Abstract
Circular data are encountered throughout a variety of scientific disciplines, such as in eye movement research as the direction of saccades. Motivated by such applications, mixtures of peaked circular distributions are developed. The peaked distributions are a novel family of Batschelet-type distributions, where the shape of the distribution is warped by means of a transformation function. Because the Inverse Batschelet distribution features an implicit inverse that is not computationally feasible for large or complex data, an alternative called the Power Batschelet distribution is introduced. This distribution is easy to compute and mimics the behavior of the Inverse Batschelet distribution. Inference is performed in both the frequentist framework, through Expectation–Maximization (EM) and the bootstrap, and the Bayesian framework, through MCMC. All parameters can be fixed, which may be done by assumption to reduce the number of parameters. Model comparison can be performed through information criteria or through bridge sampling in the Bayesian framework, which allows performing a wealth of hypothesis tests through the Bayes factor. An R package, flexcircmix, is available to perform these analyses.
Highlights
Eye movements are commonly used to study aspects of cognition and its development (Henderson, 2003; Itti & Koch, 2001)
One topic of interest using saccade directions investigates the existence of general directional biases (Tatler & Vincent, 2009), such as a preference for saccades along the horizontal axis (Foulsham, Kingstone, & Underwood, 2008) or a preference for leftward saccades (Foulsham, Gray, Nasiopoulos, & Kingstone, 2013)
Because the mixture model deals with peakedness by fitting multiple components for a single mode, it is impossible to compare variances of components, which is something of interest in many saccade direction studies, such as in Van Renswoude et al (2016)
Summary
Eye movements are commonly used to study aspects of cognition and its development (Henderson, 2003; Itti & Koch, 2001). One topic of interest using saccade directions investigates the existence of general directional biases (Tatler & Vincent, 2009), such as a preference for saccades along the horizontal axis (Foulsham, Kingstone, & Underwood, 2008) or a preference for leftward saccades (Foulsham, Gray, Nasiopoulos, & Kingstone, 2013) Another topic of interest is eye movement behavior when reading (Rayner, 2009). Because the mixture model deals with peakedness by fitting multiple components for a single mode, it is impossible to compare variances of components, which is something of interest in many saccade direction studies, such as in Van Renswoude et al (2016). The model that will be developed in this paper for saccade direction data has two main characteristics It will be a mixture of circular distributions.
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