c ○ 2001 Kluwer Academic Publishers. Manufactured in The Ne(21)

Abstract. A data cube is a popular organization for summary data. A cube is simply a multidimensional structure that contains in each cell an aggregate value, i.e., the result of applying an aggregate function to an underlying relation. In practical situat

LOGLINEAR-BASEDQUASICUBES275loglinearmodel.Thereasonbehindthisisthateventhoughadditivemodelsaresometimesaworse tfordatasets,theirparametershavesomenicepropertiesthatcanhelpinspeedingupqueryprocessing.Forinstance,inEq.(8),weshowanadditive,saturatedmodelfor4-dimensionaldata.Forinstance,theanswerofaquerythataggregatescellsoneverydimensionbutA,witha xedvalueforAcanbepreciselyobtainedbyusingtheγandγiAparameters,withoutlookingatthedata.Wearecurrentlytryingtoquantifythegainsinqueryexecutionthatthisideamaybring.

ABACADBCBDCDy ijkl=γ+γiA+γjB+γkC+γlD+γij+γik+γil+γjk+γjl+γkl(8)

Anotheraspectthatmeritsfurtherresearchisthatofcompressingcubeswithmanymeasures.(Thisisbecomingastandardoccurrenceinindustry;e.g.,forasalesdatacube,onemightbeinterestedinkeepingmeasuresoftotalsalesindollars,unitssold,pro t,andsoon.)Intheworstcase,everymeasurewouldrequireitsownsetofmodelsparametersandoutliers.(Noticethatthecompressionratiowouldnotsufferfromthis,sincenotcompressingthecubewouldrequiretostorevaluesforeachofthemeasuresinnon-nullcells,thusincreasingthetotalsizeofthedatacube.)However,itispossiblethatsomeofthemeasurescanbeestimatedorapproximatedfromtheapproximationsofothers,andwearecurrentlyresearchingthisissue.

Finally,weliketopointoutthatourcompressedversionsofthecorecuboidcanserveasconcisedatamartsforremoteclientsofthewarehouse.(Buildingsmallerdatamartsthatcanbeexportedtoremotemachinesoutsideofthewarehouseisbecomingastandardpracticeinindustry(MicroStrategy,http://www.//products/server/features.htm).)Theyareidealtobeusedinlesspowerfulmachinesbecauseoftheirsmallersize,andyettheyarepowerfulenoughtonotonlygiveapproximateanswerstoqueriesintheclientside,butalsotheycontainawealthofmininginformationinthechunkmodels.Thisisinlinewiththeopinionofsomedataminingexpertswhoclaimthatthetrendshouldbeinexportingmodelsandnotdatatotheclients(Fayyad,1998).

Notes

1.Thecorecuboidisthesetofcellsof nestgranularityinthelatticeforthedatacube.

2.Anyothercuboidinthedatacubecanbecomputedfromthecorecuboidbyaggregatingononeormoreofthedimentions.

3.Withoutlossofgeneralityandtokeepthepresentationsimple,weassumeinthispaperthattheactualattributevaluesareconsecutiveintegervaluesfrom0to Di 1where Di isith-dimensionattributedomainsize,forexample,theactualvalues{male,female}ofattributeSexincensusdatasetcanbemappedto{0,1}.Forthegeneralcasewheretheactualattributevaluesarenotnecessarilyconsecutive,wecanmapeachactualattributevaluetoitsrank.

References

Acharya,S.,Gibbons,P.B.,Poosala,V.,andRamaswamy,S.(1999).JoinSynopsesforApproximateQueryAnswering.InProceedingsofthe1999ACM-SIGMODInternationalConferenceonManagementofData,Philadelphia,PA.