Lifted Relational Kalman Filtering

Transcription

1 Lfed Relaonal Kalman Flerng Jaesk Cho and Abner Guzman-Rvera and Eyal Amr Compuer Scence Deparmen Unversy of Illnos a Urbana-Champagn Urbana, IL, 61801, USA {jaesk,aguzman5,eyal}@llnos.edu Absrac Kalman Flerng s a compuaonal ool wh wdespread applcaons n robocs, fnancal and weaher forecasng, envronmenal engneerng and defense. Gven observaon and sae ranson models, he Kalman Fler KF recursvely esmaes he sae varables of a dynamc sysem. However, he KF requres a cubc me marx nverson operaon a every mesep whch prevens s applcaon n domans wh large numbers of sae varables. We propose Relaonal Gaussan Models o represen and model dynamc sysems wh large numbers of varables effcenly. Furhermore, we devse an exac lfed Kalman Flerng algorhm whch akes only lnear me n he number of random varables a every mesep. We prove ha our algorhm akes lnear me n he number of sae varables even when ndvdual observaons apply o each varable. o our knowledge, hs s he frs lfed lnear me algorhm for flerng wh connuous dynamc relaonal models. 1 Inroducon Many real-world sysems can be modeled by connuous varables and relaonshps or dependences among hem. he Kalman Fler KF [Kalman, 1960] accuraely esmaes he sae of a dynamc sysem gven a sequence of conrol-npus and observaons. I has been appled n a broad range of domans whch nclude weaher forecasng [Burgers e al., 1998], localzaon and rackng n robocs [Lmkeka e al., 2005], economc forecasng n fnance [Bahman-Oskooee and Brown, 2004] and many ohers. Gven a sequence of observaons and Gaussan dependences beween varables, he flerng problem s o calculae he condonal probably densy of he sae varables a each mesep. Unforunaely, he KF compuaons are cubc n he number of random varables whch lms curren exac mehods o domans wh lmed number of random varables. hs has led o he combnaon of approxmaon and samplng e.g. he Ensemble Kalman Fler [Evensen, 1994]. hs paper leverages he ably of relaonal languages [Fredman e al., 1999; Poole, 2003; Rchardson and Domngos, 2006] o specfy models wh sze of represenaon ndependen of he sze of populaons nvolved. Varous lfed nference algorhms for relaonal models have been proposed [Poole, 2003; de Salvo Braz e al., 2005; Mlch and Russell, 2006; Rchardson and Domngos, 2006; Wang and Domngos, 2008; Cho e al., 2010]. hese seek o carry compuaons n me ndependen of he sze of he populaons nvolved. However, he key challenge n relaonal flerng of dynamc sysems s ensurng ha he represenaon does no degenerae o he ground case when mulple observaon are made. As more observaons are receved, an ncreasng number of objecs become dsngushed. hs precludes he applcaon of prevously known algorhms unless approxmaely equvalen objecs are grouped wh ensve cluserng algorhms. We propose Relaonal Gaussan Models RGMs o model dynamc sysems of large number of varables n a relaonal fashon. RGMs have as her man buldng block he parwse lnear Gaussan poenal as dealed n Secon 2. Furher, we propose a new lfed flerng algorhm ha s able o margnalze ou random varables of he prevous mesep effcenly n me lnear n he number of random varables whle mananng he relaonal RGM represenaon. hs prevens he model from beng ncreasngly grounded even when ndvdual observaons are made for all random varables. Moreover, updang he relaonal represenaon akes only quadrac me n he number of relaonal aoms ses of random varables. One key nsgh s ha, gven dencal observaon models, even when he means of he random varables are dspersed her varances reman dencal. hs s suffcen o manan a relaonal represenaon. hs paper s organzed as follows. Secon 2 nroduces defnons and he relaonal flerng problem. Secon 3 presens our man echncal resuls,.e., he recursve esmaon of he saes of random varables n a lfed fashon. Secon 4 presens our algorhm n deal ogeher wh complexy resuls. Secon 5 shows ermenal resuls wh a housng marke model. Secon 6 presens a real-world applcaon o Socal Neworks. Secon 7 dscusses prevous work. We conclude n Secon 8. 2 Model and Problem Defnons In hs secon, we defne Relaonal Gaussan Models RGMs and nroduce he flerng problem for dynamc relaonal models.

2 2.1 Relaonal Connuous Models Dependences beween varables are represened usng Parfacor models 1,.e. parameerzed facor models. A facor f s a par A f, φ f where A f s a uple of random varables and φ f s a poenal funcon from he range of A f o he nonnegave real numbers. Gven a valuaon v of random varables rvs, he poenal of f on v s w f v φ f A f. he jon probably defned by a se F of facors on a valuaon v of random varables s he normalzaon of f F w f v. We can have parameerzed ndexed random varables by usng predcaes, whch are funcons mappng parameer values ndces o random varables. A relaonal aom s a parameerzed random varable, possbly wh free varables or consans. For example, a predcae HP e.g. House Prce s used n a relaonal aom HP, where are free logcal varables or ndex. 2 Recesson S. and 7 Recesson S. are wo examples of possble parameer values for. HP2 Recesson S. s ground aom and drecly corresponds o a random varable. A parfacor s a uple L, C, A, φ composed of a se of parameers also called logcal varables or ndces L, a equaly consran C on L, a uple of aoms A, and a poenal funcon φ. Le a subsuon θ be an assgnmen o L and Aθ he relaonal aom possbly ground resulng from replacng logcal varables by her values n θ. A parfacor g sands for he se of facors grg wh elemens Aθ, φ for every assgnmen θ o he parameers L ha sasfes he consran C. A Frs-order Probablsc Model FOPM s a compac, or nensonal, represenaon of a graphcal model. I s composed by a doman, whch s he se of possble parameer values referred o as doman objecs and a se of parfacors. he correspondng graphcal model s he one defned by all nsanaed facors. he jon probably of a valuaon v accordng o a se of parfacors G s Pv 1/Z w f v, 1 g G f grg where Z s a normalzaon consan. 2.2 Relaonal Gaussan Models RGMs Relaonal Gaussan Models RGMs are a subse of Relaonal Connuous Models RCMs where poenals are resrced o be Gaussan dsrbuons. RGMs are composed of hree ypes of parfacor models: 1 Relaonal ranson Models s; 2 Relaonal Parwse Models RPMs; and 3 Relaonal Observaon Models ROMs. Suppose ha we have n relaonal aoms: 1L,..., n L where L s a ls of logcal varables. In a relaonal lnear dynamc model, relaonal aoms are lnearly nfluenced by conrolnpus U 1L,..., Un L. Smlarly, a lnear observaon model specfes he relaonshp beween observaon varables O 1 L,..., On L and oher relaonal aoms. Conrol npus and observaons are assocaed wh relaonal aoms n wo ways: 1 drec assocaon; and 2 ndrec assocaon. We provde furher deals n Secon Our represenaon s based on prevous work [Poole, 2003; de Salvo Braz e al., 2005; Mlch and Russell, 2006; Cho e al., 2010]. : SOLD : UNKNOWN $500K $450K Recesson S $370K HPO O HP a O P HP b P P HM HMO HPO HP a HP b HM HMO Sales prce of house h 1 : HPO h 1 $500K. Observed housng marke ndex: HMO +8%. Fgure 1: Example of a housng marke model. We are neresed n esmang he hdden value of houses gven observaons of house sales prces e.g. HPO 1 $500K. Boh, he hdden value of a house and he observed sales prces are affeced by several facors, e.g., house values ncrease by a ceran rae every year and are also nfluenced by a housng marke ndex HM. Relaonal ranson Models s model he dependence of relaonal aoms a he nex mesep, j a, on relaonal aoms a he curren mesep, a, and when avalable conrol-npu nformaon. hey ake he followng form, j a B,j a + B,j U U a + G,j, 2 where G,j N0, σ,j and Nm, σ2 s he normal dsrbuon wh mean m and varance σ 2. B,j and B,j are he U ranson models, marces or a consans, correspondng o wo relaonal aoms. For unvarae sae varables, we can represen he ranson model wh a lnear Gaussan, φ j a a, U a j a B j a Bj U U a2 2 σ j 2. 3 he mos common ranson s he ranson from he curren sae a o he nex a. I s represened as follows, a B a + B U U a + G. 4 Relaonal Observaon Models ROMs represen he relaonshps beween he hdden sae varables, a, and he observaons made a he correspondng mesep, O a, O a H a + G ROM, 5 where G ROM N0, σ ROM. H s he observaon model, a marx or a consan, beween he hdden varables and he observaons. In he lnear Gaussan represenaon, hey ake he followng form, φ ROM O a a O a H a2 2 σ 2. 6 ROM Relaonal Parwse Models RPMs represen Gaussan dependences beween pars of relaonal aoms whn he same mesep as follows, a R,j j a + G,j RPM, 7

3 Fgure 2: hs model has hree relaonal aoms,, whch may represen any number of random varables. he relaonal represenaon dramacally elmnaes he need for redundan poenals. Hence, represenaon and flerng become much more effcen han n he proposonal case. Noe ha he convenonal KF represenaon s no sued for effcen.e. lfed nference. where G,j N0, σ,j. R,j RPM RPM s he parwse coeffcen, a marx or a consan, beween he wo relaonal aoms. Noe ha s and ROMs are dreced models whle RPMs are undreced. he dreced models represen he naure of dynamc sysems e.g. he sae a he nex mesep depends on he curren mesep. he produc of RPMs s an effcen way o represen a mulvarae Gaussan densy over all he sae varables A Relaonal Flerng Problem Gven a pror or curren belef over he sae varables, he flerng problem s o compue he poseror afer a sequence of meseps. he npu o he problem s: 1 Relaonal Gaussan Model s, RPMs and ROMs; 2 curren belef over he relaonal aoms represened by a produc of 0 relaonal Gaussan poenals; 3 sequence of conrol-npus U 1,..., U ; and 4 sequence of observaons O 1,..., O. he oupu s he relaonal Gaussan poseror dsrbuon over he relaonal aoms a mesep. 2.4 Inpu and Observaon Assocaon A every mesep he conrol-npus and observaons mus be assocaed wh he random varables hey affec. he deas n hs secon apply o conrol-npus and observaons bu we llusrae hem for observaons. We dsngush wo ypes of observaons: drec and ndrec. Drec observaons are hose made for a specfc random varable. For nsance, f we make an observaon for each random varable n a subse A of he ground subsuons of relaonal aom, we are lookng a he followng model, φ ROM o a j a j. 8 a j A In he example of Fgure 1, observng he sellng prce of a house would dramacally reduce he varance of he hdden varable ha represens he rue value of ha house. Smlarly, mulple drec observaons, O o,1, o,2,..., o, O, could be made for each varable n 2 Noe ha a mulvarae Gaussan densy of sae varables s a quadrac onenal form. he quadrac onenal form can always be decomposed no erms nvolvng only sngle varables and pars of varables. We provde addonal deals n Secon 7. some se of random varables, φ ROMk o,k a j a j. 9 a j A o,k O Gven some noon of neghborhood e.g. a resdenal neghborhood or a block of houses, ndrec observaon allows he possbly ha observaons made for a random varable, o a, would nfluence nearby random varables, a j, a a j, φ ROM o a a j. 10 a j A For example, hs allows he possbly ha he observaon of he sellng prce of a house would reduce he varance of he rue values of neghborng houses. Curren exac lfed nference algorhms e.g. [Kersng e al., 2006; Cho e al., 2010] handle observaons by paronng he relaonal aoms no groupngs of groundngs for whch dencal observaons and observaon models apply. In conras, our approach parons a relaonal aom no ses accordng o he number of dfferen ypes of observaons assocaed wh each random varable. For nsance, f an ndvdual observaon of he same ROM ype s made for each random varable hen no paronng a all s necessary. he nuon for hs s ha he flerng process wll assgn he same varance o any wo hdden varables for whch he same number of observaons s made a he curren mesep. Here, he paron wll deermne new RPMs, he parwse parfacors whch manan he varances and covarances. In parcular, he number of new RPMs s quadrac n he sze of he paron. Snce ndvdual observaons cause he means of he random varables o dffer we sore he mean nformaon n he pror and poseror P and P new n Secon 3. Hence, he number of prors and poserors s lnear n he number of random varables. However, hs wll no affec he compuaonal complexy of nference as long as he RPMs do no degenerae. Furher deals are gven n Secons 3.3 and 4. Formally, gven a paron Π M 1, M 2,..., M Π of a relaonal aom,, he observaon model akes he form, φ ROMk o,k a j a j, 11 M l Π a j M o l,k O l

4 where we om he me subscrp and where O s he se of l observaons relevan o par l. 3 Lfed Relaonal Kalman Fler he Lfed Relaonal Kalman Fler LRKF, jus lke he convenonal Kalman Fler, carres wo recursve compuaons: predcon sep and updae/correcon sep. 3.1 Lfed Predcon In he predcon sep, our curren belef over he saes of he relaonal aoms ogeher wh he s, RPMs and conrolnpus are used o make a bes esmae of sae whou observaon nformaon. Frs, he produc of poenals n he s and RPMs s bul. Second, he varables from he prevous mesep are margnalzed resulng n new RPMs and esmaes of he relaonal aoms n he curren mesep. We call hs esmaes he nermedae poseror, he npu o he updae sep.,j j φ a a, U a P a φ,j RPM a, j a 1,...,n 1 <j n a A,a A j j a B,j a B,j U U a 2 1,...,n 1 <j n a A,a A j σ 2 a µ P a 2 σ 2 a R,j j a 2 12 P σ,j 2 RPM a R,j j a 2 1 <j n a A,a A j σ,j 2 a µ P a 2 σ 2 RPM P φ,j RPM a, j a P a <j n a A,a A j 1 n a A Here, φ,j RPM, P and P are respecvely he updaed RPMs, he prors and he nermedae poserors. More deals of he negraon are gven n Appendx A. 3.2 Lfed Updae In he updae sep, he nermedae poseror P and ROMs are used o correc our esmae of he relaonal aoms. When a sngle observaon, o, s assocaed wh all varables n a relaonal aom, we calculae he poseror for one random varable a and use he resul for he res of he groundngs of he same relaonal aom, P a φ ROM o a a µ P a 2 σ 2 a 2 o P σ 2 ROM a2 + 2µ P a a µ P a 2 + a2 + 2o a o 2 σ 2 P σ 2 ROM c a µ Pnew 2 σ 2 P new a. 14 Pnew In he case of mulple observaons O o, O o,1, o,2,..., we may also do he compuaon of he poseror for a sngle random varable a and use he resulng poseror for all oher groundngs of he relaonal aom o whch he same se of observaons apples. he calculaon s smlar o he above, excep ha mulple observaons need o be consdered, P a φ ROM o a o O a µ P a 2 σ 2 a o2 P o O σ 2 ROM c a µ Pnew a2 σ 2 P new a. 15 new 3.3 Lfed Inference wh Indvdual Observaons One of he key challenges n lfed nference s handlng ndvdual observaons. Curren mehods ground a relaonal aom when dfferen observaons are made for s random varables. I s usually he case ha models shaer combnaorally fas and hus forfe he benefs of a relaonal represenaon and he applcably of lfed nference. We solve hs problem n he LRKF by nong ha he varances and covarances n he model are no affeced by ndvdual observaons. We are hus able o represen he varances and covarances n a relaonal way whle allowng varables o carry ndvdual means. Furher, he lfed predcon operaon apples unmodfed o hs represenaon. Lemma 1 he varances of wo random varables a, b n an RGM are equal afer a flerng sep Lfed Predcon and Lfed Updae f he followng condons hold before he flerng sep: 1 boh random varables are n he same relaonal aom; 2 he varance of boh varables s he same; 3 observaons are made for boh varables or none of hem. Proof Gven condons 1 and 2, we frs prove ha he varance of boh random varables s he same afer he Lfed Predcon sep. Noe ha condon 3 s no relevan o hs sep. WLOG we assume a and b have dfferen means, µ a and µ b. Moreover, s easy o see ha he varance of a and b s he same afer margnalzng all random varables of mesep due o he followng wo reasons: a and b are n he same relaonal aom and hus share he same relaonshps wh oher random varables; he means are no nvolved n he margnalzaons see Secon 3.1. I follows ha we can represen he poenals relevan o he margnalzaon of a and b as follows: a µ a 2 σ 2 φ a a, U a φ RPM a, b a b µ b 2 σ 2 φ b b, U b φ RPM a, b b c a 2 a 2 + c a a 2B σ 2 a a a b σ 2 RPM c b 2 b 2 + c b b 2B σ 2 b b φ oher a, b, where c refers o he coeffcen of he erm. 3 3 For he sake of oson he s here represen dependences

5 Afer a and b are margnalzed we ge a poenal on a and b. he varances of he random varables are he nverses of he coeffcens of her squares n he resulng poenal. hus, all we need o show s ha he coeffcens of he square of he random varables, a 2 and b 2, are he same afer margnalzaon. he wo coeffcens can be represened as follows, B 2 c B 2 b 2 c c a 2 σ 2 a 2 2, c b 2 σ c σ 2 a c b c c σ 2 a b where, c σ 2 σ 2 σ 2 RPM Condon 2 σ 2 a σ2 b mples c a 2 c b 2 whch n urn mples c a 2 c b 2. hs s enough o prove ha he varance of wo random varables a and b wh dfferen means s he same afer he Lfed Predcon sep. We now prove he resul for he Lfed Updae sep. Regardng condon 3 here are wo cases: a observaons were made for boh varables; or b no observaons were made for eher varable. In he case of b he proof s complee. In he case of a, he updae sep for a can be represened by, a µ a 2 σ 2 a o a 2 a µ + σ 2 a ROM a 2 σ + a 2 where, σ +2 a σ2 a σ2 ROM, µ + σ 2 + σ 2 a σ2 ROM µ a + σ 2 o a a a ROM σ 2 ROM + σ 2 a Lkewse, afer he updae sep he varance of b s, σ + 2 b σ 2 b σ 2 ROM σ 2 b + σ 2 ROM By condon 2 and he proof for he predcon sep, σ a σ b. hus, σ+ a σ + b. Lemma 2 he covarances of wo pars of varables a, b and a, c n an RGM are equal afer a flerng sep Lfed Predcon and Lfed Updae f he followng condons hold before he flerng sep: 1 he hree random varables are n he same relaonal aom; 2 he covarance of boh pars of varables s he same; 3 observaons are made for he hree varables or none of hem. Proof he mehod used n he proof of Lemma 1 can be employed n hs proof: he erms nvolvng he ndvdual observaons do no affec erms whch deermne he covarance of wo random varables. 4 Algorhms and Compuaonal Complexy Le be he se number of all random varables n he model and 1,..., be he se of relaonal aoms from sae varables a me o he same sae varable a me + 1 e.g. from a o a. However, he general s e.g dependences from a o b produce smlar forms. also, a paron of. In hs secon we speak of he relaonal aoms as ses of random varables. Fgure 3 presens our Lfed Kalman Flerng algorhm. he npus o he algorhm are: relaonal aoms, ; he RGM, s M, RPMs M P and ROMs M O ; he pror over he relaonal aoms, P 0 ; and he conrol-npus, U [1,...,], and observaons, O [1,...,], for each mesep. he algorhm compues he poseror recursvely. Spl parons he domans of each relaonal aom as nduced by he conrol-npus U. Lfed Predc calculaes new RPMs, M 4 P, and nermedae poseror, P n, based on he ranson models, M, and he conrol-npus, U. hen, Spl Obs parons he domans of each relaonal aom as nduced by he observaons, O. Lfed Updae calculaes he new poseror, P cur, based on he nermedae poseror, P n, he observaon models, M O, and he observaons, O. Gven he conrol-npus, Spl parons relaonal aoms as done n prevous work: e.g. Spl [Poole, 2003] and SHA- ER [de Salvo Braz e al., 2005]. If he conrol-npus are allowed o dffer for he varables n a relaonal aom, he model wll be proposonalzed. Hence, here s lle advance n how we handle ndvdual conrol-npus wh respec o prevous algorhms [Cho e al., 2010]. 5 Algorhm Spl Obs parons a relaonal aom based on he observaons. However, Spl Obs wll only paron a relaonal aom n case he condons of Lemmas 1 and 2 do no hold,.e., when dfferen number of observaons are made for he relaonal varables. If he condons of Lemmas 1 and 2 hold, he effcency of he relaonal represenaon wll be preserved even f mulple observaons are made for all varables n some or all of he relaonal aoms. PROCEDURE LRKF,M,M P,M O, P 0, U [1,...,], O [1,...,] Aoms, 1,..., ;, M, RPM, M P, and ROM M O ; pror, P 0 ; conrol-npus, U [1,...,] ; observaons, O [1,...,]. 1. P cur P 0, cur 2. For 1 o a [ cur, M, M P, M O ] Spl cur, U, M, M P, M O b [P n, M p ] Lfed Predc cur, P cur, M, M P, U 3.1 c [ cur, M O ] Spl Obs cur, O, M O 3.3 d [P cur ] Lfed Updae cur, M O, P n, O Reurn cur, P cur Fgure 3: Algorhm Lfed Relaonal Kalman Fler for Relaonal Gaussan Models. Lemma 3 he complexy of Lfed Predc s O + 2. Where + s he se of relaonal aoms oupu by Spl. Proof hs sep corresponds o he margnalzaon Equaon 13 and Appendx A of he varables n. For every 4 In our represenaon he number of relaonal aoms deermnes he number of RPMs whch s equal o E, 2 he number of 2- combnaons of wh repeon. 5 However, we conjecure ha echnques smlar o he ones we used for ROMs can be appled o s. Any wo random varables n he same aom wll have he same varance afer he Lfed Predc sep f hey receve he same ypes of conrol npus. ha s, s of he same ype wll ncrease he varances of he random varables by he same amoun.

6 OSNs has demonsraed ha relaonal paerns can be loed o mprove predcve models of lnk srucure and behavor. Furher, he accurae esmaon of relaonshp srenghs has applcaons o undersandng human behavor, predcng human behavor e.g. fraud, prvacy conrol, nformaon prorzaon, recommender sysems, search and vsualzaon see e.g. [Glber and Karahalos, 2009], [an e al., 2010] and references heren. Fgure 4: Average flerng me wh ncreasng number of houses. Noe he cubc ncrease n flerng me for he Ground Kalman Fler and he lnear ncrease for our Lfed Relaonal Kalman Fler LRKF. he y-axs s shown n logarhmc scale. o show ha LRKF performs lnearly, we added markers a he measuremens on he LRKF curve. varable ha s negraed he parameers of all, E +, 2, parwse neracons beween relaonal aoms mus be updaed. Lemma 4 he complexy of Lfed Updae s O +o O max. Where +o s he se of relaonal aoms oupu by Spl Obs and O max s he larges se of observaons assocaed wh a relaonal aom. Proof For each relaonal aom n +o he compuaon n Equaon 15 eraes over all relevan observaons. Our man resul follows, heorem 5 he compuaonal complexy of LRKF s O o O max where s he number of meseps,, +, +o and O max are as above wh he ndcang he larges se across all meseps. 5 Expermenal Resuls We compare he average flerng me of LRKF and a convenonal Kalman Fler by varyng he number of random varables. We mplemened boh he LRKF and he convenonal KF whch handles random varables ndvdually n Perl. hs makes he manpulaon of he dynamcally changng srucure convenen. For he housng marke model n Fgure 1, we randomly choose he parameers of he models prors, s, RPMs, and ROMs and provde observaons for HMO and HPO. o emphasze he dfference n scalably, we assume ha some se of houses has ndvdual observaons n each mesep, HPO, whle he res of he houses do no. We ran he wo flers over 50 meseps. he resuls n graph 4 confrm our heorecal resuls conrasng he lnear me complexy of LRKF wh he cubc me complexy of he Kalman Fler. 6 Applcaon o Onlne Socal Neworks For decades socal scenss have suded how dfferen ypes of relaonshps mpac ndvduals and organzaons. More recenly, research on analyzng onlne socal neworks Prevous work on OSNs has been characerzed by wo major lmaons: 1 Followng socal meda has focused on bnary frendshp relaons,.e., wo people can be eher frends or srangers. However, n realy relaonshps may fall anywhere along a connuous specrum - an observaon made n he socal scences snce 1973 [Granoveer, 1973] wh he nroducon of he noon of e srengh. 2 Works on lnk predcon and relaonshp srengh esmaon have gven lle aenon o he dynamc naure of socal neworks SNs. However, has been demonsraed ha he level of neracons beween ndvduals vares wdely over me. For nsance, ermens on he Facebook OSN showed ha, on average, 55% of he lnks ha are acve durng a gven monh are no longer acve durng he followng monh [Vswanah e al., 2009].,j U,j o k, j I l ok O l l Fgure 5: Model wh plae noaon and nduced paron. Gven he relaonal naure of SNs and he fac ha neracon daa s noably nosy a probablsc and relaonal approach o predcon s bes. However, predcng relaonshp srenghs on a sngle nework-snapsho has remaned oo ensve for curren exac nference algorhms [Sen e al., 2008]. o exacerbae he above lmaons, curren OSNs are poenally very large and rapdly growng. Here we show ha our LRKF s able o carry accurae relaonshp srengh predcon on large dynamc neworks whle usng sae-of-he-ar modelng feaures and echnques. Followng he modelng decsons made by [an e al., 2010] we propose a probablsc model where hdden varables represen relaonshp srengh. he model s composed of wo pars Fgure 5: 1 A generave componen op models he condonal probably of relaonshp srengh gven profle smlares and; 2 a dscrmnave componen boom models he condonal probably of he neracon acvy beween users gven he srengh of her relaonshp. In [an e al., 2010] he hdden varables are esmaed eravely gven a snapsho of he OSN. In sharp conras, we are able o fler he sae of he nework as observaons abou user neracons are made. We exend prevous work by nroducng a lnear Gaussan model of relaonshp evoluon Equaon 16 ha relaes sae varables across meseps. For hs purpose, he conrol-npus are derved from a lnearcombnaon of profle smlares. he model has he followng componens,

7 P, j, j c, j 2 2σ 2 P φ Ul, j, U, j l, j α U,l β U,l U, j l 2 2σ 2 l U l U U,l φ ROMk o k,, j ok α O,k β O,k, j 2 2σ 2 o k O o k O O,k φ, j,, j, U, j, j w U, j β, j 2 2σ 2 16 where, j ranges over all pars of ndvduals, are he relaonshp srengh varables, U are he profle smlary feaure vecors and he o k are neracon observaons. One lmaon of hs model, whch nhers from s sac predecessor, s ha relaonshp srenghs are consdered ndependen of each oher. A beer approach s o consder he dependences beween edges by nroducng RPMs of he form, φ RPM, j,, k. 17 We envson a mulvarae Gaussan verson of probablsc ransvy beng represened n hs fashon. 7 Relaed Work he KF [Kalman, 1960; Rowes and Ghahraman, 1999] s a mehod for esmang he sae of a dynamc process gven a sequence of nosy observaons. I s resrced o lnear dynamc and lnear measuremen models boh wh addve Gaussan nose. he Exended Kalman Fler EKF [Sorenson and Subberud, 1968] exends he KF o non-lnear sysems. For hgh dmensonal daa, a samplng mehod has been devsed, he Ensemble Kalman Fler [Evensen, 1994]. Exac Kalman Flerng for hgh dmensonal daa s no feasble because exac flerng requres marx nversons whch ake me cubc n he number of random varables. Our RGMs represen he probably densy as a produc of node and edge facors. Any mulvarae Gaussan s a quadrac onenal and can hus be wren n hs form. hs s relaed o he nformaon form of he Gaussan densy and s he bass of oher models such as Dreced Gaussan Models DGMs [Cowell, 1998] and Gaussan Markov Random Felds GMRFs [Rue and Held, 2005]. However, RGMs are relaonal whle DGMs and GMRFs are no. hus, he prevous models do no have a compac relaonal represenaons and, more mporanly, an effcen lfed exac nference algorhm. Relaonal probablsc models allow he specfcaon of models wh sze ndependen of he szes of he populaons n he model [Fredman e al., 1999; Poole, 2003; Rchardson and Domngos, 2006]. Lfed nference algorhms [de Salvo Braz e al., 2005; Mlch and Russell, 2006] aemp o carry as much of he compuaons whou proposonalzng he model. [Poole, 2003], solves nference problems by dynamcally splng and unfyng ses of ground aoms. [de Salvo Braz e al., 2005] FOVE nroduced counng elmnaon o effcenly elmnae aoms wh dfferen parameerzaons. [Mlch e al., 2008] C-FOVE ake a slghly dfferen approach wh he nroducon of counng formulas. However, all of he above lfed nference algorhms are no applcable o models wh connuous varables. [Kersng e al., 2006] nroduced Logcal HMMs ha combne deas from Sascal Relaonal Learnng and dynamc models. Indeed her work, as ours, pursues he benefs ha he relaonal approach brngs o nference and learnng. However, her work s nherenly dscree and furher, hey assume specfc ranson and observaon models. For relaonal models wh connuous varables, recen advances have made nference possble. [Wang and Domngos, 2008] s an approxmae algorhm based on samplng, search and local opmzaon. [Cho e al., 2010] s an exac varable elmnaon algorhm for connuous domans. he laer algorhm s smlar o he margnalzaon problem ha s par of he predcon sep n flerng. However, none of hese algorhms have been devsed wh dynamc models n mnd nor do hey address he problem of ndvdual observaons. 8 Concluson and Fuure Work We propose Relaonal Gaussan Models o represen and model dynamc sysems n a relaonal frs-order way. Furher, we presen he frs algorhm for flerng or rackng a he frs-order level. Our heorecal analyss and emprcal ess show ha our approach leads o sgnfcan gans n effcency and enables flerng for sysems wh very large numbers of random varables. We also make he case for he applcably of lfed nference o address real-world problems by akng a recenly proposed model of socal relaonshp srengh and exendng o large dynamc neworks. A lmaon of our exac flerng s ha we shaer he model when he random varables n a relaonal aom receve dfferen numbers of observaons because her varances and covarances become dfferen. Our curren undersandng s ha approxmae re-groupng of random varables s he only general recourse n hs case. Acknowledgemens We wsh o hank Jhye Seong, Jeongkeun Lee and he anonymous revewers for her valuable commens. hs work was suppored n par by NSF award IIS RI: Scalng Up Inference n Dynamc Sysems wh Logcal Srucure and NSF award ECS Improvng Predcon of Subsurface Flow and ranspor hrough Exploraory Daa Analyss and Complemenary Modelng. Abner Guzman-Rvera was suppored by he C2S2 Focus Cener, one of sx research ceners funded under FCRP, a Semconducor Research Corp. eny. A Deals of Lfed Predcon he negraon s done usng he followng rule, A a 2 + 2B a C π B 2 A A C. 18 a

8 where A s a consan, B a lnear form of random varables excep a, and C s a quadrac form of random varables excep a. he negraon of one random varable n Equaon 12 can be represened as follows, a 1 j n a A j j a B,j a B,j U U a 2 a R,j j a 2 σ 2 a µ P a 2 19 σ,j 2 RPM A a 2 + c + j c j a + c j j a a C a 1 j n a A j A a 2 + c + c j j a + c j j a a C a 1 j n a A j a A j A a2 + c + c j j + cj j a C, 20 a 1 j n when c, c j and cj represen consans calculaed from Equaon 19, and j represens a A j j a. Noe he quadrac form n Equaon 18 ncludes he followng ypes of resson, σ 2 P + 2 [ 2 ] + 2[] [ 2 ] + 2[ ], 21 where [ 2 ] s a A a 2, and [] s a,a A,a a aa. Now, Equaon 20 s negraed as follows, 2 π 1 A A c + c j j + cj j C 1 j n π 1 [ ] c A j 2 A j 2 + 2c j 2 [ ] j j j + 2cc j C 1 j n 1 [ ] c j 2 j 2 + 2c j [ ] j A 2 j j + 2cc j 1 j n 1 2c j A 1 j<j cj j j + 2c j cj j j + 2cj cj j j 22 n 2 a R j,j j a j 1 j<j n a A j,a A j σ j,j 2 a µ j P a 2 j 2 σ a a RPM P f j a a f j 1 j<j n a A j,a A j 1 j,j n a A j,a A j a a f j j a R j,j j a σ j,j 2 RPM j a R j,j, j a σ j,j 2,RPM 2 2. j a µ j P a 2 σ j P 2 Here, R, R, R,, µ, µ, σ RPM and σ are new RPM consans derved from Equaon 22. References [Bahman-Oskooee and Brown, 2004] Mohsen Bahman- Oskooee and Ford Brown. Kalman fler approach o esmae he demand for nernaonal reserves. Appled Economcs, 3615: , [Burgers e al., 1998] Gerr Burgers, Peer Jan van Leeuwen, Ger Evensen, Gerr Burgers, and Gerr Burgers. On he analyss scheme n he ensemble kalman fler. Monhly Weaher Revew, 126: , [Cho e al., 2010] Jaesk Cho, Davd J. Hll, and Eyal Amr. Lfed nference for relaonal connuous models. In Proceedngs of he Conference on Uncerany n Arfcal Inellgence, pages , [Cowell, 1998] Rober Cowell. Advanced nference n bayesan neworks. In Mchel I. Jordan, edor, Learnng n graphcal models, pages MI Press, Cambrdge, MA, [de Salvo Braz e al., 2005] Rodrgo de Salvo Braz, Eyal Amr, and Dan Roh. Lfed frs-order probablsc nference. In Proceedngs of he Inernaonal Jon Conference on Arfcal Inellgence, pages , [Evensen, 1994] Ger Evensen. Sequenal daa assmlaon wh a nonlnear quas-geosrophc model usng mone carlo mehods o forecas error sascs. Journal of Geophyscal Research, 99: , [Fredman e al., 1999] Nr Fredman, Lse Geoor, Daphne Koller, and Av Pfeffer. Learnng probablsc relaonal models. In Proceedngs of he Inernaonal Jon Conference on Arfcal Inellgence, pages , [Glber and Karahalos, 2009] Erc Glber and Karre Karahalos. Predcng e srengh wh socal meda. In Proceedngs of he Conference on Human Facors n Compung Sysems, pages , [Granoveer, 1973] Mark S. Granoveer. he srengh of weak es. he Amercan Journal of Socology, 786: , [Kalman, 1960] Rudolph Eml Kalman. A new approach o lnear flerng and predcon problems. ransacons of he ASME Journal of Basc Engneerng, 82Seres D:35 45, [Kersng e al., 2006] K. Kersng, L. De Raed, and. Rako. Logcal hdden markov models. Journal of Arfcal Inellgence Research, 25: , [Lmkeka e al., 2005] Benson Lmkeka, Ln Lao, and Deer Fox. Relaonal objec maps for moble robos. In Proceedngs of he Inernaonal Jon Conference on Arfcal Inellgence, pages , [Mlch and Russell, 2006] Bran Mlch and Suar J. Russell. Frs-order probablsc languages: Ino he unknown. In Proceedngs of he Inernaonal Conference on Inducve Logc Programmng, pages 10 24, [Mlch e al., 2008] Bran Mlch, Luke S. Zelemoyer, Krsan Kersng, Mchael Hames, and Lesle Pack Kaelblng. Lfed probablsc nference wh counng formulas. In Proceedngs of he AAAI Conference on Arfcal Inellgence, pages , [Poole, 2003] Davd Poole. Frs-order probablsc nference. In Proceedngs of he Inernaonal Jon Conference on Arfcal Inellgence, pages , [Rchardson and Domngos, 2006] Mahew Rchardson and Pedro Domngos. Markov logc neworks. Machne Learnng, 621-2, 2006.

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