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# Partition Models As A Framework For Multiplicative Models For Categorical Data

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66Partition Models As A Framework For Multiplicative Models For Categorical DataAnna Klimova, Department Of Statistics, University Of WashingtonJoint Work With Tamas Rudas, Adrian DobraJune 4, 20091 Partition Models 6 Markov Chains 8 Example: Squirrel Monkeys Data (Ploog, 1967)A novel framework for the analysis of multiplicative models for categorical Rows of matrix D with integer entries is a lattice basis of matrix A.2 The genital display data in six squirrel monkeys is summarized as a 5 6data. They can be transformed into Markov moves for MCMC. table with 5 structural zeroes (a monkey does not display toward itself) :0A generalization of log-linear models, quasi-independence model, models Use moves to sample from the space of tables n that have the same asCounts R S T U V Wwith structural zeroes. observed subset (e.g. marginal) totals and structural zeroes under theR 1 5 8 9 0uniform distribution.Lattice basis for MCMC testing is easy to nd.S 29 14 46 4 0Choose the test statistic. For example,2 Partition: De nition And Examples2 U 2 3 1 38 2X (n(i) n^(i))Let X = (X ;X ;:::;X ) be a set of categorical variables cross-classi ed 21 2 k (n) = : V 0 0 0 0 1in a contingency tableI. A cell of the table i = (i ;i ;:::;i ) has a cell n^(i)1 2 ki2I W 9 25 4 6 13probability p(i).Here n^ =fn^(i)g are the expected cell values under the partition model. 2i2I Fienberg (2007). The quasi-independence model. = 168:05, d.f.=15.A partition q of a tableI is a ...

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Exrait Partition Models As A Framework For Multiplicative Models For Categorical Data Anna Klimova, Department Of Statistics, University Of Washington Joint Work With Tamas Rudas, Adrian Dobra June 4, 2009
7 A novel framework for the analysis of multiplicative models for categorical Rows of matrixD2with integer entries is alattice basisof matrixA.8eExaenmitpalled:i )Squirrel Monkeys Data (Ploog, 196 data. They can be transformed intoMarkov movesditsowargThelfMCMr.Coftr5stuucleabthwiasadt5ammuezirnkeysissuirrelmoiasnxiqspsaldyta×6: A generalization of log-linear models, quasi-independence model, models Use moves to sample from the space of tablesn0 zeroes (a monkey does not display t )that have the same as ral with structural zeroes. observed subset (e.g. marginal) totals and structural zeroes under the Counts R S T U V W Lattice basis for MCMC testing is easy to ﬁnd. uniform distribution. R1 5 8 9 0 2 Partition : Deﬁnition And ExamplesChoose the test statistiχc.2F(nr)=oexaXple(mn(,i)nˆ(i290VS2U2401403160432810 LetX= (X1,X2, . . . ,Xk ))) be a set of categorical variables cross-classiﬁed in a contingency tableI. Acellof the tablei= (i1,i2, . . . ,ik) hasa cellnˆ(i). probabilityp(ie.)erHnˆ ={nˆ(i)}i∈Iare the expeciteItrtioimnedtrehapvaluesundcell.ledomecnedn3Fienberg(2007)T.ehuqsa-iniedep.ledo29W1645χ2= 168.05, d.f.=15. A partitionqof a tableIis a division of its cells into subsetsSusuch that : For the observed tablepn(,nMCMC estiXmate ofthe exact p-value2(0i0hCne==ee.00asvulpp--luva.27)9.T9±is-EaconheDi(nsamplednteabmlielsl)i.ooitivlannorfdnoct.umoleset Su6=∅ ∀u;Su\Sv=∅ ∀u6=v;[Su=I.) ={2( )χ2(n)}P(n0).0.0006 o u n0:χn0 Examples : A marginal partition C- all cells belonging to the same marginal cell7onEsxidaemr apl2e :×R3acpoanltlion,ge2n0cy06table with a structural zero in the cell (1,2)9Dov-pSblraquauei(r=2r00e.0l399)M.8oT±nh0ek.e0Byo0sun.06DdastSamOplinrgAAlpgporriotahcmh(BSA). form a subset ; (Rapallo, 2006) and frequencies :a : u A partition induced by a cell- this cell forms a subset. 3 Partition Model : Matrix Representation23 A partition modelPM(Q) for a classQof partitions ofI: logp=A0V.The class of partitions for themµµo12d11elµ02o2f cµµo13nditional independence with a:elLodMittaBecPaasritsitic0sioomnnost00011trtiovoineaP.ledso5mf1E..ssnidg00u0e:c.bd0ym1100a1rgina1ls00and zero cells. HereA= (aij) is a model matrix,aij=ISi(j). 0structural zero : 11 0 0 0 ISiis the indicator function of the subsetSi.Vis the vector of parameters.q0is the trivial partition,q1is the partition induced by row marginal,q2is 0 0 01 1 0,0 0 0 0 0 0 olumn m pPM(Q)D2logp= 0.cerlal(p1eht,giarl,naycdbc2ed)u.innoititq3is the partition induced by the000000000000 000000000000D2is the matrix with rows that form a basis of Null (A).001011111111Method :sealresavemoovrkMamorfdelpmExacttiMhMC.Cettsniwg Theorem :LetQbe a class of partitions andPM(Qontitiaparttalehtecha.FivasisicebnoasliilnOmeni.spmbatdese1l1.0h1c0a0rnBun1-i0i00rateitno.seb) model. The sum of observed probabilities of the cells from the same subset of partition is equal to the sum of the MLEs of the probabilitiesp(i) underA.1a0mlexirt0010001001=medosihtrevnoCeRustl:g.epn-cvealupelo=t0.f9o8r±ex0a.0c0t46p.-values : thePM(Q) model. 0 1 0 0 1 0 Thedegrees of freedomequal dim Null (A) (the number of rows inD2). 5 Iterative Proportional Fitting001001001111PM(Q) is a partition model. The model of independence :D2logp= logp11p23= 0, Let (Sj,j= 1, . . . ,J) be all the subsets of the partitions inQandp21p13 {n(i)}i∈Ibe the set observed frequencies. Setm0(i where) = 1 and iterateD2=1 011 0 1. cyclically overjJ. Withd+ 1 =lJ+jA lattice basis consists of only one move : md+1(i) =md(i)PP0i0SjSjnmd(i(0i)0),ifiSj,Partition model leads to theMs1mae=m10a0esvo1a1R1p.allo obtained, but without i md+1(i) =md(i) ifi/Sj.special tools of computational algebra.
Klimova et al (UW)
email : klimova@u.washington.edu
Partition Models As A Framework For Multiplicative Models For Categorical Data
Applications of partition models : topological models etc.
Sampling strategies from lattice basis
Partition models and Neumann structure
June 4, 2009 1 / 1