Dangers and uses of cross-correlation in analyzing time series in perception, performance, movement, and neuroscience: The importance of constructing transfer function autoregressive models

Abstract

Many articles on perception, performance, psychophysiology, and neuroscience seek to relate pairs of time series through assessments of their cross-correlations. Most such series are individually autocorrelated: they do not comprise independent values. Given this situation, an unfounded reliance is often placed on cross-correlation as an indicator of relationships (e.g., referent vs. response, leading vs. following). Such cross-correlations can indicate spurious relationships, because of autocorrelation. Given these dangers, we here simulated how and why such spurious conclusions can arise, to provide an approach to resolving them. We show that when multiple pairs of series are aggregated in several different ways for a cross-correlation analysis, problems remain. Finally, even a genuine cross-correlation function does not answer key motivating questions, such as whether there are likely causal relationships between the series. Thus, we illustrate how to obtain a transfer function describing such relationships, informed by any genuine cross-correlations. We illustrate the confounds and the meaningful transfer functions by two concrete examples, one each in perception and performance, together with key elements of the R software code needed. The approach involves autocorrelation functions, the establishment of stationarity, prewhitening, the determination of cross-correlation functions, the assessment of Granger causality, and autoregressive model development. Autocorrelation also limits the interpretability of other measures of possible relationships between pairs of time series, such as mutual information. We emphasize that further complexity may be required as the appropriate analysis is pursued fully, and that causal intervention experiments will likely also be needed.Keywords Autocorrelation . Autoregression . Cross-correlation . Continuous responses . Time series analysis . Granger causality . Prewhitening . Transfer function modelingThe study of relationships between simultaneous time series, particularly those involving continuous human perceptions and performance, has been ongoing in many fields of psychology for several decades (e.g., Brunsdon & Skinner, 1987;Gregson, 1983;Pressing, 1999). To illustrate the ubiquity of time series data in these fields, consider the categorization provided by Pressing (1999) in his synthesis of "the referential dynamics of cognition and action". Pressing enunciates "referential behavior theory," which is a "general dynamical approach to psychological . . . systems that operate through a control or referencing process" (p. 714)-specifically, processes that operate continuously. The analytical approach to such continuous processes is most commonly discrete, in that the data are samples of the process spaced regularly in time, and this is what concerns us here. The resulting "discrete control equation" is normally an example of a vector (multivariate) autoregression time series analysis model of the paired (or several) continuous response/performance time series, w...

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