Modeling animal movement with directional persistence and attractive points
MModeling animal movement with directional persistence andattractive points
Gianluca MastrantonioPolitecnico di Torino, Dipartimento di Scienze Matematiche
Abstract
GPS technology is more accessible to researchers and, nowadays, animal movement data arewidely available. To analyze such data, different approaches have been proposed, and among them,hidden Markov models with the Ornstein-Uhlenbeck or the step-and-turn emission distribution arethe most commonly used. The former characterizes movement with the use of a center of attraction,while the latter has directional persistence.In this work we propose a new emission distribution that posses the defining characteristicsof the two aforementioned approaches, and at any given time, an animal can exhibit a differentdegree of directional persistence and attraction to a point in space.Hidden Markov models based on our proposal, the Ornstein-Uhlenbeck, and the step-and-turn,are estimated on a real data example, where GPS locations of a Maremma Sheepdog are recorded.We show that our proposal has the richest output and better describes the data.
The use of statistical models, to understand animal movement, has become increasingly popular. Thedata is generally in the form of a time series of 2-dimensional spatial locations recorded using a GPSdevice attached to the animal, with time-interval between observations programmed by the researcher(Cagnacci et al. , 2010). These data, often called ”trajectory tracking data”, allow investigating featuresof the animal movement, such as habitat selection (Hebblewhite and Merrill, 2008), spatiotemporalpatterns (Morales et al. , 2004a; Fryxell et al. , 2008; Nathan et al. , 2008; Frair et al. , 2010) and animalbehavior (Merrill and David Mech, 2000; Anderson and Lindzey, 2003); for a detailed review thereader may refer to Hooten et al. (2017). These approaches can be grouped into three main categories:point processes (Johnson et al. , 2013; Brost et al. , 2015), continuous-time dynamic models (CTM)(Blackwell, 1997; Johnson et al. , 2008; Fleming et al. , 2014) and discrete-time dynamic models (DTM)(Morales et al. , 2004a; Jonsen et al. , 2005; McClintock et al. , 2012).In the CTM, the observed locations are seen as the finite-dimensional realization of a stochasticprocess, often the Ornstein-Uhlenbeck (OU) (Blackwell, 1997; Dunn and Gipson, 1977). In this ap-proach, the animal is attracted to a specific spatial location, called center of attraction , which can beused to model a tendency toward a patch of space (McClintock et al. , 2012) or the home range (Christ et al. , 2008). In the DTM, the movement is described using the movement-metrics , i.e., the step-length and the turning-angle (ST), that are proxies of the speed and the direction between consecutive loca-tions, observed at discrete time intervals; this approach allows to have directional persistence betweenconsecutive steps (Jonsen et al. , 2005). Models based on the movement-metrics are often referred toas step-and-turn (ST) models.Animal movement is a continuous process, and thus CTMs have a stronger theoretical justifica-tion, but from an interpretative point of view, it is easier to interpret and analyze movements usingmovement-metrics. On the other hand, movement-metrics have a coherent meaning only if the time-interval between observations is fixed (Codling and Hill, 2005; Patterson et al. , 2017). For a discussion1 a r X i v : . [ s t a t . A P ] D ec f the differences and similarities between the two models, the reader may refer to McClintock et al. (2014).The need of capturing heterogeneity, which reflects changes in the behavior over time, has led tocouple an observational model , either a DTM or CTM, with a state model , that describes the timeevolution of the behavior (Patterson et al. , 2008). The introduction of switching between behavioralmodes, which is generally assumed to be temporally structured, is limited in the context of CTMand in most cases is defined on a discrete-time scale (Hanks et al. , 2011; Kranstauber et al. , 2014),defeating the purpose of using a CTM; some exception exists (see for example Harris and Blackwell,2013). On the other hand, DTM can be easily exploited in a state-space model, with discrete statesin discrete time. In both approaches, conditionally to the state, the data are distributed accordinglyto a parametric distribution, whose parameters depend on the latent state. In a DTM, independencebetween the movement-metrics is generally assumed, see for example Morales and Ellner (2002) orPatterson et al. (2017), and rarely they are dependent (Mastrantonio, 2018; Mastrantonio et al. ,2019). If the temporal switching between behaviors is assumed to follow a Markov process, as in manyapplications (see for example Michelot et al. , 2016; Langrock et al. , 2012), the model is said to be ahidden Markov model (HMM). In the HMM context, the observational model is also called emissiondistribution. This class of models is used due to the easiness of implementation and interpretation.In this work we propose an HMM with a state-specific density that has as special cases the OU andthe ST. To define our proposal, we envision the step-length and turning-angle as the polar coordinatesrepresentation of Cartesian coordinates. The latter is assumed to be normally distributed, whichinduces a normal distribution on the spatial locations. Since the distribution of the OU is also normal,we can combine the two by introducing a parameter ρ ∈ [0 , ρ = 0 and the ST ones if ρ = 1. Then,the movement is biased toward the center of attraction and has a directional persistence for any valuein between. Indeed, the closer is ρ to 0, and the stronger is the attraction and weaker the directionalpersistence, while the opposite is true, if it is close to 1. We call our proposal the step-and-turn withan attractive point (STAP) model . The STAP is used as emission distribution of the HMM, whichallows us to detect, for any behavior, if it follows an OU dynamic, a ST dynamic, or a mixture of thetwo. Other proposals tried to combine the defining characteristics of the OU and ST, as McClintock et al. (2012) and Barton et al. (2009), but this is generally done using a ST model with direction meanthat points to a spatial location, i.e., the attractive point.To estimate the model, the number of latent behaviors is generally set at priori, see for exampleLangrock et al. (2012) or Morales et al. (2004b), and model selection is performed using informationalcriteria, e.g., the Integrated Classification Likelihood (ICL) (Biernacki et al. , 2000). Our HMM isformalized in a Bayesian framework, using the sticky hierarchical Dirichlet process HMM (sHDP-HMM)of Fox et al. (2011). The sHDP-HMM allows estimating, along with all the other model parameters,the number of latent behaviors, avoiding the use of informational criteria. The sHDP-HMM, with ourproposal as emission distribution, is indicated as STAP-HMM.We use our proposal to model the trajectory tracking data of a Maremma Sheepdog. For centuries,these dogs have been used in Europe and Asia to protect livestock from predators, and only recentlyin Australia (Gehring et al. , 2017; van Bommel and Johnson, 2016). They generally work with theshepherd in keeping the stock together, but, due to the properties extension, this is not always possiblein Australia. Since the owner is often unaware of the dogs behavior, it is of interest to understandand characterize it. To study its behavior, we analyze a dataset, taken from the movebank repository( ), where the spatial locations of a dog, used to protect livestock in a property inAustralia, are recorded using a GPS device (van and Johnson, 2014; van Bommel and Johnson, 2014).In this work we want to show that the model we propose is able to describe the behavior of this animalbetter than possible competitors and giving better insight into its movement patterns.On the same dataset, sHDP-HMMs with OU (OU-HMM) and ST (ST-HMM) emission distribu-tions are also estimated. We provide a comparison of the models results, highlighting differences andsimilarities, interpreting the behaviors found, the spatial distributions, and the hours of the day wherethe behaviors are more likely to be observed. We also show that our proposal produces a richer output,2howing that two of the five behaviors found follows a ST dynamic while the other 3 are OU.The paper is organized as follows. In Section 2 we formalize the OU and ST models, that are usedto build our proposal. In Section 3 the proposal is introduced as well as the sHDP-HMM while thereal data application is in Section 4. The paper ends with some conclusive remarks, in Section 5. Let suppose to have a set of coordinates s = ( s t , s t , . . . , s t T ) , with s t = ( s t, , s t, ) ∈ D ⊂ R andwhere t is a temporal index, assuming a fixed temporal distance between consecutive coordinates, i.e.,they are regularly spaced in time. The time dynamic of a bivariate OU process can be written using the conditional density of the spatiallocation at time-point t i + ∆ t given the previously observed ones, where ∆ t ∈ R + , in the following way s t i +∆ t | s t i ∼ N (cid:16) µ + e B ∆ t ( s t i − µ ) , Λ − e B ∆ t Λ e B ∆ t (cid:17) , where B is a 2 × Λ is a covariance matrix (Dunn and Gipson, 1977). Notice that theprocess is first-order Markovian. Under the assumptions that lim ∆ t ⇒∞ e B ∆ t = 0, B is said to be stable , and the long term distribution of the process is s t i ∼ N ( µ , Λ ) . If B is stable, µ can be interpreted as the central tendency and B controls its form and strength. Tosimply inference, and also interpretation, matrix B is often assumed to be isotropic, i.e. B = b I , with b ≤
0, as in Iglehart (1968). Under this assumption, the drift to the central coordinate µ depends onlyon the spatial distance || s t i − µ || ; for a more discussion of the different parametrization of B we referthe reader to Blackwell (1997).Owing to the fixed time interval between observations, we can reparametrize the process in a waythat is more suitable for our proposal: s t i +1 = s t i + ν ( µ − s t i ) + (cid:15) t i , (1) (cid:15) t i ∼ N ( , Σ ) , where ν I = − e b I ( t i +1 − t i ) , (2) Σ = Λ − e b I ( t i +1 − t i ) Λ e b I ( t i +1 − t i ) , or, equivalently s t i +1 | s t i ∼ N ( s t i + ν ( µ − s t i ) , Σ ) . (3)From equation (2), we have ν ∈ [0 , ν = 0, the process is a two-dimensional random walkwith no central drift, while ν = 1 defines a model where the spatial locations are independent. Theformulation given in (3) is the one that we are using in our proposal. Notice that it is easy to find therelation between ν and b , that is b = log(1 − ν ) t i +1 − t i . l l y t i ,1 y t i ,2 s t i−1 s t i s t i+1 q t i r t i f t i+1 f t i −1.0−0.50.00.51.0−1.0 −0.5 0.0 0.5 1.0 Figure 1: A graphical representation of the relation between the spatial locations, the movement-metrics and the displacement-coordinates.
In the discrete-time framework, a probability distribution is generally defined for the movement-metricsinstead of modeling directly the coordinates. Movement-metrics are a representation of the speed ofthe movement and an angle that measure how the animal turns. The most commonly used are the step-length r t i = || s t i +1 − s t i || ∈ R + , which is the distance traveled between two consecutive locations,and the turning angle θ t i ∈ [ − π, π ), which is the angle between s t i +1 and s t i onto the direction of theprevious increment s t i − s t i − , computed as θ t i = φ t i +1 − φ t i , where φ t i = atan ∗ ( s t i , − s t i − , , s t i , − s t i − , ) , (4)and atan ∗ ( · ) is the two-argument tangent function (Jammalamadaka and Kozubowski, 2004). InFigure 1 we depict the relation between the spatial coordinates and the movement-metrics. With thisapproach is possible to model directional persistence, e.g, if the distribution of θ t i has small varianceand circular mean equal to − π/
2, between consecutive observations, the animal tends to turn to theright with an angle of − π/ θ t i and r t i are assumed to be independent, see for example Parton and Black-well (2017) or Morales et al. (2004a), but some exceptions exist (Mastrantonio, 2018; Mastrantonio et al. , 2019). Here, instead of modeling the movement-metrics directly, we see them as the polarcoordinates representation of the Euclidian coordinates y t i = ( y t i , , y t i , ) , y t i , = r t i cos( θ t i ) ,y t i , = r t i sin( θ t i ) , called displacement-coordinates , see Figure 1, and we assume y t i = η + (cid:15) t i , (cid:15) t i ∼ N ( , Σ ) , where η ∈ R is a vector of length 2. Indeed, a distribution over y t i induces a distribution for θ t i and r t i , which will be discussed in Section 3.If we introduce the rotation matrix R ( ω ) = (cid:18) cos( ω ) − sin( ω )sin( ω ) cos( ω ) (cid:19) , (5)4e have the following relations between the displacement-coordinates and the spatial locations: y t i = R ( φ t i ) (cid:0) s t i +1 − s t i (cid:1) , (6) s t i +1 = s t i + R ( φ t i ) η + R ( φ t i ) (cid:15) t i . Notice that, since φ t i is computed using s t i and s t i − , see equation (4), the process is Markovian ofthe second order and s t i +1 | s t i , s t i − ∼ N ( s t i + R ( φ t i ) η , R ( φ t i ) ΣR ( φ t i ) ) . (7) The OU and ST models have different features, useful in the description of animal movement, thatare, respectively, the attractive point and the directional persistence. Our idea is to combine the twoto have a model that possess both features with different degree, e.g., the movement can be highlybiased toward an attractive point with a moderate directional persistence.As we can see from equations (1) and (7), due to the distributional assumption on y , both ap-proaches model the conditional distribution of s t i +1 as a normal. We combine the two in a more generalmodel, assuming s t i +1 | s t i , s t i − ∼ N ( M t i , V t i ) , where we allow ( M t i , V t i ) to have the values of equation (1) and (7). We first focus on parameter V t i , which is the variance of the conditional distribution, that is equal to Σ and R ( φ t i ) ΣR ( φ t i ) for,respectively, the OU and the ST. If the rotation matrix argument is 0, we have R (0) = I (see equation(5)) and R ( φ t i ) ΣR ( φ t i ) reduces to Σ . We then introduce the parameter ρ ∈ [0 , φ t i and changes the argument of R ( · ). The covariance matrix V t i is then defined as V t i = R ( ρφ t i ) ΣR ( ρφ t i ) . (8)For the mean vector we follow a similar approach. Here, again, we want to have a parameter thatallows us to move from the mean of the OU to the one of the ST, respectively equal to s t i + ν ( µ − s t i )and s t i + R ( φ t i ) η . We use the same parameter ρ of equation (8), and define M t i = s t i + (1 − ρ ) ν ( µ − s t ) + ρ R ( ρφ t i ) η . The conditional distribution of our proposal, that we call step-and-turn with an attractive point, isthen s t i +1 | s t i , s t i − ∼ N ( s t i + (1 − ρ ) ν ( µ − s t i ) + ρ R ( ρφ t i ) η , R ( ρφ t i ) ΣR ( ρφ t i ) ) . (9)If ρ = 0, equation (9) is equal to (3), while (9) reduces to (7) if ρ = 1. For ρ ∈ (0 , µ , as in the OU model, but, since through the rotation matrix R ( ρφ t i )the movement is affected by the previous direction, it shows also some degree of directional persistence,as in the ST model. Notice that our proposal adds only a single parameter to be able to model OUand ST behaviors.The displacement-coordinates of our model can be computed as R ( φ t i ) (1 − ρ ) ν ( µ − s t i ) + ρ R ( φ t i ) R ( ρφ t i ) η + R ( φ t i ) R ( ρφ t i ) (cid:15) t i , see equation (6), and they are normally distributed.It is well known that the angle of the polar coordinates of a bivariate normal random variable,i.e., θ t i , is projected normal distributed (Wang and Gelfand, 2013), which is one of the most flexibledistributions for circular data, see for example Mastrantonio et al. (2016) or Maruotti et al. (2016).The projected normal can be very helpful since is the only circular distribution that can be bimodal.On the other hand, the distribution of the step-length can be computed in closed-form only in specialcases, e.g., if Σ is diagonal, then r t i has a Rice distribution (Kobayashi et al. , 2011), while if the meanis 0, the step-length is Hoyt distributed (Hoyt, 1947).5 TAP-HMM OU-HMM ST-HMM -7523283.903 -7760842.59 -7689036.136
Table 1: ICL index of the estimated models. The best value is in bold.
In equation (9), parameters do not change with time. This hypothesis is unrealistic, since it is rea-sonable to assume that an animal exhibits different behaviors over time, which results in differentmovement characteristics and likelihood parameters. To introduce heterogeneity, it is generally as-sumed that the data come from a mixture-type model. Different approaches have been proposed, seefor example Patterson et al. (2017), Harris and Blackwell (2013) or Mastrantonio et al. (2019), butthe most commonly used is the HMM. Under this model, the behavior is coded by a discrete randomvariable z t i ∈ Z ⊆ N whose value indicates the behavior that the animal is following at time t i , andthe temporal evolution is generated by a first-order Markov process. We decide to define the HMMunder the non-parametric Bayesian framework, using the the sHDP-HMM (Fox et al. , 2011). Then,let M t i ,j = s t i + (1 − ρ j ) ν j (cid:0) µ j − s t i (cid:1) + ρ j R ( ρ j φ t i ) η j and V t i ,j = R ( ρ j φ t i ) Σ j R ( ρ j φ t i ) , the proposed model is: s t i +1 | s t i , s t i − , M t i , s ti , V t i , s ti ∼ N (cid:0) M t i , s ti , V t i , s ti (cid:1) , i = 1 , . . . T − , s t ∼ Unif( D ) ,z t i | z t i − , π z t ∼ Multinomial(1 , π z ti − ) , i = 1 , . . . T − , (10) z t = 1 , π j | α, κ, β ∼ Dir (cid:18) α + κ, α β + κδ j α + κ (cid:19) , j ∈ N , β | γ ∼ GEM ( γ ) , ξ j ∼ H, j ∈ N , where π j and β are infinite-dimensional probability vectors, ξ j = ( ρ j , ν j , µ j , η j , Σ j ), GEM( · ) indicatethe Griffiths-Engen-McCloskey distribution (Ishwaran and Zarepour, 2002), and δ j is the Dirac-delta.To define the conditional distribution of the observations we need s t , that here is assumed to comefrom a uniform distribution over the observed domain D . Notice that in (10) the number of possiblebehaviors is infinite, since π j is infinite-dimensional but in the observed time-window only a finitenumber of them can be observed. The K unique values assumed by { z t i } T − i =1 is a random variablethat can be used to estimate the number of latent behaviors. Maremma Sheepdogs are dogs of large breeds, originated from Italy, used for centuries all over Europeand Asia to protect livestocks from predators and thieves. These dogs, from an early age, are trainedto live with the livestock, creating a strong bond and then adult dogs view the livestock as theirsocial companions, protecting them from threats for the rest of their life. Recent studies proved thatSheepdogs are effective in protecting livestock from a wide range of potential predators (Gehring et al. ,2017; van Bommel and Johnson, 2016). The dogs can be fence-trained, to remain in proximity of thepaddock where the livestock are confined, but they are generally allowed to cross fences, move freely,and roam over large areas. 6 ll lllllllllllllllllllllll l llllllllllllllllllllllllllllllllllllll lllll lllll l lllllllllllllllllllllllllllllllll llll l l lll llll ll lllllllllllllllllllllllllllllllllllllllll lll lllllllllllllllllllll llll lll llllllllllllll lllllllllllllllll l l lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllll lllllllll llllllllllllllll lllllll llllllllllllllllllllll lllllllllll llllll lllllllllllll lllllllllll lll l lllllllllllllllllllllllllllllllll llllll lllllllll llllllllllllllllllllll llll llllll llllllll llllll l lllllllllllllllllllllllllllllll l lllllllllll lllllll l llllllllllllllllllllllllllll llll llllllllllllll lllllllllllllllllllllllll llll ll llllll lllllll lllllllllllllllllllllllllllllllllll l llllllllllllll l l llllllllllllllllllllllllllllllllll 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lllllllllllllllllllll lllllll llllllllll lllllll lll lll llll lllllll llllllllllllll l llllllllllllllllll l lllll lllllllllllllll lllllll lllllll lllll lllll lllll l l lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllll ll l l lllllll ll l llllllllllllllll lllllllllll lllllll llllll lllll l ll l llllllllllllllllllllll llll ll l lll llll l ll llll ll llllllllllllllllllllllllll l l llllllll llll lll l ll llll llllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllll l llll l llllllllllllll llll l lllllllllllllllll lllllllllll l ll llll l lllll lllllllllllllllllllllllllllllllll ll ll lllllll lll lllllllllll lllllll llll lllll lllll lll llll ll l lllllllllll lllllll llllllllllllllllllll lllllllllllllll lllllllllll ll llll ll llllllll llllllllll llll llllllllllll −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (a) Sheepdog llllllllll l llll llll llllllllllllllllllll ll l llllllllllllllllllllllllll lll lllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l llllll lllll l l lllllllllllllllll l llllllllllllll ll l llllllllllllllllllllllllll l l lllll lllllllllllllllllllll l lll llllllllllllllllllllllllllll lllllllllllllllllllll llllllll ll llllllllllllllllllllllllllllll lllllllllllll lllllllllllllllll l llll llllllllllllllllllllllllllllll l llllllllllllll llllllllllllllll l llllll llllllllll llllllllllll l l lll l llllllllllllllllll llllllllll llllllllllllllllllllllllllll llllll lllllllllllllllllllll l l llllllllllllllllllllllllll l llllllllllllllllllllll llllll l l llllllllllll llllllllllllllllll ll lllllllllllllllllllllllllllllllllll l llllllllllllllllllll l llllll llllllllllllllllllllll l l llllllllllllllllllllllll llllllllll lll lllllllllllllllllllllll l llll llllll llllllllllllllllllll l l llllllllllll llllllllllllll l l lllllllll l lllllllllllllllllll l llllllllll l llllllllllllllll l l l lllllllllllllllllllllllllllllll llll llllllllllllllllllllll l l lllllllllllllllllllllllllllll l llllllllllllllllllllllllllll l llllllllllllllllllllllllllllll l l l llllllllll llllllllllllll l l llllllllllllllllllllllllllll l l llllllllllllllllllllllllll l l lllllllllllllllllllllllll l ll lllllllllllll lllllll lllll llll lllllll lllllllllllllllll l llllll lllllll lllllllllllllllll l l lllllllllll l l lllllllllllllllllll ll llllll l llllllllllllll l l llllllll l llllllllll lllllll lll lll lllllll lllllllllllllll l l lllllllllllllllllllllllll l lll lllllllllllllllllll l l llllllllllllllllllllllllllllllllllll l llllllll llllllll l llllllll l lllllllllllllll l llll lllll llllllllllllll lllll lllll llllllll l lllllll l l l lllllllllllllllllllllllll l llll lllllllllllllllllllllll lllllllllllllllllllllllllll l ll lll lll llllllllllllllllllllllllllllll lllllllllllllllllllll l lll llllllll llllllllllllllllll l llllllllllllllll lllllllllllll l l l lllllllllllllllllllllllllll l l lll llllllllllllllllllllllllll l lllllll llllllllllllllllll lllllll l lllllllllllllllllllllllllllll l llll lllllllllllllllllllllllllll llllllllllllllllllllllllll l llllllllllllllllllllllllllll l l llllll l llllllll lll ll llllll l l llllllllllllllllllll lllllllll l ll llllllllllllllllllllllllll llll lll lllll l lllllllllllllllll ll l lllllllllll l l lllllll ll lllll l llllllllllllllllllllllllllll l lll llllllllllllllllll lllllllll l l lllllll lllllllllllllllllllll l llll lllllllllllllllllllllllllll llllllllllllllllllll lllllll ll l llll lllllllllllll lllllllll l llllllllllllllllllll lllllll l llllllllllllllll llllllll l llllllllllllllllllllll lllllll l lllll lllllllllllllllllllllll l lll llllllllllll ll lllllllll l lllllllllllllllllllll lllllll l ll lllllllllllllllllllllllllllll llllllllllllllllllllllllllllll l ll lllllllllllllllllllllllllllll l l lllllllllllllllllllllllllll l llllll l llllllllllllllllllllllll l lll llllllllllllllllllllllllll l ll llllllllllllllllllllllll l llllllllllllllllllllllllllllllll l llllllllllllllllllllllll l llllllllllllllllllllllllll l lllll ll lllllllllllllllllll l l llllllllllllllllllllll l llllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllll llllllllllll lllllllll lllllllll llll llllllllllllllllllllllllll l llllllllllllll lllllllllllll ll llllllllllllllllllllllllllll lllllllllllllllllllll llllllll lllllllllllllllllllllllllllllll llllllllllllllllllllllllllllll l l l llllllllllll lllllll lllllll l lllllllllllllllllll ll l lllllll llllllllllllllllllllll l lllllll l lllllllllllllllllllllll l lllll llllllllllllllllllllll llllll l lllllllllllllllllllllllllllll l llllllllllllllllllllll llllll llllllllllllllllllllllllllll lllllllllllllllllllllllllllll l llll lllllllllllllllllll lllllllllllllllllllllll l llllllllllllllllllllllllllllllll l l lllllllllllllllllllllll l lllllll l llllllllllllllllllll lllllll l llll llllllllllllllllllllll lllllll llllllll l llllllllllllllllllllll ll l llllllllllllllllllll l lllll llllllllllllllllllllllllll l l llllllllllllllllllllllllllllllll lllllllllllllllllllll lllll lllllllllllllllllllllllll lllllll llll llllllllllll l lllllll llllllllllllllllllllllllll lllllllllllllllllllllllllllllllll l lllllllllllllllllllll llllll l lllllllllllllllllllllllllllllll llllll llllllllllllllllllllllllllll l llllll lllllllllllllllllllllll l l lllllllllllllll lllllllllllllllllllllll llllllllllllll l llll llllllllllllll llllllllll l l l lllllllllllllll llllllllll l lllllllllllllllllllllllllllll l l llllllllllllllllllllllll lll llllll llllllllllllll lllllll l l llllllllllllllll llllllll l llllllllllllllllllllllllllll llllllllllllllllllllllll ll l llll llllllllllllllllllllll lll llllllllllllllllllllllllll l l l lllllllllllllllllllllllllllll l l lllllllllllllllllllllllllll llll llllllll lllllllllllllllll lllllllllllllllllllllllllllll ll ll llllllllllllllllllllll llllllllllllllllllllllllll ll llllllllllllllllllllllllllll lllllllllllllllll llllllllllll ll l lllll lllllllllllllllll l llllllllllllllllllllllllll llllll lll llllllllllllllllll llll lllllllllllllllllllll ll ll lllllllllllllllllllllllllll lllllllllll llllllllllllllll l l ll lll ll lllllllllllllllllll l lllllllllllll lllllllllllllllll lll ll lllllllllllllllllllllll l llllllllllllllllllllllll lll llll lllllllllllllllllllllllll l lllllllllllllllllllllllllllllll lllllllllllllllllll l l llllllllllllllllllllllllllll l l lllllllllll lllllllll l llllllllllllllllllllllll llllllllllllllllllllllllllllll ll l lllllllllllllllllllllll l lllllllllllllllllllllllllllllllll l lllll lllllllllllllllll l llll llllllllllllll llllllllllll l lllllllllllllllllllllllll lllllllllllllllllllllllllllllllllll llllllllll llllll lllll ll lll ll llllllllllllllllllllll lllllllllll l lllllllllllllllllll l lllllllllllllllllll ll l llllllllllllllllllllllllll l llllllllllllllllllllllllll llllllllllllllllllllllllll llllllllll llllllllllllllll lllllllll l lll lllllllllllllllllllllllllllllllllllllll lllllllllllll l l llllllllllllllllllllllll lllll llllllllllllllllllllllll lll l lllllllllllllllllllllllll ll lll lll llllllllllllllllllllllll llllllllllllllll lllllllllll lll lllllll llllllllllllllllll l llll lllllllllllllllllllll ll ll llllllllllllllllllllllll ll lllllllllllllllllllllllllllll ll lllllllllllllllllllll llllllllllllllllllllllllllll l lllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll llll llllllllllllllllllllllll lllllllll lllllllllllllllllllllll l lllll l l lllllllllllllllllll llllllllllllllllllllllllllllll lll llllllllllllll lllllllllllllllllllllllllllllll lll llllllllllllllllllllll l l lllllllllllllllllllllllll ll l lllllll llllllllllllllllllll lllllllllllllllll llllllllll l llll l lllllllllllllllllllll l lllllllllllllllllllllllll ll l lllllllllllllllllllllllllll l l llllllllllllllllllllllllll llll lllll l lll l lllllllll lllllllllllllll lllllllllllllllllllllllllll llllllll l lllllllllllllllll l llllllll l lllllllllllllllllllll ll l llllllllllllllllllllllllllllll l llllll llllllllllllllllllllllllll lll llllllllllllllllll lllllllllllllllllllllllllllllllllll lllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllll lll lllllllll lllll ll llllllllllll lllll ll llllll ll l lllllllllllllllll ll lllll l l lllllllll llllllllllllll llllll ll l lllllllllllllllllllllllllll l l l llllllllllllllll lllll ll l lll ll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l lllllllllllllllllllllllllllll l ll l llllllllllllllllllllllllll l llllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllll llll l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllll l llllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllll lll llllllllllllllllllllllllllll l lllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllll l lllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllll lllllllllllllllllllllllllll l lllllllllllllllllllllllll llll llllllllllllllllllllllllllllllllll lllllllllll l lll l lllllllllllllllllllllllllllllll llllllllllllllllllll llllllll lllllllllllllllll lll llllllllllllllllllllllllllllllllllll ll lllllllllllllllllll lllllll ll llllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllll llll ll ll llllllllll l llll llll llllll llllllllllll ll l lllllllllll ll lllllllllll llllllllllllllllll llllllllllll lllllllllllll lllllllllllll lllllllllllllllllllllllllllllll llllllllllllllll lllll llllll l lllllllllll lll lllllllllll l llll llllllllllllllllllllll l lllllllllllllllllllllllllll l lllllllllllllllllllllllllllll llllllllllllllllllllllllll l l lllllllllllllllllllllllllll lllllllllllllllllllllllllllllll l lllllllllllllllllllllllllll l llllllll lllllllllllllllll l l llllllllllll lllllllllllllll l l lllll lllllllllllllllllll lllllll llllllllllllllllllllllllllllll l lllllllllllllllllllllll lllllll l llllllllllllll llllllllllllllll l lllllllll llllllllllll llllll l l lllllllllllllllllllllllllllll l ll lllllllllllllllllllllllllllllllllllllllll l lllllllllllllllllllll l llll l llllllll llllllllllllll l lllllllllllll lllllllllllll llllll l l l lllllllllllllllllllllll l llll llllll l llllllllllllllllllllll l l llllllllllllllllllllll l l lllllllllllllllllllllllllll l l lllllllllll lllll lllllllllllll ll l l lllllllllllllllllllllllllll l lll lllllllllllll lllllllllllllllll l l lllllllllllllllllllllllllll l l llllllllllllllllllllllllll l llllll lllllllllllllllllll l l llllllllllllllllllllllllll lllllllllllllllllllllllllll l llllllllllllllllllllllllll l llllllllllllllllllllll l l llllllllllllllllllllllllllllll l l llllllllll llllllllllllll l ll llllll lllllllllllllll lllllllllllllllllllllllllllll l l lllllllllll lllllllllllllll l llllllllllllllll llllllllll l llllllllllllll lllllllllll l l lllllllllllllllllllllllllllll l llllllllllllllllllllllllllllll l lll lllllllllllllllllllllllllllllllll l llll llllllllll lll lllllll lllllllllllllllllllll lllllllll lllll lllllllllllllllll lllll lllll ll llllllllllllll llllllllll llll llllllllllll l lll l lllll lllllllllllllllllllllll l l llllllll lllllllllllllllllllll l l lll ll llllllllllllllllllllllllllll llll ll llllllllllllllll l l llll llllllllllllllllllllll l l llllll lllllllllllllllllllllllll l llllllllllll llllllllllllllllll l l ll lllllllllllllllllllll l l llllll llllllllllllll lllllll l lllllllll lllllllllllllllll l ll lllllllllllllllllllllllll l llllllllllllllllllllllllllllll l lllllllllllllllllllllllllllll l lllllllllllllllllllllllllllllllllll llllllllllllllllllllll llllllllllllllllllllll llllllll llllllllllll l lllllllllllllllll l lll llllllllllllllll lllllll l llllllllllllllllllll lllllll ll llllllllllllllllllllllllllll l lll llllllllllllllllllllllll l llllll lllllllllllllllllllllll llllllllllllllllll lllllll l l llllllllllllllllllllll l lllllllllllllllllllll lllllll l l lll llllllllllllllllllllllllllll l llllllllllllllllllllllllllll l lllll llllllllllllllllllllllll l llllll lllllllllllllllllllllllll l llllllllllllllllllllll llllllll lll llllllllllllllllllllllllllllllll lllllllllllllllllllllllllll l lllllllllllllllllllllllllllllll llllllllllllllllllllllllll llllllllllllllllllllllllll l llllllllllllllllllllllllllll l ll llllllllllllllllllllllllllll l llllllllllllllllllllllllll ll lllllllllllllllllllllllllllll l lllllllllllll lllllllllllllll l llllllllllllllllllllllll lllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll l llllllllllllllllllll lllll l l lllllllllllllllllll lllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllllllll l llllllllllllllllllllllllllll lll lllllllllllllllllllllllll l lllllllllllllllllllllllllllll l llllllllllllllllllllllllll l lllllllllllllllllllllll lllllll l llllllllllllllllllllllllllll llllllllllllllllll lllllllll llllllllllllllllllllllll lllll lllllllllllllllllllllllllllll l llllllllllllllllllllllllllllllll ll lllllllllllllllllllllllllllll lllllllllllllllllllllllll lllllllllllllllllllllllllllll l 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llllllllllllllllllllllllllll l l lllllllllll llll llllllllllll lll l lllllllllllllllllllllllllll l lllllllllllllllll llllllllll l l lllllllllllllll lllllllllllll lllll lllllllllllllllllllllllllllll ll llllllllllllllllllllll lllllllll lllllllll lllllll lllllllllll l llllllll llllllllllllllllllllll l l l lllllllllllllllllllllllllllll l l lllll lll lll lllllllllllllll l lllllll l llllllllllllllllllllllll lll llllllllllll lllllllllllllll lll lllllll lllllllllllllllll l l lll lllll llllllllllllllllllllllll l llllllll l lllllllllllllllll lll ll llll lllllllllllllllllllllll l llll llllllllllllllllllll ll l lll lllllllllllllllllllllllllll l llllll lllllllllllllllllllllllllll l llllllllllllllllllllllll ll l ll llllll llllllllllllllllllll l l llllll llllllllllllllllllll l lllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllll llllllllllllllllllllll l lllll llll lllllllllllllllllllll l lllll llllllllllllllll llll ll lllllllllllllll llllllllllllll l llllllllllllllllllll llll lllllllllll lllllllllllllll lll lllllllll llllllllll llllll llll l llll lllllllllllllllllllllll llllllll l lllll llllllllllllllllllllllll l lllllllllllllllllll l llllll lllllllllll llllllll l llllllllllllllllllllllllllllllll l llllllllllllllllllllllllllll l l lll lllllllllllllllllllllll llll llll llll lllllllllllllllll lllllllllllllllllllll llllllllllllll l lllll llllllllllllllllll llllllll llllllllllllllllllllll llll l l lllllllllllllllllllllll l ll l ll lllllllllllllllllllllllll llll llllllllllllll llllllllllll l l l lllllllllllllllllllllllllllll l lllll lllllllllllllllllllll l l lllllllll lllllllllllllllllllll l l llllllll llllllllllllllllll l lll l llllllllllllllllllllll l lll llllll lllllllllllllllll l llllllllllllllllllllllllllllll llll llllllllllllllllllllllllllll lll llllllllllllllllllllllllllll llll l lllllllll lllllllllllllllllll l llll lllllllllllllllllllllll lll lll l l llll llllllllllllllllllll l llll l lllllllllll lllllllllll l lllll l lllllllllllllllllllllll ll l lllllll llllllllllllllll l llll llll lllllllllllllllllllll l l lllll ll lll lllllllllllllllllllll l l lllllllllll llllllll lllllllll ll lllllll llllllllllllllllllll l l lllll lllllllllll lllllllllllll lll llllll lllllllll llllllllllllll ll l lllll lllllllllllllllllllllll llllllllllllllllllllllllllll ll l l lllllllllllllllllllllllll l llllllllllllllllllllll llll l llllllllllllllllllllllllllll l llllll llllllllllllllll lll lllllll ll lllllllllllllll l l llllllll lllllllllllllllllll llll l ll llllllllllllllllllllllllllll llll l l lllllllllllllllllllllllll l llllll llllllllllllllllllllll l l l lllll llllllllllllllllll llll l llllllllllllllllll l lll lllllllllllllllllllllllll llll llllllllll l llllllllllllll llll l lllllll l lllllllllllllllll llll llll l llll l lllllllllllllllllllllll l lll l l lllllllllllllllllllll l l l lllllllll l l l lllllllllllllllll l lll l llllllllllllll llllllllll lllllllllll l lllllllllllllllllllllllll l l l lllllllllllllllllllllllllllll l l lllllllllllllllllllllllllllllll l lll lllllllllllllllll l llllllll llllllllllllllllll llll ll l l l lllllllllllllllllllllllll l llll llllllllllllllllllllll llllll l llllllllllllllllllllllll llllll l llllllllllllllllllll lllllll l l lllllllllllllllll lllll l lll llll lllllllllllllllllllllllll l lllllll llllllllllllllllllllllll lll lllllllll lllllllllllllllll ll llll ll ll lllllllllllll llllllll llllll llllllllllllllllllll lllllllll lllllllllllllllllllllll llll llll l llllllllllllllllllllllllll llll llllllllllllllllllllll l ll l l llllll lllllllllllllllllll lll ll l l llllllllllllllllllllllllll llllll lllllllllllllllllll l llllllll ll ll llllllllllll llll l llllllllllllllllllllllllllllll l l l lllllllllllllllllll l lll l l l lllllllllllllllllll lllll ll l llllllllllllllllllllllllllll l lll llllllllllllllllll lll l llllllllllllllllllllllllll ll l l lllllllllllllllllllllllll l lllllllllllllllllllll lllll l lll lll −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (b) Sheep Figure 2: Spatial locations of the Maremma Sheepdog (a) and sheep (b). j=1 j=2 j=3 j=4 j=5ˆ µ j, µ j, -0.338 0.251 6.235 -0.322 -0.404(CI) (-62.685 63.304) (-62.459 62.19) (-21.542 41.287) (-0.685 -0.21) (-0.413 -0.395)ˆ η j, -0.005 -0.043 -0.389 2.093 -0.555(CI) (-0.006 -0.004) (-0.057 -0.028) (-62.227 60.293) (-60.5 63.858) (-64.229 58.681)ˆ η j, ν j ρ j Σ j, , Σ j, , Σ j, , π j, π j, π j, π j, π j, Table 2: Posterior means and CIs of the STAP-HMM parameters, under K=5.In Australia, and outside the country of origin, the use of Maremma Sheepdogs (or general livestockguardian dogs) is relatively new and the interest in their use is increasing (van Bommel and InvasiveAnimals Cooperative Research Centre, 2010). Owing to the properties extension, which can be severalthousand hectares in area, is hard, or even impossible, for the owner to supervision the dogs (vanBommel and Johnson, 2012), which are visited rarely, sometimes only once a week. The owner is,therefore, unaware of the dog movements and behavior (van Bommel and Invasive Animals CooperativeResearch Centre, 2010).In this section, using data taken from the movebank repository ( ), which con-tains datasets, freely available, of GPS coordinates of a wide range of animals, we want to model andunderstand the behavior of a Maremma Sheepdog in Australia. The dataset contains GPS locationsof Maremma Sheepdogs and sheep on three properties (van and Johnson, 2014). The dataset has been available at the address .02.55.07.5 0 1 2 3 4Step−length D e n s i t y (a) D e n s i t y (b)
012 −2.5 0.0 2.5 5.0Coordinate X D e n s i t y (c) D e n s i t y (d) Figure 3: Distributions of the observed movement-metrics ((a) and (b)) and coordinates ((c) and (d)). j=1 j=2 j=3 j=4 j=5ˆ µ j, -3.168 -0.105 17.062 -0.356 0.18(CI) (-52.457 43.823) (-22.525 23.693) (-15.255 59.094) (-0.423 -0.295) (0.17 0.19)ˆ µ j, -0.294 3.153 5.021 -0.314 -0.404(CI) (-46.638 45.392) (-7.564 31.502) (-25.564 41.621) (-0.389 -0.244) (-0.412 -0.395)ˆ ν j Σ j, , Σ j, , Σ j, , π j, π j, π j, π j, π j, Table 3: Posterior means and CIs of the OU-HMM parameters, under K=5.previously analyzed by van Bommel and Johnson (2014), and the authors, using tests and descriptivestatistics, estimated home-ranges, activity patterns and path tortuosity.Among the available data, we select a subset of temporally-contiguous observations of one dog,called “Bindi”, which belongs to the “Rivesdale” property, situated in North-East Victoria. Its locationsare recorded every 30 minutes, starting from 2010-03-13 at 18:30, to 2010-07-23 at 17:33. The data,that consist of 6335 time-points with 196 missings, are plotted in Figure 2 (a); notice that, to facilitate8 =1 j=2 j=3ˆ η j, -0.005 -0.044 0.026(CI) (-0.006 -0.004) (-0.059 -0.029) (-0.021 0.073)ˆ η j, Σ j, , Σ j, , Σ j, , π j, π j, π j, Table 4: Posterior means and CIs of the ST-HMM parameters, under K=3.the priors specification, the coordinates are centered and divided by a pooled variance. The observedsheep coordinates are shown in Figure 2 (b).In Figure 3, we depicted the kernel density estimates of the observed movement-metrics and coor-dinates. Both OU and ST models can be fitted to the data since a clear bimodality can be seen in theturning-angle distribution, which suggests a ST-HMM, as well as different modes on the coordinates,that can be attractive points.
We estimate our proposal (STAP-HMM) and the competitive models OU-HMM and ST-HMM. Forall models, we use weakly informative priors for the likelihood parameters, namely N ( , I ) for µ j and η j , Σ j ∼ IW (3 , I ), a uniform over [0 ,
1] for φ j . For parameter ρ j we define a prior that is amixture of a uniform over (0 , ρ j to assume the value 0and 1 with probability greater than 0.We also assume ( α, κ, γ ), i.e., the parameters of the latent classification, to be random variables.Following Fox et al. (2011), closed-form expression for the update of these parameters can be achievedif we define λ = α + κ and λ = κ/ ( α + κ ) with the following priors: λ ∼ G ( a , b ), λ ∼ B ( a , b ),and γ ∼ G ( a γ , b γ ), where B ( · , · ) stands for the beta distribution and G ( · , · ) for the gamma. Thepriors are λ ∼ G (0 . , λ ∼ B (10 , γ ∼ G (0 . , et al. , 2011) by assuming a prior over K , evaluated throughsimulations, that is almost fully concentrated on 1. This means that we are assuming (a-priori) thatonly one behavior is observed, i.e., the model is not a mixture, and if the data strongly support thehypothesis of different behaviors, the posterior of K will move away from the prior.The model is estimated using an MCMC algorithm with 50000 iterations, burnin 30000, thin 4.The MCMC implementation is straightforward since, with the exception of the likelihood parameters,we can use the work of Fox et al. (2011) to obtain posterior samples. On the other hand, owing to thenormal (conditional) distribution of the observed locations, the full conditional of µ j , η j , and Σ j canbe obtained in closed-form using standard results, while ν j and ρ j are updated with Metropolis steps.For all models, convergence was established using the coda package (Plummer et al. , 2006) of R . Themodels are implemented in Julia 1.3 (Bezanson et al. , 2017) and the code to replicate the results areavailable at https://github.com/GianlucaMastrantonio/STAP_HMM_model .9 .00.10.20.3 −2 0 2Turning−Angle D e n s i t y Behavior12 (a) D e n s i t y Behavior12 (b) D e n s i t y Behavior123 (c) D e n s i t y Behavior123 (d)
Figure 4: Turning-angle (first column) and step-length (second column) distributions for the first twobehaviors of the STAP-HMM (first row) and all behaviors of the ST-HMM (second row).
We compare the models performance with the ICL (Biernacki et al. , 2000) and in Table 1 the resultsare shown. The index suggests that our model fits the data better. As suggested by Pohle et al. (2017),we think that is also important to compare models from an interpretative point of view to see whichone gives a better insight on the animal movement. Then, we first describe the results obtained withour proposal, and then we compare the models.For each model and time-point, we compute the maximum at posterior estimate of z t i , indicatedwith ˆ z t i , and we call it the MAP behaviors . The number of unique values of ˆ z t i is the estimate ofthe number of behaviors K . For both STAP-HMM and OU-HMM, K is equal to 5 (behaviors), while K = 3 for the ST-HMM. The posterior means (indicated using the hat notation ˆ · ) and CIs of all modelparameters are shown in Tables 2, 3 and 4. To simplify the discussion, we indicate with LB j the j − thlatent behavior under our model, while LB j,OU and LB j,ST are the ones under the OU-HMM andST-HMM, respectively. STAP-HMM results
Parameter ρ , in both LB and LB , has CI (1,1], suggesting that they are STstates. The posterior densities of turning-angles and step-lengths for the ST part of these behaviors,i.e. R ( φ t i ) η + R ( φ t i ) (cid:15) t i , are depicted in Figure 4 (a) and (b), showing that LB has slower speed,with respect to LB , and both have a bimodal circular distribution with major mode at around − π and a smaller one at 0. These modes indicate that the dog tends to move on the opposite direction ofthe previous movement (mode at − π ) or on the same direction (mode at 0); the latter is less likely.In Figure 5, for each behavior is shown the proportion of time that the associate value of ˆ z t i isobserved at a given time-point. We can see that LB is more probable during the central hours andless during the night. On the other hand, LB has larger values during the night. From Figure 6 (a)and (b), we see that LB and LB share, approximatively, the same spatial locations. We don’t have10 .000.020.040.0600:00 06:00 12:00 18:00Time Behavior12345 (a) STAP-HMM (b) ST-HMM Figure 5: For each behavior, the line represents the proportion of times (y-axis) that is selected at agiven hour (x-axis).the coordinates of the property boundaries, but they can be seen in van Bommel and Johnson (2014),Figure 1, second row, and it is clear that LB and LB are inside the property boundaries.These two behaviors represent the boundary patrolling or scent-marking, which is a common be-havior of Maremma Sheepdog (Black and Green, 1985; I and M.E., 1999), where the higher speed ofthe second, is mostly related to the predator incursions (van Bommel and Johnson, 2014), which ismore likely during the night. The behavior LB , owing to the very low speed, can also represent theresting behavior.LB is an OU-type behavior (the CI of ρ j is [0 , ν is almost 0, and the CIs of µ , and µ , contain the entire spatial domain. It is moreprobable to observe it in the first hours of the morning, and the animal goes outside the propertyboundaries, as we can see from Figure 6 (c). This can be interpreted as exploratory behavior.The fourth behavior, LB , is an OU-type ( ρ ≈ µ ) located in the area where thelivestock paddock is, see Figure 2 (b). The parameter ν has posterior mean equal to 0.625, whichshows a moderate attraction to ˆ µ , and it is more likely during the night. This is the behavior thatrepresents the dog attending livestock at the core of the property.The last behavior is again an OU-type (ˆ ρ = 0) with a strong attraction to µ (ˆ ν = 0 . and the behavior is more likely during the central hoursof the day, with two spikes in the first hours of the morning and before sunset. Differently from allthe others, where ˆ π jj > ˆ π jj for j = j , here ˆ π < ˆ π = 0 . µ , i.e., the attractive point of LB .This can be interpreted as the feeding behavior and, after feeding, the dog switches to a boundarypatrolling/resting behavior, which is represented by LB . OU-HMM results
We can easily compare the MAP behaviors under the STAP- and OU-HMM,showing the proportion of time that the j − th behavior of the STAP-HMM is observed at the sametemporal indices of the l − the behavior of the OU-HMM. The results are depicted in Figure 7 (a). Fromthe figure is clear that the animal behaviors under the two models have the same temporal dynamic,i.e., the animal exhibits behavior LB j and LB j,OU at the same temporal points. This almost one-to-onerelation explains why the parameter estimates of LB , LB and LB , i.e. the OU-type behaviors ofour proposal, are very similar to LB ,OU , LB ,OU and LB ,OU , respectively (see Tables 2 and 3).Having the same temporal evolution of the STAP-HMM, the behaviors have the same spatial loca-tions and time of the day where they are more likely to be observed. This means that the interpretation11 lllllllllllllllllllllllllllllllllllllllllllllllll llll lllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll lllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllll llllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllll lll llllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllll lll llllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll 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lllllllllllllllllllllllllllll lllllllllllllllll lllllllllllll −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (a) First behavior llllll llllll l l llllll lllll lll llllll llllll lllll llllll lllll l lllll llllll llllll ll ll lll lllll lllllllllll lllllllll l llllllllll lll lllllllll lllllllll ll lllll lll lllllllll l llll ll ll lll l lllll l lll ll lllll ll ll ll llll llll lll llllll ll lll l lllllllllllllllllllll lll lllll ll llllllll llllllllllll ll lllllll llllll llllll lllllllll llllll lllllllllllllll lll l lllllllllllllllllllllll l lllllllllllllllllllll lllllll llll lllllll lllllllllllllllllll lll ll lll ll llll llll l lll lllllllllll l llll llllll llll llllllllllll llllllll lllllll llllll lllllllllll lllll llllll l l lllllllllll llllllllll llll llllll llll lllllllllllllllllllllllll llll lll lll lllllll l lll lll llllllllllll l lllllllllllllll lllll lllllll lll lllll llll lll llllll llll llllllllllllllllll llllllllllll lll lllllllllllll l llllll llll l l ll lllllll l lllll lllll llllllll l lll lll llll l ll lllllll l llllllllllllllllllll lll lllll ll llllllllllllllllllll l −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (b) Second behavior ll ll l ll ll l ll ll l l l lll l llll lll ll ll llll ll ll l l lllll l l l l lll llll ll l ll l l lll lllll l lllll l l llllllll ll ll l ll lllll ll lllll ll l lll ll lll llll lllll ll lll llllll l l l ll l lll ll lll ll ll lllll l ll llll llll llll lllll l l lllll llll ll lll ll l l l ll llllll llll l ll ll l ll ll l l lllll l l l l ll llllllll l l ll l l lll l lll ll llllllll l l llll l ll l lllll l ll lll l ll llll lllll ll lll l ll lllll ll lll llll l lll lll lll ll lllll ll l llll l ll l l lllll llll llllll lll llll lll lllll −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (c) Third behavior −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (d) Fourth behavior −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (e) Fifth behavior Figure 6: Spatial locations of the MAP behavior under the STAP-HMM model. In (d) and (e) theblack dot represents ˆ µ and ˆ µ , respectively.is similar, but, on the other hand, the OU-HMM fails to recognize the directional persistence that arepresent in the first and second behavior, and estimates a random walk with no attractive point, i.e.,ˆ ν = 0 and ˆ ν = 0 . z t i = 1 or z t i = 2, while inFigure 9 the turning-angle distributions are computed using the simulated paths, which are composedof 50000 time-points. As we can see the turning-angles of Figure 8 closely resemble the ones estimatedby the STAP-HMM, while in the OU-HMM case the turning-angles are almost uniformly distributed. ST-HMM results
The ST-HMM estimate only 3 behaviors, and the posterior densities of themovement-metrics are shown in Figure 4 (c) and (d). As we can see from Figure 7 (b), the dog followsLB and LB ,ST at the same temporal points, which is also shown by the spatial location of LB ,ST ,that can be seen in Figure 10 (a). Posterior estimates of the likelihood parameters are almost identical,and this can be also verified by the posterior densities in Figure 4 (c) and (d).LB ,ST has mostly the same temporal indices of LB , similar spatial locations (Figure 10 (b)),frequencies during the day (Figure 5 (b)), step-length distribution (Figure 4 (d)), but a differentcircular one (Figure 4 (c)). With respect to LB ,ST , this behavior has a higher speed, a bimodalcircular distribution with the major mode at − π and a smaller one at 0. As LB , this behavior can beinterpreted as boundary patrolling or protection from predator incursions.12 S T A P − H MM STAP−HMM12345 (a) S T A P − H MM STAP−HMM12345 (b)
Figure 7: Graphical representation of the two-way table between MAP behaviors under differentmodels: (a) STAP-HMM and OU-HMM; (b) STAP-HMM and ST-HMM. The area is proportional tothe frequency. D e n s i t y (a) D e n s i t y (b) Figure 8: Observed turning-angle distributions computed using the data that belong to the temporalindices of the first (a) and second (b) OU-HMM MAP behaviors.The last behavior, LB ,ST , has the highest speed and circular distribution with two modes, ap-proximatively with the same density, at 0 and − π , see Figure 4 (c) and (d). It is more likely duringthe night or the first hours of the morning. Since the spatial distribution is similar to the one of LB ,we can interpret this behavior as the exploring one. General comments
Our proposal is able to estimate behaviors that are similar, in interpretationand posterior inference, to LB ,OU , LB ,OU , LB ,OU and LB ,ST , as well as to find a behavior, LB ,that is not estimated by the others two. The ST-HMM finds three behaviors but, since none of themhas spatial distribution concentrated on the livestock paddock, as the LB ,ST AP and LB ,ST AP , itdoes not identify the dog attending livestock. The OU-HMM fails to find structure in LB ,OU andLB ,OU , where it estimates a simple random walk, while we shows (see Figures 8 and 9) that there isdirectional persistence.From an interpretative point of view, our model is able to find 5 behaviors. The first one, owing tothe low speed, is the resting behavior and, as the second one, the dog remains inside the boundaries(see Figure 6 (a) and (b)). The second one has higher speed, as can be seen in Figure 4 (b), and sinceit is more likely during the night, see Figure 5 (a), is the behavior where the dog is protecting livestock13 .00.10.20.3 −2 0 2Turning−angle D e n s i t y (a) D e n s i t y (b) Figure 9: Turning-angle distributions of the first (a) and second (b) OU-HMM MAP behaviors. llllllllllllllllllllllllllllllllllllllllllllllllll llll lllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll lllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllll llllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllll lll llllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l lllllllll lll l l lllllllllllllllllllllllll llllllllllllllllllllllll lll llllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll lllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllll lllllll lllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllll lllllllllllllllllll lllllllllll llllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllll llll 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lllllllllllllllllllllllllllllll llllllllllllllllllllllllll ll llllllllllllllllll llllllll lllllllllllllllllllllllll llllllllllllllllllllllllllllll lll lllllllllllllll lllllllllll llllllllllllllllllllllllllllllllllllllllllllllll lll ll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllll lllllllllllllllllllllllllll lllllllllllllllllllllll lllllllllllllllllllllllllllll lllllllllllllllll lllllllllllll −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (a) First behavior lllll lll llll lllllll l l llllll lllll lllllllllllllllll lllllllllll lllllllll lllllll lllll l lllll llllll l llllllll llllllll lll lllll lllllllll l l l llllll l l l llllllllllll llll l lllll lllll lllllllllllll ll ll lllllllll lll lllllllll l lllll ll ll llll l llll l lllll l lll ll llllllllllll ll ll lllllll llllll lllllll ll lll l lllllllllllllllllllllllllll llll lllllll ll llllllll llllllllll ll l ll llllllllll llllllllll llllllllll lllllllllllll l lllll lllllllll ll l lllllll lll ll lllll l llllllll l llllll ll llll ll lllllllllll l lll l ll l ll lll l lllllll llllllllllll lll lll l ll l l l lll l ll llll lllll lllll lllllllllll l llll lll l ll llll lllllllllllll lllllll ll ll lllllll llllll llll lllllllll lll llllll llllll ll ll l lllllllllll lllllllll llllll lllllllll l llll lll llll lllllllllllll lllll lllllllllll llllll lll llllll llllllllllll llll lllll llllllllllllllll l llllllllllllllllllll lllll llllllll ll ll llll llll lll lllllll l llll lllllllllllllllll llllll lllllllllll lll llll ll ll llllllllllllllllll llll l llllllll llll l l llll lllllll ll lll llllll llllllll l lllll lll llllll ll lllll llllllll l llllllllllllllllllll l llll lllll ll lllllllllllllllll lllll lll l −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (b) Second behavior l lll lll ll lll ll ll lll ll ll l l llll lll llll ll ll ll l lll ll l llll l l lll l l llll ll l l l l llll lllll lll l lll l lll ll ll llll lllll lllll ll lll ll ll lllll ll l llllll l l l l llllll llll lll l lll ll llll l llll l l lll l l llllll llll l ll ll l lllll l l lll lll llll l l llllllll l ll llll ll llll l ll ll lll llll l ll ll lll l l ll ll lll l l lll l ll lll ll l l llll ll ll lll l lllll l lllll l l llll ll ll l llllll lll l l ll ll lll ll l l llllllll lll l l ll lll lll lllll l l ll l l lllll ll l ll llll ll llll l lllll ll l ll lllll lll ll ll ll llll l ll llllll l lllllll l lllll lll lllll l ll llll llll l lllllllll l l llllll lll lll ll llll lllll ll lll l lll lll lll lll ll l llllllll lll l ll llll lll ll l llllll lll lll ll llll llll lll ll l lll ll ll l lllll l l lllll l lll lllll l l l llllllllll l l l l lllllll l lll lll ll ll ll ll lll l l ll l l llll llll l llll l lll l lll llll llll ll l lll ll l lll l lllllll llllll llll l llllll l l ll l lllllll l ll l l lll ll ll llllllll llll llll ll l ll llll llll l lll ll lll lll lllllll ll llllll lllll ll lllllllll l llll llllll l lll llll ll l ll lllll l ll lll llll lll lll ll ll ll lll ll llll llllll llll l lll llll l l ll ll lll llll l l llllll l llll l l lll ll lll l ll lll ll l lll lll l llll l l lllll llll l llll llll l lll l lll lll llllll lll l lllll ll lll l ll lllllll llll l l lllll l l ll ll lll lllll ll −4−2024 −2.5 0.0 2.5 5.0Longitude L a t i t ud e (c) Third behavior Figure 10: Spatial locations of the MAP behavior under the ST-HMM model.from predator incursions. In the third one the dog is exploring, since it goes outside the propertyboundaries, and it is the behavior that has most of the spatial locations far away from the livestock.As already pointed out by van Bommel and Invasive Animals Cooperative Research Centre (2010),Maremma Sheepdogs tend to leave the livestock in the first morning hours, and this is confirmed byFigure 5 (a). In the last two, the dog remains in the proximity of the livestock, with differences inthe time of the day where they are more likely: LB ,ST AP during the night, LB ,ST AP during the day.When the dog is in LB ,ST AP , there is a high probability that in the following time-point, the behaviorwill switch to the resting one, i.e., LB ,ST AP (see row ˆ π j, in Table 2).The results clearly show that our proposal has a richer output, better ICL, and is able to identifythe different types of movements that the animal exhibits, indicating clearly when this is based on anOU or a ST model. For this reason, we believe that our proposal is the one that better describes thedata. In this work we introduced a new model that can be used to analyze animal movement data. Ourproposal can be seen as a generalization of the two state-of-the-art models used, generally, in thiscontext, namely the OU and the ST. In our proposal the animal can be attracted to a spatial point, asin the OU, but showing directional persistence, as in the ST, at the same time. We were able to do thisby introducing only a further parameter, other than the ones related to the ST and OU approaches.The proposed models, the OU-HMM, and ST-HMM, were estimated on a real data example,where the spatial locations of a Maremma Sheepdog are recorded. We compared the results of the14hree approaches, showing differences and similarities. While all of them gave a good description ofthe animal movement, our proposal was the one with the richest output, better interpretation andgoodness-of-fit.Before this proposal, we tried to combine the ST and OU approaches using a likelihood defined asa mixture of the two but, although it seemed promising, it worked only when used without the HMM.When multiple behaviors were assumed, identifiability problems arose between the mixtures in thedata likelihood and the mixture defines by the HMM. In the future, we will investigate further thistype of model to see if the identification problems can be solved. Moreover, we will extend our modelto incorporate multiple animals, as in Langrock et al. (2014), and we will use more flexible temporaldynamics for the latent behavior switching, as in Harris and Blackwell (2013) or Mastrantonio et al. (2019).
Acknowledgement
This work has partially been developed under the MIUR grant Dipartimenti di Eccellenza 2018 - 2022(E11G18000350001), conferred to the Dipartimento di Scienze Matematiche - DISMA, Politecnico diTorino.
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