Gaussian Process Regression for Trajectory Analysis

Gaussian Process Regression for Trajectory Analysis Gregory E. Cox (grcox@indiana.edu) George Kachergis (gkacherg@indiana.edu) Richard M. Shiffrin (shiffrin@indiana.edu) Department of Psychological and Brain Sciences, Indiana University 1101 E. Tenth St., Bloomington, IN 47405 USA Abstract Cognitive scientists have begun collecting the trajectories of hand movements as participants make decisions in experi- ments. These response trajectories offer a fine-grained glimpse into ongoing cognitive processes. For example, difficult deci- sions show more hesitation and deflection from the optimal path than easy decisions. However, many summary statistics used for trajectories throw away much information, or are cor- related and thus partially redundant. To alleviate these issues, we introduce Gaussian process regression for the purpose of modeling trajectory data collected in psychology experiments. Gaussian processes are a well-developed statistical model that can find parametric differences in trajectories and their deriva- tives (e.g., velocity and acceleration) rather than a summary statistic. We show how Gaussian process regression can be im- plemented hierarchically across conditions and subjects, and used to model the actual shape and covariance of the trajecto- ries. Finally, we demonstrate how to construct a generative hi- erarchical Bayesian model of trajectories using Gaussian pro- cesses. Keywords: Trajectory analysis; Gaussian processes; Bayesian statistics. Introduction Cognitive scientists are gradually turning toward more fine- grained measures to gain more insight into the continuous nature of the cognitive processes that underly behavior. Per- haps the most widespread of these measures is eye tracking, in which we assume that where people gaze is the current fo- cus of attention and processing. For example, when reading a syntactically ambiguous sentence, people tend to make eye movements back toward the function word or pronoun that best helps resolve the ambiguity (Frazier & Rayner, 1987). Or, when hearing continuous speech, people will tend to look more at objects whose names are consistent with a partially- heard word (e.g., people will look at either a “ball” or a “bear” if they have just heard the syllable “b”), indicating that peo- ple make continuous predictions about the content of speech based on partial information (Spivey, Grosjean, & Knoblich, 2005). Thus, a continuous measure of behavior, like eye tracking, appears to provide insight into ongoing cognitive processes. More recently, researchers have begun to collect explicit continuous behavioral measures in the form of mouse or sty- lus movements (e.g., Freeman & Ambady, 2010). These may easily be used in place of any task that requires an explicit choice on the part of the participant, which includes most ex- perimental paradigms in cognitive psychology. Rather than simply pressing a key to make their response, a participant can instead move their hand (as well as an attached mouse or stylus) toward the option of their choice before selecting (clicking) it. Similar to eye tracking, the trajectories of these continuous motor movements provide a way of measuring the ongoing cognitive processes that lead to the participant’s final choice. A major hurdle with any new measure is the need for ap- propriate analytical tools and statistical tests that allow re- searchers to draw inferences from trajectory data. Due to the richness of this data, many measures are possible and can lead to principled inferences (for an overview, see Freeman, Dale, & Farmer, 2011). When moving their hand while making a decision, people may deviate more from a straight trajectory if there is a tempting alternative, making viable such mea- sures as maximum deviation, curvature area, and switches in direction. In this paper, we introduce a new method for analyzing tra- jectory data. Our method is based on treating trajectories as a Gaussian process, for which there is much well-developed statistical theory. We begin by providing a brief overview of Gaussian process regression and show how it may be ap- plied to motor response trajectories and—more fruitfully, we argue—their derivatives. Finally, we show how Gaussian pro- cess regression can be incorporated into a generative hierar- chical Bayesian model of trajectories. Gaussian Process Regression Gaussian process regression (GPR) is a statistical tech- nique with a long history in spatial statistics, and more re- cently in function estimation and prediction (Griffiths, Lucas, Williams, & Kalish, 2009). The interested reader is directed to the excellent text on Gaussian processes by Rasmussen and Williams (2006). Gaussian Processes A Gaussian process (GP) is simply a collection of random variables, all of which are jointly Gaussian distributed. What differentiates a Gaussian process from the more familiar mul- tivariate Gaussian distribution is the fact that a Gaussian pro- cess may have an infinite index set, that is, it may specify an infinite number of jointly Gaussian variables. Thus, it is possible to define a Gaussian process over a continuous vari- able, like time. Just as a multivariate Gaussian distribution is defined entirely by its mean vector and covariance matrix, a Gaussian process is defined by its mean function m(x) and covariance kernel, k (x, x 0 ), where x and x 0 are two (possibly multidimensional) values of some predictor variable X (e.g., time). We denote the fact that a function f (x) is a Gaussian