Variational Inference for Graphical Models of Multivariate Piecewise-Stationary Time Series

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1 18th Internatonal Conference on Informaton Fuson Washngton, DC - July 6-9, 2015 Varatonal Inference for Graphcal Models of Multvarate Pecewse-Statonary Tme Seres Hang Yu and Justn Dauwels School of Electrcal and Electroncs Engneerng, Nanyang Technologcal Unversty, 50 Nanyang Avenue, Sngapore Abstract Graphcal models provde a powerful formalsm for statstcal modelng of complex systems. Especally sparse graphcal models have seen wde applcatons recently, as they allow us to nfer network structure from multple tme seres (e.g., functonal bran networks from multchannel electroencephalograms). So far, most of the lterature deals wth statonary tme seres, whereas real-lfe tme seres often exhbt non-statonarty. In ths paper, we focus on multvarate pecewse-statonary tme seres, and propose novel Bayesan technques to nfer the change ponts and the graphcal models of statonary tme segments. Concretely, we model the tme seres as a hdden Markov model whose hdden states correspond to dfferent Gaussan graphcal models. As such, the transton between dfferent states represents a change pont. We further mpose a stck-breakng process pror on the hdden states and shrnkage prors on the nverse covarance matrces of dfferent states. We then derve an effcent stochastc varatonal nference algorthm to learn the model wth sublnear tme complexty. As an mportant advantage of the proposed approach, the number and poston of the change ponts as well as the graphcal model structures are nferred n an automatc manner wthout tunng any parameters. The proposed method s valdated through numercal experments. I. INTRODUCTION Inferrng sparse graphcal models s currently en vogue, snce such models can represent the dependence between a great number of varables n a succnct manner [1], [2]. For example, gven a large collecton of genes, a sparse graphcal model (a.k.a a gene regulatory network n ths example) can be used to automatcally detect the gene pars wth strong correlaton, thus greatly reducng the tme and effort for further expermental analyss. As a result, there s substantal lterature on learnng graphcal models from varous types of data, such as Gaussan [3], [4], non-gaussan [5]-[7], and dscrete [8]. Whle the prevous works are lmted to estmatng a sngle statc graphcal model from ndependent and dentcally dstrbuted (..d) samples, real data are often assocated wth nonstatonarty, and proper consderaton of t wll greatly help nterpret the data. In [9], for nstance, the functonal bran network s shown to evolve through a dstnct topologcal progresson durng the sezure, thus provdng new nsghts nto the mechansms of sezures and novel nterventon strateges. However, the authors smply defned a tme wndow wth fxed length and nferred a network for each wndow, ntroducng artfacts to the analyss. To resolve the ssue, change ponts detecton s requred, yet modelng changng dependency structure n multvarate tme seres has only receved lmted attenton so far. Below, we present a bref revew of multple change ponts detecton for multvarate tme seres. Xuan et al [10] extended the Bayesan change pont detecton approaches for unvarate tme seres to the multvarate settng: they adopt a geometrc pror on the tme segment lengths, and then terate between MAP segmentaton and graphcal model nference. Despte the tme-consumng Monte Carlo Markov chan (MCMC) method used n the paper, the man restrcton s that the graph for all segments must be decomposable. As an alternatve, a greedy bnary segmentaton scheme s proposed n [11]. A change pont s nserted such that the Bayesan nformaton crteron (BIC) of the two graphcal models of the data before and after the change pont s mnmzed; ths procedure s repeated untl no further splts reduce the BIC score. Unfortunately, besdes the hgh computatonal complexty, the bnary segmentaton can be msleadng and overestmate the number of change ponts, as ponted out n [12]. To overcome the problem, dynamc programmng s appled n [13], leadng to jont estmaton of all the change ponts. However, the method has computatonal complexty of order O(T 3 ) n the number of fxed ponts T, whch s mpractcal for most real-lfe tme seres wth length of hundreds or thousands. In our prevous work [14], we formulate the change pont detecton problem as maxmzng the log-lkelhood of all segments wth a penalty on the number of change ponts. The optmzaton problem s then solved usng a pruned dynamc programmng method wth lnear tme complexty [15]. Graphcal models assocated wth each segments are nferred va convex optmzaton technques proposed n [3], [4]. We also put forward adaptve methods to choose the penalty parameters for both change ponts detecton and graphcal model nference. Ths method s stll computatonal demandng n practce though, snce the algorthm has to be run on every possble choce of the penalty parameters n order to fnd the best ones. To address the abovementoned concerns, we propose n ths paper a novel varatonal Bayes method to nfer the abruptly changng graphcal models for multvarate pecewsestatonary tme seres. Specfcally, we descrbe the pecewsestatonary tme seres usng a hdden Markov model (HMM), n whch the emsson dstrbuton of each state s gven by a Gaussan graphcal model (GGM). We further assume that the transton matrx s upper trangular such that the resultng HMM s equvalent to the classcal change pont detecton models n whch data of dfferent tme segments are ndependent of each other. Consequently, the number of ISIF 808

2 states equals the number of change ponts n the tme seres, and a transton between dfferent states represents a change pont between two tme segments. In order to nfer the number of hdden states, we mpose a stck-breakng process on the transton probabltes. Such a pror automatcally selects a proper number of hdden states to express the data. On the other hand, to obtan a sparse graphcal model for each hdden state, we put a shrnkage pror on the correspondng precson (nverse covarance) matrx of the GGM. The resultng Bayesan model s then learnt usng a stochastc varatonal nference approach [16]. To be more explct, we borrow the dea from [17] and compute the stochastc (natural) gradents n each teraton based on a mnbatch of subchans samplng from the HMM. Therefore, the tme complexty can be reduced to sublnear, and the resultng model can be applcable to tme seres wth length of thousands or mllons. Note that Wulsn et al. [18] utlzes a smlar framework when descrbng changng correlatons n bran recordngs. However, n ther model, the structure of the graphcal model s fxed to be the neghborng structure of the electrodes. Furthermore, n order to automatcally nfer the number of hdden sates, a herarchcal Drchlet process (HDP) pror s leveraged. Due to the lack of conjugacy between the two levels of Drchlet process, t s not straghtforward to derve fast varatonal Bayes algorthms. Instead, they apply MCMC methods, and hence ths method s computatonally ntensve. On the other hand, n [19], a HMM wth an upper trangular transton matrx s ntegrated n a Bayesan framework to dentfy change ponts n unvarate tme seres. The model s nferred by MCMC methods. Dfferent from these models, the proposed model deals wth abruptly changng graphcal models of multvarate tme seres. Moreover, we develop low-complexty stochastc varatonal nference algorthms to learn the model such that the proposed model s applcable to large scale data. Expermental results show that the proposed algorthm can nfer the number and poston of the change ponts as well as the sparse graphcal model of each state n an automatc manner. Ths paper s structured as follows. In Secton II, we present the proposed graphcal models for multvarate pecewsestatonary tme seres n length. We then derve the stochastc varatonal nference algorthm n Secton III. Numercal results for both synthetc data are presented n Secton IV. We close the paper by offerng concludng remarks n Secton V. II. GRAPHICAL MODELS OF MULTIVARIATE PIEICEWISE-STATIONARY TIME SERIES Let us suppose that we have an ordered tme sequence of data y = (y t ), where t = 1,,T and y t R P. We am to partton the T samples nto K statonary tme segments, thus ntroducng K 1 change ponts τ 1:K 1 = (τ 1,,τ K 1). Each change pont s an nteger between 1 and T 1. We further defne τ 0 = 0 and τ K = T, therefore, the k-th segment s gven by y (1:P) (τ k 1+1:τ k), where k = 1,,K. A hdden Markov model (HMM) defnes a probablty dstrbuton of y by ntroducng another sequence of hdden states s = (s t ) T t=1, where s t {1,,N s } and N s K s the number of states. The sequence of hdden states s a Markov process. Gven the states t at tme t, the observedy t s ndependent of other varables n the model. As a consequence, the model s well defned by three sets of parameters, ncludng the ntal dstrbuton p(s 1 ), the transton matrx A such that A j = p(s t+1 = j s t = ), and the emsson dstrbuton p(y t s t = ) = N(y t ;0,(J ) 1 ), where we assume that the mean s zero and J s the precson matrx (nverse covarance matrx) characterzng the Gaussan graphcal model (GGM) ndexed by the state at tme t (.e., s t = ). We further defne p(s 1 ) to be a unform dstrbuton over all possble states for smplcty. Note that the HMM ntroduces a change pont automatcally when s t+1 s t. In prevous works [10]-[14], the product partton model (PPM) s often utlzed to dentfy change ponts, n whch data s ndependent across dfferent tme segments. In other words, n the PPM, we can never enter an old state once we have left the correspondng tme segment. Therefore, to resemble the PPM, we assume that the transton matrx of the HMM s upper trangular. As a result, n our model, the number of hdden states equals the number of change ponts plus one. Here, our objectve s to nfer the state varables s, thereby obtanng the change ponts, as well as the precson matrx J correspondng to all states s t =. For the problem of change ponts detecton, we am to use the smallest number of change ponts (.e., number of states) that can well explan the pecewse-statonary property of the tme seres. As mentoned n Secton I, mposng a nonparametrc herarchcal Drchlet process pror on the transton matrx has proven effectve to nfer the number of states automatcally [18]. However, ths pror does not extend to fast varatonal algorthms due to the non-conjugate ssue. Instead, we resort to the stck-breakng process [20] that s conjugate to the transton probabltes. For the upper trangular transton matrx, the stck breakng process can be defned as follows: of the P varables y (1:P) t j 1 A j = V j (1 V k ), (1) k= V j Beta(α j,β j ). (2) The process can be nterpreted as teratvely breakng the porton of V j from the remanng of a unt-length stck j 1 k= (1 V k). Accordng to the defnton, the process s nfnte, that s, we may allow a countably nfnte number of hdden states as the length of the tme seres ncreases. Moreover, although the state space s nfnte, the resultng posterors p(a : y) wll only have large probabltes n a fnte number of states that are useful n explanng the observed data, whereas all others are nearly equal to zero [20]. As such, the stck-breakng process pror can effectvely select a proper number of states. We can fnd from (2) that the stck-breakng process requres an nfnte parameterzatons. To smplfy ths, we follow [20] to fx α j = 1 and put a conjugate gamma pror Gamma(β j ;a,b) on β j. The hyper parameters a and b are set to be 10 6 and 0.1 respectvely, 809

3 such that the state transton s properly encouraged to detect the entre state structure [20]. On the other hand, we wsh to nfer a sparse J for every possble state, thus we can unvel the changng network structure of the observed data. We thus assocate the offdagonal elements Jjk of J wth Gaussan prors wth zero means and precsons λ jk,.e., ( p(jjk λ jk) λ jk exp 1 ) 2 λ jkjjk2, (3) for all j > k. As shown n our numercal experments, many of the precsons λ jk wll take very large values durng the learnng process, and consequently, the pror can successfully shrnk most elements of J to zero, and yeld sparse graphcal models. We further mpose conjugate Gamma hyperpror on the precsons λ jk : p(λ jk) = Gamma(λ jk;c,d) λ jka 1 exp( bλ jk ). (4) The parameters c and d are set to small values (e.g., ) to obtan a flat non-nformatve pror. Note that p(jjk λ jk)p(λ jk)dλ jk = Γ( ) c+ 1 ( ) c+ 1 2 Γ(c) 2 1 2, 2πd d J jk whch s a t dstrbuton. Therefore, we essentally put a t pror on Jjk. Such shrnkage pror s often used n the Bayesan framework to promote sparsty [22], [23]. Note that n the lterature of learnng graphcal models [3], Laplace prors are often used snce they amount to l 1 norm penaltes on the precson matrx and the resultng optmzaton problem s convex. Although Laplace prors can also be regarded as a scale mxture of Gaussan, the hyperpror on precsons λ j s the nverse Gamma dstrbuton that s not conjugate to the Gaussan dstrbutons parameterzed by precsons [24]. As a result, we employ t pror here snce t s more tractable for Bayesan nference. We now turn our attenton to the overall model, whch can be expressed as: p(y,s,v,α,β,j,λ) = p(y s, J)p(V β)p(β)p(j λ)p(λ) T T = p(s 1 ) p(s t s t 1,V) p(y t s t,j st ) =1j=1 t=1 [ p(vj β j )p(β j ) ] P P =1k=1j=k+1 [ p([j ] jk λ jk)p(λ jk) ]. III. STOCHASTIC VARIATIONAL INFERENCE In ths secton, we derve a stochastc varatonal nference algorthm [16] to learn the model parameters. Concretely, we seek a varatonal dstrbuton q(s, V, α, β, J, λ) that maxmzes the evdence lower bound L: logp(y) E q [logp(y,s,v,α,β,j,λ)] E q [logq] = L. (6) (5) Maxmzng L s equvalent to mnmzng the KL dvergence between the varatonal dstrbuton q and the ntractable posteror p(s, V, α, β, J, λ y) as measured by KL(q p) = qlog(q/p). Here, we apply the mean-feld approxmaton, and therefore, the varatonal dstrbuton can be factorzed as: K q(s,v,α,β,j,λ) =q(s) where K 1 =1 j= K K P q(j ) =1 [ q(vj )q(β j ) ] P =1k=1j=k+1 q(λ jk), (7) q(v j ) = Beta(V j ;W 1j,W 2j ), (8) q(β j ) = Gamma(β j ;W 3j,W 4j ), (9) P q(j ) = δ(jj:p,j Jj:P,j ), (10) j=1 q(λ jk) = Gamma(λ jk;w 5 jk,w 6 jk ), (11) δ(jj:p,j J j:p,j ) s a delta functon whch equals 1 when Jj:P,j = Jj:P,j and 0 otherwse, and J j:p,j denotes the jth to P th elements n the jth column. Snce p(j λ) s not conjugate to p(y J) n the proposed model (5), there s no closed-form varatonal dstrbuton q(j) n the framework of mean-feld varatonal nference. Instead, t s convenent to use a pont estmate of J (10) as n [23], [24]. Furthermore, as the algorthm proceeds, many of the precsons λ jk wll become very large, and then the delta functons can well approxmate the true posteror dstrbuton. Addtonally, n expresson (7), we truncate the varatonal stck-breakng process to yeld K levels, snce the nfnte large state space s computatonally ntractable for varatonal nference. In other words, the varatonal transton probablty à s gven by: j 1 V j k= (1 V k) for j < K à j = j 1 k= (1 V k) for j = k, (12) 0 for j > K V j q(v j ). (13) We emphasze that usng the truncaton level s qute dfferent from settng a fnte state-space n a statstcal perspectve. The proposed model s stll a full stck-breakng process and s not truncated. K should be suffcently large to ensure the accuracy of the approxmaton. Fnally, note that makng a full mean-feld approxmaton of the latent states q(s) = T t=1 q(s t) would lose crtcal nformaton about the hdden Markov chan requred for accurate nference. Instead, we wll nfer a jont varatonal dstrbuton q(s) over the states of all tme ponts. The stochastc varatonal nference algorthm ams to fnd the varatonal parameters W ( = 1,,6) and J that maxmzes the evdence lower bound L. To ths end, the algorthm proceeds by updatng the varatonal parameters n 810

4 the drecton of the stochastc natural gradent. More precsely, n each teraton, W, for example, can be updated as: W (κ+1) = W +ρ κ W L(W ), (14) where W L(W ) denotes the stochastc natural gradents of L w.r.t. W at the value of W. In the sequel, we frst lst the natural gradents of all varatonal parameters, and further elaborate on the stochastc verson of the gradents. Natural gradents have a convenent form f the pror and the complete-data lkelhood correspondng to the varatonal dstrbuton are a conjugate par of exponentals famly dstrbutons. Ths condton s satsfed by q(v), q(β), and q(λ). Specfcally, 1) For the varatonal parameters of V, W1j L(W 1 j ) =1+E q(s) [ logp(s t s t 1 )] W 1 j =1+ = W 3 j W 4 j = W 3 j + W 4 j q(s t 1 =,s t = j) W 1 j, (15) W2j L(W 2 j ) +E q(s) [ 2) For the varatonal parameters of β, logp(s t s t 1 )] W 2 j q(s t 1 =,s t > j) W 2 j. (16) W3j L(W 3 j ) = a+1 W 2 j, (17) W3j L(W 3 j ) = b E q(v j)[log(1 V j )] W 3 j, (18) 3) For the varatonal parameters of λ, 1 W5 jk L(W 5jk ) = c+ 2 W 5 jk, (19) Jjk W6 jk L(W 6jk ) = d+ 2 2 W 6 jk. (20) Note that natural gradents are closely related to tradtonal varatonal Bayes (VB) algorthms [25]. By settng the natural gradents to zero n each teraton, we obtan the update rules of the VB algorthm. In other words, the natural gradents can be regarded as the dfference between two consecutve VB updates. Ths approach s employed to compute the gradent of L w.r.t J. In ths case, the correspondng pror p(j λ) s not the conjugate to the data lkelhood p(y s, J). Specfcally, we frst sequentally set the gradent of the L w.r.t Jj:P,j to zero, as n the VB framework. As such, we can update Jj:P,j TABLE I: SVI of Graphcal models for Multvarate Pecewse-Statonary Tme Seres. Input: observed multvarate tme seres y, maxmum number of possble change ponts K, number M = 50 and length L s = 2 of subchans drawn from the HMM n each teraton Iterate the followng steps untl convergence. 1) draw M subchans from the entre Markov chan, each wth length L 2) For each subchan s sub m, compute q(st) and q(s t 1,s t) for t s sub m as follows: a) Intalze q old (s t) (t s sub m ) by runnng the forward-backward n s sub m. u = 1. b) Augment s sub m n each drecton by u observatons and compute q new (s t) (t s sub m ) usng the augmented subchan. c) If q new (s t) q old (s t) ǫ, return q(s t) = q new (s t); otherwse, set u = u+1, q old (s t) = q new (s t) and go back to Step 2b. 3) Compute the (natural) gradents for all parameters of varatonal dstrbutons followng Eq. (15) to Eq. (23). 4) Update the parameters followng Eq. (14). as: [ [ ] Jj+1:P,j (κ+1) = Sjj J j, j 1 j:p 1,j:P 1 ] 1 ( + dag(e q(λ) [λ j+1:p,j]) Sj+1:P,j [ ] ) +Sjj J j, j 1 J1:j 1,j, (21) j:p 1,1:j 1 J jj (κ+1) = N /[S ] jj +J j, j J j, j 1 J j,j. (22) where N = T t=1 q(s t = ) and S = N E q(s) [y t y T t ]. Next, we can calculate the correspondng gradent as: J L(J ) = J (κ+1) J. (23) Note that J s always postve defnte durng the update procedure gven that ts ntal value s postve defnte, as proven n [24]. We can tell from Eqs. (15), (16), (21), and (22) that n order to compute the accurate natural gradent for W 1, W 2 and J, we need to run forward-backward algorthm on the entre Markov chan to get q(s t ) and q(s t 1,s t ). Although the tme complexty of the message passng algorthm s lnear, t may stll be prohbtve for very long tme seres. Therefore, we nstead borrow the dea from [17] and compute nosy stochastc natural gradents usng the q(s t ) and q(s t 1,s t ) from subchans s sub contanng consecutve observatons. As a consequence, the tme complexty of the proposed model s sublnear. In order to consder the forward and backward messages passng nto the subchan, when computng the local belefs q(s t ) (t s sub ), the subchan s augmented adaptvely to nclude enough extra observatons on each end, untl further augmentaton of the subchan wll not sgnfcantly change the local belefs. In other words, the local belefs are wthn an ǫ-ball of the optmal q (s t ) resultng from the entre chan. Fot et al. [17] has proven that such stochastc natural gradent ascent method converges to a local maxmum as long as step szes ρ κ satsfy κ ρ2 κ < and κ ρ κ =. We adopt the 811

5 Fg. 1: 25-dmensonal synthetc data, change ponts (red lnes), and the true graphcal models of all the segments. automatc methods proposed n [26] to tune step szes. The entre algorthm s summarzed n Table I. IV. NUMERICAL RESULTS In ths secton, we present our results on synthetc data; we benchmark the proposed Bayesan model wth the optmzaton method-based model [14],and the graphcal lasso method [3] that s used to nfer GGMs from statonary data. The penalty parameters n the second and thrd model are chosen by adaptve methods, cf [14]. The synthetc dataset has 25 varables and 5850 samples. The true value of change ponts are {1000, 2800, 3900, 4800}. The sgnals and the graphcal models of the fve tme segments are shown n Fg. 1. We then test the three models usng ths dataset. More specfcally, we compare the accuracy of change pont detecton, the accuracy of graphcal model nference, and the runnng tme. For graphcal model nference, we consder three crtera, that s, precson, recall, and F 1 - score. Precson s defned as the proporton of correctly estmated edges to all the edges n the estmated graph; recall s defned as the proporton of successfully estmated edges to all the edges n the true graph; F 1 -score s defned as 2 precson recall/(precson+recall), whch s a weghted average of the precson and recall. We set K = 10, L s = 2, and M = 98 when runnng the proposed stochastc varatonal nference algorthm. Due to the stochastc natural of the algorthm, the runnng tme may vary dependng on the varance of the stochastc gradent n each teraton. Therefore, we average the value of the runnng tme as well as other crtera over 100 trals. All the smulatons are runnng on a 20-core 3GHz CPU. Parallel computng s mplemented for the penalty parameter selecton procedure of the optmzaton model and the graphcal lasso, as well as the forward-backward algorthm n the M subchans n the proposed model. The results are summarzed n Table II. TABLE II: Quanttatve comparson of dfferent models Models Bayesan Model Optmzaton Model Graphcal Lasso Change Ponts {995.44, 2800, 3900, 4800} {995, 2800, 3900, 4800} N.A. Precson Recall F 1-score Runnng Tme We can fnd from the table that both the proposed Bayesan model and the optmzaton-based model performs well n terms of the accuracy of change pont detecton. The Bayesan model yelds slghtly better estmates of the frst change pont n some trals. The estmated poston s 997, whch s more close to the ground truth For graphcal model nference, the recall of the Bayesan model s much larger than that of the optmzaton model, ndcatng that the proposed Bayesan model can relably recover the true graph. On the other hand, the precson of the Bayesan model s slghtly lower than the optmzaton, mplyng that the Bayesan model ntroduces few more false postves. In summary, compared wth the optmzaton model, the proposed Bayesan model yelds a relatvely dense graph, successfully recoverng the true graph 812

6 at the cost of ncludng a few extra edges. Such slghtly dense graphs are often favored n practce, as false postves can be dentfed n further analyss whereas false negatves are bured by the other absent edges. Another obvous advantage of the proposed model s that t s much faster than the optmzaton model. Fnally, we notce that graphcal lasso gves based estmaton to the graphcal models of all segments, because of the wrong assumpton that the data s statonary. It s therefore necessary to develop specfc methods for non-statonary tme seres. V. CONCLUSION AND FUTURE WORK In ths paper, we focus on change pont detecton and graphcal model nference for pecewse-statonary tme seres. We formulate the problem as nferrng the hdden states and the emsson dstrbutons of a HMM. We further develop a low-complexty stochastc varatonal nference algorthm to learn the model. Numercal results show that the proposed model can automatcally estmate the number and poston of change ponts as well as the sparse graphcal models wthout tunng any parameters. One of our future work s to apply the proposed model to real data, such as mult-electrode bran recordngs and fnancal tme seres. It s also nterestng to nvestgate an onlne verson of the proposed algorthm, snce t s straghtforward to ncorporate new nformaton n the stochastc optmzaton framework. REFERENCES [1] A. T. Ihler, S. Krshner, M. Ghl, A. W. Robertson, and P. Smyth, Graphcal Models for Statstcal Inference and Data Assmlaton, Physca D vol. 230, pp , [2] H.-A. Loelger, J. Dauwels, J. Hu, S. Korl, P. L, and F. Kschschang, The factor graph approach to model-based sgnal processng, Proceedngs of the IEEE 95(6), pp , [3] J. Fredman, T. Haste, and R. Tbshran, Sparse nverse covarance estmaton wth the graphcal lasso, Bostatstcs, vol. 9, no. 3, pp , [4] V. Chandrasekaran, P. A. Parrlo, and A. S. 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