Internal Representation of III-Defined Perceptual Categories

Internal Representation of 111-Defined Perceptual Categories PETERQUAAS and WINFRIED HACKER

1. Introduction

Classification is essential for coping efficiently with the manifold influences from the external world. Classifications are the basis of nearly all human decisions. They are thus basic to the selection of actions. In recent times, cognitive psychologists have been concerned with the so-called illdefined or fuzzy categories. They play a more essential role in real life than the traditional, better-defined categories. Modern research on concept formation increasingly deals with fuzzy concepts (viz. e.g., Klix, 1976; Kukla, 1976, Strobel, 1976; Das-Smaal & de Swart, 1981). However, in the present chapter, we shall concentrate on ill-defined perceptual categories, whose investigation has become a center of modern research on perception (viz. e.g., Evans & Arnoult, 1967; Posner, Goldsmith, & Welton, 1967; Posner & Keele, 1968; Hacker, 1974; Homa, 1978; Hacker, Dilova, & Kunze, 1979; Hacker & von Sucro. 1979). Analogies to the structure of natural semantic categories cannot be overlooked (e.g., Rosch, 1973). The membership 01’ O I ~ ~ L X111L \~cll-clcl’ined S categories is simply defined by whether or not an object is contained in the category. Therefore these categories must have precisely defined criteria of membership, and all instances of a category are equally good examples of the category. In contrast to this, Rosch (1973) characterizes natural categories by the fact that not all instances of a category are equally good examples of their category. Members of such categories differ in typicality. The best examples or prototypes form the centers of the categories. Already in 1965, Zadeh proposed a mathematical description of so-called fuzzy sets, which fit to natural concepts as well as to ill-defined perceptual categories. A fuzzy set A is characterized by a membership function . J ~ ( . Y ) , which associates with the physical description x of a n object a “grade of membership” between 0 and 1. The nearer the value of&(x) to unity, the higher the grade of membership of x in A. Especially with ill-defined perceptual categories of complex configural stimuli (following the definition by Garner, 1976), the membership in a category depends on many relevant features not all of which must be present; different features may thus be equivalent as classification criteria (Klix, 1971, p. 590; Kukla, 1976). Classification is mainly based on the perceived similarity between the objects. Such categories can only be learned from examples. However, the same new object may be classified by example into more than one category with different grades of membership (overlapping). The relations obtaining are given in Table 1. The efficiency of human activities based on such classifications heavily depends on their mental or internal representation. However, different theories postulate different kinds of such internal representations, depending on different conditions. In this chapter, we try to test this alternative theories in a series of psychological experiments with complex configural visual objects. According to Rosch (1 973), the internal structure of natural concepts or, more general125

Tab. I ; Comparison of well-defined and ill-defined perceplual categories well-defined categories

illdefined categories

I . elements of the category

equally good examples

differing in typicality, prototypes as best examples

2. membership

containment x k A

grade of membershipf,(x)

3. criteria

precisely defined, i.e. all relevant features must be present

based on perceived similarity, i.e. many relevant features, not all must be present, different features are equivalent

4. learning

from examples or criteria

only from examples

5 . transfer after learning

unique

not necessarily unique, i.e. overlapping possible

~

~~

ly, the internal structure of illdefined categories is formed by prototypes as centers of the categories surrounded by instances of decreasing grade of membership. However, the internal representation of the categories is not directly accessible. There are three ways to infer these internal representations: 1. Subjects can judge the grade of membership of different objects in a category (Rosch, 1973). We suppose that these judgments are derived from a judgment of the similarity between the internal representation of the object to be classified and the prototypes or other kinds of category representations. 2. Statements on the prototypes of the categories can be derived from presented objects with maximal grade of membership (central objects). The term prototype has been ambiguously used in the literature. We have to differentiate three aspects: a) the central objects, b) the actual internal representation of the centre of a category, and c) the model of this center, e.g., as a mathematical average or a constructed prototype. We propose to use the term prototype only for aspect b). From the variation of the central objects under different conditions and from the comparison with different models we can infer the internal representation of the center of a category. 3. The internal structure of the categories affects the cognitive processing of the categories (Rosch, 1973). Therefore, inferences about the prototype can be drawn from processing characteristics such as frequency of assignment to a category and the time required for assignment. In the remainder of this chapter, we shall describe different models of internal representation of categories. After some remarks on the material and design of our experiments with illdefined perceptual categories we shall describe relevant data from the literature and then our own results. After that we shall discuss to what extent these results can be explained by the competing models. We conclude with our own conception of the classification process.

2. Competing models of internal categorial representations The different models of internal categorial representations cannot be discussed without ideas about the classification process. We shall discuss the following models: M 1. Integrutiveprototype model: The representation of the center of a category is formed in the course of handling the objects by an integrative, averaging process. This central representation is called prototype. These prototypes need not correspond to represen126

tations of real objects. Objects are classified according to their similarity to the prototypes of different categories. In dimensional models the objects are described as points in a multidimensional feature space. Here the prototypes are modelled by the centroids of the points belonging to the category (see e.g., Posner, Goldsmith, & Welton, 1967; Posner & Keele, 1968).In featural models the prototypes are modelled by lists of common differentiating features per category (e.g., Homa & Chambliss, 1975). M 2. Object prototype model: The center of a category can be a representation of distinguishedpresented objects. According to Rosch (1973), prototypes are those instances that have the most in common with other members of the same category while sharing the least with contrasting categories. However, there are several other ways of distinguishing objects: 1 ) by their mean position in the categorial range of variation, 2) by a more frequent presentation in comparison with other objects, 3) by presentation as examples of the category, or 4) by semantic context. Objects are classified according to their similarity to these distinguished objects. M 3. Average model: All objects belonging to the category are internally stored. Objects are classified according to their average similarity to the representation of all objects of the category (e.g., one of the models discussed by Reed, 1972). M 4. Range model: All admissible transformations for the objects of a category are stored. Objects are classified by a comparison with the transformations stored. (As it is based on random transformations, this model cannot be tested for systematic transformations within the experimental framework of this paper. Systematic transformations were investigated e.g., by Franks & Bransford, 1971 and Geissler, 1976.) In the case of random transformations on one dimension, the admissible transformations can be characterised by the borders of the range of variation within a category. Objects are then classified by a comparison with these border values. This idea proved useful for predicting the classification of unidimensional stimuli (e.g., W. Quaas, 1974; Petzold, 1976) as well as for two-class problems using multidimensional objects (Kukla, 1976). The question arises, whether or not this is applicable also with multiclass problems.

3. Material and design of the experiments The objects to be classified were irregular octagons, constructed by different radial distances of the corners from an imagined center. The variation of the objects can be physically described by variations of the eight corner distances, i.e. the objects are multidimensional and configural. Categories were constructed as clusters in a feature space, the corner distances taken as describing features. The objects belonging to one category were constructed by small radial displacements of the corners in relation to a selected typical object (see Fig. I). The averages of displacements within a constructed category were zero for each corner. Thus the selected typical objects coincide with the mathematical average or centroid within the constructed category, i.e. prototypes according to a special mathematical model. Without additional information, sets of objects so constructed were to be grouped by subjects into the constructed categories on the basis of perceived similarity. In contrast to the more frequently investigated two-class problem, we have investigated object sets with at least three categories. This causes some problems for the representation of empirical results. The perceptual grouping of such a great set of complex objects develops in the course of repeated handling of all objects. However, from the beginning the subjects had to express their reactions as unambiguous classifications. The variation of these unique classification judgments reflects the formation of the perceptual grouping. We assume that this formation is finished, if on the basis of the 127

v

Fig. 1 : Selected typical objects (prototypes) for a three-class problem.

perceptive grouping a stationary unique assignment of all objects occurs. Frequently, investigators compare the categories formed by the subjects with the constructed “objective” categories, defining the misfits as “classification errors”. There are objections against such an approach, because - at least in the case of classification tasks without external feedback there is actually no correct classification. Each subject may form his own grouping, and deviations from the constructed categories have to be interpreted as a measure for the fit of a model. We think that it is better to describe the performance of the subjects by goodness of judgment such as concordance of judgment between the subjects or consistency of judgment across successive trials (Quaas, 1980). Nevertheless, from the frequency of assignment to a given category we can deduce the grade of membership of objects in the category. We assume that the frequency with which an object is assigned to a category is correlated with its grade of membership. The grade of membership of an object depends on its similarity to the center and to other elements of the category, and on the similarity to contrasting categories. For the following discussion a simplified representation of the results is convenient, where the frequency of assignment to the constructed category is regarded as a function of the similarity to the constructed prototype, taking into account only those objects belonging to the constructed category and averaging over all categories. With this representation we can verify whether or not the constructed prototypes are on average the centers of the categories. It is also possible to discuss the influence of the variability within the category and the influence of context and additional information. However, the averaging of frequencies permits statements only on the overall effect of the contrasting categories, which can be sufficient for two-class problems, but requires additional considerations for multiple classes. Classification time per object was recorded as another dependent variable. ~

4. Experimental results on the classification into illdefined perceptual categories The dependence of the grade of membership and the classification time on the following experimentally independent variables is of intrinsic interest but may be used at the same time as a means of choosing between different kinds of internal categorial representation. 128

The influence of the internal structure of a category on the classification of aa individual object can be regarded as an internal context effect. This effect depends on the similarity to the center of the category and on the variability of the objects within the category. The variability can be manipulated experimentally by varying category range, category size, and distribution of objects over category range. The influences of the contrasting categories on the classification can be called external context effects. We shall consider the effect of the number of categories, the grade of differences between the categories and the saliency of differentiating categorial features. Finally, classification depends heavily on additional information about the constructed categories. We shall compare the additional presentation of individual examples differing in their position within the range of variation of the constructed categories, and the effects of repeated classification of all objects with feedback according to the constructed categories.

4.1. Effects of internal context 4.1.1. Similarity of the presented object to the prototype

How does the grade of membership for different objects depend on the similarity to the prototype? By way of answer, we shall single out one special experiment (Zedler, 1978; Quaas, 1980), which will ,be discussed in more detail later on. Similarity differences between two irregular polygons were measured physically by city-block distance. i.e. the sum of the absolute differences between corresponding corner distances.' Stirriiili: Three categories of 14 objects each were constructed from the prototypes in Fig. 1. The rules of deviation were the same for all categories and are demonstrated in Fig. 2. The objects belonging to a category were homogeneously distributed over a constant range of distances from the prototype. Procedure: The 42 objects were presented in random order under instruction to classify them into three categories, with 17 subjects given yes-no feedback; the remaining 18 subjects got no feedback, but the constructed prototypes for each category were displayed for them during the whole trial. In each case the subjects had to continue classifying all objects until reaching the criterion of the same responses for two successive blocks of 42 objects. Results: Fig. 3 shows the relative frequency of assignment to a category averaged over the subjects, the trials, and the different categories as a function of the city-block distance from the constructed pi (1iotypes. Zero distance thus equals the constructed prototype itself. The corresponding y iiclients are supposed to reflect the different degrees or grades of membership. Here we discuss only two facts common to both feedback and no feedback: I ) Central objects with maximal grade of membership coincide with the objects constructed as means of the category. 2) The grade of membership decreases I The ocpagons were constructed by connecting points on eight radii where the angle between neighbouring radii had a constant value of 45 degrees. Different octagons were produced by varying the radial corner distances from the center. The city-block distance dAB between two octagons A and B was then calculated by 8

dAB

=

i= 1

b A t

- xBil

'

The correlation between city-block distance and the subjective similarity is extremely high (rank correlation r = 0.80 _ _ _ 0.90 for 20 objects). 9

Geissler. Modern Issues

129

pro t o type

a a a

Fig. 2 : A category of objects as an example.

with increasing distance from the prototype. This confirms that the categories formed by the subjects are really illdefined ones. 4.1.2. Cutegory range

The category range characterizes the maximal distortion or distance from the prototype, up to which objects are predominantly assigned to the given category. It may be affected by variatim of the mean distortion within the constructed categories, and it may depend upon the mean similarity of all objects belonging to the category. I n the above experiment category range was held constant, but the literature includes some interesting results. Brown, Walker, and Evans (1967) asked subjects to assign randomly generated histoform patterns to three categories. They varied the variability of the column heights within categories, which was described by a redundancy measure. Without any additional information, classification was easier with lower varialiility within the categories. This effect was not found for learning with feedback. Also Posner, Goldsmith, and Welton (1967) observed that (with four categories of randomly distorted dot patterns) classification is easier with smaller variability. Finally the same effect was found by Aiken and Brown (l971), with randomly generated octagons classified without additional information. Homa and Vosburgh (1976) demonstrated that the subjective category range can be changed by feedback. Categories of dot patterns with low distortion are learned 130

0

1

2

3

4

5

6

d

Fig. 3 : Mean relative frequency .7 of assignment to a category for learning with prototypes and with feedback as a function of the city-block distance d to the constructed prototype (relative units).

faster than categories with low, medium and high distortions. But.the transfer of the learned classification to new patterns with low, medium and high distortions is worse with low distortions, i.e. narrow categories are not so easily generalized. Summarizing the results, we may state the trivial fact that it is easier to classify as members of the same category objects that have more in common. Moreover, it is easier to classify a distorted pattern as a member of a learned category with higher variability (i.e. greater category range). We suppose that the internal representations of categories also depend on the variability of objects within the category. Therefore, we now take up additional possibilities for modifying this variability. 4.1.3. Cutegorv .size

The influence of category size, i.e., the number of objects belonging to a constructed category, has been investigated from two different points of view. Some papers have investigated clussijicurion learning with feedback using random dot patterns (Homa, Cross, Cornell, Goldman. & Schwartz, 1973), artificial faces (Goldman & Homa, 1977), and random polygons (Homa, 1978). The subjects had simultaneously to learn several categories differing in category size. After learning, the subjects had to classify new objects constructed from the same categories. Transfer was better for categories with greater category size. Thus the more the subjects get informed on possible variations within the category during learning, the better the transfer. Our own experiments (Richter, 1975; Breitenfeld, 1977; Zedler, 1978; Hacker & von Sucro, 1979; Quaas, 1980) studied classification with knowledge of the prototypes but without feedback. Material: From three irregular octagons 7, 14 and 20 distorted objects were constructed; and from 10 other octagons, 3 and 6 distorted objects were constructed. In all cases, the objects were distributed homogeneously over a constant category range. The construction rules are illustrated by Fig. 2. 9'

131

Procedure: We used criterion-matched groups of subjects. The prototypes were present throughout the experiment. Subjects classified the whole set of objects three times. Results: Fig. 4 shows the mean relative frequency of assigning objects to the constructed categories from five comparable studies :

50 ..

I

a-0

10 categories

L

10

20

n

Fig. 4: Mean relative frequency Jof assignment to the constructed categories for repeated classification with knowledge of prototype as a function of category size n with 3 and 10 categories.

With three categories the frequency of assignment decreases with increasing category size, whereas there is no difference in the case of 10 categories. With the constructed prototype present, classification deteriorates with increasing category size. This seems to contradict the results obtained in the studies by Homa and his collaborators. On the other hand, there are essential differences in method. An experimental variation of category size may not only change the variability within categories but also provide different information about the categories as well. This will be discussed in Section 4.3.2. 4.1.4. Distribution of the objects over the category range

An inhomogeneous distribution of the objects can be generated by an inhomogeneous frequency of presentation of several objects. This also changes the variability of objects within a category though the range and the size of the categories remain constant. Kukla (1976) had subjects classify irregular polygons (earlier used by Stenson, 1968) into two categories. The polygons could be subjectively ordered on one complex feature dimension. By means of feedback different groups learned to classify objects differently positioned on the feature dimension and with differing frequencies (see Fig. 5 for one example); the border (mb) of the constructed categories was set at the center between the two extreme values of the feature dimension. He obtained the following results: 1. Central objects with a maximal subjective grade of membership and minimal classification time were objects with extreme feature values. Their position was constant in spite of different frequency distributions (the one from Fig. 5 and a mirrored one). 2. Classification of new peripheral objects with feature values near (mb) depended on the special distribution of frequencies (i.e. different functions of memberships were formed during learning) : The subjective category borders with equal subjective grade of membership in both categories and maximal classification time moved to the category K with lower variability, i.e. the range of the complemental category f? with higher variability was increased. I32

category i? with high within- variabifify

category K with low

mb

m

Fig. 5 : Relative presentation frequency p for objects with different values m of the complex feature dimension for two categories. m,-border value of the constructed categories (adapted from Kukla, 1976).

Hacker, Dilova, and Kunze (1979) investigated effects of a more frequent presentation of individual objects, the so-called frequency accentuation. Mutcviul: Sixty objects were constructed by choosing six prototypes and generating nine distorted objects, each similar to Fig. 1. The distorted objects were homogeneously distributed over a given range of distortion. Procedure: Accentuation was produced by presenting particular objects 10 times more frequently than other objects. The effect of a homogeneous frequency distribution was compared with accentuation of the prototype P, of the so-called border object B, subjectively extremely dissimilar to the prototype, or both of them. Three groups of seven subjects each repeatedly classified the 60 objects into six categories without feedback. Results: Fig. 6 shows the mean frequencyJof assignment to the constructed categories as a function of decreasing similarity to the constructed prototype for the third trial: I . Clearly, the accentuation of individual objects modifies the function of membership. In each case the grade of membership of the more frequent objects is increased (as compared to homogeneous frequency). The effect on the other objects shows the following tendencies : a) With accentuation of the prototype the grade of membership is increased for central objects and decreased for peripheral objects. b) With accentuation of the border object the grade of membership is greatly decreased, and the function of membership is inverted. Almost all objects are randomly assigned. c) With accentuation of both prototype and peripheral object, the grade of membership increased for all objects. In each case the assignment frequency can be compared with the chance value of random assignment. 2 . Subjects’ verbal reports suggest two phases of classification (which may occasionally be simultaneous): I n the beginning there seems to take place a comparison with the objects identified best. Gradually, an averaged categorial representation is formed that takes account of other elements within the category. For a more detailed discussion see Hacker et al. (1979). The interaction between accentuation of individual objects and category size was investigated by Hacker and von Sucro (1979). M u t e r i d : SI\and 10 irregular octagons, respectively, were selected as prototypes. Nine objects each were constructed by distortions of these prototypes (as in Fig. 1) 133

80

-4

50 homogeneous distribution accentuation of P

30

0)

(PI 1

2 3 4

5 6

8

7

9 rank of dissimilarity to P

A 80

8

-

accentuation of B

homogeneous distribution random assignment

(PI 1

b)

2 3 4

8

7

5 6

9 rank of dissimilarity to P

accentuation of P and B

30

homogeneous distribution

-

201 lo

-

0

__-------------random I

I

I

I

1

1

,

I

I

assignment

-

Fig. 6: Mean relative frequency Jof assignment to the constructed categories for repeated classification with frequency accentuation of a) prototype P. b) border object B, c) prototype P and border object B as a function of ranked dissimilarity to the constructed prototype in contrast to classification without accentuation (Iiomogmeous distribution) and random assignment (adapted from Dilova, 1976).

134

and ordered according to their subjective similarity to the prototype. In the case of six categories, the prototype and all nine distortions were the memkrs of each constructed category. In the case of 10 categories, the prototype and two distortions homogeneously distributed over the range of all 10 objects were selected. Procedure: Ten subjects each repeated the classification of all corresponding objects for three times with the constructed prototypes for each category presented on cards. The experimental design varied the category size (3 and 10 elements) and the distribution of the presentation frequency of individual objects (homogeneous distribution and accentuation of the border objects by presenting them four times more often than the other objects). Fig. 7, adapted from Hacker and von Sucro (1979), shows the mean relative frequency

-

40 30

-

10 -

occentuation of B

20

0)

P

0-0

homogeneous distribution 0--0 ,

1

1

1

1

1

1

2 3 4

1

5

6

7

1

,

8 9=B

rank of similarity to P

homogeneous

-

distribution 0--

0

accen tuotion o f 8-0

b)

P

1

2

3

4

5

6 7 8 9-8

ronk of similarity

Fig. 7 : Mean relative frequency S o f assignment to the constructed categories for repeated classification with presented prototypes for a) category size 3, b) category size 10, as a function of ranked dissimilarity to the presented prototypes with homogeneous frequency distribution (0-0) and accentuation of the border object U ( 0 - 0 ) .

I35

17)of assignment to the constructed categories as a function of ranked

dissimilarity to the presented prototypes with gradients analogous to those in Fig. 6. We found: 1. In the case of smaller category size (3 elements) there is no significant difference between the grades of membership with and without accentuation of the border objects. 2 . In the case of greater category size (10 elements), the grade of membership of the other objects is decreased by the accentuation of the border object. Thus the negative effect of the accentuated border object only spreads out over the other objects in the case of greater category size.2 3. The verbal reports of the subjects differ with category size: With small categories, the subjects more often compared the objects to be categorized with the representations of other particular objects already assigned ; with large categories, the objects are compared with integrative internal representations of the categories.

4.2. Effects of external context 4.2.1. Number of categories

Several of our investigations studied the overall effect of the number of categories on the frequency of assignment to a given category. Material: In each case a corresponding number of irregular octagons were taken as prototypes and a corresponding number of distorted objects were generated from them in the way discussed in Section 4.1.1. All constructed categories had the same range, and the distorted objects were homogeneously distributed over the range of each category. Procedure: Smaller category sizes (n) with different number of categories ( k ) were investigated by Quaas (1980, n = 7 with k = 3), Hacker and von Sucro (1979, n = 6 with k = lo), and Richter (1975, n = 5 with k = 12). Greater category sizes were investigated by Hacker and von Sucro (1979, n = 10 with k = 6) and Zedler (1978, ti = 14 with k = 3). In each case subjects repeated the classification of the whole set of objects three times, with the constructed prototype for each category always present. Results: Fig. 8 shows the mean relative frequency of assignment to the constructed categories as a function of the number of categories ( k ) , averaged over the objects of each constructed category, the corresponding categories, the subjects, and the three trials. 1. For smaller as well as for greater category sizes, the frequency of assignment decreases with increasing number of categories. The same tendency was found by Homa and Chambliss (1975) for classification learning with feedback using distorted dot patterns. 2 . Classification time per object increases with increasing number of categories. -The.number of categories has its greatest effect upon decision time: the more alternatives, the longer the decision time. The degree of membership to other categories also varies: the more categories, the greater the membership in other categories. Whether or not the decreasing frequency of assignment is a result of a decreasing grade of membership cannot be discussed on the basis of the above results.

Since only homogeneous and accentuated distributions are compared in each case, we assume that the difference between the number of categories (chosen to prevent extreme differences in the size of the whole object set) does not matter.

136

5

10

k

Fig. 8 : Mean relative frequency,rof assignment to the constructed categories for repeatcd classification with knowledge of prototypes as a function of the member of categories k with smaller category size ( n = 7 with k = 3. n = 6 with k = 10. ti = 5 with k = 12) and greater category size ( n = 10 with k = 6 . n = 14 with k = 3).

4.2.2. Differences between categories

In all our experiments with irregular polygons, we tried to hold constant the similarity distances between the alternative prototypes. Now, we will discuss the influence of differences between categories. Our research on this problem is based on letter recognition. Usually, letter recognition is investigated with short displaying-time or decreased figure-ground contrast (e.g., Townsend, 1971). Such conditions of so-called state constraint (Norman & Bobrow, 1975) given, the internal representations of the presented letters are incomplete as compared to more easily perceptible letters. The subjects have learned the letters as natural categories with varying grade of membership and have formed categorial memory representations. They can just solve the task by comparing the incomplete internal representation of the presented letters with the categorial representations. From an analysis of the confusion matrix in letter recognition tasks (Quaas, 1980), we only cite the following result : The frequency of assignment to a category is the greater, the less similar a prototype is to the prototypes of all other categories; i.e. objects of a category with an extreme position (in the similarity structure of all categories) are classified more easily. 4.2.3. Saliency of differen t ia ti jig categorial ,feutures

The influence of the saliency of differentiating categorial features was discussed in more detail elsewhere (Quaas, 1982). Here we describe two of the conclusions. Material: For two kinds of material, three categories of 20 irregular octagons each were constructed by distortions from suitable octagons shown in Fig. 9 in the mode described in Section 4.1. With the so-called feature material the octagons selected as prototypes for category construction mainly differed in salient features, i.e. the presence of typical internal angles (marked by bold lines). With the sinzilarity material, the prototypes differed mainly in their global appearance. However, the city-block distances between the constructed categories were constant, independent of the kind of material, feature vs. similarity. 137

Fig. 9: Prototypes for the two kinds of material. Typical interior angles are marked by thick lines; a) feature material. b) similarity material.

Procedure : The experimental design combined the two kinds of material with two kinds of learning. All objects of the three categories constructed were presented in random order. Ten subjects each had to classify them into three categories. Half the subjects were given yes-no feedback. For the other half, the objective prototypes were always present. In each case, subjects had to repeat the classification of all objects to a criterion of two successive trials without change. Results: 1. The number of trials to reach criterion was significantly smaller with feature material. 2. The mean frequency of assignment to the constructed categories during learning was smaller with feature material than with similarity material for learning with prototypes, but not for learning with feedback. This can be explained by the strategies of information processing (Quaas, 1982): with presented prototypes, a similarity comparison with the prototypes seems to prevail ; whereas with salient differentiating features, feature-comparison seems to prevail. Thus, learning feature-material with prototypes first induces comparisons with the prototypes, but during learning the strategy is changed to a feature comparison more suitable to salient differentiating features. This change may enlarge temporarily the uncertainty of assignment and decrease the frequency of assignment.

138

4.3. Effects of additional information The constructed categories correspond to clusters in a multidimensional feature space.

As already pointed out, subjects succeed in classifying such clusters without any addi-

tional information, but additional information will facilitate classification. 4.3.I . Prescritation of e.uanipli>.s

The additional presentation of examples for each category differing in their position in the range of the constructed categories was discussed by Hacker (1974) and Hacker and Quaas (1977). Murerial: Seventy one complex geometrical line patterns each consisted of 5 connected line elements, 4 straight and 1 curved line, varying in length and mutual position. For each of eight arbitrarily selected “prototypes”, eight variations were constructed by change of scale, rotation, reflection, distortion and random changes of lengths and angles. Subjects, uninstructed on this kind of transformation, experienced them as random variations. Procedure : Four groups of 10 subjects each classified all objects sequentially presented in random order three times under the following conditions: 1 . without any additional information (condition 0); 2. with the “prototypes” used for category construction presented as examples (condition P); 3. with the “border objects” presented as examples, i.e. that pattern of each category which is subjectively felt most dissimilar to the prototypes (condition B); 4. with the prototypes and the border objects of each category presented as examples (condition PB). Results: We assume the increase of the frequency of assignment and the decrease of classification time caused by additional examples in contrast to condition0 to be the greater the more the examples facilitate the formation of suitable categorial representations. Tab. 2 shows the mean classification time t per object and the mean relative frequency f of assignment to the constructed categories (averaged over all members per category, all categories and all subjects in the first and third trial): 1. If only one example per category is presented, the increase of assignment frequency is the example. This-suggests that it is the actual constructed prototype that best corresponds to the subjective representation of the category center. Tab. 2: Mean relative frequency f of assignment and mean classification time t for classification without additional in/ormation (0).with border objects ( B ) .prototypes ( P ) and both ( PB) as examples, dijferences

off

to condition 0 and their approximate proportions

1. trial :

condition

0

B

P

PB

3

55.3

59.7 % 4.4 % 8.9 sec

67.8 % 123% 3 7.9 sec

80.6 % 253% 6 16.6 sec

76.7 % 0.9 % 1 5.4 sec

87.8 % 12.0% 13 7.1 Sec

95.5 % 19.7% 22 8.3 sec

.f -.A

proportions 1

3. trial:

f

f

~

8.7 sec 75.8 %

-f o

proportions 1

-

-

5.6 sec

1

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2. Simultaneous presentation of prototypes and border objects further increases the assignment frequency, and this increase exceeds the summed increase by presentation of both individual objects separately. Simultaneous presentation of both examples provides additional information on the variation within the constructed categories. Classification time for condition PB on the first trial is approximately twice that for the other conditions, whereas this difference disappears by the third trial. This suggests that on the first trial both examples are separately taken into account but that by the third trial an integrative categorial representation has been f ~ r r n e d . ~ 4.3.2. Kinds of’ learning As discussed in Section 4.1.3., the two approaches to the classification of ill-defined categories show essential methodological differences. Homa and coworkers (e.g., Homa, 1978) made subjects learn to classify with feedback and used categories with differing size on the same trial. Our subjects repeated the classification of an object set a few times with the prototypes being presented, and the category size was varied between experimental groups. To eliminate these differences and to compare the two approaches using the same material, we employed the design described in Section4.1.1. (Zedler, 1978, Quaas, 1980). Procedure : Subjects had to classify 52 irregular polygons into three categories. Seventeen subjects were given a yes-no feedback. Eighteen other subjects classified the object set with knowledge of the constructed prototypes for each category. In each case the subjects had to repeat the classification of all objects until the classification of each object was the same for two successive trials. Results: Here we discuss the differences shown in Fig. 3. The mean relative frequency of assignment is averaged over all trials. 1. As could be expected, the relative frequency of assignment for the constructed prototypes (zero distance) is nearly 100 per cent for subjects classifying with presented prototypes. Subjects learning with feedback have to extract information on the center of the category themselves. This is obviously not as effective as with presented prototypes. 2. I n the course of learning, there are further differences: in the first trial feedback has nearly no effect with very small assignment frequencies, but reaches 100% effectiveness in the last trials. On the other hand, classificdtion with prototypes starts with higher assignment frequencies, judgments stabilize quicker, but still show systematic differences from the constructed categories after learning. 3. Therefore, we investigated the classification of new objects constructed from the same prototypes and distributed homogeneously on the category range (transfer). As can be seen in Fig. 10, there is no difference in the mean frequency of assignment. But the frequency of assignment for central objects with presented prototypes is higher, whereas with feedback it is higher for peripheral objects. Therefore, we conclude that presented prototypes will improve the internal representation of the category center and that it is more difficult to gain this information by feedback. On the contrary, by feedback subjects get informed also on the variability of the category; therefore, they are better at classifying peripheral objects than are prototypic subjects. Finally, it is possible to discuss the influence of category size for the two kinds of learning under comparable conditions (Breitenfeld, 1977 ; Zedler, 1978 ; Quaas, 1980). Material: From each of the three octagons shown in Fig. 2, nineteen distorted objects were constructed in the manner described above. For the variation of category size, The effect of different informational components contained in the presented examples is analysed by a decomposition method (Hacker & Quaas, 1977).

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7 , 14, or all 20 objects were used, each set homogeneously distributed over the constant category range. Procedure: Different groups of subjects classified all objects with feedback (n = 14 and 20) or with the constructed prototypes presented on cards (n = 7, 14, and 20). Classification was repeated up to the criterion of equal judgments on two successive trials. Results: Fig. 11 shows the regression for the mean relative frequency f of assignment to the constructed categories, averaged over all trials as a function of the increasing city-block distance (4 from the constructed prototypes. We can deduce the following: 1. With presented prototypes, the grade of membership decreases with increasing category size. This decrease itself enhances with greater distances from the constructed prototypes. With feedback the degree of membership increases with increasing category

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.

size, and this increase remains nearly constant with increasing distance from the constructed prototype. 2. We have to take into account that with presented prototypes the additional information is not affected by category size; but with feedback, the additional information increases with category size. 3. Another fact is related to the multidimensionality of the objects and to the special construction of categories with differing numbers of objects homogeneously distributed over category range: the distance from the prototype is not the only dimension varying. With increasing category size, variation increases within categories. Measuring withinvariability by the mean mutual city-block distance C?, of all objects we found no difference between n = 14 (C?, = 4.93) and n = 20 (a, = 4.80), but variation was 20 percent lower with n = 7 (a, = 3.93). Thus, we can summarize the influence of category size on the grade of membership as follows: There was no difference of within-variability between the two experimental groups with feedback. Therefore the increase of the grade of membership with increasing category size must be due to the simultaneous increase of information. For classification with presentedprototypes, there was no difference in variation between n = 14 and n = 20 and, due to constant additional information, no difference in grade of membership. The increase of variation in relation to n = 7 produces the decrease of the grade of membership. From feedback the subjects may learn which variations belong to the category, but feedback gives less information on the category center. On the contrary, prototypes yield good information on the category center independently of category size but no information on variation. Without additional information, increasing variation yields a decrease in grade of membership.

5. Discussion and consequences What can we infer about internal representations of ill-defined perceptual categories? We prefer models that explain most of the results of a series of experiments, for there are too many factors to be controlled by a single experiment. Let us begin with some general remarks about the models to be discussed.

5.1. Competing models for categorial representation M 1 . Zntegrative prototype model: The grade of membership is described by the similarity to an integrative representation of the category center. We think that on the basis of our experimental results it is not possible to choose between dimensional or feature models. It is always possible to find suitable features which offer viable alternative interpretation to dimensional models: The center can be described by the centroid or by extreme values of common features. There are several suitable similarity measures for dimensions and features (Quaas, 1980). However, in the centroid differences between the categories are not taken into account, whereas Homa and coworkers define the prototype by common differentiating features. To appreciate the experimental findings, those dimensions that completely describe the physical variation of the objects must be employed (e.g., differences of corner distances). Equivalent descriptions may exist (e.g.. polar coordinates). Internal processes like feature extraction and assessment of similarity can be discussed on the basis of such an objective description, but for validation further experiments are necessary.

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M 2. Object prototype model: The grade of membership is described by the similarity to the representation of a salient presented object. Neither the influence of perceptual saliency of objects, nor the symmetry and the semantic context have been investigated in the experiments discussed. M 3. Average model: The grade of membership of an object is described by the average similarity of this object to the representations of all objects belonging to the category. The average similarity decreases with increasing distance from the center. The center is described by the object representation with the least average similarity to all other objects. But the essential problem is to define which objects do belong to a category. In the case of classification without feedback, additional assumptions are necessary. From the experimental results we may include e.g. all obiects with assignment frequencies greater than the chance value of random assignment. Earlier we also proposed models with a similarity threshold in order to predict which objects belong to a category (Quaas, 1974). In the case of learning more than two categories with yes-no feedback, we could ask whether only those objects that were correctly assigned belong to the category. M 4 . Range model: Here the categorial representation consists of the multidimensional borders of the range of variation. We may assume that objects within the category range belong to the category and that their grade of membership decreases when approaching the border (because of the increasing uncertainty of assignment near the border). Thus, central objects with maximal grade of membership are those objects within the range that are maximally distant from all borders.

5.2. Effects of internal context on the grade of membership 5.2.1. Positioti of’ the center

Let us first summarize results accordihg to the position of central objects: 1. With two-class problems, central objects are objects with extreme values of a differentiating feature. I n each case, they differ from the most frequent object in their category (Section 4.1.4.). 2. With more than two categories, the central objects coincide with the mean of the constructed categories, i.e. objects with maximal average similarity to all other objects (Section 4.1 . I .). In the same way, the grade of membership of all objects is increased by the additional presentation of examples provided that these coincide with the mean of the constructed categories (Section 4.3.1 .). 3. Frequency accentuation of the border object (i.e. the object most dissimilar to the mean of the constructed category) makes classification impossible: only objects close to the border are recognized as belonging together, all others being assigned randomly (Section 4.1.4.). 4. With frequency accentuation, subjects reported that a comparison with the accentuated objects takes place first; but in the course of repeated classification, an averaged category-representation is formed (Section 4.1.4.). All models mentioned describe the decrease of the grade of membership with increasing distance from the center. The difference in position of central objects for two-class problems (extreme values) from that for multiclass problems (mean values) may be due to subjects redefining the classification task originally intended by the experimenter: subjects look for a differentiating feature in order to discriminate between two categories so that extreme values are most easily discriminated; with more than two categories, the category means are most representative; moreover in multiclass but not in twoclass problems, representations of the mean might serve to diminish the demands upon 143

working memory, and thus upon effort. If we continue to believe in a simultaneous integration and differentiation following Rosch (1973) and Homa and Chambliss (1979, the following arguments appear cogent : 1. The grade of membership in different categories dominates the similarity between individual objects. Consequently, the subjective similarity between objects of different categories increases after classification in the same category. This was found with the classification of unidimensional stimuli (e.g., W. Quaas, 1974) but remains to be experimentally verified for the multidimensional objects discussed in this chapter. All objects of a category may be equally good examples with respect to differences from other categories, but the mean is most representative of the common properties within the category. 2. Differences between categories may only affect the decision process if the grades of membership in all contrasting categories are compared. 3. We suppose that the classification process is sequentially organized (Quaas, 1980). In a first stage, global similarity comparisons take place. Thereby, in some cases, two categories may remain with nearly equal grade of membership. In that case, in a second stage, these two categories are discriminated on the basis of differentiating features. In this way the different strategies mentioned in Section 4.2.3. may be explained. The other experimental results are equally consistent with each of the alternative models. There remains the question of whether the center of the category is a representation of a salient presented object or an integrated prototype. In the early stages of classification tasks that employ either a more frequent object (frequency accentuation) or the presentation of examples, comparison with the representation of these objects will take place. Eventually, an integrative prototype is formed. When this happens, frequency accentuation and presentation of particular examples are no longer effective unless the objects concerned are similar to the category mean. The integrative prototype is also confirmed by the fact found (e.g., by Posner and Keele. 1968) that after learning, objects at the mean of a constructed category are assigned with highest frequency, even if they had not been presented during learning. 5.2.2. Variation within categories

We shall begin with a summary of essential results: a) General result 1. The grade of membership of objects with equal distance from the center may be changed depending on the differing variation: - connected with a change of the distribution of presentation frequencies (shown for learning of two categories with feedback) (Section 4.1.4.); - connected with a change of category size (exemplified for classification into three categories with presented prototypes) (Section 4.3.2.); - connected with different frequency accentuations (exemplified for more than two categories) (Section 4.1.4.).

b) Effect of variation without additional information 2. The frequency accentuation of central objects can be interpreted as a decrease of variation as compafed to a homogenous distribution, if variation is measured by the mean city-block distance of all objects including the repetition of the accentuated object. This causes an increasing grade of membership for central objects and a decreasing grade for peripheral objects (Section 4.1.4.). 144

3. With presented prototypes, the increase of variation by an increase of category size causes a decrease of the grade of membership for all objects. This decrease is greater with greater distances from the center (Section 4.3.2.). 4. The frequency accentuation of border objects corresponds to an increase of variation and causes a decrease of the grade of membership, but not for categories of small size. With small categories, subjects reported mainly comparisons with all of the individual objects ; with larger categories, they reported mainly comparisons between integrative categorial representations (Section 4.1.4.). c) Effects of additional information on variation

5. Learning categories of greater variation with feedback will extend the subjective category range (Section 4.1.4.).Therefore, broad categories may be generalized more easily (Section 4.1.3.). 6. If categories of different category size but constant variation are learned with feedback, the increase of information on the variation is connected with an increase of the grade of membership for all objects (Section 4.3.2.). 7. The simultaneous presentation of constructed prototypes and border objects can also be interpreted as yielding information on variation. Consequently, the grade of membership of all objects increases. With repeated blocks of trials, classification-time was reduced by half. This may be interpreted as an integration of information on both the center and the variation into a complex integrative categorial representation (Section 4.3.1.). The frequency accentuation of both objects effects an increase of the grade of membership, too (Section 4.1.4.). The influence of variation on the grade of membership cannot be explained by the two prototype models. The decrease of the grade of membership with increasing variation would be attributed by the average model to an average increase in distance to all other objects, i.e., smaller subjective similarity. This decrease of average distance enhances with ,greater distances from the center, thus exactly explaining our statement 3. The decrease of the grade of membership with accentuation of central objects (statement 2) can be explained in the same manner. However, here the average model does not predict the decrease of the grade of membership for peripheral objects of the categories, since their mean distance to all other objects remains constant. On the other hand, in the range model the decrease of the grade of membership with increasing variation could be explained by a flattening of the membership function near the subjective border (see Fig. 12).

1

low variability

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Fig. 12 : Comparison of the grade of membershipf, in category A for high and low variation of the category as a function of the distance d from the category centre. 10 Geissler. Modern Issues

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The decrease of variation associated with accentuation of prototypes could then be used to explain the increase of the grade of membership for most of the objects. Also a shift of the subjective border could be postulated. However, the range model has to be elaborated further, since (in contrast to the average model) it does not describe the relationship between the physical description of the objects and the membership function. In summary, the average model seems to yield the simplest explanations in the sense that it entails the fewest number of assumptions. However, the average model presumes that all individual objects are stored. This seems unlikely with large categories, but it is suggested by statement 4 of Section 5.2.2. for small categories. Memory load connected with large categories can be diminished by use of prototypes. However, those cannot account for the effects of variation within categories. Therefore, the combination of integrative prototypes and subjective ranges appears to be an attractive alternative, but such a model remains to be elaborated mathematically.

5.3. Effects of external context on classification There are only a few results on the effects of external context: 1. Frequency of assignment varies inversely with similarity to other categories (Section 4.2.2.). 2. With feedback, narrow categories are easier to learn (Section 4.1.2.). We also found that salient differences with mathematically equal city-block distances facilitate classification learning both with feedback and with presented prototypes (Section 4.2.2.). If we assume a weighting of features for calculating subjective similarity, then salient differences will correspond to subjectively greater distances between the categories. 3. With increasing number of categories, frequency of assignment decreases and classification-time increases. The mode discussed so far only describe the internal representation of categories, i.e. the grade of membership. In order to explain the assignment of an individual object we assume a'comparison of the grades of membership of the object in all possible categories. The less the grade of membership to other categories, the greater the frequency of assignment to the highest-grade category. This accounts for statements 1 and 2, because the grade of membership in other categories decreases with decrease in similarity. The effect of an increasing number of ill-defined, partly overlapping categories can be explained by the possible membership of objects in an increasing number of other categories and consequently an increasing uncertainty in the assignment decision. However, the results reviewed here give no hints on how the comparisons are realized. Two possibilities are discussed in the literature :

Is

Assignment according to maximal grade of membership (extreme value model) :

So far, the disadvantage of this decision model has been that the uncertainty of a decision could not be predicted. However, on the basis of Zadeh's theory of fuzzy sets, we have developed a model (see appendix) which describes the uncertainty of a decision by the maximal grade of membership in one of the other categories. This must be tested experimentally. However, without further assumptions this model cannot predict the influence of increasing member of categories. Assignment by comparing the grades of membership with a threshold value (threshold model) :

In the threshold model, the decision uncertainty does not depend on the grade of member-

ship in contrasting categories but rather on the difference between the grade of member146

ship and a threshold. This threshold value could reflect individual differences in risk behaviour. A decision on the basis of the threshold model is not necessarily unique, i.e. the grade of membership can be above threshold for two different categories. Then a differentiation of the two categories could follow (see Section 5.2.1.). Maybe there is a relation between the threshold and the subjective border emphasized by the range model. I n both decision models the comparison with the individual caiegories can be performed either in parallel or sequentially in time. It may be possible that the extreme value model describes the behaviour in the initial period of learning. With increasing experience with the object set to be classified, there could be a transition to processes described by the threshold model. We think the threshold model to be effective only when uncertainty is as low as possible. This could be gained as Tollows: after classification a change in the subjective grade of membership might cause assimilation (increase of the grade of membership in the extreme value category) and contrast (decrease of the grade of membership in the remaining categories). An examination of these ideas requires measurement of the subjective grade of membership and also testing the predictions of the alternative decision models (based on these grades of membership) against empirical classification data.

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