Inferring Dynamic Dependency with Applications to Link Analysis

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1 Inferring Dynamic Dependency wih Applicaions o Link Analysis Michael R. Siracusa Massachuses Insiue of Technology 77 Massachuses Ave. Cambridge, MA 239 John W. Fisher III Massachuses Insiue of Technology 77 Massachuses Ave. Cambridge, MA 239 Absrac Saisical approaches o modeling dynamics and clusering daa are well sudied research areas. This paper considers a special class of such problems in which one is presened wih muliple daa sreams and wishes o infer heir ineracion as i evolves over ime. This problem is viewed as one of inference on a class of models in which ineracion is described by changing dependency srucures, i.e. he presence or absence of edges in a graphical model, bu for which he full se of parameers are no available. The applicaion domain of dynamic link analysis as applied o racked objec behavior is explored. An approximae inference mehod is presened along wih empirical resuls demonsraing is performance. I. INTRODUCTION Consider a scene in which muliple objecs such as people or vehicles are moving around in an environmen. Given noisy measuremens of heir posiion over ime, a quesion one may ask is, which, if any, of hese objecs are ineracing a each poin in ime. The answer o his ype of quesion is useful for social ineracion analysis, anomalous even deecion and auomaic scene summarizaion. For example, Heider and Simmel [] showed ha when describing a caroon of simple shapes moving on a screen, one will end o described heir behavior wih erms like chasing, following or being independen of one anoher. These semanic labels describe he dependency beween objecs and allow humans o provide a compac descripion of wha hey are observing. We view his problem as a specific example of a general class of problems which we refer o as dynamic dependency ess. A dynamic dependency es answers he following quesion: given muliple daa sreams, how does heir ineracion evolve over ime? Here, ineracion is defined in erms of changing graphical srucures, i.e., he presence or absence of edges in a graphical model. For example, in he objec ineracion case, X is following Y implies a casual dependency beween Y and X. We cas a dynamic dependency es as a problem of inference on a special class of probabilisic models in which a laen sae variable indexes a discree se of possible dependency srucures on measuremens. We refer o his class of models as dynamic dependence models and inroduce a specific implemenaion via a hidden facorizaion Markov model (HFacMM). This model allows us o ake advanage of boh srucural and parameric changes associaed wih changes in he sae of ineracion of a se of objecs. We furher show how inference and learning on an HFacMM can be used in an approximae inference procedure on a more expressive swiching linear dynamic sysems (SLDS) model. The approach presened in his paper fis ino he general caegory of daa clusering and dynamic modeling. Two classic examples in his caegory are fiing mixure models and raining hidden Markov models (HMMs) using he EM algorihm [2]. Typically hese models assume fixed dependency srucure for he observed daa. The sudy of models whose graphical srucure is coningen upon he values/conex of he nodes in he graph can be raced back o Heckerman and Geiger s similariy neworks and mulines [3]. This class of models has been furher explored and formalized by Bouilier e al. s Conex-Specific Independence [4] and more recenly Milch e. al s Coningen Bayesian Neworks [5]. An HFacMM fis ino his class of models and is closely relaed o Bilmes s Dynamic Bayesian Mulines [6]. The focus of [6] was o show how learning sae-indexed srucure using labeled raining daa can yield beer models for classificaion asks. In conras, here he dependency srucures are defined by he problem and no labeled daa is required. HFacHMMs are also closely relaed o SLDS models used by he racking communiy [7]. SLDS models are combinaions of discree Markov models and linear sae-space dynamical sysems. The hidden discree sae chooses beween a predefined number of sae-space models o describe he daa a each poin in ime. SLDSs are primarily used o help improve racking and rack inerpreaion by allowing changes o he sae-space model parameers. Exac inference and learning for such models is difficul. However, in his paper we are only ineresed in changes o he dependency srucure of he observaions and consider he laen dynamic sae in he sae-space

2 model a nuisance variable. An HFacMM explicily models varying dependence srucure over ime and allows for efficien learning and inference. This paper explores he radeoff beween using his simple model and incorporaing a laen dynamic sae using an SLDS model when performing a dynamic dependency es. We begin by describing HFacMMs and laer show how inference on an HFacMM can be used as a subrouine in an approximae inference algorihm for SLDS models. II. HIDDEN FACTORIZATION MARKOV MODEL Le O = {o, o 2,..., o N } be an observaion of N random variables a ime wih o i R di. Le O :T represen O from ime o T. Given O :T, he goal is o label he sequence according o he dependency among he N random variables a each ime. To his end, we propose a ph order hidden facorizaion Markov Model, HFacMM(p), in which we assume ha he observaion O depend on he pas p observaions and a hidden sae S. The saes S :T are Markov yielding p(o :T, S :T ; Θ) = p(s :T ; Θ) TY = p(o O ( p):( ), S ; Θ), () where Θ are he parameers. This model is an HMM when p = and a Markov swiching auo-regressive model (MSAR) for p >. However, i has he special propery ha he value k [...K] of he hidden sae variable S indicaes one of K possible facorizaions F k and paramerizaions Θ k such ha p(o (O ( p):( ), S = k; Θ) = C Y k i= p(f k i, F k i,( p):( ); Θ k ) p Θ k,p(f k ), where F k specifies a pariioning of a full se of N random variables ino C k subses such ha C [ k i= (2) F k i = {o,..., o N } and F k i \ F k j = (3) i, j [...C k ] when i j. For example, given hree objecs, p = and he facorizaion for sae k, F k = {{o, o 2 }, {o 3 }}, yields p Θ k,(f k ) = p(o, o 2 o, o 2 ; Θ k )p(o 3 o 3 ; Θ k ) as he sae condiional disribuion of he observaion a ime. Figure shows an example order HFacMM wih wo possible ineracions (K = 2) in which F = {{o, o 2 }, {o 3 }} and F 2 = {{o 2, o 3 }, {o }}. Noe ha he value of he sae S deermines he probabilisic srucure of he observaions a ime. We consider siuaions in which he model parameers are no known a priori. This necessiaes boh a learning and inference sep. The Baum-Welch/EM algorihm can be used wih a sligh modificaion for learning he parameers of an HFacMM, subsequenly Vierbi decoding can be used for exac inference [2]. We consruc and uilize an HFacMM model in he following way: o S S = S = 2 o 2 o 3 o + S + S = + S = o + + o + Fig.. Example HFacMM(). Condiionally labeled edges are only presen when he condiion is rue, following he noaion in This graph noaion follows ha of CBNs [5]. ) Define he K possible dependency srucures and parameerizaion of he HFacMM for your ask. 2) Learning: Esimae ˆΘ = arg max Θ p (O :T ; Θ) (via EM) 3) Inference: Find ŝ :T = arg max s:t p s :T O :T ; ˆΘ This approach does no use any raining daa and performs boh learning and inference on he daa being analyzed. We make he usual assumpion ha all K saes are visied a leas once (and ypically muliple imes) during he observed sequence. A. Learning Le Θ = {π, Z, Θ,..., Θ K } be he parameer se for he model where π k = p (S = k) are he prior sae probabiliies, Z is a K K marix wih Z ij = p (S + = i S = j), and Θ k are he parameers for facorizaion F k (i.e., parameers for p Θ,p(F k )). As wih k ypical HMMs and MSAR models, he EM algorihm can be applied o models wih his srucure in order o find he parameers, ˆΘ, ha maximize he likelihood funcion [2]. While he E-sep is unchanged, he HFacMM requires a minor change o he M-sep of EM. Since he sae condiional model p Θ,p(F k ) breaks up ino he C k k facors of F k, he srucure of he M-sep updaes simplify accordingly yielding a more srucured learning procedure wih savings in sorage and compuaion. B. Inference Having learned he parameers, ˆΘ, he daa sequence is labeled by finding he mos likely sae sequence {ŝ :T } = arg max s :T p(s :T O :T ; ˆΘ). (4) This can be done efficienly wih he Vierbi algorihm [2]. Vierbi decoding implicily performs an M-ary hypohesis es comparing all M = K T possible sae sequences. This alernaive view exposes how sae sequences disinguish hemselves via boh srucural and general saisical model differences beween he learned sae condiional disribuions. Consider a binary hypohesis es beween wo differen sae sequences S H :T parameers ˆΘ. The es has he form p O :T S H :T ˆL ; ˆΘ H,H 2 log p O :T S H 2 :T ; ˆΘ and SH2 :T H H 2 log given he p p S H 2 :T ; ˆΘ S H :T ; ˆΘ A. (5)

3 Taking he expeced value of he log likelihood raio when H is rue yields i X E H hˆlh,h 2 = D p H F SH ˆΘS,p s.. S H S H 2 + X s.. D S H S H 2 p H F ˆΘS,p p H F p H 2 ˆΘS,p ˆΘS,p F SH 2, where F is he common dependency srucure (se of edges) shared by all F k (e.g., if {o, o 2 } is common o all K srucures hen i will also appear in F ). Noe ha any parameer se Θ k can be applied o he F facorizaion by marginalizing over p Θ k(f k ) appropriaely. An equivalen break down occurs under H 2 (wih a negaion). Noice ha he expeced log likelihood raio decomposes ino wo erms. The firs erm concerns purely srucural differences. I compares he rue srucure wih he common srucure, under he rue parameers. The second erm conains boh srucural and parameer differences. I compares he common srucure wih he rue parameers o he incorrec hypohesis. Typically HMMs and MSAR models assume a fixed dependency srucure and hus only he second erm is non-zero and depends only on parameric differences. The oher exreme occurs when performing windowed dependency ess which only ake advanage of he firs erm [8]. III. INCORPORATING A STATE-SPACE MODEL A HFacMM wih order p > explicily models he dynamics of he observaion sequence. If we assume a ph order Gaussian auo-regressive model he sae condiional disribuion will have he form p(o O ( p):( ), S ) = N(O A S O ( p):( ) ;, Σ). (7) i.e., he observaion a ime will be a linear combinaion of he previous p wih he addiion of Gaussian noise. A S deermines he linear combinaion when he sae is S. This is a simple model in which he dynamic sae of sysem is being direcly observed. However, i is common o assume here is a laen dynamic sae ha is only parially observed hrough a noisy measuremen process. Such sysems are ofen assumed o be linear wih Gaussian noise following a sae-space model of he form X = AX + V O = Y = CX + W, where X is a hidden dynamic sae variable a ime and V and W are independen Gaussian noise sources. An SLDS assumes here is an addiional hidden discree sae S ha deermines A, C, and he covariance of he noise sources. The high level srucure of such models is shown in Figure 2(a). For he problem of objec ineracion analysis S indexes how he disribuion of he dynamic sae X facorizes via he srucure of A and he noise (6) (8) v - x - o - S - S S + X - O - S - S - = 2 v - 2 x - 2 o - v x o X X + O O + (a) S S = (b) Fig. 2. High level SLDS srucure (a) and deailed srucure of model used in experimens (b) source V. Noe ha he dynamics on he observaions are induced via he dynamics on he laen dynamic sae X. Learning and exac inference on an SLDS model is difficul. For example, if S can ake K discree values which indexes K differen Gaussian models for X, he disribuion p(x O : ) is a mixure of K Gaussians, each one associaed wih a possible sae sequence S :. This exponenial growh can be deal wih in many ways from collapsing he K mixure ino a smaller mixure a each [9], [], o using a variaional approach which fis a simpler model o maximizing a lower bound on he likelihood of he daa [7]. As an alernaive we ake an ieraive coordinae ascen approach. If one were given he sae sequence S :T EM can be used o learn he parameers of he model and inference on he dynamic sae X :T can be performed using Rauch-Tung-Srieber smoohing wih he appropriae model parameers (which are specified a each ime by S ). Similarly if one were given he dynamic sae X :T learning and inference for he S :T is simple. Tha is, condiioned on X :T, O can be ignored and one is lef wih an HFacMM of order which reas X :T as is observaion. Thus we begin by firs iniializing o a random sae sequence Ŝ:T, hen perform learning and inference on X :T o produce an esimae ˆX :T. We nex condiion on his esimae o updae Ŝ:T and repea. I is imporan o remember ha for he applicaion of ineres S indexes a paricular srucure via facorizaion F S. For his saespace model his facorizaion describes how he dynamic sae X :T evolves. We will give a specific example in he nex secion. 2 v 2 x 2 o v + x + o + S + S + = 2 v + 2 x + 2 o +

4 IV. EXPERIMENTS Here we consider he N = 2 case in which we have noisy observaions of posiion for wo objecs. We assume here are wo saes of ineracion specified by he hidden discree sae S. If S = we assume hey are moving independenly and if S = hey are ineracing. We begin wih a consan velociy model where a laen dynamic sae evolves via x A =, (9) x 2 A 2 x x 2 + v v 2 where he dynamic sae is posiion and velociy in 2D, x i = [ T, x i y i ẋ i ẏ] i and A = A 2 I I =, () I where I and are 2x2 ideniy and zero marices. Noisy observaions of posiion are obained via o C x o 2 = w C 2 x 2 + w 2, () where C = C 2 = [ I 2 ] and he observaion noise is independen of he sae S, i.e., p(w, w 2 S ) = p(w, w 2 ) = N w ;, σ wi N w 2 ;, σ wi (2) Furhermore, in his simple model we assume ha all informaion abou he ineracive sae is capured via he driving noise. Tha is, when S = he driving noise is independen and facorizes as p(v, v 2 S = ) = N v ;, Σ v N v 2 ;, Σ v (3) and when S = he noise is fully dependen p(v, v 2 S = ) = N v v 2 ; Σv ρσ ;, D v ρσ v Σ v (4) wih correlaion ρ and a facor D more variance. Here ρ describes he amoun of dependency when hey are ineracing. I is he only parameer ha affecs he firs erm in Equaion 6. The parameer D changes only he second erm in Equaion 6. The srucure of his model is shown in Figure 2(b). This give us a simple model wih parameers ha are easy o inerpre. I is imporan o noe ha he HFacMM and SLDS models used in his paper are no limied o his form and can capure oher ypes of ineracion (e.g., via changing a A marix). A. Illusraive Example For he purpose of esing he approach under known condiions, we draw muliple sample pahs from he model described above wih various seings of ρ, D and observaion noise variance σ w. We do his wih a fixed dynamic on he sae S wih π = π =.5, Z = Z =.95 and Z 2 = Z 2 =.5. For each seing of ρ, D, and σ w we draw T = 4 samples (of boh X and O ) from he model. This is performed 5 imes for each seing and he average performance of a dynamic dependency es using 3 differen approaches is recorded. As a baseline, he firs approach is given he rue dynamic sae X :T as is observaions and uses an HFacMM() model o do learning and inference. The full se of parameers used for his baseline model are Θ = {π, Z, Θ, Θ } where he independen (S = ) parameers are Θ = {A, A 2, Σ v, Σ 2 v}, and he ineracing/dependen (S = ) parameers are Θ = {A 2, Σ 2 v }. The independen model parameers mach Equaion 9, while a more generic model is used for he dependen case wih X = A 2 X + V and V drawn from a zero mean Gaussian wih full covariance Σ 2 v. Given X :T he baseline model learns he mos likely se of parameers ˆΘ and hen oupus he mos likely sae sequence. The second approach is given only he observaion sequence O :T and performs coordinae ascen on he full SLDS model. I has he same parameer se as he baseline approach wih he addiion of he observaion marix C and observaion noise variance σ w. For simpliciy we assume hese observaion process parameers are known. We se a maximum number of ieraions of coordinae ascen o bu found ha i usually converges in less han 5 ieraions for our experimens. The hird approach is given only he observaion sequence and fis a higher order HFacMM(3). While his model yields efficien inference and learning i does no model he laen dynamic sae X. I models he effecs of any laen dynamics hrough a higher order auoregressive model on he observaions. Again, noe ha for he problems of ineres in his paper, X is a nuisance variable. Our primary focus is on accuraely esimaing he ineracive sae S and no on producing accurae esimaes of he dynamic sae X. A quesion we wish o explore is wheher no his simple model is useful even when here is a rue underlying laen dynamic sae. Figure 3(a) shows for fixed D and σ w he average probabiliy of error for hese hree echniques as a funcion ρ. As expeced all hree approaches improve dramaically as ρ increases. A similar rend occurs for a fixed ρ and σ w and an increasing D in Figure 3(b). Figure 3(c) shows how observaion noise degrades he performance of he approaches ha do no have he benefi of direcly observing he dynamic sae. In all of he figures i is clear even hough he coordinae ascen echnique comes wih more compuaional complexiy and approximae inference i ouperforms he simple HFacMM(3) by incorporaing a laen dynamic sae. However his performance gap disappears when he sae condiional models are easily disinguishable (i.e., high ρ or D). B. Ineracing People Nex we analyzed he ineracion of wo objecs moving in a real environmen. These wo objecs are individuals wearing has designed o be easy o visually rack. The

5 5 4 D=2, Noise Var=. Coord Descen HFacMM(3) 5 4 rho=.6, Noise Var=. Coord Descen HFacMM(3) 5 4 rho=.95, D=2 Coord Descen HFacMM(3) Fig ρ D (a) (b) (c) (d) Avg. probabiliy of error for synheic daa as a funcion of (a) ρ (b) D (c) σ w. (d) Sample frame from ineracing people daa. Obs Noise Var TABLE I RESULTS FOR INTERACTING PEOPLE DATA. CV=CONSTANT VELOCITY. B=BROWNIAN MOTION Probabiliy of Error Sequence Coord. Asc. CV Coord. Asc. B HFacMM() wo individuals move around an enclosed environmen and are randomly insruced o inerac for a limi period of ime. When ineracing hey are eiher following or rying o mirror each oher s acions. Each sequence was over 2.5 minues long and conained beween 8 o 2 ransiions beween ineracing and moving independenly. In he firs sequence he individuals move faser when ineracing. This is no he case for he second sequence in which individuals mainain he same speed when ineracing and moving independenly. Boh sequences were recorded using a camcorder a 2 fps. Simple background subracion and color filering wih smoohing provided a simple blob racker ha was furher labeled/racked by hand o provide (x, y) posiions for each individual for each frame. Ground ruh of ineracion sae sequence was also esablished by hand labeling. Figure 3(d) shows a sample frame from one of he sequences. We ran dynamic dependency ess on boh of hese sequences using he coordinae ascen approach assuming a consan velociy model (4 dimensional dynamic sae iniialized wih an A marix wih consan velociy srucure). The resuls are shown in he second column of Table I. Afer analyzing he learned parameers for each sequence we found ha he dynamic sae ransiion marix A was close o diagonal indicaing ha a Brownian moion model may be more appropriae. Rerunning he dependency es assuming a simple Brownian moion model (he sae is posiion only) we obained he improved resuls in column hree of Table I. This indicaes ha a consan velociy model may no be correc for his daa in which individuals make many urns in a confined space. Lasly if we apply a simple firs order HFacMM o he daa we do jus as well. This indicaes ha his daa has srong dependency informaion when he individuals are ineracing (i.e., like he high ρ case above). V. CONCLUSION In his paper we looked a he problem of inferring laen dynamic dependency srucure o deermine he ineracion among muliple moving objecs. We have shown ha by modeling dependency as a dynamic process one can exploi boh srucural and parameer differences o disinguish beween hypohesized saes of dependency/ineracion. We discussed he use of an HFacMM for such asks and showed how i can be used o perform approximae inference on an SLDS model which incorporaes a sae-space descripion of objec dynamics. Experimens were performed on boh conrolled synheic and recorded daa of people/objecs ineracing. Empirical performance demonsraed he uiliy of approximae inference based on coordinae ascen for deermining ineracion saes. REFERENCES [] F. Heider and M. Simmel, An experimenal sudy of apparen behavior, in American Journal of Psychology, 944, vol. 57, pp [2] L.R. Rabiner, A uorial on hidden markov models and seleced applicaions in speech recogniion, in Proc. IEEE, 989, vol. 77 No. 2, pp [3] D. Geiger and D. Heckerman, Knowledge represenaion and inference in similariy neworks and bayesian mulines, in Arificial Ineglligence, 996, vol. 82, pp [4] Craig Bouilier, Nir Friedman, Moises Goldszmid, and Daphne Koller, Conex-specific independence in Bayesian neworks, in UAI, 996, pp [5] B. Milch, B. Marhi, D. Sonag, S. Russell, D.L. Ong, and A. Kolobov, Approximae inference for infinie coningen bayesian neworks, in h Inl. Workshop on Arificial Inelligence and Sas., 25. [6] J. A. Bilmes, Dynamic bayesian mulines, in Proc. of 6h conf. on Uncerainy in Arificial Inelligence, 2, pp [7] Z. Ghahramani and G. E. Hinon, Variaional learning for swiching sae-space models, in Neural Compuaion, 998, vol. 2, pp [8] A.T. Ihler, J.W. Fisher, and A.S. Willsky, Nonparameeric hypohesis ess for saisical dependency, in Trans. on signal processing, special issue on machine learning, 24. [9] Y. Bar-Shalom and X. Li, Esimaion and Tracking: Principles, Techniques and Sofware, Arech House, 993. [] C-J. Kim, Dynamic linear models wih markov-swiching, in Journal of Economerics, 994, vol. 6, pp. 22. This maerial is based upon work suppored by he AFOSR under Award No. FA Any opinions, findings, and conclusions or recommendaions expressed in his publicaion are hose of he auhor(s) and do no necessarily reflec he views of he Air Force.