Collective Inference on Markov Models for Modeling Bird Migration

Transcription

1 Collective Inference on Markov Models for Modeling Bird Migration Daniel Sheldon Cornell University M. A. Saleh Elmohamed Cornell University Dexter Kozen Cornell University Abstract We investigate a family of inference problems on Markov models, where many sample paths are drawn from a Markov chain and partial information is revealed to an observer who attempts to reconstrct the sample paths. We present algorithms and hardness reslts for several variants of this problem which arise by revealing different information to the observer and imposing different reqirements for the reconstrction of sample paths. Or algorithms are analogos to the classical Viterbi algorithm for Hidden Markov Models, which finds the single most probable sample path given a seqence of observations. Or work is motivated by an important application in ecology: inferring bird migration paths from a large database of observations. 1 Introdction Hidden Markov Models (HMMs) assme a generative model for seqential data whereby a seqence of states (or sample path) is drawn from a Markov chain in a hidden experiment. Each state generates an otpt symbol from alphabet Σ, and these otpt symbols constitte the data or observations. A classical problem, solved by the Viterbi algorithm, is to find the most probable sample path given the observations for a given Markov model. We call this the single path problem; it is well sited to labeling or tagging a single seqence of data. For example, HMMs have been sccessflly applied in speech recognition [1], natral langage processing [2], and biological seqencing [3]. We introdce two generalizations of the single path problem for performing collective inference on Markov models, motivated by an effort to model bird migration patterns sing a large database of static observations. The ebird database hosted by the Cornell Lab of Ornithology contains millions of bird observations from throghot North America, reported by the general pblic sing the ebird web application. 1 Observations report location, date, species and nmber of birds observed. The ebird data set is very rich; the hman eye can easily discern migration patterns from animations showing the observations as they nfold over time on a map of North America. 2 However, the ebird data are static, and they do not explicitly record movement, only the distribtions at different points in time. Conclsions abot migration patterns are made by the hman observer. Or goal is to bild a mathematical framework to infer dynamic migration models from the static ebird data. Qantitative migration models are of great scientific and practical import: for example, this problem arose ot of an interdisciplinary project at Cornell University to model the possible spread of avian inflenza in North America throgh wild bird migration. The migratory behavior for a species of birds can be modeled as a single generative process that independently governs how individal birds fly between locations, giving rise to the following inference problem: a hidden experiment simltaneosly draws many independent sample paths from

2 a Markov chain, and the observations reveal aggregate information abot the collection of sample paths at each time step. For example, the ebird data estimate the geographical distribtion of a species on sccessive days, bt do not track individal birds. We discss two problems within this framework. In the mltiple path problem, we assme that M independent sample paths are simltaneosly drawn from a Markov model, and the observations reveal the nmber of paths that otpt symbol α at time t, for each α and t. The observer seeks the most likely collection of paths given the observations. The fractional path problem is a frther generalization in which paths are divisible entities. The observations reveal the fraction of paths that otpt symbol α at time t, and the observer s job is to find the most likely (in a sense to be defined later) weighted collection of paths given the observations. Conceptally, the fractional path problem can be derived from the mltiple path problem by letting M go to infinity; or it has a probabilistic interpretation in terms of distribtions over paths. After discssing some preliminaries in section 2, sections 3 and 4 present algorithms for the mltiple and fractional path problems, respectively, sing network flow techniqes on the trellis graph of the Markov model. The mltiple path problem in its most general form is NP-hard, bt can be solved as an integer program. The special case when otpt symbols niqely identify their associated states can be solved efficiently as a flow problem; althogh the single path problem is trivial in this case, the mltiple and fractional path problems remain interesting. The fractional path problem can be solved by linear programming. We also introdce a practical extension to the fractional paths problem, inclding slack variables allowing the soltion to deviate slightly from potentially noisy observations. In section 5, we demonstrate or techniqes with visalizations for the migration of Archilochs colbris, the Rby-throated Hmmingbird, devoting some attention to a challenging problem we have neglected so far: estimating species distribtions from ebird observations. We briefly mention some related work. Carana et al. [4] and Phillips et al. [5] sed machine learning techniqes to model bird distribtions from observations and environmental featres. For problems on seqential data, many variants of HMMs have been proposed [3], and recently, conditional random fields (CRFs) have become a poplar alternative [6]. Roth and Yih [7] present an integer programming inference framework for CRFs that is similar to or problem formlations. 2 Preliminaries 2.1 Data Model and Notation A Markov model (V, p, Σ, σ) is a Markov chain with state set V and transition probabilities p(, v) for all, v V. Each state generates a niqe otpt symbol from alphabet Σ, given by the mapping σ : V Σ. Althogh some presentations allow each state to otpt mltiple symbols with different emission probabilities, we lose no generality assming that each state emits a niqe symbol to encode a model where state v otpt mltiple symbols, we simply dplicate v for each symbol and encode the emission probabilities into the transitions. Of corse, σ need not be one-to-one. It is sefl to think of σ as a partition of the states, letting V α = σ 1 (α) be the set of all states that otpt α. We assme each model has a distingished start state s and otpt symbol start. Let Y = V T be the set of all possible sample paths of length T. We represent a path y Y as a row vector y = (y 1,..., y T ), and a collection of M paths as the M T matrix Y = (y it ), with each row y i representing an independent sample path. The transition probabilities indce a distribtion λ on Y, where λ(y) = T 1 t=1 p(y t, y t+1 ). We will also consider arbitrary distribtions π over Y, letting Y = (Y 1,..., Y T ) denote a random path from π. Then, for example, we write Pr π [Y t = ] to be the probability nder π that the tth state is, and E π [f(y )] to be the expected vale of f(y ) for any fnction f of a random path Y drawn from π. Note that Y (boldface) denotes a matrix of M paths, while Y denotes a random path. 2.2 The Trellis Graph and Viterbi as Shortest Path To develop or flow-based algorithms, it is instrctive to bild pon a shortest-path interpretation of the Viterbi algorithm [7]. In an instance of the single path problem we are given a model (V, p, Σ, σ) and observations α 1,..., α T, and we seek the most probable path y given the observations. We call path y feasible if σ(y t ) = α t for all t; then we wish to maximize λ(y) over feasible y. The problem 2

3 p(, ) c(, ) s p(s, ) v V 0 V 0 V 0 p(, w) s c(s, ) v V 0 V 0 V 0 c(, w) w V 1 V 1 V 1 w V 1 V 1 V 1 start 0 Observations 0 1 start 0 Observations 0 1 (a) (b) Figre 1: Trellis graph for Markov model with states {s,, v, w} and alphabet {start, 0, 1}. States and v otpt the symbol 0, and state w otpts the symbol 1. (a) The bold path is feasible for the specified observations, with probability p(s, )p(, )p(, w). (b) Infeasible edges have been removed (indicated by light dashed lines), and probabilities changed to costs. The bold path has cost c(s, ) + c(, ) + c(, w). is conveniently illstrated sing the trellis graph of the Markov model (Figre 1). Here, the states are replicated for each time step, and edges connect a state at time t to its possible sccessors at time t + 1, labeled with the transition probability. A feasible path mst pass throgh partition V αt at step t, so we can prne all edges incident on other partitions, leaving only feasible paths. By defining the cost of an edge as c(, v) = log p(, v), and letting the path cost c(y) be the sm of its edge costs, straightforward algebra shows that arg max y λ(y) = arg min y c(y), i.e., the path of maximm probability becomes the path of minimm cost nder this transformation. Ths the Viterbi algorithm finds the shortest feasible path in the trellis sing edge lengths c(, v). 3 Mltiple Path Problem In the mltiple path problem, M sample paths are drawn from the model and the observations reveal the nmber of paths N t (α) that otpt α at time t, for all α and t; or, eqivalently, the mltiset A t of otpt symbols at time t. The objective is to find the most probable collection Y that is feasible, meaning it prodces mltisets A 1,..., A T. The probability λ(y) is jst the prodct of the path-wise probabilities: M M λ(y) = λ(y i ) = i=1 Then the formal specification of this problem is T 1 i=1 t=1 p(y i,t, y i,t+1 ). (1) max Y λ(y) sbject to {i : y i,t V α } = N t (α) for all α, t. (2) 3.1 Redction to the Single Path Problem A naive approach to the mltiple path problem redces it to the single path problem by creating a new Markov model on state set V M where state v 1,..., v M encodes an entire tple of original states, and the transition probabilities are given by the prodct of the element-wise transition probabilities: p( 1,..., M, v 1,..., v M ) = M p( i, v i ). A state from the prodct space V M corresponds to an entire colmn of the matrix Y, and by changing the order of mltiplication in (1), we see that the probability of a path in the new model is eqal to the probability of the entire collection of paths in the old model. To complete the redction, we form a new alphabet ˆΣ whose symbols represent mltisets of size M on Σ. Then the soltion to (2) can be fond by rnning the Viterbi algorithm to find the most likely seqence of states from V M that prodce otpt symbols (mltisets) A 1,..., A T. The rnning time is polynomial in V M and ˆΣ, bt exponential in M. i=1 3

4 3.2 Graph Flow Formlation Can we do better than the naive approach? Viewing the cost of a path as the cost of roting one nit of flow along that path in the trellis, a minimm cost collection of M paths is eqivalent to a minimm cost flow of M nits throgh the trellis given M paths, we can rote one nit along each to get a flow, and we can decompose any flow of M nits into paths each carrying a single nit of flow. Ths we can write the optimization problem in (2) as the following flow integer program, with additional constraints that the flow paths generate the correct observations. The decision variable x t v indicates the flow traveling from to v at time t; or, the nmber of sample paths that transition from to v at time t. (IP) min,v,t c(, v)x t v s.t. x t v = x t+1 vw for all v, t, (3) w x t v = N t (α) for all α, t, (4) V α,v V x t v N for all, v, t. The flow conservation constraints (3) are standard: the flow into v at time t is eqal to the flow leaving v at time t + 1. The observation constraints (4) specify that N t (α) nits of flow leave partition V α at time t. These also imply that exactly M nits of flow pass throgh each level of the trellis, by smming over all α, x t v = x t v = N t (α) = M.,v α V α,v V α Withot the observation constraints, IP wold be an instance of the minimm-cost flow problem [8], which is solvable in polynomial time by a variety of algorithms [9]. However, we cannot hope to encode the observation constraints into the flow framework, de to the following reslt. Lemma 1. The mltiple path problem is NP-hard. The proof of Lemma 1 is by redction from SET COVER, and is omitted. One may se a general prpose integer program solver to solve IP directly; this may be efficient in some cases despite the lack of polynomial time performance garantees. In the following sections we discss alternatives that are efficiently solvable. 3.3 An Efficient Special Case In the special case when σ is one-to-one, the otpt symbols niqely identify their generating states, so we may assme that Σ = V, and the otpt symbol is always the name of the crrent state. To see how the problem IP simplifies, we now have V = {} for all, so each partition consists of a single state, and the observations completely specify the flow throgh each node in the trellis: x t v = N t () for all, t. (4 ) v Sbstitting the new observation constraints (4 ) for time t+1 into the RHS of the flow conservation constraints (3) for time t yield the following replacements: x t v = N t+1 (v) for all v, t. (3 ) This gives an eqivalent set of constraints, each of which refers only to variables x t v for a single t. Hence the problem can be decomposed into T 1 disjoint sbproblems for t = 1,..., T 1. The tth sbproblem IP t is given in Figre 2(a), and illstrated on the trellis in Figre 2(b). State at time t has a spply of N t () nits of flow coming from the previos step, and we mst rote N t+1 (v) nits of flow to state v at time t+1, so we place a demand of N t+1 (v) at the corresponding node. Then the problem redces to finding a minimm cost roting of the spply from time t to meet the demand at time t + 1, solved separately for all t = 1,..., T 1. The problem IP t an instance of the transportation problem [10], a special case of the minimm-cost flow problem. There are a variety of efficient algorithms to solve both problems [8,9], or one may se a general prpose linear program (LP) solver; any basic soltion to the LP relaxation of IP t is garanteed to be integral [8]. 4

5 (IP t ) min c(, v)x t v,v s.t. x t v = N t+1 (v) v, (3 ) x t v = N t (), (4 ) v x t v N, v. (a) Nt( ) Spply v 1 c(v1, v1) v 2 v 3 v 1 v 2 v 3 t t + 1 (b) Nt+1( ) Demand Figre 2: (a) The definition of sbproblem IP t. (b) Illstration on the trellis. 4 Fractional Paths Problem In the fractional paths problem, a path is a divisible entity. The observations specify q t (α), the fraction of paths that otpt α at time t, and the observer chooses π(y) fractional nits of each path y, totaling one nit, sch that q t (α) nits otpt α at time t. The objective is to maximize y Y λ(y)π(y). Pt another way, π is a distribtion over paths sch that Pr π [Y t V α ] = q t (α), i.e., q t specifies the marginal distribtion over symbols at time t. By taking the logarithm, an eqivalent objective is to maximize E π [log λ(y )], so we seek the distribtion π that maximizes the expected log-probability of a path Y drawn from π. Conceptally, the fractional paths problem arises by letting M in the mltiple paths problem and normalizing to let q t (α) = N t (α)/m specify the fraction of paths that otpt α at time t. Operationally, the fractional paths problem is modeled by the LP relaxation of IP, which rotes one splittable nit of flow throgh the trellis. (RELAX) min,v,t c(, v)x t v s.t. x t v = x t+1 vw for all v, t, w x t v = q t (α) for all α, t, (5) V α v V x t v 0 for all, v, t. It is easy to see that a single-nit flow x corresponds to a probability distribtion π. Given π, let x t v = Pr π [Y t =, Y t+1 = v]; then x is a flow becase the probability a path enters v at time t is eqal to the probability it leaves v at time t + 1. Conversely, given x, any path decomposition assigning flow π(y) to each y Y is a probability distribtion becase the total flow is one. In general, the decomposition is not niqe, bt any choice yields a distribtion π with the same objective vale. Frthermore, nder this correspondence, x satisfies the marginal constraints (5) if and only if π has the correct marginals: x t v = Pr [Y t =, Y t+1 = v] = Pr [Y t = ] = Pr [Y t V α ]. v V v V V α V α V α Finally, we can rewrite the objective fnction in terms of paths: π(y)c(y) = E π [c(y )] = E π [ log λ(y )].,v,t c(, v)x t v = y Y By switching signs and changing from minimization to maximization, we see that RELAX solves the fractional paths problem. This problem is very similar to maximm entropy or minimm cross entropy modeling, bt the details are slightly different: sch a model wold typically find the distribtion π with the correct marginals that minimizes the cross entropy or Kllback-Leibler divergence [11] between λ and π, which, after removing a constant term, redces to minimizing E λ [ log π(y )]. Like IP, the RELAX problem also decomposes into sbproblems in the case when σ is one-to-one, bt this simplification is incompatible with the slack variables introdced in the following section. 5

6 4.1 Incorporating Slack In or application, the marginal distribtions q t ( ) are themselves estimates, and it is sefl to allow the LP to deviate slightly from these marginals to find a better overall soltion. To accomplish this, we add slack variables δ t into the marginal constraints (5), and charge for the slack in the objective fnction. The new marginals constraints are x t v = q t (α) + δα t for all α, t, (5 ) V α v V and we add the term α,t γt α δ t α into the objective fnction to charge for the slack, sing a standard LP trick [8] to model the absolte vale term. The slack costs γ t α can be tailored to individal inpt vales; for example, yo may want to charge more to deviate from a confident estimate. This will depend on the specific application. We also add the necessary constraints to ensre that the new marginals q t(α) = q t (α) + δ t α form a valid probability distribtion for all t. 5 Demonstration In this section, we demonstrate or techniqes by sing the fractional paths problem to help visalize the migration of Archilochs colbris, the Rby-throated Hmmingbird, a common bird whose range is relatively well covered by ebird observations. We work in discretized space and time, dividing the map into grid cells and the year into weeks. In addition to the Markov model for transitions between locations (grid cells) in sccessive weeks, we reqire estimates q t ( ) for the weekly distribtions of hmmingbirds across locations. Since the actal ebird observations are highly non-niform in space and time, estimating weekly distribtions reqires significant inference for locations with few or no observations. Althogh or focs is on migration inference, this is an important part of the overall problem. In the appendix, we otline one approach based on on harmonic energy minimzation [12], bt or methods are compatible with any techniqe that prodces weekly distribtions q t () and slack costs γ t. Finally, althogh or final observations are distribtions over states (locations) and not otpt symbols i.e., σ is one-to-one we cannot se the simplification from section 3.3 becase we incorporate slack into the model. 5.1 ebird Data Lanched in 2002, ebird is a citizen science project rn by the Cornell Lab of Ornithology, leveraging the data gathering power of the pblic. On the ebird website, birdwatchers sbmit checklists of birds they observe, indicating a cont for each species, along with the location, date, time and additional information. Or data set consists of the 428,648 complete checklists from 1995 throgh 2007, meaning the reporter listed all species observed. This means we can infer a cont of zero, or a negative observation, for any species not listed. Using a USGS land cover map, we divide North America into grid cells that are roghly 225 km on a side. All years of data are aggregated into one, and the year is divided into weeks so t = 1,..., 52 represents the week of the year. 5.2 Migration Inference Given weekly distribtions q t () and slack costs γ t (see the appendix), it remains to specify the Markov model. We se a simple Gassian model favoring short flights, letting p(, v) exp( d(, v) 2 /σ 2 ), where d(, v) measres the distance between grid cell centers. This corresponds to a sqared distance cost fnction. To redce problem size, we omitted variables x t v from the LP when d(, v) > 1350 km, effectively setting p(, v) = 0. We also fond it sefl to impose pper bonds δ t 2q t () on the slack variables so no single vale cold increase by more than a factor of 3. Or final LP, which was solved sing the MOSEK optimization toolbox, had 78,521 constraints and 3,031,116 variables. Figre 3 displays the migration paths or model inferred for the for weeks indicated. The top row shows the distribtion and paths inferred by the model; grid cells colored in darker shades of red have more birds (higher vales for q t()). Ble arrows indicate flight paths (x t v) between the week shown and the following week, with line width in proportion to x t v. In the bottom row, the raw data is given for comparison. Yellow dots indicate negative observations; ble sqares indicate 6

7 Week 10 Week 20 Week 30 Week 40 3/5 5/14 7/28 10/01 Figre 3: Rby-throated Hmmingbird migration. See text for description. Observations... Sink t 1 t t+1 Figre 4: Range of Rby-throated Hmmingbird. From BNA [15]. Figre 5: Illstration of local connectivity for lattice point t (appendix). positive observations, with size proportional to cont. Locations with both positive and negative observations appear dark gray. The inferred distribtions and paths are consitent with both seasonal ranges (Figre 4), and what experts crrently know abot migration rotes. For example, in the smmary paragraph on migration from the Archilochs colbris species accont in Birds of North America, Robinson et al. write Many fly across Glf of Mexico, bt many also follow coastal rote. Rotes may differ for north- and sothbond birds. Acknowledgements We are gratefl to Daniel Fink, Wesley Hochachka and Steve Kelling from the Cornell Lab of Ornithology for sefl discssions. This work was spported in part by ONR Grant N and by NSF grant CCF The views and conclsions herein are those of the athors and do not necessarily represent the official policies or endorsements of these organizations or the US Government. 7

8 References [1] L. R. Rabiner. A ttorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2): , [2] E. Charniak. Statistical techniqes for natral langage parsing. AI Magazine, 18(4):33 44, [3] R. Drbin, S. Eddy, A. Krogh, and G. Mitchison. Biological seqence analysis: Probabilistic models of proteins and ncleic acids. Cambridge University Press, [4] R. Carana, M. Elhawary, A. Mnson, M. Riedewald, D. Sorokina, D. Fink, W. M. Hochachka, and S. Kelling. Mining citizen science data to predict prevalence of wild bird species. In SIGKDD, [5] S. J. Phillips, M. Ddík, and R. E. Schapire. A maximm entropy approach to species distribtion modeling. In ICML, [6] J. Lafferty, A. McCallm, and F. Pereira. Conditional random fields: Probabilistic models for segmenting and labeling seqence data. ICML, [7] D. Roth and W. Yih. Integer linear programming inference for conditional random fields. ICML, [8] V. Chvátal. Linear Programming. W.H. Freeman, New York, NY, [9] A. V. Goldberg, S. A. Plotkin, and E. Tardos. Combinatorial algorithms for the generalized circlation problem. Math. Oper. Res., 16(2): , [10] G. B. Dantzig. Application of the simplex method to a transportation problem. In T. C. Koopmans, editor, Activity Analysis of Prodction and Allocation, volme 13 of Cowles Commission for Research in Economics, pages Wiley, [11] J. Shore and R. Johnson. Properties of cross-entropy minimization. IEEE Trans. on Information Theory, 27: , [12] X. Zh, Z. Ghahramani, and J. Lafferty. Semi-spervised learning sing Gassian fields and harmonic fnctions. In ICML, [13] P. G. Doyle and J. L. Snell. Random walks and electric networks. abstract?id=oai:arxiv.org:math/ , [14] D. Aldos and J. Fill. Reversible Markov Chains and Random Walks on Graphs. Monograph in Preparation, [15] T. R. Robinson, R. R. Sargent, and M. B. Sargent. Rby-throated Hmmingbird (Archilochs colbris). In A. Poole and F. Gill, editors, The Birds of North America, nmber 204. The Academy of Natral Sciences, Philadelphia, and The American Ornithologists Union, Washington, D.C., A Estimating Weekly Distribtions from ebird Or goal is to estimate q t (), the fraction of birds in grid cell dring week t. Given enogh observations, we can estimate q t () sing the average nmber of birds conted per checklist, a qantity we call the rate r t (). However, even for a bird with good ebird coverage, there are cells with few or no observations dring some weeks. To fill these gaps, we se the harmonic energy minimization techniqe [12] to determine vales for empty cells based on cells that are nearby in space and time. To start, we bild a 3-dimensional lattice on the points t, where t represents cell dring week t. Edges are weighted, with weights representing similarity between points. The connections for point t are illstrated in Figre 5. We exclde the edge between cells t, v t when the line connecting the centers of and v is more than half water. All edges have weight 1, except for edges connecting t to t 1 and t+1, which have weight 1/4, to prioritize spatial similarity over temporal similarity. Given the lattice, harmonic energy minimzation learns a fnction f on the lattice; the idea is to match r t () on points with sfficient data and find vales for other points according to the similarity strctre. To this end, we designate some bondary points for which the vale of f is fixed, while other points are interior points. The vale of f at an interior point t is determined by the expected vale of the following random experiment. Perform a random walk starting from t, following otgoing edges with probability proportional to their weight. When a bondary point is reached, terminate the walk and accept the bondary vale. In this way, the vales at interior points are a weighted average of nearby bondary points, where nearness is interpreted as the absportion probability in an absorbing random walk. We derive a measre of confidence in the vale f( t ) from the same experiment: let h( t ) be the expected nmber of steps for the random walk from t 8

9 to hit the bondary (the hitting time [14] of the bondary set). When h( t ) is small, t is close to the bondary and we are more confident in f( t ). Rather than choosing a threshold on the nmber of observations reqired to be a bondary point, we create a soft bondary by designating all lattice points t as interior points, and adding one bondary node connected to t for each observation, with vale eqal to the nmber of birds observed. As t gains more observations, the probability of reaching an observation in the first step approaches one, and f( t ) approaches r t (), the average of the observations. As a conservative measre, each node is also connected to a sink with bondary vale 0, to prevent vales from propagating over very long distances. We compte h and f iteratively [13]. Since f( t ) approximates the rate r t (), we mltiply by the land mass of cell to get a (relative) estimate ˆq t () for the nmber of birds in cell at time t. Finally, we normalize ˆq for each t, taking q t () = ˆq t ()/ ˆq t(). For slack costs, we set γ t = γ 0 /h( t ) to be inversely proportional to bondary hitting time, with γ chosen in conjnction with the transition costs in section 5.2 so the average cost for a nit of slack is the same as moving 40 pixels. 9