The dispersion of gaze within a fixation was computed using a measure of inertia, a metric used to quantify the spread of a cloud of data points with respect to a fixed point, usually its empirical mean. Here, we used a similar but generalized formula based on the mean quadratic distance from an arbitrary reference point. As such, in the case of stimulus motion, we can compute inertia with respect to the stimulus’ center of gravity. Let
\(\overline{\boldsymbol{q}}_U\doteq \frac{1}{N}\sum _{i=1}^N \boldsymbol{q}_U^i\) be the empirical mean of a signal whose samples (
\(i = 1, \ldots , N\)) are given by
\(\boldsymbol{q}_U^i=\left[x_U^i, y_U^i\right]^{\top }\). We will use
\(U=G\) for the observed gaze and
\(U=S\) for the coordinates of the stimulus (center of gravity). Gaze inertia
\(I\) was computed over the stimulus trajectories over a trial as follows:
\begin{eqnarray}
I &=& \frac{1}{N} {\sum _{i=1}^{N}} \left(\boldsymbol{q}_{G}^{i}-\boldsymbol{q}_{O}^{i}\right)^{\top } \left(\boldsymbol{q}_{G}^{i}-\boldsymbol{q}_{O}^{i}\right)\nonumber\\
& = &\frac{1}{N} \sum _{i=1}^N \Vert \boldsymbol{q}_{G}^{i}-\boldsymbol{q}_{O}^{i} \Vert ^{2}
\end{eqnarray}
where
\(N\) represents the total number of frames in the trial,
\(\boldsymbol{q}_G=[\boldsymbol{x}_G,\boldsymbol{y}_G]^{\top }\) the measured monocular bivariate gaze signal coordinates, and
\(\boldsymbol{q}_O=[\boldsymbol{x}_O,\boldsymbol{y}_O]^{\top }\) the origin reference point coordinates in the screen plane—however, one can compute inertia with respect to other points in space, for example, stimulus center of gravity or the fixation’s mean gaze position. Inertia quantifies gaze displacement as does BCEA (
Epelboim & Kowler, 1993) and box-count measures (
Engbert & Mergenthaler, 2006). Its key advantage over the former two is that inertia is a more intuitive measure of spatial displacement over a fixation period. The box-count metric is simple and provides similar insight in gaze dispersion over an epoch; it is dependent on the size of the box in space and time used for analysis. Hence, it corresponds to a down-sampling measurement of inertia over a fixed time window. Finally, inertia provides the advantage of being a metric relative to a chosen origin or reference point—box count being independent of the origin—and thus it can be used to look at spatial displacement in the following three contexts: (1) absolute inertia (
\(I_{\text{screen}}\)) is obtained by choosing the center of screen as a reference (absolute, like box count;
\(\boldsymbol{q}_O = [0,0]^{\top }\)), (2) relative retinal image instability (
\(I_{\text{stimulus}}\)) by choosing the stimulus’ center of gravity (for pursuit;
\(\boldsymbol{q}_O = \boldsymbol{q}_S = [\boldsymbol{x}_S, \boldsymbol{y}_S]^{\top }\)), and (3) general relative fixational eye movement instability (
\(I_{\text{fixation}}\)) by referring to the fixation center of gravity (obtained by choosing
\(\boldsymbol{q}_O = \overline{\boldsymbol{q}}_G = [\overline{\boldsymbol{x}}_G , \overline{\boldsymbol{y}}_G]^{\top }\) with
\(\overline{\boldsymbol{q}}_G\), the empirical mean of the gaze for an
\(N\) samples fixation epoch).