Entropy, visual diversity, and preference

Entropy, visual diversity, and preference

Arthur E. Stamps, III

THE PURPOSE OF THIS STUDY was to investigate relationships between psychological responses of pleasure and diversity and the stimulus feature of entropy. The concept of entropy has captivated the imaginations of many researchers across a wide range of fields. Originally created as a measure of physical disorder, entropy was reinvented in 1949 as a measure of disorder in information (Shannon & Weaver, 1949). Researchers have used entropy to measure basic cognitive limits (Miller’s [1956] magical number seven plus or minus two) and, as a stimulus property, to predict aesthetic preferences (Berlyne, 1974b). The fundamental equation for statistical entropy is Equation 1, in which H is the entropy, p is the probability of occurrence of a level of a factor, and the summation is over the levels of the factors. More detail on entropy can be found in Cover and Thomas (1991) or Renyi (1984).

[H.sub.factor] = – [summation over (nlevels/i=1)] [p.sub.i][log.sub.2][p.sub.i]

Entropy is relevant to applications in environmental aesthetics because more than 90% of the cities in the United States, as well as cities in many other countries, regulate architectural aesthetics (Calderon, 1994; Lightner, 1993; Loew, 1994; Nelissen & de Vocht, 1994; Nystrom, 1994; Pantel, 1994; Uzzell & Jones, 1996; Vignozzi, 1994), and a very common regulatory criterion is visual diversity. Typical goals include promoting visual diversity, avoiding monotony, avoiding chaos, or a combination of all three (Duerksen & Goebel, 1999; Lightner, 1993). In scientific terms, these goals become the hypotheses that pleasure is a positive linear, negative linear, or inverted U function of visual diversity. However, without both an accurate way to measure visual diversity and evidence relating visual diversity to preferences, the validity of those regulations is unknown.

The most relevant psychological work was done by D. E. Berlyne and his colleagues (Berlyne, 1960, 1971, 1974b; Berlyne & Madsen, 1973). The model presupposed by Berlyne’s work is that dependent variables are psychological responses of affect and independent variables are stimulus properties. This model raises two important subissues: how to measure affect and how to measure stimulus properties.

Measuring Affect

In Berlyne’s model, there are two categories of affect: arousal and pleasure. Arousal typically is measured in terms of subjective ratings, such as stimulated/relaxed, excited/calm, and wide-awake/sleepy. Often the scales are used to define a factor of arousal. Examples include an activity factor (Osgood, May, & Miron, 1975; Osgood, Suci, & Tannenbaum, 1957) and an arousal factor (Mehrabian & Russell, 1974). Pleasure also typically is measured by ratings scales, such as happy/unhappy, pleased/annoyed, and satisfied/unsatisfied. Examples of the corresponding factor can be found in the work of Osgood et al. (1957, 1975) and Mehrabian and Russell, The experimental database for the measurement of affective responses is extensive. A review of 40 studies, involving 7,168 participants and 1,768 stimuli, indicated that three factors accounted for most affective responses (Stamps, 2000, pp. 80-85). The factors that accounted for most of the affective-response variance were Pleasure and Arousal (60% and 36%, respectiv ely). That finding subsequently was replicated specifically for physical environments (Russell, Lewicka, & Nut, 1989; Russell & Pratt, 1980; Russell, Ward, & Pratt, 1981). The third factor (Power) has accounted for less variance in affective responses (about 9%). In addition, the literature contains an extensive list of the average pleasure, arousal, and power associated with 3,215 stimuli, ranging from words, emotional concepts, events, and environmental descriptors to lifestyle changes (Stamps, 2000, pp. 82-83).

There are other ways to measure pleasure and arousal. Ratings of pleasure have been correlated with looking time and willingness to stay in an area, for instance. Ratings of arousal have been correlated with physiological responses. See Mehrabian and Russell (1974) and Berlyne (1973) for discussions of physiological measurements. In this article, I use the terms pleasure and arousal to indicate basic types of affect. Other terms are used as subsets of these basic types of affect. Thus, happiness is a subset of pleasure, and excitement is a subset of arousal. Terms that correlate highly with the basic factors are taken to be synonyms (e.g., pleasure, like, preference; aroused, jittery, interested).

Measuring Stimulus Features

Stimulus properties can be described in two ways: with subjective ratings or with physical features. Examples of subjective descriptions include ratings of complexity, interest, or diversity. Physical descriptions, in contrast, are defined in terms of materials and spatial relations. For example, a square can be described in terms of one line, four translations, and four rotations. A circle can be described in terms of a point and a distance. A physical measurement of “circleness/squareness” can be created from the equation

d(x,y)=[([summation over (n/i=1)][[x.sub.i]-[y.sub.i].sup.p]).sup.1/p], (2)

where d(x,y) is the distance from point x to pointy and n is the number of dimensions (n = 2 for two-dimensional shapes; n = 3 for three-dimensional shapes, etc.). When p = 1, the resulting shape is a square; when p = 2, the resulting shape is a circle (Lu, 1997, p. 19). Physical descriptions become quite useful when one is trying to apply research results. For instance, suppose the literature contains experiments that reported findings of rated pleasure in terms of rated circleness/squareness. The researchers who conducted these experiments suggested that rated pleasure was maximized at a moderate degree of circleness. Now suppose one were asked to modify a shape to have the most pleasing circleness. Is there anything in the ratings measures of circleness/squareness that would enable one to adjust the shape appropriately? Clearly not. The best possible advice would be to “create a shape with neither too much nor too little circleness.” In contrast, suppose the literature contained findings that subjective im pressions of circleness/squareness were strongly related to the exponent in Equation 2 and that pleasantness was maximized at p = 1.25. If those data were available, then, when faced with the task of changing a shape to have the right amount of circleness, one would have only to adjust the shape to have a p of 1.25.

The point is this: Subjective impressions of physical features, such as rated complexity, uniformity, or diversity, do not adequately describe the physical features that correspond to those impressions. Therefore, if experimental findings are to be applied to physical environments, the physical features should be defined in terms of materials and spatial relations, not in terms of rating scales. Phrased differently, psychophysical experiments should have psychological responses as dependent variables and physical descriptions as independent variables. In regard to visual diversity, subjective impressions of diversity or pleasure are appropriate as dependent variables, but the independent variables must be physical measures of visual images. Because entropy can be defined solely in terms of frequencies of spatial units (as indicated in Equation 1), entropy is an appropriate candidate for describing a stimulus feature that may correspond to subjective impressions of diversity.

Relations Between Pleasure and Diversity

In Berlyne’s theory (1960), positive affect (hedonic value, aesthetic preference, Osgood’s evaluation factor (1957), or any correlates thereof) is related to degrees of arousal. Degrees of arousal are elicited by stimulus properties. Therefore, aesthetic preference should be related to stimulus properties that elicit specific degrees of arousal. Researchers have suggested that arousal is elicited by stimulus novelty, conflict, complexity, or uncertainty. Many researchers have tried to apply Berlyne’s theory by using ratings of novelty, conflict, complexity, or uncertainty as independent variables, but, as was noted previously, that is a mistake. Stimulus properties need to be described in physical terms. One physical measure of uncertainty is statistical entropy (Berlyne, 1960, pp. 18-44). Entropy is zero if everything is the same, and entropy is maximized if each thing is different. Because total sameness is uniformity and each thing being unique is the maximum possible amount of diversity, entropy should b e a strong candidate as a physical measure of subjective impressions of diversity. Just what degrees of diversity are pleasing appears to be an open question. One plausible hypothesis is that both low and high degrees of diversity are displeasing (boring and confusing, respectively), although moderate degrees of diversity are pleasing, generate positive feedback regarding attention, and thus encourage further attention (Berlyne, 1974a). This is the familiar inverted U function.

A second possibility is suggested by Miller’s (1956) article on mental storage capacity. The concept here is that perceptual processes, cognitive processes, or both have limited capacity. The capacity can be measured in terms of distinct, recognizable categories (e.g., seven) or, equivalently, in terms of bits of statistical entropy. The relationship between equally probable categories and bits of information is simple: bits = [log.sub.2] (categories) or categories = [2.sup.bits] so storage capacity of 7 corresponds to 2.8 bits of entropy. Studies of multiple-stimulus features (Miller, 1994) indicated that adding more and more stimulus features increased the amount of recognized entropy, but at a decreasing rate. Consequently, the results for multiple-factor experiments showed an asymptotic function between stimulus entropy and perceived entropy. The possible application of Miller’s magical number to psychological responses is simply that above a certain level of entropy, there will be no further increases i n reported diversity or pleasure. In reference to responses of pleasure, the implication is that scenes with entropies above some value will appear busy and neither more nor less pleasant than other scenes with high entropies.

Literature

Searches in the PsycINFO database in the summer of 2001 generated 438 references for the key word “entropy,” and 173 references for “visual diversity,” but no references for both “entropy” and “visual diversity.” Early work on entropy and psychology is reviewed in Attneave (1959). Vitz (1964) created sequences of 2, 4, and 8 acoustic tones that were presented at 2, 4, and 8 tones per second. There were significant fits for linear and asymptotic functions between pleasure and entropy. In another study, Vitz (1966) created tone sequences that varied in pitch, duration, and loudness. There was a linear fit between “variation or unexpected change” and entropy, and an inverse U relation between pleasure and entropy. Eisenman (1966) investigated the relationships of rated interest, rated pleasure, and complexity of polygons. It was reported that a monotonic relationship existed between rated interest and complexity, but there was no relationship between rated pleasure and complexity. Vitz and Todd (1969) reported findings on the relation between entropies of groups of letters and arousal responses. There were strong linear fits between complexity and entropy of groups of letters. Crozier (1974) also used sound sequences. There were strong linear relations between ratings of simple/complex and entropy. There were significant inverse U relations between pleasure and entropy. Normore (1974) investigated the relationship between entropy of location, brightness, and duration of sequences of dots and responses of uninteresting/interesting and displeasing/pleasing. There was a strong linear fit between ratings of uninteresting/interesting and entropy. No function provided a significant fit between displeasing/pleasing and entropy. Berlyne (1974b) investigated entropies of arrays of 9 and 36 visual elements and responses of both uncertainty/arousal and hedonic tone. There were strong linear fits between uncertainty/arousal and entropy. No functions fit the relationship between hedonic tone and entropy. Hare (1974) investigate d entropies of colors (factors of area covered and number of colors) on responses of complexity, interestiagness, and pleasure. There were strong linear relations between complexity and entropy and between interestingness and entropy. There was no significant fit for the relationship between pleasure and entropy. Saklofske (1975) investigated the relationship of rated attractiveness and complexity for 15 paintings of human figures and found a significant inverted U relationship.

In the environmental literature, Kaplan, Kaplan, and Wendt (1972) investigated the relationship between rated pleasure and rated complexity for natural, urban, and suburban scenes. The researchers reported that the type of scene mediated the relationship between looking time, pleasure, and rated complexity. Wohlwill (1968) investigated relationships among exploratory behavior, preference, and scaled complexity for scenes of environments and pictures of nonrepresentational art. It was reported that exploratory behavior was monotonically related to rated complexity and that the relationship between preference and rated complexity was an inverted U function. Wohlwill (1975) investigated the relationship between rated pleasure and rated diversity for pictures of environments and pictures of stamps. For the environmental stimuli, the researcher reported that looking time had a monotonic relationship with rated diversity and that pleasure had an inverted U relationship with rated diversity. Thayer and Atwood (1978 ) also investigated the relationship between rated pleasure and rated complexity, but they used five different types of scenes (residential, industrial, urban commercial, strip highway, and park). They were unable to detect an inverted U relationship between pleasure and complexity. Krampen (1979) calculated the entropy of facades of tenement buildings, and obtained subjective responses of both pleasure and diversity. There were two types of facades: facades built before 1900 and facades built after 1940. Entropies were in the range of 1 to 3 bits. There was a strong monotonic relationship between rated diversity and entropy. The type of facade had a strong effect on pleasure, but entropy did not. Nasar (1981) used multiple regression to investigate the relations between preference evaluations, rated dilapidation, and rated diversity of street scenes. Preference was inversely related to both rated dilapidation and rated diversity. Nasar (1983) reported a factor analysis of 18 adjective descriptors of visual a ttributes for 60 residential scenes. Four factors were found: Diversity, Nuisances, Enclosure, and Clarity. Preferences were monotonically related to factor scores of diversity (r = .26). Nasar (1984) examined preferences for urban street scenes. Stepwise regression showed that preference was a function of two factors: Order (indicated by order, naturalness, and upkeep) and Diversity (indicated by high contrast, diversity, and few vehicles). Nasar (1989) investigated the relationships between pleasure, rated coherence, and rated complexity for different configurations of signs. Sign variables included height, shape, color, lettering style, and material. He suggested that preference was highest for high-rated coherence and a moderate amount of rated complexity. Stamps (1994) obtained responses of pleasure for rows of houses that varied in scale (two- or three-story), character (stucco box, Victorian), or both. Entropies for the factors of scale and character could be calculated from the published data. Pleasur e was negatively related to entropy (r = -.21). Elsheshtawy (1997) investigated responses of complexity for rows of commercial buildings. A complexity index was created with variables obtained from the design tradition (overall massing, secondary massing, openings, levels, and groups of levels with different gestalts, textures, widths, heights, and setbacks). A high multiple correlation was reported between the responses of complexity and the proposed complexity index.

Issues Addressed

The literature currently contains conflicting conclusions regarding possible relationships among responses of pleasure, arousal (including impressions of uniformity or diversity), and stimulus features. If psychological research is to be useful in environmental decision making regarding monotony, uniformity, and diversity, then additional work is needed to clarify the conditions under which the relationship between pleasure and visual diversity fits different functions. In addition, to gain useful information for environmental decision making, future researchers on these relationships should use physical measures, not rating scales, to describe environments.

Accordingly, to fill in the gaps in the current literature, in the present study I investigated (a) how well subjective impressions of diversity and pleasure correlate with visual entropy and (b) whether the relationship between entropy and pleasure was best explained by a linear, an inverted U, or an asymptotic function. Figure 1 shows the three curves.

EXPERIMENT 1: Entropy for a Single Factor

Method

Participants

A total of 57 participants were recruited by a professional survey-research firm from the adult population of San Francisco. The participants were not informed of the content of the experiment until after they had arrived at the testing site, so self-selection based on environmental attitudes or architectural preferences was eliminated. Thirty participants were female; 27 were male. Five participants were in their 30s; 20 in their 40s; 10 in their 50s; 6 in their 60s; and the remainder were more than 70 years old. Politically, 18 were liberal, 24 were moderate, and 15 were conservative.

Materials

Because the experiment focused on the effects of visual diversity, the construction of the experimental stimuli was very important. Each of 18 scenes contained an equal number of houses (seven), so that effects attributable to entropy would not be confounded with effects attributable to the number of houses. The scenes were created to have different entropies for different factors. Entropy was calculated according to Equation 1. I designed appropriate artificial scenes to control the factors and their levels. Three factors were chosen for investigation: color, scale, and building silhouette. The factors were selected on the basis of previous work in environmental aesthetics (Stamps, 2000). Each factor is geometrically independent. A house can have different colors without changing its apparent size or shape. It also can have different apparent sizes without changing color or shape, or different shapes without changing its apparent size or color. Colors were illustrated on two-story, simple, boxy houses. Level s for this factor were orange, yellow, green, blue, dark blue, purple, and pink. The second factor was scale. Scale was expressed in terms of setbacks. The farther away something is, the smaller it appears. The levels of setback were in increments of 5 ft (1.5 in), ranging from a minimum of 60 ft (18.2 in) up to 90 ft (27.4 in). The resulting scales, expressed in terms of percentages of the largest apparent size, were 100%, 92%, 85%, 79%, 75%, 70%, and 65%. The same house was shown at different setbacks. Shape was expressed by the creation of seven different houses in the same style (shed roof beach house) and with the same number of silhouette turns (nine). Randomization was used whenever selection was required.

According to Equation 1, entropy is a function of the distribution of the levels of a factor. Thus, if all the levels are the same, the entropy is 0.0. Visually, this would correspond to all houses having the same color, the same scale, or the same silhouette. An entropy of 0.0 implies total uniformity (e.g., all houses green). Conversely, if each house had a unique color (size, shape), then the entropy would be [log.sub.2](7) = 2.8 bits and the visual effect would be the maximum diversity (one orange, one yellow, one green, one blue, one dark blue, one purple, and one pink house). Other distributions (six green houses and one blue house; four greens, one blue, and one purple) would have entropies between 0.0 and 2.8. The technical term for this type of distribution is integer partition. Kreher and Stinson (1999, pp. 67-78) described how to list integer partitions. Once a list of the integer partitions is available, one can use Equation 1 to calculate the corresponding entropies. Table 1 lists the 15 possible integer partitions and corresponding entropies for seven units.

Thus, for the distribution of (6, 1) for colors, it was necessary to (a) select at random two levels from seven (blue and green) and (b) assign locations at random for the six blue houses and the one green house. The same protocols were used for all factors and all stimuli. Figure 2 shows stimuli at different entropies for the three factors.

Design and Procedure

The participants were seated in a small room. Each stimulus was shown as a slide. The slides were shown until all the respondents had recorded their responses. The average time required for each slide was 20 s. The participants rated each slide on an eight-level semantic differential scale of pleasant/unpleasant. Figure 3 shows an example of the response form.

Sample Sizes

Cohen (1988, chap. 2) suggested that relationships with standardized mean differences (d) less than 0.2 would be difficult to detect with the naked eye. He provided the example of trying to estimate the height difference between groups of 15- and 16-year-old women. After inspecting more than 3,000 standardized mean differences between pairs of environmental stimuli, Stamps (2000, chap. 4) suggested that a d of 0.2 or less is very difficult to detect visually. Consequently, the present experiment was designed to detect a d of 0.2. At an alpha level of .05 and with a repeated-measures experimental design, a power of .80 could be achieved with 18 stimuli and 49 respondents (Cohen & Cohen, 1993, p. 449).

Results

The variance for the linear function between entropy and pleasantness was partitioned into the following sources: participants (SS = 1,130, df = 56), entropy (SS = 467, df = 1), type of stimulus variation (color, scale, shape; SS = 134, df = 2), and residual (SS = 1,748, df = 966). The entropy factor was statistically significant, F(1, 966) = 254, P < .001. Type of stimulus (color, scale, shape) was also significant, F(2, 966) = 37, p < .001. The correlations between preference and entropy for type of stimulus were .97 for color, .94 for scale, and .96 for shape.

Functional Relationships

Plots of the results (Figure 4) suggested that either a linear or an asymptotic function might be appropriate for the data in this experiment. Accordingly, both functions were fitted for color, scale, and shape, and all three factors were put together. Nonlinear regressions (Bates & Watts, 1988; Gallant, 1987; Ryan, 1997) produced significant fits for both functions for each factor (p levels of .01 to 7 e-4). In terms of shrunken [R.sup.2]s, the asymptotic function performed better than did the linear function. For color, the respective shrunken [R.sup.2]s were .94 and .85; for scale, .88 and .58; and for shape, .98 and .86.

EXPERIMENT 2: Entropy of Multiple Factors

Method

Participants

The same men and women who participated in Experiment 1 participated in Experiment 2.

Materials

The second experiment had a total of 16 stimuli. The stimuli varied in two factors simultaneously. The factors were shape complexity and degree of facade articulation. I varied shape complexity by changing the number of turns in the silhouette. The levels of this factor ranged from 4 turns to 10 turns. I varied facade articulation by dividing the facade into nine parts and using random numbers to move each part either in or out from the facade’s surface. Figure 5 shows the seven levels of the shape and articulation factors and a row of houses in which each house has a unique shape and a unique articulation. Figure 6 shows diagrams of typical stimuli at different entropy levels.

With two factors, each stimulus has different entropies for each factor. For independent factors, the total entropy is the sum of the entropies of each factor. In this experiment, there were seven levels of entropy for two factors. The entropies for each factor were H= 0.00, 0.59, 1.14, 1.66, 2.23, and 2.80, but because there were two independent factors, the total entropy for this experiment ranged from 0.00 (all houses the same on both factors) to 5.60 (each house unique on each factor).

Design and Procedure

The participants were seated in a small room. Each stimulus was shown as a slide. Slides were shown until all respondents recorded their responses. The average time required for each slide was 20 s. The participants rated each slide on an eight-level semantic differential scale of pleasant/unpleasant and on an eight-level semantic differential scale of uniform/diverse.

Results

Rated Diversity

The data plot for rated diversity and entropy is shown in Figure 7 (top). The variance was partitioned into the following sources: respondents (SS = 1,273, df = 56), entropy of shape (SS = 532, df = 1), entropy of articulation (SS = 94, df = 1), other stimulus properties (SS = 98, df = 13), and residual (SS = 1,246, df = 840). The F values for both shape and articulation entropy were statistically significant: for shape entropy, F(1, 840) = 337, p < .00 1, and for articulation entropy, F(1, 840) = 59, p <.001. Rated diversity was strongly related to the combined entropy of shape and articulation (r = .86). The entropy of shape was more strongly related to rated diversity (r = .85) than was entropy of articulation (r = .36). The data indicate that rated diversity is strongly related to the entropy of the factors (r = .86).

Rated Pleasantness

The plot for rated pleasantness and entropy is shown in Figure 7 (bottom). The variance for pleasantness was partitioned into the following sources: respondents (SS = 1,641, df = 56), entropy of shape (SS = 9.1, df = 1), entropy of articulation (SS = 32.7, df= 1), other stimulus properties (SS = 74.1, df = 13), and residual (SS = 1,021, df = 840). The F values for both shape and articulation entropy were statistically significant: for shape entropy, F(1, 840) = 7.5, p < .006, and for articulation entropy, F(l, 840) = 27, p < .001. The correlations of total entropy, entropy of shape, and entropy of articulation with rated pleasantness were .57, .28, and .36, respectively.

Functional Relationships

Because there was no indication of a decrease in pleasantness at the upper range of entropy, only two functions were fit: linear and asymptotic. Both functions fit the data equally well: for the linear model, F(2, 14) = 6.89, p = .02, shrunken [R.sup.2]=.21 for the asymptotic model, F(3, 13) = 5.4, p = .01, shrunken [R.sup.2]=.14. For these data, the linear and asymptotic curves were nearly indistinguishable. Differences, if any, would not appear until the entropy was higher.

GENERAL DISCUSSION

The major goal of this study was to investigate two issues: (a) how well subjective impressions of diversity and pleasure correlate with visual entropy and (b) whether the relationship between entropy and pleasure was best explained by a linear, an inverted U, or an asymptotic function. The major findings of the present study may be summarized as follows:

1. The correlations (r) between responses of uniformity/diversity and entropy were .97 for color, .94 for scale, .96 for shape, and .86 for silhouette and articulation.

2. There was no support for an inverted U function between pleasure and entropy.

3. The asymptotic function fit the data for the first experiment slightly better than did the linear function. In the second set of data, there was very little difference in the fit between the asymptotic and linear functions.

The present results corroborate previous findings that there is a strong, linear relationship between entropy and subjective responses such as rated interest (Eisenman, 1966), complexity (Vitz, 1966), and uncertainty (Berlyne, 1974c). The present results did not corroborate previous findings of an inverted U function between entropy and pleasure (Crozier, 1974; Vita, 1966). One possible explanation of this discrepancy is that the inverted U function was obtained for auditory stimuli, the presentation rate of which was not under the participants’ control. When given control of the presentation rate, the participants may be able to appreciate more complex stimuli without being overwhelmed.

The contributions of this study may be considered from two perspectives: research and practice. From a research viewpoint, the results of this study have enhanced knowledge about the functional relationships between entropy and impressions of diversity or pleasure. Previous findings on the relationship between the entropies of simple laboratory stimuli and responses such as interest, complexity, and uncertainty were replicated with environmental scenes as stimuli and rated visual diversity as the response. The results of the present work also suggest a new explanation for the function between entropy and pleasure: the asymptotic function. According to this function, pleasure increases as entropy increases above 0 bits. The marginal increase in pleasure decreases, and, after a certain amount of entropy, the pleasure approaches an asymptotic value. In more subjective terms, the function suggests that totally uniform stimuli are unpleasant (e.g., boring), but after a certain amount of entropy, the stimuli appea r to be equally busy. Further increases in entropy produce stimuli that are neither more nor less pleasant. Figure 6 illustrates this idea. The difference in visual diversity between the top two rows of stimuli is clear. The change in entropy is about 1 bit. The change in entropy for the bottom two rows is also about 1 bit, but the impression of the difference in visual diversity between the two bottom rows appears to be far less than the difference between the two top rows.

From a practical viewpoint, the present work has three implications for the design of buildings and cities. First, verbal criteria such as “promoting visual diversity” or “avoiding monotony” are much less useful than specifying diversity in terms of entropy. Second, the present findings suggest that policies requiring more diversity in a given building will result in greater preference if the original design has total uniformity. Third, it is not yet known how preference is related to higher levels of entropy. If the function should prove to be an inverted U, then the level of entropy at which pleasure was maximized could provide the basis for effective design modifications or valid regulation. In contrast, if the function should prove to be asymptotic, then there would be a level of diversity above which design modifications or regulation would be ineffective.

Some potential limitations to this study must be considered. One potential limitation is that an 8-point semantic differential scale was used instead of a 7-point scale. However, a review of studies covering 1,150 stimuli indicated that many common methods of scaling (ratings, rank orders, qsorts, physically placing stimuli on a table, raw score, comparative judgment, true score, and signal-detection theory) generated findings that correlated at .99 (.05 confidence interval = .99-.99; Stamps, 2000, pp. 98-101). Consequently, it is likely that the choice of the number of levels used in semantic differential scales will have minor effects on the results.

Another potential limitation of the present work is that size of presentation room, viewing distance, ambient light levels, time of day, time stimuli displayed, and other conditions under which the stimuli were shown were not controlled for. The reason is that previous studies indicated that psychological responses to environments can be replicated without controlling for these conditions. In one pair of experiments, preferences were obtained for the same 13 scenes but different participants, different locations, different viewing conditions, and different scaling methods were used (Stamps, 1992). The preferences between the two replications correlated at .90. In another experiment on preferences for 35 houses, two sets of responses were obtained from different groups of respondents, in different cities, under different viewing conditions (Stamps & Nasar, 1997). The results again correlated at .90. Feimer (1984) reported findings from a large (1,148 participants) study of effects of experimental conditions on evaluations of environments. One of the tests compared evaluation scores obtained from two different rooms (a room in a church and another room, at a different location, in a school). The effect of interview site was very small, r = 0.006, t(103) = 0.07, p = .94. Similar results were obtained in a literature review of demographic effects on environmental preferences (Stamps, 1999). The review covered data from more than 19,000 participants and more than 3,200 environmental scenes. Of particular relevance was the contrast between results generated from the protocols used in the present article and results obtained in other laboratories. The size of the contrast was quite small, [eta] = .03, (1,36) = .048, p = .83, indicating that the protocols used in this article are quite reproducible. Accordingly, in the present experiments, viewing conditions were not controlled for.

A third potential limitation of the present study is the existence of demographic factors that may mediate the relationships between entropy and pleasure. The literature on demographic preferences for environments has recently been reviewed elsewhere (Stamps, 1999). The review covered more than 19,000 participants from 21 countries and more than 3,200 scenes. Demographically, the groups were constituted according to designers and nondesigners, ethnic affiliation, political affiliation, students, country, sex, age, and special interests. There were three categories of stimuli: scenes of nature, scenes of ordinary architecture, and scenes of avant-garde architecture. The preference correlations between most demographic groups were in the range .82 < r < .89. For instance, the correlation over political affiliation was .86. The exceptions were for participants aged 12 years or younger versus participants older than 12 years (r = .6 1), special interest groups versus others (r = .56), and designers versus nondesi gners for avantgarde architecture. In addition, the authors of three studies reported findings on the specific question of age effects on impressions of scaled environmental diversity. Wohlwill (1975), with a sample of 192 first graders, reported that looking time had a monotonic relationship to scaled diversity, that preference peaked in the middle of scaled diversity, and that age had a relatively slight effect on these functions. Bernaldez, Gallardo, and Abello (1987) had 191 eleven-year-olds and 292 sixteen-year-olds rate landscapes on many scales, including a scale of diversity. They reported that the effect of age on ratings of diversity was not significant. In Nasar's (1989) study of signscapes, judgments of excitement and calmness correlated at .95 and .92, respectively, between participants aged 18 to 34 and participants aged 35 to 55. It would appear that demographic differences, such as viewing conditions, have had relatively small effects on impressions of environmental preference or diversity, an d so demographic factors were not investigated in the present article.

Another potential limitation of the present work is that static color images may not be valid representations of actual environments. However, a review of previous research, based on more than 4,200 participants and 1,215 scenes, indicated that preferences obtained from static color simulations correlated highly (.84) with preference obtained on-site (Stamps, 1993). In addition, it was found that the specific simulation technique used in the present article also worked well. Preferences obtained from simulated buildings correlated at .78 with preferences obtained from photographs of the corresponding real buildings (Stamps, 1997).

A final potential limitation of the present work is the range of entropy that was tested. In the first experiment, the range of entropy was from 0 to 2.8 bits. In the second experiment, the range was extended to 5.6 bits. Those ranges may have been too narrow to detect a difference between the linear and asymptotic relationships between entropy and pleasure. In addition, the inverted U function may not be applicable for stimuli with entropies in the ranges tested.

Future research could be directed toward obtaining a better understanding of factors that may mediate the relationships between entropy and pleasure. One obvious factor is the range of entropy. For example, a study with entropies in the range of 0 to 12 bits would provide a more definitive basis for evaluating the fits for the linear, inverted U, and asymptotic functions. A second obvious factor is whether the presentation rate is under the participants’ control. Researchers could test that factor by creating dynamic simulations of environments and having some respondents view the environments at their leisure while other respondents viewed the environments at fixed rates. (Typical examples would be a self-guided visit vs. a view from a tourist tram.) The simulation factor could then be crossed with an entropy factor, and the resulting analysis of variance would indicate whether control of presentation rate mediated the relationship between entropy and pleasure. In addition, readers interested in demographic effects may want to pursue the hypothesis that the relationships among entropy, visual diversity, and preference are mediated by individual, social, or cultural distinctions.

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TABLE 1

Integer Partitions and Entropies for Seven Units

Number of times each level is

present in the stimulus

Entropy A B C D E F

2.8 (a) 1 1 1 1 1 1

2.52 1 1 1 1 1 2

2.23 (a) 1 1 1 2 2

2.12 1 1 1 1 3

1.95 1 2 2 2

1.84 1 1 2 3

1.66 (a) 1 1 1 4

1.55 2 2 3

1.44 1 3 3

1.37 1 2 4

1.14 (a) 1 1 5

0.98 3 4

0.86 2 5

0.59 (a) 1 6

0.00 (a) 7

Number of

times each

level is

present in

the

stimulus

Entropy G

2.8 (a) 1

2.52

2.23 (a)

2.12

1.95

1.84

1.66 (a)

1.55

1.44

1.37

1.14 (a)

0.98

0.86

0.59 (a)

0.00 (a)

Note. These partitions are used to create stimuli with the desired

amount of entropy. Thus, in order to create a stimulus with an entropy

of H = 1.44, it would be necessary to use three levels. The first level

would be applied to one unit, and the second level would be applied to

each of three units.

(a)The partition was used in this study.

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Address correspondence to Arthur E. Stamps III, Institute of Environmental Quality, 290 Rutledge Street, San Francisco, CA 94110; artstamps@ worldnet.att.net (e-mail).

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