Statistical Approaches for Hidden Variables in Ecology. Nathalie Peyrard

Чтение книги онлайн.

Figure 1.5. Masked Booby (Sula dactylatra) Photo: Sophie Bertrand. For a color version of this figure, see www.iste.co.uk/peyrard/ecology.zip

      1.3.1. Data

      This case study concerns the behavior of the masked booby (Sula dactylatra).

The data used here consist of three trajectories of three masked boobies around the Ilha do Meion, an island in the Fernando de Noronha archipelago (Brazil) in the Atlantic Ocean (Figure 1.6). Positions were recorded by GPS, with data collected every 10 s.

Figure 1.6. Area of study (shown in red on the map) and three trajectories obtained by tracking three different red-footed boobies. Data were acquired in time increments of 10 s. For a color version of this figure, see www.iste.co.uk/peyrard/ecology.zip

      1.3.2. Projection

      The recorded data for the boobies were provided in the form of latitude and longitude measurements, that is, in terms of angles with respect to an origin point on the Earth’s surface. The methods presented earlier use a notion of distance (such as step length). While it is possible to calculate distances traveled over the Earth’s surface using latitude and longitude coordinates, this requires the use of specific formulas for movement on a sphere. Instead, data are often projected onto a plane, enabling the use of Euclidean distance. Due to the spherical nature of the globe, the actual projection used depends on the zone of interest². In this case, projection is carried out using UTM coordinates for zone 25 – south. In R, the sf library may be used to facilitate geographical data processing (and, notably, projection).

      1.3.3. Data smoothing

      In this case, the frequency of data acquisition was high (one point every 10 s). While there were few errors in the data (obtained using GPS), the temporal proximity of observations may result in a somewhat erratic-looking trajectory. This erratic effect is even more pronounced in the movement metrics used to detect different activities.

      To correct errors, let us take a Gaussian linear hidden Markov model, as described in section 1.2.1. Taking the equations in model [1.1], matrices A and B are taken as known and equal to the identity, while vectors μ and ν are known and equal to 0. Matrices Σ^m and Σ^o are presumed to be diagonal, but unknown. The unknown variables, represented the actual position, in this model are estimated using an EM algorithm from the MARSS package. The estimated parameters are then used to reconstruct the real trajectory by means of Kalman smoothing.

Figure 1.7 shows an example of trajectory smoothing. This smoothing process greatly reduces the irregularities present in the trajectory. It is important to note that we are now working with processed data. This transformation is shown here for illustrative purposes, although its relevance in this specific case is somewhat debatable.