Academic journal article Memory & Cognition

Learning and Extrapolating a Periodic Function

Academic journal article Memory & Cognition

Learning and Extrapolating a Periodic Function

Article excerpt

Published online: 9 March 2013

© Psychonomic Society, Inc. 2013

Abstract How people learn continuous functional relationships remains a poorly understood capacity. In this article, I argue that the mere presence of nonmonotonic extrapolation of periodic functions neither threatens existing theories of function learning nor distinguishes between them. However, I show that merely learning periodic functions is extremely difficult. It is only when stimuli are presented numerically, rather than as numberless quantities, that participants learn anything like a periodic function. In addition, I show that even then, people do not regularly extrapolate periodically. The lesson is that careful methodologies will be required to understand a psychological capacity that is as idiosyncratic as the learning of complex functions appears to be.

Keywords Function learning · Individual differences

(ProQuest: ... denotes formulae omitted.)

Functions, like categories, are an essential aspect of our environment and our relationship to it. We must know how much force to use to lift a bucket as a function of how much water it holds, how long to water the lawn as a function of the day's temperature, and how slowly to drink our wine to enable our safe drive home given how long we intend to stay at the party and how much we have been eating. We all must be able to quickly learn how hard to press the gas pedal to get a given amount of acceleration each time we get into a rental car, while plumbers may learn over time to estimate how far from a junction a leak is by the sound of the turbulent water.

Theories of function learning must explain how people become sensitive to the structure of a causal relationship between metric dimensions, such as frequency and distance or temperature and water loss, from a few examples. The functional form of a causal relationship does not have to be explicitly computed by the learner; indeed, the vast majority of people who ever have and ever will learn functions cannot do even simple algebra. Theories of function learning recognize this and look to describe the cognitive mechanism that enables function learning as being one or another sort of formal system. There are generally assumed to be two possibilities for the kind of computation such a system must perform: Either it hypothesizes one of a small number of explicit functions and times their parameters to fit the data (e.g., Carroll, 1963; Koh & Meyer, 1991; McDaniel & Busemeyer, 2005), or it generalizes from the training examples on the basis of their similarity to novel items (Busemeyer, Byun, DeLosh, & McDaniel, 1997; DeLosh, Busemeyer, & McDaniel, 1997). Precisely which functions should be available to such a mechanism defines various parametric models, and precisely how a mechanism generalizes defines instance-based, nonparametric models. Importantly, these same two possibilities are reflected in normative approaches to the problem of regression. Parametric (e.g., Bayesian regression), on the one hand, and kernel-based (e.g., Gaussian process prediction) techniques, on the other, can both be used to approximate training data to arbitrary precision.

Parametric regression is familiar to all. Given an expression such as /(.r) = ax + bx + c, it is a simple matter to find the values of the parameters {a, b, c} that minimize the error, [f(x) -y]2 associated with the expression when given a set of (a",v) pairs. Bayesian regression (Williams, 1998) similarly uses parametric functions but predicts the posterior distribution of y given the history of (.y,v) pairs by integrating over a set of candidate functions weighted by their posteriors, which are themselves dependent on both the data and the prior distribution over candidate functions. Early approaches to the cognitive processes of function learning (Carroll, 1963; Koh & Meyer, 1991) were concerned largely with identifying the set of candidate functions people (or, ambiguously, their learning mechanisms) may have access to and the facility they (or their mechanisms) have with finding the best-fitting parameter values. …

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