Finally, the
amplitude Ξ
n of microsaccades is modeled as a mixture of two elliptically distributed Student-r laws,
Display Formula
, with 0 ≤
α ≤ 1 and where the Student-components (or Pearson type IIIa - see Kotz, Balakrishnan, & Johnson,
2000) write:
where
ν corresponds to the degrees of freedom,
R is the covariance matrix of the bivariate random vector modeling the spatial microsaccadic component, (.)
+ is equal to max(.,0), and .
t stands for the transposition of a vector. For large values of
ν, a Student-r law behaves like a Gaussian (it tends to a Gaussian as
Display Formula
→∞), but it has a bounded support (note that a gamma distribution could also be used here). The term
elliptical comes from the fact that the iso-probability contours (in
Equation 4) are ellipses given by the equation
ξtR−1ξ =
constant (see Zozor & Vignat,
2010, for details). For
R proportional to the identity, the distribution is isotropic while for the nonidentity, the main direction given by the eigenvectors of
R−1 is privileged: This allowed us to model a
directional bias. For
α = 0 or
α = 1, only one component remains. Otherwise, for appropriate choices of
R0 and
R1 (provided that
R0 and
R1 are not proportional) the bias is quadrimodal. Note that to obtain snapshots following such a mixture in the simulation, we computed each sample according to the first or second components with the probabilities
α and 1 −
α, respectively. In our simulation, we set
Display Formula
= 4 while the covariance matrices were chosen such that the large axes of each component were orthogonal in the mixture case.