproblem,inthiscase(?gure5)from10major-elementratiologe-transformedvariablestotwodimensionsofdiscriminantfunctionsDF1andDF2.Inthisfour-groupdiscriminantanaly-sis,whichresultedin?gure5,threediscriminantfunctionswereobtained,butthe?rsttwofunc-tionsaccountedfor94.16%ofbetween-groupvari-ability(table6),thethirdfunctionbeingrelatively
insigni?cant(capableofexplainingtheremain-ing5.84%ofthisvariability).Forthree-groupdiagrams,thebetween-groupvariabilityiscom-pletelyrepresentedbytwodiscriminantfunctions(table7).
Theclassi?cationrulesin?gure5arebasedononlytwodiscriminantfunctions,DF1andDF2asfollows(notethecoe?cientsorthemulti-plicationfactorsforeachloge-transformedratioafterroundingtofourdecimalplacesarelistedintable6):
DF1=?4.6761·ln(TiO2/SiO2)adj
+2.5330·ln(Al2O3/SiO2)adj?0.3884·ln(Fe2O3/SiO2)adj+3.9688·ln(FeO/SiO2)adj+0.8980·ln(MnO/SiO2)adj?0.5832·ln(MgO/SiO2)adj