Partition Models As A Framework For Multiplicative Models For Categorical Data

Anna Klimova, Tamas Rudas, Adrian Dobra

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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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• Publié par: Udru
• Nombre de lectures: 16
• Langue: English

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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 matrixD 2 with integer entries is alattice basisof matrixA. 8 e Exa en m it p a l l e d : i  )Squirrel Monkeys Data (Ploog, 196 data. They can be transformed intoMarkov moves ditsowar gTh elf MCMr.Cof tr5stuucleabthwi asad t 5ammuezirnkeysissuirrelmoiasnxiqspsaldyta×6 : A generalization of log-linear models, quasi-independence model, models Use moves to sample from the space of tablesn 0  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.  R−1 5 8 9 0 2 Partition : Definition And Examples Choose the test statisti χ c. 2 F (n r )= oexa X ple ( m n( , i)−nˆ(i 2 9 0V S2 U2 −4 0 14 0 31− 6 0 4 3 − 28 1 0 LetX= (X 1 ,X 2 , . . . ,X k  )) ) be a set of categorical variables cross-classified in a contingency tableI. Acellof the tablei= (i 1 ,i 2 , . . . ,i k ) hasa cellnˆ(i) . probabilityp(i e .) erHnˆ ={nˆ(i)} i∈I are the expec i t ∈ e I trtioimnedtrehapvaluesundcell .ledomecnedn 3 Fienberg(2007)T.ehuqsa-iniedep .ledo 29W1645− χ 2 = 168.05, d.f.=15. A partitionqof a tableIis a division of its cells into subsetsS u such that :  For the observed table p n ( , n MCMC esti X mate ofthe exact p-value 2(0 i 0hCne = = e e . 0 0a s vul p p - - luva.2 7) 9 .T 9± is-EaconheDi (nsamplednteabmlielsl)i.o oitivlannorfdnoct.umoleset S u 6=∅ ∀u;S u \ S v =∅ ∀u6=v; [ S u =I. ) = { 2 ( )≥χ 2 (n)} P(n 0 ). 0.0006 o u n 0 :χn 0 Examples : A marginal partition  C - all cells belonging to the same marginal cell 7 on E s x id a e m r a pl 2 e : × R 3 a c p o a n l t l i o n , ge 2 n 0 cy 06 table with a structural zero in the cell (1,2) 9 Do v-p S b l r a q u a u e i ( r = 2 r 0 0 e . 0 l 39 9) M . 8 o T ± n h 0 e k . e 0 B y o 0 s un .06 D d a s t Sam O plin r g A Al p g p o r ri o t a h c m h (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 Representation 23 A partition modelPM(Q) for a classQof partitions ofI: logp=A 0 V. The class of partitions for the  m µ µ o 1 2 d 1 1 el µ 0 2 o 2 f c µ µ o 13 nd  itional independence with a :el L odM ittaBec P a a s r i t s iti  c 0 si o o m n no st −00011 trtio vo in e aP.led so5mf1  E.. s s nid  g 00 u 0 e : c . bd 0− ym 1 1− 00 a 1 rgina 1 ls 00  and zero cells. HereA= (a ij ) is a model matrix,a ij =I S i (j).  0 structural zero :  1−1 0 0 0 I S i is the indicator function of the subsetS i .Vis the vector of parameters. q 0 is the trivial partition,q 1 is the partition induced by row marginal,q 2 is − 0 0 01 1 0 , 0 0 0 0 0 0 olumn m p∈PM(Q)⇔D 2 logp= 0. ce r l a l( p 1 eht , giarl,naycdbc 2 ed ) u . innoititq 3 is the partition induced by the   000 000000 000     000000 000000   D 2 is the matrix with rows that form a basis of Null (A).  001011 111111  Method :s e a l resavemoovrkMamorfdelpmExacttiMhMC.Cettsniwg Theorem :LetQbe a class of partitions andPM(Qontitiapar ttalehtecha.FivasisicebnoasliilnOmeni.spmbatdese 1 l 1 . 0 h 1 c 0 a 0 rnBun 1 -i 0i00rateitno.s eb) 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) under A. 1 a 0 mlexirt 0010 0010 01 =medosiht revnoC eRustl: g . e p n - c v e alu p e lo = t 0. f 9 o 8 r ± ex 0 a .0 c 0 t 46 p . -values : thePM(Q) model.  0 1 0 0 1 0 Thedegrees of freedomequal dim Null (A) (the number of rows inD 2 ). 5 Iterative Proportional Fitting   00 10 0100 1111   PM(Q) is a partition model.  The model of independence :D 2 logp= log p 11 p 23 = 0, Let (S j ,j= 1, . . . ,J) be all the subsets of the partitions inQand p 21 p 13 {n(i)} i∈I be the set observed frequencies. Setm 0 (i  where ) = 1 and iterate D 2 =  1 0−1−1 0 1  . cyclically overj∈J. Withd+ 1 =lJ+j A lattice basis consists of only one move : m d+1 (i) =m d (i) P P 0 i 0 ∈ ∈ S j S j n m d (i ( 0 i ) 0 ) ,ifi∈S j , Partition model leads to the M s 1 mae =  m −10 a 0 esvo 1− a 1 R 1  p . allo obtained, but without i m d+1 (i) =m d (i) ifi/∈S j . special tools of computational algebra.

Klimova et al (UW)

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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

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