Distintas posiciones ante los intereses de los animales

3.2.1 Yarbus Model

The truncated cosine model suggested by Yarbus (1967) was chosen as a simple,

relatively realistic model with clearly defined spectral minima (Harris, 1998). Equations

1 and 2 describe the position profile and Fourier energy spectrum respectively.

0 r < 0

%(r) = —[1 — c o s (^ /7 ’)] 0 < / < Z" (1)

C0^{7TC0T)\ C0{7t" - CO^T^) 'J (2) 3 5 0 1 10 3 0 0 - 8 2 5 0 - 6 > ₂₀₀ - O O 150 - > 100 - 4 2 5 0 - 0 0 10 20 30 40 50 0 10 20 30 4 0 50 Tim e (ms) Tim e (ms) 1 e + 2 1 e - 1 1 e - 2 1 e - 3 1 e - 4 >. a> <ü 1 e - 5 1 e - 6 1 e - 7 1 e - 9 1 e - 1 2 1 10 10 0 F r e q u e n c y (Hz)

Figure 3.1: Analytical Yarbus model. A) position-timc plot of Equation 1 with amplitude A=10°, duration T= 50ms (i.e. 1/T =20Hz). B) velocity profile of A. C) position energy spectrum as described in Equation 2 with first four minima M l, M2, M3 and M4 at frequencies 30, 50, 70, and 90Hz.

This function tends to zero energy at frequencies of 1.5/T, 2.5/T, 3.5/T...(n+I/2)/T

(ne Z ) and declines at high frequencies |X(ry)|^ oc . The parameters studied were

the frequencies of the first 4 minima and the maximum 3 values of the energy function

between the 4 minima. Saccades with amplitudes of 5°, 10°, 20° with respective

durations of 38, 50 and 75ms were modelled.

3.2.2 Monte Carlo Simulations

Instrument noise was simulated as zero-mean, additive, gaussian (‘white’) perturbations

to each sample of the model saccade, taken from distributions with standard deviations

(ctnoise) of Iminarc, 2minarc, 0.05°, 0.1°, 0.175°, 0.25°, 0.375° and 0.5°. Successive

sample perturbations were independent of each other, simulating a stationary stochastic

process. Simulations were repeated 1000 times at each Onoise with a different random

number generator seed each time. Energy spectra were calculated from the perturbed

saccade position simulations. Similarly, the power spectral density (PSD) of the noise

alone was estimated from averaging sets of 1000 simulations of the same duration (38-

75ms) at each o noise-

The identification of the smooth start and end points of a saccade in the presence of

noise is a non-trivial problem. To avoid truncating the movement and thereby

introducing artificial discontinuities, it is desirable to extend the region for Fourier

analysis beyond the estimated start and end points. However, extending the data region

adds more noise to the finite FFT padding (8192 points). The effect of this source of

increased noise was examined by perturbing an additional 0 ,3 ,5 , 10 and 20 ms beyond

the end of the model saccade with white noise.

3.2.3 Real Saccades

Real saccade data obtained from four of the main eye movement recording devices are

analysed to establish the instrument spectral noise characteristics of the equipment used

in this thesis and elsewhere, and to check on the applicability of the simulation

First, subject 1 (male, 37 years old) made saccades recorded with the infrared limbus

eye tracker (IRl) and the laboratory setting used in all but Chapter 7. ERl was sampled

at 1000 Hz, has a stated optimal resolution of 2minarc, and Bode plot -3dB bandwidth

of 185Hz. Saccades from a second, but identical model, infrared tracker (IR2) were

recorded simultaneously with bi-temporal dc-EOG (1090Hz) in the laboratory used in

Chapter 7 from subjects 2 and 3 (M, 44; F, 23). The IR2-identified start and end times

were used in the Fourier analysis of both IR2 and EOG traces and allowed a direct

comparison of spectra from identical movements but recorded with very different

instrument noise levels.

A fourth subject (M, 38) was recorded separately using two of the common video

oculography systems, the infrared IS CAN camera at 240Hz (VOGl) and the Eyelink II

at 500Hz (V0G2). For completeness, an old data set from subject 2 on a scleral search

coil system (1000 Hz) was also analysed. These three systems (VOGl, V0G2, COIL)

were recorded in different laboratories.

3.2.4 Data analysis

The Fourier methods used were as described in Chapter 2. The first 4 minima (M l, M2,

M3 and M4) and 3 maxima (M xl, Mx2 and Mx3) were measured to a resolution of

0.12Hz. Although durations for the simulated saccades were known, they were also

calculated using the best central difference derivative filter for each given Onoise and

amplitude (see 3.3.3).

Regression analyses were performed among successive minima, and between the

on the simulation data when the unperturbed durations were assigned as the independent

variable (Sokal and Rohlf, 1981). When both variables had variance associated with

them, such as when the simulated or real saccade durations were estimated, better

values of slope and intercept are provided by Model II regressions. Throughout this

thesis these Model II regressions, in which x-on-y as well as y-on-x sums of squares are

taken into account, are referred to as bivariate regressions. The Yarbus SMS go through

the origin as the shape is fixed by definition (Equation 1), but real saccades can and do

change their shape with increasing amplitudes. Hence, unconstrained regressions were

performed on the real data and also, as a check on the methodology, on the Yarbus

saccades as well.

Linear regressions are*biased when the data contain subsets with unequal variances. At

high Onoise the variance in M2, M3 and M4 was found to be considerably uneven across

amplitude ranges (see 3.3.1). Rather than transform each data set so that the data had

even variance across all amplitudes and minima, the effect of the uneven variances

could be assessed by observing the biases introduced by regressions on the analytic

Yarbus model (see Fig. 3.10). In addition, the bivariate regressions were compared to

weighted regressions that are robust to outliers and thus are less affected by uneven

variances.

The robust regression algorithm used was the ‘robustfit.m’ program from the Statistics

4.0 Matlab Toolbox. This is an iteratively re-weighted least-squares algorithm whose

weights at each iteration are calculated by applying the bi-square function W = { I - r^Ÿ

to residuals from the previous iteration. The value r in the weight expression is a

from their median, and the leverage of a least-squares fit (i.e. a measure of the influence

of a given observation on a regression).

To estimate the PSD of the real fixation noise, pre- and post-saccadic fixation periods of

the same duration as the interceding saccade were averaged over at least 200 trials for

each subject. To avoid pre-saccadic EOG spike artefacts and post-saccadic glissadic

forward/reverse drift, each fixation period was chosen at least 50ms before or 400ms

after the saccade and such that no velocity exceeding 107s was included. Temporal

estimates of noise standard deviations were calculated using the same criteria and a

moving window algorithm.