FUNCTIONAL LEARNING: THE LEARNING OF CONTINUOUS FUNCTIONAL MAPPINGS RELATING STIMULUS AND RESPONSE CONTINUA
Authors: Carroll J. Douglas 
Published: August 2014

A general model was proposed in which it was assumed that, in learning situations involving scaled stimuli and responses, subjects will tend to establish continuous functional relations between stimuli and responses. In particular, it was assumed that each subject has available a general “functional form” dependent only on certain parameters (pi), and that in learning the subject effectively assigns specific values to these parameters, thus establishing a specific function defining a unique mapping of stimuli into responses. In the specialized case of the model, the more restrictive assumption is made that the general form is constituted of linear combinations of more basic functions, so that the parameters (pi) may be identified with the weights assigned to each of these primitive functions in establishing a particular stimulus‐response mapping. This “specialized” model was assumed throughout the present study.

Three hypotheses were derived from the postulates of the proposed model, and an experimental study was undertaken designed to test these hypotheses, as well as to answer a number of related questions. The hypotheses were:

Subjects will reproduce responses which bear a continuous relation to stimuli (according to an index proposed as an operational measure of continuity) even when the stimulus‐response pairs they are given to learn are randomly related.

A set of stimulus‐response connections related by a continuous function will be learned more efficiently than a randomly related set. A secondary hypothesis was that “simple” functional relations (defined by few parameters) will be learned more effectively than more “complex” functions (defined by a greater number of parameters).

Subjects will respond to stimuli to which no response has been explicitly associated in learning by interpolating or extrapolating the functional relation to these stimuli.

The experimental paradigm used consisted of a paired associates task involving 26 scaled stimuli (“V” marks varying along the length of a narrow rectangle), only 15 of which were used in the learning phase of each trial, the remaining 11 being included in the reproduction phase to allow observation of interpolation and extrapolation effects. Six conditions were used, in three of which the response (a vertical mark on a line below the rectangular slot) was related to the stimulus according to a simple continuous function (linear in two cases, quadratic in the third), while in the other three conditions stimuli and responses were randomly related. All three hypotheses outlined above were verified. In addition, an analytic technique utilizing Fisher's method of orthogonal polynomials was applied, enabling determination of which polynomials significantly related to mean responses (averaged over six trials), and of which polynomials exhibited significant trial to trial variation in slope. It was found that the first four orthogonal polynomials were sufficient to account for most of the reliable variance in mean responses. Trial to trial variance was slight, but significantly present, while tending to be somewhat more heavily concentrated in the higher degree polynomials. The residual variance in the means, once significantly fitting polynomials were extracted, was generally non‐significant, and no evidence was found that the residuals tended to represent discrete “correction” toward the veridical S‐R pairings.

The data were subjected to an Eckart‐Young analysis with a rotation aimed at finding continuous structure. Three factors were found, very nearly identical with the first three orthogonal polynomials, but bearing a slightly closer resemblance to sinusoidal curves of varying frequency. These accounted for about 88% of the variance in the mean responses, a fact taken as supporting the adequacy of the “specialized” model.