Concept Generalization in Seperable and Integral Stimulus Spaces
C oncept G eneralization in Separable and Integral Stim ulus Spaces N icolas D avidenko (ndaviden@ psych.stanford.edu) Joshua B. Tenenbaum (jbt@ psych.stanford.edu) D epartm ent of Psychology Stanford U niversity, Stanford, CA M any m odels of hum an concept learning are built around a hypothesis space of possible concepts w ith an associated probability distribution (Tenenbaum , 1999). These hypothesis spaces are difficult to describe, even in the case of tw o-dim ensional stim ulus spaces. G arner (1974) distinguished tw o types of stim ulus spaces: separable (w here sim ilarity judgm ents follow a city- block m etric) and integral (w here they follow a Euclidean m etric). To explain w hy generalization contours from one exem plar are diam onds in separable spaces and circular in integral spaces, Shepard (1987) suggested tw o corresponding hypothesis spaces: the space of axis-aligned rectangles (for separable spaces), and the space of circular discs (for integral spaces). H ow ever, the generalization contours from one exem plar do not uniquely determ ine a hypothesis space (e.g. the hypothesis space of squares under all possible rotations produces the sam e generalization contours as the hypothesis space of circular discs). In this w ork, w e attem pt to constrain the choice of hypothesis spaces by analyzing concept generalization from m ultiple exem plars. O ur prelim inary findings are generally consistent w ith Shepard’s form ulation except for one significant difference: people show correlational concept generalization even in separable spaces. M ethod Fourteen subjects (ages 15 to 49) participated in a tw o- part experim ent. Each part included 6 trials, and the order of the parts w as counterbalanced across subjects. In each trial in part I (concept generalization in a separable space), subjects observed on a com puter screen a group of five stim uli said to be “representative of a larger set.” The stim uli w ere circles of variable size w ith radial lines of variable orientation. The param eters w ere chosen to fall on line segm ents em bedded in the Size-O rientation space and w ere either axis-aligned (varying only in one dim ension) or correlational (varying in both dim ensions sim ultaneously). The five exem plars w ere evenly spread in the stim ulus space. A fter observing the exem plars, subjects rated each of eight test item s according to the perceived probability that it belonged to the sam e set represented by the exem plars, on a 0-10 scale. The test item s w ere placed either on the linear extension of the concept or perpendicular to it. Part II (concept generalization in an integral space) follow ed the sam e design except that the stim uli w ere discs varying in tw o integral color dim ensions: saturation and brightness. R esults and D iscussion Figure 1 show s the predictions using Shepard’s hypothesis spaces of the 50% generalization contours for axis-aligned and correlational stim ulus sets (dark dots). Figure 1 Figure 2 G eneralization in a separable space (top row ) is orientation-dependent; in particular, adaptation to the one-dim ensional extent of the concept occurs only w hen the concept is axis-aligned (A ). In an integral space (C and D ), concept orientation is irrelevant. Figure 2 sum m arizes the experim ental data. Probability ratings of the test item s (averaged across subjects) are show n as num bers. The contour regions w ere obtained by extrapolating the 50% probability boundary. In the separable space (top row ), linear adaptation occurs m ore noticeably in the axis-aligned concept (A ), but also occurs to som e extent in the correlational concept (B), contrary to an axis-aligned rectangular hypothesis space. This finding suggests that the hypothesis space for separable dim ension stim ulus spaces is m ore com plex than originally form ulated by Shepard, perhaps including rectangular regions of all possible orientations w eighted by som e prior probability distribution. W e are currently exploring this possibility through hum an experim ents and m athem atical m odeling. R eferences G arner, W . R. (1974). The processing of inform ation and structure. Experim ental Psychology Series, Potom ac, M aryland, Law rence Erlbaum A ssociates. Shepard, R. (1987). Tow ards a universal law of generalization for psychological science. Science, Tenenbaum , J.B. (1999). Bayesian m odeling of hum an concept learning. Advances in Neural Inform ation Processing System s, 11.