Statistical Approaches for Hidden Variables in Ecology. Nathalie Peyrard

      where p is a generic notation for probability density. In this case, the expression of likelihood implies the calculation of an integral in very high dimensions, as it must be integrated across all hidden states. However, given a known sequence of real positions X_0:n, we would have an explicit expression of the full log-likelihood:

[1.2]

      As all of the densities in this model are Gaussian, maximization of the log-likelihood would be simple. The expectation–maximization (EM) algorithm uses this full likelihood to maximize likelihood. Based on an initial parameter value θ⁽⁰⁾, the algorithm produces a series of estimations as follows:

       – Step E calculates:[1.3]

       – Step M takes:

The series converges to a local maximum likelihood (Dempster et al. 1977). Equation [1.3] consists of calculating the expectation of [1.2] with respect to the distribution of missing data given the existing observations, that is, the distribution of X_0:n| {Y_0:n = y_0:n), using a “real” parameter The smoothing distribution for the parameter must, therefore, be calculated as part of this step; this is done using Kalman smoothing. A solution is then obtained explicitly in step M thanks to the Gaussian linear nature of the problem.

      1.2.1.3. Filtering and smoothing a trajectory

      As we have seen, the reconstruction of a trajectory is reliant on the determination of a smoothing distribution, that is, for all 0 ≤ t ≤ n, the distribution of Zt|Y_0:n. Note that the inference of the real position at a time t takes account of all observations. As this distribution is Gaussian in the context of the model [1.1], this corresponds to calculating [Zt|Y_0:n] and [Zt|Y_0:n] using Kalman recursions.

      The name Kalman is more often encountered in the context of Kalman filtering, rather than Kalman smoothing. In these contexts, the Kalman filter is used to determine the filter distribution, that is, the distribution of Zt|Y_0:t. It is, thus, the distribution of the position at time t on the basis of the observations up to time t.

Intuitively, smoothing gives a better estimation than filtering, as the future can be taken into account when estimating a position at time t. Using filtering, the t^th position is corrected using positions observed up until time t, while in the case of smoothing, all of the available information is taken into account. Figure 1.2, taken from Lopez et al. (2015), illustrates the advantages of smoothing. A precise, frequent recording of the movement of an elephant seal, obtained using GPS (the reference curve), is shown alongside a reconstruction of the same real trajectory obtained using Argos data, filtering and smoothing.

1.2.2. Activity reconstruction model

      1.2.2.1. Overview

      As we indicated earlier, an individual alternates between different activities, and these are reflected in different modes of movement. For example, an individual who is looking for food will move slowly, with frequent changes of direction as potential food sources are detected. An individual traveling back to the colony, on the other hand, will travel relatively quickly and in a relatively straight line.

Subjacent (hidden) activities may be reconstructed by analyzing a trajectory, using a model that connects activities and movement. In this case, the observations y_0:n = (y₀, . . . , yn) are measures of a metric, which is presumed to be affected by an animal’s activity (typically, this metric represents speed; other examples are discussed in the following section). Taking 0 ≤ t ≤ n, zt is used to represent the unobserved activity of an individual at an instant t. This activity is encoded as an integer between 1 and J, where J is a known integer, representing the number of expected activities. Observations and hidden activities are considered as realizations of random variables. Let Z := (Z₀, . . . , Zn) be the series of hidden states (subjacent activities) and Y := (Y₀, . . . , Yn) the series of movement measurements.

Figure 1.2. Figure extracted from Figure 4 in Lopez et al. (2015). The black line shows a precise recording of the movements of an elephant seal. The green line was obtained by filtering positions recorded using the Argos system, and the purple line shows a smoothed version of the same data. The trajectory reconstructed using smoothing corresponds more closely to the reference data than the version obtained by filtering. For a color version of this figure, see www.iste.co.uk/peyrard/ecology.zip

      Using a classic activity reconstruction approach, the sequence Z is modeled by a Markov chain, that is, the series of random variables Zt verifies the Markov property; in other terms, for any series of integers z_0:t with values in {1, . . . , J}i⁺¹:

      Furthermore, if we consider that this probability of transition is independent of the instant t, the Markov chain is said to be homogeneous¹.

      The model draws on the idea that the distribution of Yt is dependent on the activity. The modeler must, therefore, specify the distribution of Yt|{Zt = j}. This specification is generally carried out using a parametric distribution (typically a normal distribution). Activity identification is based on the ways in which the parameters of this distribution change (the mean and variance change as the activity changes).

      The full model is formulated as follows:

[1.4]

      From top to bottom, these three equations define:

       – The initial distribution: this the probability distribution for the first activity, and is thus a vector of probabilities ν0 = (ν0(1), . . . , ν0(J)). In the common case where only one trajectory is observed, the