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Advanced Statistical Methods for Eye Movement Analysis and Modelling: A Gentle Introduction

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Eye Movement Research

Abstract

In this Chapter we consider eye movements and, in particular, the resulting sequence of gaze shifts to be the observable outcome of a stochastic process. Crucially, we show that, under such assumption, a wide variety of tools become available for analyses and modelling beyond conventional statistical methods. Such tools encompass random walk analyses and more complex techniques borrowed from the Pattern Recognition and Machine Learning fields. After a brief, though critical, probabilistic tour of current computational models of eye movements and visual attention, we lay down the basis for gaze shift pattern analysis. To this end, the concepts of Markov Processes, the Wiener process and related random walks within the Gaussian framework of the Central Limit Theorem will be introduced. Then, we will deliberately violate fundamental assumptions of the Central Limit Theorem to elicit a larger perspective, rooted in statistical physics, for analysing and modelling eye movements in terms of anomalous, non-Gaussian, random walks and modern foraging theory. Eventually, by resorting to Statistical Machine Learning techniques, we discuss how the analyses of movement patterns can develop into the inference of hidden patterns of the mind: inferring the observer’s task, assessing cognitive impairments, classifying expertise.

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Notes

  1. 1.

    Actually, the first who noted the Brownian motion was the Dutch physician, Jan Ingen-Housz in 1794, in the Austrian court of Empress Maria Theresa. He observed that finely powdered charcoal floating on an alcohol surface executed a highly random motion.

  2. 2.

    Freely available at https://web.math.princeton.edu/~nelson/books/bmotion.pdf.

  3. 3.

    More precisely, they used the latest version of Itti’s salience algorithm, available at http://www.saliencytoolbox.net (Walther & Koch, 2006), with defaults parameters setting. One may argue that since then the methods of saliency computation have developed and improved significantly so far. However, if one compares the predictive power results obtained by salience maps obtained within the very complex computational framework of deep networks, e.g., via the PDP system (with fine tuning) (Jetley, Murray, & Vig, 2016), against a simple central bias map (saliency inversely proportional to distance from centre, blind to image information), one can read an AUC performance of 0.875 against 0.780 on the large VOCA dataset (Jetley et al., 2016) (on the same dataset, the Itti et al. model achieves 0.533 AUC). Note that a central bias map can be computed in a few Matlab lines (Mathe & Sminchisescu, 2013).

  4. 4.

    \(\arg \max _{x} f(x)\) is the mathematical shorthand for “find the value of the argument x that maximizes \(f(\cdot )\)”.

  5. 5.

    Given a posterior distribution \(P(X \mid Y)\) the MAP rule is just about choosing the argument \(X=x\) for which \(P(X \mid Y)\) reaches its maximum value (the \(\arg \max \)) ; thus, if \(P(X \mid Y)\) is a Gaussian distribution, then the \(\arg \max \) corresponds to the mode, which for the Gaussian is also the mean value.

  6. 6.

    This gives an intuitive insight into the notion of P(1, 2) as a density.

  7. 7.

    If we have a function of more than one variable, e.g., \(f(x,y, z,\ldots )\), we can calculate the derivative with respect to one of those variables, with the others kept fixed. Thus, if we want to compute \(\frac{\partial f(x,y, z,\ldots )}{\partial x}\), we define the increment \(\Delta f= f(\left[ x + \Delta x\right] , y,z,\ldots ) - f(x,y, z,\ldots )\) and we construct the partial derivative as in the simple derivative case as \(\frac{\partial f(x,y, z,\ldots )}{\partial x}=\lim _{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x}\). By the same method we can obtain the partial derivative with respect to any of the other variables.

  8. 8.

    This is physicists’ preferred notation, which you are likely to most frequently run into when dealing with these problems. In other more mathematically inclined papers and books you will find the expectation notation \(E\left[ f(X)\right] \) or Ef(X).

  9. 9.

    A salience map, and thus the potential field V derived from salience, varies in space (as shown in Fig. 9.32). The map of such variation, namely the rate of change of V in any spatial direction, is captured by the vector field \(\nabla V\). To keep things simple, think of \(\nabla \) as a “vector” of components \((\frac{\partial }{\partial x}, \frac{\partial }{\partial y})\). When \(\nabla \) is applied to the field V, i.e. \(\nabla V=(\frac{\partial V}{\partial x}, \frac{\partial V}{\partial y})\), the gradient of V is obtained.

  10. 10.

    Matlab software for the simulation is freely downloadable at http://www.mathworks.com/matlabcentral/fileexchange/38512.

  11. 11.

    http://antoinecoutrot.magix.net/public/code.html.

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  1. Department of Computer Science, Universitá di Milano, Milano, Italy

    Giuseppe Boccignone

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    Christoph Klein

  2. Department of Psychology, University of Bonn, Bonn, Nordrhein-Westfalen, Germany

    Ulrich Ettinger

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Boccignone, G. (2019). Advanced Statistical Methods for Eye Movement Analysis and Modelling: A Gentle Introduction. In: Klein, C., Ettinger, U. (eds) Eye Movement Research. Studies in Neuroscience, Psychology and Behavioral Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-20085-5_9

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