MATH 697 ITERATIVE PROPORTIONAL SCALING

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1 MATH 697 ITERATIVE PROPORTIONAL SCALING REN BETTENDORF 1. Abstract The purpose of ths project s to show observatons taken from the Iteratve Proportonal Scalng algorthm when used wth data taken from Herarchcal Log- Lnear Models. We show wth emprcal proof that the dfference between decomposable and non-decomposable graph models used s not exceedngly dfferent. For the purpose of ths project we looked at the propertes of the two, three, and four varable case of graph models. 2. Introducton For many years, statstcans and probablsts struggled to fnd an effectve way to maxmze lkelhood estmators. It was through ths work that these mathematcans dscovered a way to maxmze these lkelhood estmates. The Iteratve Proportonal Scalng algorthm (IPS) was frst ntroduced n 1940 by W. Edwards Demng and Frederck Stephan. Demng and Stephan used t manly to maxmze lkelhood estmators gven a fnte dscrete set of data wth fxed margnal totals. After ts ntroducton, Darroch and Ratclff gave an algorthm for generalzed loglnear models when the expected frequences are restrcted by equaltes. It was from ths result that the current verson of the algorthm exsts and was used for ths project. The goal of usng the Iteratve Proportonal Scalng algorthm s to maxmze a set of lkelhood estmators n the form of a contngency table subject to the margnal probabltes. Ths s acheved through solvng a lnear system by takng the data from the table and puttng t nto the form of a data vector and a matrx generated from a graph or graphcal model. 3. Herarchcal Log-Lnear Models In ths project, we specfcally use graph models that come from herarchcal loglnear models. These models are parametrzed by the dfferent levels that each vertex can take and generate a jont dstrbuton gven the set of edges {X 1, X 2,, X n } that have the followng levels [1, 2,, d ]: p = p 1,..., n = X 1 X 2 X n We can also look at ths model through the followng parametrzaton n =1 p u Date: May 28,

2 2 REN BETTENDORF Where u s the parametrzaton. Example 1. Consder the followng trangle model: We wll call the edge connectng 1 and 2 to be α whch takes values, j, the edge connectng 2 and 3 to be β whch takes values j, k, and the edge connectng 1 and 3 to be γ whch takes values, k. So the followng model s: p = α j β jk γ k Now f we fx a matrx, A whose columns all sum to the same value k. The log-lnear model assocated wth A s the set of probablty tables takng values of 0 or 1. M A = {p = (p ) : log(p) rowspan(a)} where rowspan(a) s the lnear space spanned by the rows of A and also log(p) denotes the vector whose -th coordnate s the logarthm of the postve real number p. It should be noted also that torc model s used to denote the model used. Example 2. A d k probablty table p = (p j ) can be generated from the followng graph model, where the vertces both take values from {1, 2, 3}. Therefore as long as each p j can be factored nto a product of margnal probabltes, p + and p +j then the followng s the log-lnear model log(p j ) = log(p + ) + log(p +j ) Therefore we have the followng matrx generated from the row span of the vectors that generate A A = The frst three numbers to the left of the matrx refer to the frst number on top of the matrx and the second set of three refers to the second number. One mportant part to recognze s that the dmensons of A s a (d+k) dk matrx. It s from ths aspect that t s easy to develop an algorthm that generates ths matrx and s gven

3 ITERATIVE PROPORTIONAL SCALING 3 n the followng pseudocode after ths example. Another mportant observaton s that n ths case we used two random varables and j and the sum of every column s 2 whch s the same as the cardnalty of the set of random varables. The followng s the algorthm that was used to generate all the A matrces for ths project. Algorthm 1. Input: Array B of edge connectons, Array C wth levels of all vertces Output: Matrx A (1) Determne number of columns, N = k C =1 (2) Determne rows for every edge through multplyng cardnaltes together then summng {r 1, r 2,, r l } (3) Set A wth these dmensons wth every entry 0. (4) Set counter varables for number of vertces (5) For = 1 : n For j = 1 : r 1 f counters are same as and j, A(, j) = 1, else A(, j) = 0 Repeat untl through every edge Increase counter varables startng wth the last For ths project, we used graph models usng two, three, or four varables that generated the A matrx. These models could ether be descrbed as decomposable or ndecomposable. In the case of graph models, a model s consdered decomposable f t upholds the followng propertes when gven a graph G = (V, E) (1) G s complete (2) We can express V as V = A B C (3) A,B, and C are dsjont (4) A and C are non-empty (5) B s complete (6) B separates A and C n G (7) A B and B C are decomposable From ths defnton we were able to label the models that we used as beng ether decomposable or ndecomposable. The followng are pctures of the models that we used and a descrpton f they are decomposable or not.

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5 4 REN BETTENDORF 4. Iteratve Proportonal Scalng Algorthm We know from Lectures on Algebrac Statstcs, that f we are gven a matrx A wth row and column dmensons d, k and u N k s a vector of postve counts. The maxmum lkelhood estmate of the frequences of û n the log-lnear model M A s the unque non-negatve soluton to the smultaneous system of equatons Aû = Au We know that ths has a soluton due to Brch s Theorem. In the next secton we wll show how ths system can be expressed usng ths defnton. The Iteratve Proportonal Scalng algorthm s defned through the followng: Let there be n random varables denoted as such S = {X 1, X 2,, X n } where these take a value from the set D = {1, 2,, C}. As well each element n S combnes wth each other to create a set of probabltes P 1,, n = P (X 1 = 1,, X n = n ) where each p has a correspondng frequency u. Therefore, we can denote the lkelhood functon as we dd at the begnnng of the Herarchcal Log-Lnear Models secton. p = P 1,, n = P (X 1 = 1,, X n = n ) = x 1 x 2 x n Ths leads to the followng problem when we parametrze our model as a product of each p usng u as the parametrzaton: n Maxmze: =1 p u Subject to n p = 1 =1 p = Where usng log-lkelhood functon s equvalent to the followng n ln(p) = u log(p ) n =1 =1 Now f we let each p can be expressed as the unque combnaton of values taken from S and take n partal dervatves we see that we have n lnear equatons whch can be expressed as a matrx Aˆp. Gven that A s the matrx generated from the log-lnear graph model and each column sums α. Now ntroducng Lagrangan multplers from the boundary condton, these are set equal to Aˆp and are denoted as Au. Whch gves us the defnton we sought. n naˆp = Au p u Now we wsh to show how the algorthm solves ths system. We start by lookng at the th row and multplyng both sdes by 1ˆp (Au) = n(aˆp) ˆp ˆp Where (Au) refers to the th row n the multplcaton whch gves us the margnal total and the same for (Aˆp). As well we are gong to let the rght ˆp be the

6 ITERATIVE PROPORTIONAL SCALING 5 kth teraton and the left wll be the k + 1st teraton. Therefore, multplyng both sdes by ˆp k ˆpk+1. (Au) ˆp (k) ˆp k+1 = n(aˆp) ˆp (k+1) = ˆp k (Au) n(aˆp) Therefore we have defned the man part of the algorthm, but are not complete because we need to take the α-th root because we are multplyng these numbers α tmes. Now all we have to do s run ths through a loop untl the dfference between ˆp (k+1) ˆp (k) ɛ where ɛ s defned by the user. Rewrtng the above nto pseudocode Algorthm 2. Input: The matrx A N dxk, a table of counts u N k, and a tolerance ɛ > 0. Output: Expected counts ˆp. (1) Intalze p = u 1 1 k (2) Intalze α = sum of one column. (3) Whle Av Au 1 > ɛ do: (4) Output ˆp. ( ) 1 For all [k], set p := p (Au) α n(aˆp) 5. Observatons At the end of ths project we found four observatons that defned how the Iteratve Proportonal Scalng algorthm works. These four observatons are the dfference between absolute and relatve error make a huge dfference, expansons along one varable does have a maxmum lmt such as n the case {3; 3; k} case, the convergence between decomposable and ndecomposable models, and how tolerance affects the algorthms convergence. The followng teraton data was found through generatng a random table of ntegers rangng from 0 to 100. Ths process was done 100 tmes and then the algorthm proceeded to take the average of these teratons. Before the average teratons though, the A matrx was created through consderng the number of levels for each edge n the graph models. The random data table was then used for each model to see the dfference. 1) When I frst started usng the Iteratve Proportonal algorthm, I followed the pseudocode from Lectures on Algebrac Statstcs but I found that at a tolerance of the algorthm was falng to converge whle usng absolute error n combnaton wth a basc two ndependent varable model. By makng the swtch we were able to decrease the algorthm from falng to converge at 50 to convergng n 7. So for the rest of the project, I contnued to use the relatve error whch led to much faster convergences. 2) The next observaton of ths project came from lookng at what happens when all but one level of a vertex of a graph model s kept constant. We then tracked

7 6 REN BETTENDORF what happened as we ncreased the level by 1 n 2 and 3 vertex models. In the followng data table we looked at what happened when we kept the number of edges the same where we saw nterestng behavour n both decomposable and ndecomposable graph models. After lookng at ths behavour, we compared the average teratons for models that had the same number of vertces. Table 1. Indecomposable Lne Graph Model k 1.A 1.B As we see the rght model s decomposable as t follows the 7 propertes defned n the herarchcal log-lnear model secton and we see that as k ncreases we get the followng graph whch shows the dfference between the two models. Decomposable 3 Edge Model Indecomposable 3 Edge Model

8 ITERATIVE PROPORTIONAL SCALING 7 From the graphs we see a bump n the ndecomposable model whch follows from a theorem that there s a pont n whch no new models are generated after a certan pont. Ths pont emprcally s shown through the model at the {2; 2; 5} and {2; 2; 2; 5}. In the case of the decomposable case, we see that t also reaches ths pont, but at a qucker rate. We then recreated ths affect by changng the data to {3; 3; k} and {3; 3; 3; k}. Table 2. Indecomposable Lne Graph Model k 1,2;1,3;2,3 1,2;2,3;3, Decomposable 3 Edge Model Indecomposable 3 Edge Model

9 8 REN BETTENDORF 3) Smlarly to the prevous case, we see that there s a bump n the ndecomposable model, but for ths case we see that when k = 6 we get the bump and then slopes back down. Agan n the decomposable case, we see a gentle slope untl k = 4 then a more aggressve downward slope whch follows from the prevous theorem. Ths shows that there s a fundamental dfference between the decomposable and ndecomposable case, but when we compare number of varables we see more of a dfference. Table 3. Indecomposable Lne Graph Model k 1,2;1,3;2,3 1,2;2,3;3,

10 ITERATIVE PROPORTIONAL SCALING 9 4)We see that the models whch are decomposable follow the same rules as above, as do the ndecomposable models. It s easy to see from these three cases whether comparng number of edges or number of vertces, we see that the dfference s not that dfferent. It has a maxmum dfference of 0.20 teratons. Ths dfference s because usng the relatve error means that we are lookng at the relatve dfference whch means that by comparng smlar number of edges or vertces s rrelevant. Whle t s easy to see that as we ncrease the levels the number of teratons seems to scale equally for the dfferent cases dependng on decomposable or not. If we move onto the tolerance then we see that as the tolerance ncreases we see an asymptotc boundary whch approaches 10 whch s shown n the followng graph and table. Table 4. Indecomposable Lne Graph Model k 1,2;1,3;2,3 1,2;2,3;3,4 5e e e e e e e e e e e e As we have shown n ths graph and data table the dfference between decomposable and ndecomposable models s not that dfferent as the tolerance approaches 1. Yet, as the tolerance decreases we see a lnear ncrease n the number of teratons for each model. Ths ncrease appears to be about 3 teratons for every ncrease n toleraton. I m not sure why ths ncrease happens lnearly, but t does suggest that there mght be a lnear relatonshp between the two.

11 10 REN BETTENDORF 6. References (1) Bartlett, Peter. Undrected Graphcal Models: Chordal Graphs, Decomposable Graphs, Juncton Trees, and Factorzatons. October (2) Drton, Mathas, Bernd Sturmfels, and Seth Sullvant. Lectures on Algebrac Statstcs. Basel: Brkhäuser, 2009