A feature common to Argos and other types of remotely sensed tracking data is that locations are observed irregularly through time. In order to link the process model (eqn 1) to the observations we need a method to ‘regularize’ the data because the state transitions are assumed to occur over regular time intervals. We let i be an index for the locations observed during each day or night period, i.e. i = (0, 1, 2, … , nt,j), where nt,j is the number of locations observed during the time interval between t and t + 1 for turtle j. Our observation equation is given by:
where yi,t,j are the locations observed during the day or night period t and ɛi,t,j is a random variable representing the estimation error associated with a location's quality class. This formulation allows for the possibility of having multiple observation equations for each transition equation. A particular challenge for analysing many kinds of movement data is the need to deal with extreme observations (Fig. 2, see anomalous, large deviations in tracks) in an objective fashion. Extreme observations can be removed a priori by filtering on a maximum travel rate (e.g. McConnell, Chambers & Fedak 1992). This approach can lead to loss of information because the filters remove suspect observations; more importantly, it does not deal with biologically plausible but none the less erroneous observations. Fortunately, the Argos satellite system classifies each location into one of six quality classes and we can use this information to account for uncertainty in all the observations without resorting to a priori filtering methods (Jonsen et al. 2005). Jonsen et al. (2005) show that Argos location errors are non-Gaussian and that a t-distribution provides a far better fit to the location errors. The t-distribution is robust because it has the effect of making extreme values less unlikely, thereby ensuring that outliers do not have undue influence. We therefore choose to use t-distributions to model the estimation error in eqn 3. That is, for estimation errors in latitude or longitude of quality class q (q = 1, … , 6, see below) we let ɛq(i),t,j ∼t(0,τq(i),t,j, νq(i),t,j), where τq(i),t,j is the scale parameter and νq(i),t,j is the degrees of freedom. Rather than estimate the parameters of the 12 t-distributions within the state–space model, we make use of an independent data set (Vincent et al. 2002) to obtain maximum likelihood estimates for each of the six Argos error classes in the two directions (see Jonsen et al. 2005, Appendix A). This allows us to fix the parameters of each estimation error distribution within the state–space model, thereby reducing the total number of parameters to be estimated. Because we limited our analysis to individuals with similar tags and settings, one could assume that the estimation errors are constant among tags. However, there may be variability in performance even among identically configured transmitters of the same model type. Therefore, we include a scaling factor ψj that adjusts the scale of the estimation error τ up or down for individual transmitters:
Figure 2. Hierarchical structure of the state–space model for estimation of day-time travel rate αp(t),j, the log ratio of day-to-night travel rates δj, and the process variability σj. Note that xt are the unobservable states and yt are the data. The notation I(0,) in the prior for σ denotes the constraint σ > 0, i.e. a half-normal distribution. The transition and observations sampling densities are sampled from t = 1, … , Tj, where Tj is the maximum number of time steps for the ith individual. We do not show the priors for the ψjs to simplify the figure.
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Note that ψj scales each of the τs identically. We examine two alternate assumptions; first, that ψj = 1 for all transmitters, and secondly, that ψj differs between latitude and longitude (see ‘Alternate models’, below).