Читать книгу Statistical Approaches for Hidden Variables in Ecology - Nathalie Peyrard - Страница 22

In cases where there are errors in observed positions, data can be smoothed in order to recreate the real trajectory. To smooth errors, all collected data points are combined with a movement model in order to “straighten out” outlying observations and thus correct positioning errors.

Different ways of taking account of observation errors in movement models have been discussed at length in the literature (Freitas et al. 2008; Johnson et al. 2008; Patterson et al. 2010). For initial, simple trajectory reconstructions, however, a linear Gaussian hierarchical model can be used as a first data exploration. This approach draws on the notion that the observed position is a noisy version of the real position, and that the noise around this position is Gaussian. In formal terms, take n noisy observations, y_0:n = (y₀, . . . , yn), of an animal’s position. Generally speaking (and throughout this chapter), we presume that each observed position is a vector of ℝ². These observations are presumed to be realizations of random variables Y_0:n, the distribution of which depends on the real position of the animal. Moreover, the real position of an animal at a given instant (unknown) is dependent on its real position for the previous instant (also unknown). In formal terms, these positions themselves can be seen as a sequence of non-independent random variables, noted Z_0:n = (Z₀, . . . Zn), with values in ℝ².

We consider that all of these random variables obey the following hierarchical model:

[1.1]

From top to bottom, these three equations define:

 – The initial distribution: the a priori initial position of the individual. In this case, we have a normal distribution (in dimension 2) about an initial position μ0, with a variance–covariance matrix Σ0.

 – The transition distribution (or dynamic model): in this case, a model of the individual’s movement. We consider that the current position is given by a random Gaussian variable, centered about an affine transformation of the previous position, with a variance–covariance matrix Σm. The affine transformation is obtained from two parameters: a matrix A (of size 2 × 2) and a vector μ of dimension 2. The most common approach is to consider that μ = 0 and to take A as the identity matrix. The resulting model is a random walk.

 – The emission distribution (or observation model): the observation is taken to be a random Gaussian variable centered about an affine transformation of the current position, with variance–covariance matrix Σo. The affine transformation is given by two parameters: a matrix B (of size 2 × 2) and a vector ν of dimension 2. The most common approach is to consider that ν = 0 and to take B as the identity matrix. The observation is thus presumed to be centered about the real position.