heuristic approaches to spatial analysis in archaeology

Society for American Archaeology

Heuristic Approaches to Spatial Analysis in ArchaeologyAuthor(s): Keith W. Kintigh and Albert J. AmmermanSource: American Antiquity, Vol. 47, No. 1 (Jan., 1982), pp. 31-63Published by: Society for American ArchaeologyStable URL: http://www.jstor.org/stable/280052 .

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HEURISTIC APPROACHES TO SPATIAL ANALYSIS IN ARCHAEOLOGY

Keith W. Kintigh and Albert J. Ammerman

This article discusses an approach to spatial analysis which is more closely tailored to archaeological objec- tives and archaeological data than are more "traditional" quantitative techniques such as nearest neighbor analysis. Heuristic methods, methods which make use of the problem context and which are guided in part by intuitively derived "rules," are discussed in general and with reference to the problem of spatial analysis in archaeology. A preliminary implementation of such a method is described and applied to artificial settlement data and artifact distributions from the Magdalenian camp of Pincevent. Finally, the prospects for further development of heuristic methods are elaborated.

SPATIAL ANALYSIS MAY BE SEEN as a process of searching for theoretically meaningful patterns in spatial data. Of course, this problem has been approached by archaeologists in several ways. The most obvious method of spatial analysis is the visual examination of a point distribution on a map with relevant background information in mind.

This intuitive approach has been forsaken (and berated) by many archaeologists with greater aspirations to rigor, in favor of quantitative techniques of spatial analysis, such as nearest neighbor analysis. These techniques generally yield a summary statistic which attempts to characterize the spatial pattern with a single number and perhaps test its significance. The summary statistic is commonly compared from period to period or from area to area.

This article reports the progress of an experiment in an alternative approach to the analysis of spatial patterns. This approach, the heuristic approach, is synthetic in that it attempts to open the way for the use of contextual knowledge and human expertise within a formal (computer- executed) procedure for aiding human-directed spatial analysis.

This presentation starts with a brief review of "traditional" quantitative approaches to spatial analysis. It is followed by a discussion of heuristic approaches to problem solving and their application to spatial analysis in archaeology. In the next section, heuristic procedures that have been developed are applied to artificial data sets and then to an analysis of actual data from the Magdalenian camp of Pincevent. The article closes with a discussion of the conclusions of this experiment and prospects for further development.

"TRADITIONAL" APPROACHES TO SPATIAL ANALYSIS IN ARCHAEOLOGY

During the last ten years, there has been a growing interest in the development of formal methods of spatial analysis in archaeology. This interest stems, in part, from dissatisfaction with informal characterizations of spatial patterns. It is also related to the increasing emphasis now being placed on the explicit statement of archaeological research problems and their quantitative evaluation. There is a widespread belief that we should go beyond the "eyeballing" of spatial distributions and develop more objective approaches to the recognition of spatial patterns.

At the same time, the size of the data sets and the effort involved in searching for patterns by hand have led to the use of computer-associated methods of analysis and display. This is especially true for very large data sets and for those in which multiple levels of patterning may be in-

Keith W. Kintigh, Arizona State Museum, University of Arizona, Tucson, AZ 85721 Albert J. Ammerman, Department of Anthropology, State University of New York, Binghamton, NY 13901

Copyright ? 1982 by the Society for American Archaeology 0002-7316/82/010031-33$3.80/1

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volved. Hodder and Orton (1976) provide a comprehensive review of the state-of-the-art methods of spatial analysis used in archaeology (see also Clarke [1977]; Flannery [1976]; Hietala and Stevens [1977]; Whallon [1973, 1974]; Pinder et al. [1979]).

It is apparent that rather than develop methods and models specifically tailored to the theoretical problems and forms of information of their own discipline, archaeologists have borrowed ex- tensively from other fields, such as human geography (e.g., Dacey 1963, 1973; Haggett et al. 1977) and ecology (e.g., Pielou 1969; Kooijman 1979). While borrowed methods such as nearest neighbor analysis and various forms of quadrat analysis have made it possible for archaeologists to con- duct quantitative analyses, the results obtained from these applications have not proved to be nearly as productive as had been hoped.

Some of the problems involved can be illustrated by a caricature of the path followed by an archaeologist embarking upon a spatial analysis for the first time.

(1) The analyst begins with a complex distribution of points, along with a good deal of background information, and perhaps some ideas concerning the processes that may have been responsible for the observed distribution.

(2) With regard to these data, the analyst formulates some basic questions which are to be answered by the analysis. At a minimum these probably include the identification and interpretation (explanation) of patterns in the distribution.

(3) The data are visually inspected in hopes of gaining insights required to interpret them satisfactorily. While there may be a perception of some patterning in the data, it is difficult to characterize adequately the regularities perceived.

(4) The time spent in visually examining distribution maps while waiting for a flash of insight, and the difficulties met in attempting the evaluation of even a few intuitive hypotheses, lead inex-

orably to the belief that there must be a better way. (5) After a perusal of the literature, the analyst may make an initial attempt to carry out a

nearest neighbor analysis using a hand calculator. However, it is likely that this laborious task will soon be turned over to the nearest available computer. The results of computer analysis will

probably indicate primarily whether the spatial distribution should be characterized as clustered, random, or uniformly spaced.

(6) At this point, there is the alternative of reporting results which do not directly address the

analyst's original interpretive questions or of returning to further intuitive interpretation. This account of the frustrations encountered by the archaeologist is admittedly exaggerated.

Methods such as nearest neighbor analysis may be used productively in some situations. In the 1970s a variety of attempts was made to tailor these methods more closely to the demands of ar-

chaeological situations (Pinder et al. 1979) and to extend their scope along interpretive lines (Han- son and Goodyear 1975). But even with these modifications, it does not appear that our ambitions with regard to spatial analysis are likely to be achieved.

It is worth noting that human geographers trying to develop so-called "form to process" arguments in spatial analysis are aware that there are still many questions which cannot be

readily answered by existing statistical methods. Among these geographers, there is evident an active interest in the development of new analytical methods (Cliff and Ord 1975; Haggett et al.

1977). Seen from an archaeological perspective, the existing methods are impaired by the fact that

they operate in ways which are fundamentally independent of the context of a given problem. We use context to refer to that knowledge which we as archaeologists have about a particular set of

data, other than the information typically used by these analytical methods. For example, nearest

neighbor analysis uses only point locations in space; however, in formulating an interpretation of a settlement map, human analysts employ the site locations displayed on a map along with general and specific knowledge of the environment, geography, and human behavior.

In many fields of problem solving there is an increasing realization that the incorporation of contextual information is essential to the recognition of complex patterns (see Toussaint [1978]). In "traditional" quantitative approaches to spatial analysis the problem context does not play a

significant role in the formulation of statistical measures or in their evaluation. It was with the

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SPATIAL ANALYSIS IN ARCHAEOLOGY

limitations of such approaches in mind that we became interested in exploring the feasibility of heuristic approaches to spatial analysis in archaeology.

AN ALTERNATIVE APPROACH TO SPATIAL ANALYSIS

Informal approaches to spatial analysis have often been hindered by a lack of systematic rigor and information processing capacity, while quantitative approaches to this problem have been crippled by the naivete of their assumptions. The alternative we are advocating, which may be called heuristic spatial analysis, attempts to combine the intellectual sophistication of intuitive approaches with the information processing capacity and systematic benefits of quantitative treatments.

In our consideration of spatial analysis as a search for underlying structure in a spatial distribution, we are making the distinction between patterning as seen in the spatial distribution of items on a map, and structure as related to more abstract organizing principles which are held to be responsible for the generation of a given pattern. It is assumed that there are more or less orderly cultural processes (most likely of a complex, multiform nature) that generate the spatial distributions that we examine for patterns. At the outset what is required are methods of direc- ting a search through a large set of complex spatial relations among the objects of interest, in order to "recognize" patterning. This is done with the awareness that to engage in spatial analysis requires some prior theoretical ideas or models of possible spatial structures through which the essentially content-free patterns can be related to the culturally interesting structures which produced them.

At the same time, however, the complexity and the unordered components of a distribution serve to obfuscate the patterns. of relationships among the pointber of relationships among the points (sites, artifacts, or whatever) is generally much too large to permit a simple exhaustive check of all the combinations against a list of possible patterns. Even if the number of possible patterns. Even if the number of points is fairly small, the number of spatial relationships among the points could be astronomical. Thus, the only alternative is a more sophisticated search guided by heuristic rules. Heuristic rules may be thought of as rules of thumb, rules which facilitate discovery but which are not absolute rules leading directly to the solution of a problem. In this kind of heuristic search, context plays a basic role in the evaluation of the relationships between distributions, patterns, and structural models.

HEURISTIC METHODS

The heuristic approach has been pioneered in a branch of computer science called artiicial intelligence and the related multidisciplinary field of cognitive science (Simon 1980). Heuristic approaches have been applied to such diverse problem areas as natural language understanding (Winograd 1980; Erman et al. 1980), game playing (notably chess), mass spectroscopy, robotics, mathematical theorem proving, and the diagnosis of human illness.

Heuristic methods grew out of the intractability of many complex problems attacked by cognitive scientists. (See Simon [1969] for a discussion of the nature of complex problems and their solutions.) Even well-defined problems like winning at chess have not yielded to computa- tional brute force, and analytic solutions to these problems have not emerged. It is quite clear, however, that relative to the most advanced computer efforts, people are extraordinarily adept at solving some of these sorts of problems. The best human chess players can still beat the best computer programs, and compared with current computer abilities, even grade school children do quite well at understanding written English (to say nothing of spoken English).

Studies of human strategies for solving complex problems have led cognitive scientists to the notion of a heuristic search directed toward some abstract goal. Successful (or "intelligent") problem solving computer programs in artificial intelligence are often modeled, at least in part, on heuristic methods used by human problem solvers (Newell and Simon 1976; Simon 1979).

For our purposes a heuristic analysis will have three components. First, there is the problem space: the data and the relationships among them which are to be investigated. The second com- ponent is a set of goal models toward which the search is directed. Third, there is a body of

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knowledge about the problem domain which is not sufficiently complete to lead us directly to the solution, nor so inadequate that all possible solutions must be examined in order to find the best one. This knowledge about the problem domain is formulated in terms of heuristic rules, or simply heuristics. Such guides to discovery are used to direct the search toward its goal. Although heuristic rules are not infallible in their application-they may direct the search to a dead end-they are nevertheless considered to be indispensable in helping to solve problems which do not have more direct solutions. It may be pointed out that Tversky and Kahneman (1974) have illustrated some of the systematic biases in judgment which result from errors in typical human heuristics. However, their paper should not be construed as an argument against the use of heuristics, but as a warning to be aware of some of the errors they may contain.

Finally, the notion of successive refinement of heuristic search procedures should be noted. Heuristic methods direct a search for the goal by choosing at each step the most profitable avenues to explore. During the process of searching, information can be extracted concerning the

problem which will further improve the ability of the search procedure to converge on its goal in successive steps.

A HEURISTIC APPROACH TO SPATIAL ANALYSIS

It seems to us that spatial analysis in archaeology lends itself particularly well to a computer- assisted heuristic approach. Archaeologists engaged in spatial analysis examine spatial distributions and relate them to theoretical constructs; yet, while their intuitive methods of pattern recognition may be admirably insightful, human analysts have limited capacities to "try out" alternative ideas or models mentally, especially as the number of points involved becomes large. Insofar as a computer program can simulate these human processes, it can enormously expedite the analysis because of the speed with which it can evaluate alternative models and process large quantities of data efficiently.

What we are attempting to do is to build a number of intuitively derived heuristics into an

analytically useful computer program. Our immediate goal is to focus on heuristics which extract information from the problem space, not to develop a full-blown artificial intelligence program capable of an essentially complete analysis. That is, we wish to construct a program that will examine spatial data and derive concise higher level information concerning their distribution, information which will help identify underlying structures in the data. Such a procedure would provide a step-by-step display of its results for consideration by the archaeologist.

Given the nature of archaeological data, it appears that the initial filtering and condensation of data is the most serious weakness in human analytical processes. Even if the program were to meet our wildest expectations, it would not replace the human analyst: rather, it would assist in the time-consuming initial analysis, leaving the more difficult interpretive problems to the ar-

chaeologist. Furthermore, there would always be the opportunity to intervene at any step in the

analysis and redirect the search along lines suggested by results of a previous step. The three elements required by this outline of a heuristic problem-solving strategy are: the

specification of a data base, a statement of goal models, and the formulation of heuristics. The data will consist of a set of point locations within a specified area. In an archaeological study such points might represent either site locations in a region or artifacts on a living floor. Of

course, associated with these points are other dimensions of information such as site size or artifact type. (It should be clear that our concern in this article is with the analysis which begins after data have been collected in the field. Of course, the ultimate quality of any analysis depends on the research questions and on the methods under which the data were collected.)

To date, four basic point patterns have been employed as goal models: clustered, uniform, linear, and random. Figures la-d show examples of these basic patterns. Our objective is not

simply to characterize a pattern in a reductive manner and stop there. Instead, we wish to evaluate the degree of fit between some distribution and one or more of these models, to identify the components within a pattern, and to explore and perhaps explain their structure. Due atten-

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Kintigh and Ammerman]

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tion will also be paid to those portions of the distribution which do not fit the pattern. Once the basic patterns have been identified, there are several questions that can be asked:

(1) If the pattern is divisible into components (clusters or linear segments), what configurations are found among the components? Are the clusters randomly or uniformly spaced? In the case of linear patterning, can the components be said to form a network?

(2) What spatial characteristics (e.g., size, shape) do the components have? (3) What is the internal organization of each of the components? Are the points in a cluster

uniformly or randomly spaced? (4) What can be said about the points which do not fit the pattern? How are they located with

respect to each other and the pattern components? For example, the procedure could look at the clustered distribution shown in Figure la and pro-

vide us with a number of heuristic indicators showing a pattern of relatively similar, uniformly spaced clusters with about 10 points per cluster and a cluster radius of about 1.2 units. The points appear to be somewhat uniformly spaced within the clusters. It could further tell us which points are in each cluster and which points do not fit in any cluster.

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Figure lb. Uniform site distribution.

While this sort of analysis tells us a great deal about the distribution, there is still much left for the archaeologist to do. Knowledge of the geographical and cultural situation will have to be used by the analyst to explain the patterns recognized. In particular, the archaeologist might propose a structural model which would account for such patterns.

One might argue that in the simple case of the clustered distribution (Figure la), what the procedure discovers is patently obvious. In this case the point would be well taken, although to tabulate the information provided by the program would be a substantial task if done by hand. However, what if the pattern were not so clear? What if the clusters overlapped, or if there were

actually a linear pattern superimposed over a clustered pattern? Most archaeologists would welcome a program which could propose an intuitively satisfying series of objectively derived and

explicitly justified levels of patterning. Such program results could be accepted or modified to

help form a satisfactory interpretation or explanation of the pattern. The archaeological utility of such an approach is borne out, for example, by the results obtained in the Pincevent analysis presented below.

The advantages of such a program at a more advanced level of development are even more evi-

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Figure lc. Linear site distribution.

dent. Suppose each site in a settlement distribution is assigned a site type, such as hamlet, village, town, or center. The program could individually consider, as described above, the distribution of sites according to type for each of the different levels of patterning. The same thing could readily be done with many different variables. In this way a large number of possible determinants of settlement pattern could be examined and evaluated.

CURRENT IMPLEMENTATION

We have made a general argument for the utility of a heuristic approach to spatial analysis and we have outlined some aspects of such a procedure. It remains to be seen, however, if such an approach will work in a real situation. We have therefore constructed a prototype procedure in terms of which the utility of our heuristic approach can be evaluated. The development of this program began with the lower level steps which extract information concerning the problem data, and we are now working toward their greater integration.

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AMERICAN ANTIQUITY [Vol. 47, No. 1,1982

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Because of the frustrations with current quantitative techniques discussed in the second section of this paper, we decided to start from scratch in developing a heuristic program to help iden-

tify point patterns. In the abstract, the procedure should start with a map of a distribution and iteratively search for patterns within that map. Because we assumed that there could simultaneously exist many levels of patterns for such a procedure to identify, it was necessary to have an ongoing display (map) of the search as it proceeded.

In developing this procedure we started by identifying intuitively satisfying patterns in actual settlement maps in which the patterns were relatively clear. While working with these maps we kept a record of how we went about finding the patterns and attempted to identify their salient properties. In this way we developed a fairly clear idea of how the procedure should unfold and what results it should yield. After this abstract design was developed, we found that a k-means cluster analysis essentially operationalized the procedure we had independently defined. As will be seen, this procedure proved capable not only of identifying intuitively obvious patterning but also of providing analytical insight when no obvious patterning could be discerned.

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Operation of the K-Means Procedure

The k-means program is a nonhierarchical divisive cluster analysis which attempts to minimize the intracluster variances while maximizing the intercluster distances. We have found that, modified to compute some additional statistics, it provides a great deal of information about point distributions. The method is divisive because it starts with a single group (cluster) and subdivides that group to obtain more clusters. It is nonhierarchical in that the clustering stages are not strictly nested (i.e., it is not necessarily possible to draw a "tree" diagram of the clusters). This is in contradistinction to the commonly used hierarchical agglomerative techniques such as single link average link cluster analysis. In addition, k-means operates directly on the input data, in contrast to other commonly used clustering methods which employ a similarity matrix. Because of the nature of the algorithm, k-means is able to process a much larger number of items than is practical with these other methods. K-means also deals directly and explicitly with the level of overall clustering, as well as with individual cluster centers and cluster compactness. All of these characteristics were advantageous for the sort of analysis and development we were an- ticipating. (In archaeology, the k-means procedure has been used under very different circum- stances by Hodson 1971; see Doran and Hodson 1975:180-184, 218-251], who provided us with a copy of his computer program. A related method has been used by Johnson and Johnson [1975]).

Although the program will process data of arbitrary dimensionality, we are here concerned only with the clustering of two-dimensional data, i.e., the locations of objects on a surface. For clari- ty, the discussion and formulae are presented for the two-dimensional case, although (except for r2), all are easily generalizable to any number of dimensions.

The program accepts as data the x and y coordinates of a set of objects (e.g., sites, stone tools). Its goal is to find (locally) optimal cluster configurations from one to some user-specified maximum number of clusters. There is no set rule for choosing this maximum. However, there are usually a priori reasons why no more than some number of clusters is likely to be of interest. The maximum can then be set somewhat higher than this a priori limit. The k cluster configuration (also referred to as the kth clustering stage) is simply a division of all of the objects in the objects in the analysis into one of k different clusters. Each cluster in a configuration is defined by its centroid (or center of gravity) and by the objects assigned to that cluster. The centroid is the point with the mean x and mean y values of all of the objects in the cluster. The clustering criterion the program is designed to minimize is the sum squared error, or SSE. The SSE is calculated as the sum over all objects in the analysis of the squared Euclidean distance from each object to the centroid of the cluster to which it is assigned:

N SSE = [(xi-xc)2 + (Yi-Yci)2]

i= 1

Where (xi, Yi) is the location of the ith point of an N point data set and (xci, Yci) is the centroid of the cluster containing point i. (If the cluster configuration is highly clustered, the average distance from an object to its cluster centroid will be relatively small and the SSE will be correspondingly low.)

The program starts with all points included in a single cluster. Then the SPLIT procedure, which creates new clusters by subdivision, is repeatedly executed (forming 2, 3, 4, etc. clusters) until the maximum specified number of clusters has been formed.

SPLIT. The distance from each point to its cluster centroid is computed. Next, the point farthest from its assigned cluster centroid is split off to form a new cluster. After this, each point which is closer to the new cluster centroid than it is to its own centroid is assigned to the new cluster and the affected centroids are recomputed. Then, another pass is made through the points to reallocate each point to the cluster with the closest centroid. At each reallocation, the cluster centroids are recomputed. The result of this procedure is that one additional cluster is formed and the SSE is reduced.




AMERICAN ANTIQUITY

LUMP. Next, the two clusters with the closest centroids are lumped together and a new centroid is recomputed. Again a pass is made through the points to reassign each point to the cluster with the nearest centroid (and again, at each reassignment the centroids are recomputed). Although the number of clusters is now reduced by one, the resulting configuration is not necessarily the same as the previous configuration with the same number of clusters.

CONTROL. If the current configuration is less well clustered (has a higher SSE) than the previous configuration with that number of clusters, then the program rejects the current configuration and accepts the previous configuration and goes to the LUMP procedure to again reduce the number of clusters. However, if the current configuration is an improvement over the previous configuration (has a lower SSE), the program keeps the current configuration and then executes the SPLIT procedure to try to find a better configuration with one more cluster. After each new configuration is produced by either the SPLIT or the LUMP procedure, the program returns to the CONTROL procedure (which then decides whether to continue splitting or start

lumping). If the SPLIT procedure ever produces an improved configuration with the maximum number of clusters, the program automatically executes the LUMP procedure.

This basic loop continues until all of the points have been lumped back into a single cluster at which time the program stops and reports the best achieved configuration for each number of

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clusters up to the set maximum. This process as applied to the clustered points (Figure la) is illustrated in Figures 2a, b, and c. These figures show the distribution divided successively into 3, 6, and 12 clusters.

It remains to describe the heuristic statistics we have developed and how they permit in-

ferences concerning the point pattern. Statistics are computed for two distinct levels of analysis during this procedure: statistics describing the current cluster configuration as a whole (such as the configuration of six clusters shown in Figure 2b) and statistics describing the individual clusters.

Individual Cluster Statistics

The most obvious individual cluster information is the cluster assignment of each point and, of

course, we know the number of points, n, in each cluster. Another datum is the location of the cluster centroid (xc, Yc). This point is the center of gravity of the points in each cluster, the point whose x and y coordinates are simply the means of the x and y coordinates of each point in the cluster.

The cluster radius as measured by the RMS, root mean squared deviation, is an index of cluster size. This statistic is the square root of the mean of the squared distances from each point in the

41 Kintigh and Ammerman]



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cluster to the centroid. This value is equivalent to the square root of the sum of the variance of the x and the variance of the y coordinate. Formally:

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Where (xi, Yi) is the location of the ith point of an n point cluster and (xc, Yc) is the location of the cluster centroid.

Two final individual cluster statistics are the coefficient of determination, r2 (the square of the product moment correlation coefficient), and b, the SLOPE obtained from a least squares linear regression of y on x for the points in the cluster. If the r2 statistic is high (i.e., close to 1.0), it indicates that there is a strong linear trend in the cluster. If r2 is near 0, it tells us that there is not a strong linear trend. However, the r2 statistic is not so sensitive to the linear patterns as we would like, and it has obvious difficulties in application to small numbers of points. Thus, we are search-

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ing for a heuristic measure which better indicates linearity or simple curvilinearity. (If r2 is undefined it is printed as 1.0; if the SLOPE is undefined it is printed as 999.99.)

Thus, the individual clusters can be characterized according to their constituents, the number of points, the radius of the cluster, and any linear trend in the point distribution. The graphical meaning of these indices is shown in Figure 3; the corresponding cluster statistics are shown in Table 1.

Cluster Configuration Statistics

The statistics discussed in the preceding paragraphs describe the individual clusters in a particular cluster configuration (such as the six-cluster configuration of Table 1 and Figure 2b). However, since the objective of this procedure is not just to describe the data but to identify useful patterns, we need a way to locate the configurations which show the clearest patterning. The configuration statistics described below are designed to enable the evaluation of different sorts of patterning at each clustering stage without detailed examination of the individual cluster statistics. The configuration statistics for the analysis of the clustered points are shown in Table 2.

One set of configuration statistics can be obtained by simply aggregating the individual cluster statistics into a mean and standard deviation of each. For example, if the number of units in each cluster (n), has a low standard deviation (nstd), then the average number of points per cluster (nbar) is a good index of cluster size. The output illustrated in Table 3 gives a much more complete accounting of the cluster size distribution. Here, the number of clusters of each cluster size is plot-

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Figure 3. Graphic illustration of cluster statistics. In this enlargement of cluster 4 of Figure 2b: the cluster centroid is shown by an asterisk; the length of the RMS is shown by the dashed line extending to the right of the centroid; the regression line is the angled dashed line extending across the figure; the SLOPE is the tangent of the angle between the horizontal and the regression line.




AMERICAN ANTIQUITY

Table 1. Clustered Pattern-Six-Cluster Configuration.

CLUS N SSE1 SSE2 %SSE log(%SSE) 6 78 220 220 4.81 0.68

CLUS nbar nstd RMSbar RMSstd n>2 r2wbar r2wstd

6 13.0 1.6 1.64 0.26 6 0.11 0.19

CLUS n RMS r2 SLOPE

1 10 1.50 0.02 0.07 2 15 2.13 0.05 0.35 3 12 1.34 0.02 -0.15 4 14 1.60 0.51 0.39 5 13 1.77 0.00 0.03 6 14 1.49 0.00 -0.00

CLUS VAR MEAN STDDEV VAR MEAN STDDEV

1 X 1.35/ 1.33 Y 15.06/ 0.68 2 X 17.73/ 1.16 Y 13.21/ 1.79 3 X 17.27/ 0.88 Y 3.68/ 1.01 4 X 10.06/ 1.41 Y 15.47/ 0.76 5 X 9.07/ 1.31 Y 5.32/ 1.18 6 X 4.09/ 1.32 Y 9.86/ 0.70

CLUS UNITS...

1 14 15 16 17 18 19 20 21 22 23 2 26 27 43 44 45 46 47 48 49 50 51 52 53 54 55 3 56 57 58 59 60 61 62 63 64 65 66 67 4 25 30 31 32 33 34 35 36 37 38 39 40 41 42 5 28 29 68 69 70 71 72 73 74 75 76 77 78 6 1 2 3 4 5 6 7 8 9 10 11 12 13 24

ted for each stage of clustering making cluster size modalities obvious without looking at the individual cluster solutions. Looking at the twelve-cluster solution, it can be seen that six clusters have only one or two members, while the other six clusters have between nine and fourteen members. The former six clusters are, in fact, single object "clusters." The latter six clusters are the obvious clusters with several points.

Similarly, the mean and standard deviation of the RMS (RMSbar and RMSstd) yield aggregate measures of cluster radius or compactness. A high mean (weighted by the number of points in the cluster) value of r2 (r2wbar) indicates that many clusters have linear trends, suggesting a linear point pattern. Similarly, a high weighted standard deviation of r2 (r2wstd) indicates a configuration with some strongly linear and other nonlinear clusters. However, a low r2wbar can only be the result of mainly nonlinear clusters. With the aggregate r2 statistics is provided a number la- belled n > 2 which is simply the number of clusters in the configuration with more than two members. This number indicates the number of clusters over which the r2 statistics are aggre- gated (since the r2 statistic is not informative if there are only one or two objects in the cluster.)

These aggregate characterizations contain considerable information concerning the configuration; however, they are only intended to be suggestive. Insofar as these indices suggest interesting patterning in a configuration, the distribution of the individual cluster statistics and the configuration plot should, of course, be examined.

A final global statistic is the SSE which is described above. This is a measure of the degree of

clustering which is related to the individual cluster RMS. As the number of clusters increases, this measure must decrease; however, if the points are well clustered, the measure will drop rapidly; if the points are spaced with relative uniformity, it will decrease relatively slowly. This is illustrated by reference to the uniform distribution in Figure lb and the six-cluster configuration of

44 [Vol. 47, No. 1,1982




Table 2. Clustered Pattern-Run Statistics Summary.

CLUS N SSE1 SSE2 %SSE log(%SSE) 1 78 4570 4570 100.00 2.00 2 78 2260 2260 49.46 1.69 3 78 1309 1309 28.65 1.46 4 78 846 846 18.52 1.27 5 78 415 415 9.09 0.96 6 78 220 220 4.81 0.68 7 78 179 179 3.91 0.59 8 78 160 160 3.50 0.54 9 78 140 140 3.07 0.49

10 78 125 125 2.73 0.44 11 78 113 113 2.46 0.39 12 78 98 98 2.14 0.33

CLUS nbar nstd RMSbar RMSstd n>2 r2wbar r2wstd

1 78.0 0.0 7.65 0.00 1 0.06 0.00 2 39.0 10.0 5.37 0.03 2 0.03 0.01 3 26.0 1.4 4.03 0.57 3 0.28 0.17 4 19.5 6.3 2.88 0.91 4 0.27 0.20 5 15.6 3.8 2.07 0.61 5 0.27 0.26 6 13.0 1.6 1.64 0.26 6 0.11 0.19 7 11.1 3.9 1.59 0.33 6 0.11 0.19 8 9.8 4.9 1.33 0.61 6 0.05 0.07 9 8.7 5.1 1.36 0.75 6 0.12 0.12

10 7.8 5.3 1.19 0.81 6 0.17 0.12 11 7.1 5.3 1.05 0.83 6 0.14 0.11 12 6.5 5.4 0.73 0.70 6 0.14 0.11

Figure 2b. For the six clusters of the clustered distribution in Figure 2b, the average distance from a point to its cluster centroid is fairly small. However, if one imagines six clusters in the uniform distribution, the average distance to the cluster centroid would be much larger. Consequently, other things being equal, the SSE will be much larger for the uniform distribution than for the clustered one.

For most purposes we are not concerned with the actual SSE but with the fraction of maximum (single cluster) SSE formed by the current configuration's SSE. This fraction (%SSE) can be ex- pressed as the percentage of the maximum.

N MAXSSE = [(xi- xc)2 + (Yi-Yc)2]

Where N is the number of points in the analysis, and xc is the mean x value, and Yc is the mean y value. Clearly, this is equivalent to the SSE for a single cluster.

100 SSE %SSE =

MAXSSE

This percentage is conveniently scaled by a loglo transformation yielding log(%SSE) which has a maximum value of 2. If the log(%SSE) is plotted against the number of clusters (the solid line of Figures 4a-c), inflection points in the plot (e.g., the six-cluster point of Figure 4a) will indicate the configurations with better clustering. (In the unusual case that %SSE is less than 1%, the log(%SSE) is less than 0 although it is plotted as 0.)

The degree of clustering in the data can be assessed by comparing results of k-means analyses of actual and randomized data. Randomization is accomplished by taking the original data matrix and randomizing the order of the x values with respect to the y values. Thus, each x value of the




AMERICAN ANTIQUITY

Table 3. Clustered Pattern-Cluster Size (n) Distribution.

Cluster Clustering Stage Size 1 2 3 4 5 6 7 8 9 10 11 12

77-78 1

2 1

1 1 1

1 1 2 2 3

1 1 1

4 4 4 3 3 3 1 1 1 2 2 2 1 1 1 1 1 1

1 2 3 4 5 6

original data is associated with a y value chosen at random from the set of y values in the data, ef- fectively destroying the associations which result in clustering. The cluster analysis is executed again using the randomized data. Since the maximum SSE is the same as for the unrandomized data (because the variances of x and y are obviously unchanged by reordering the data columns), the degree of clustering in the data can be evaluated by comparing a plot of the log(%SSE) against the number of clusters for the actual data, with a similar plot for several analyses of randomized data (the dashed lines of Figures 4a-c).

If the points are well clustered (Figure 4a), the log(%SSE) of the actual data decreases much more rapidly (as the number of clusters increases) than that of the randomized data. Thus, the actual data line appears below the random lines in the log(%SSE) plots. However, if the actual data are fairly random (Figure 4b), the log(%SSE) plots of the actual data and the randomized data will fall in the same range and will likely cross (since a randomization of random data should still be random). Finally, if the points are uniformly spaced (Figure 4c), the log(%SSE) will decrease more

75-76 73-74 71-72 69-70 67-68 65-66 63-64 61-62 59-60 57-58 55-56 53-54 51-52 49-50 47-48 45-46 43-44 41-42 39-40 37-38 35-36 33-34 31-32 29-30 27-28 25-26 23-24 21-22 19-20 17-18 15-16 13-14 11-12

9-10 7-8 5-6 3-4 1-2

1

46 [Vol. 47, No. 1,1982




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slowly than for randomizations of the data and the actual data line will appear above the random data lines in the plot. Thus, we can derive expectations for the behavior of the log(%SSE) statistic for clustered, uniform, and random point distributions as compared to its behavior using randomized data.

47

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AMERICAN ANTIQUITY

\

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The k-means spatial analysis suffers neither from the sort of boundary problems encountered in nearest neighbor analysis nor from the problems that arise in defining the size and placement of

grids in approaches involving quadrat methods. However, it should be pointed out that by the nature of its goal of variance minimization, k-means attempts to form circular clusters wherever

[Vol. 47, No. 1,1982 48

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possible. For certain data sets, this may cause somewhat undesirable cluster formations, although if if does, the problem should be readily apparent in the graphical output.

The ability of our approach to distinguish different kinds of point patterns in noisy data rests on a series of expectations, based on the goal models, for the heuristic indices we have derived. Us-

49



AMERICAN ANTIQUITY

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ing these expectations, the various displays of the configuration statistics (Tables 2 and 3 and Figures 5a-c) can be scanned to locate the few configurations in any analysis which should be more carefully examined. These heuristic indices were designed to allow the rapid evaluation of a particular configuration by human analysts and later by a higher level heuristic program. Indeed, these indices are heuristic not only because they are informative and may facilitate discovery but also because they are not strict statistical indicators or tests of certain patterns.

Finally, it should be noted that the use of iterative mapping is a powerful heuristic tool, regard- less of the particular analytical technique employed (in our case, k-means). Analytically suggestive graphics can be a tremendous aid to human problem solving in a great diversity of situations.

RESULTS OF PROGRAM APPLICATIONS

In the main, development of this program has consisted of proposing possible heuristics which would be diagnostically or substantively useful, and then implementing them in the program and experimenting with them to see if they behave in an intuitively plausible manner. Thus, for devel- opmental purposes, we wished to use data for which our intuitive expectations were fairly clear,

50 [Vol. 47, No. 1, 1982

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Figure 5b. Linear site distribution-ten clusters.

but not so clear that the exercise would be trivial. Consequently, we decided to use artificial data, based on known archaeological patterns. These data are illustrated in Figures la-d. It should be noted that (except for the random distribution) these distributions were not tailored for this application (they were employed in the AMPRA regional sampling game which was developed by Am- merman at Stanford University in 1974). While the pattern illustrated by each of these distributions should be intuitively clear, the data do not represent "pure" examples of these patterns (again, except in the case of the random distribution). For example, although in Figure la the points are mostly clustered, some points in the distribution do not fit in a clear clustered pattern.

The main features of two completed analyses of these distributions are presented below. These examples should serve to illustrate the operation of the program and to give some idea of the power of the results obtainable at this stage of the program's development.

Clustered Distribution Analysis

The basic results of the clustered distribution analysis are shown in Figures 2 and 4a and in Tables 2 and 3. Figures 2a-c illustrate the sequence of cluster configurations as the program divides the distribution into two, three, four clusters, and so on. In reality, maps of each stage of

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AMERICAN ANTIQUITY [Vol. 47, No. 1,1982

R 3

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clustering are plotted, although only three configurations are shown here. Table 2 presents the

global configuration statistics for this analysis. The cluster size distribution is shown in Table 3 and the log(%SSE) for the data run, and three random runs is plotted in Figure 4a.

The SSE plot clearly indicates a clustered pattern. Furthermore, we see that the clusters chosen by the program are intuitively satisfactory, in addition to the fact that they are objectively derived. The RMSbar, nbar, and nstd statistics provide a radius and size in accord with our ex-

pectations. The cautionary notes concerning the interpretation of aggregate indices can be illustrated using the r2wbar statistic of Table 2. The three-cluster configuration shows a relatively high r2wbar, 0.28. However, a quick examination of the configuration plot of Figure 2a does not show intuitively interesting linear patterning. However, at the higher clustering stages (six and

above) the r2wbar and r2wstd figures drop to a low level indicating no strong linear trends. This

analysis can be contrasted with the results of a nearest neighbor analysis which yields a near- random nearest neighbor statistic of .80. (Recall that a nearest neighbor statistic of 1.0 indicates a random pattern; a statistic of 0 indicates perfect clustering; and a statistic approaching 2.15 indicates a highly uniform distribution.)

52

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CLUS

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15

N

120 120 120 120 120 120 120 120 120 120 120 120 120 120 120

SSE1

6378 3596 2278 1508 1199 872 753 629 560 491 427 375 325 297 272

SSE2

6378 3597 2279 1508 1199 873 753 629 560 491 427 375 326 298 273

%SSE

100.00 56.39 35.73 23.64 18.79 13.68 11.80 9.87 8.78 7.70 6.70 5.88 5.11 4.67 4.28

log(%SSE) 2.00 1.75 1.55 1.37 1.27 1.14 1.07 0.99 0.94 0.89 0.83 0.77 0.71 0.67 0.63

CLUS nbar nstd RMSbar RMSstd n> 2 r2wbar r2wstd

1 120.0 0.0 7.29 0.00 1 0.05 0.00 2 60.0 5.0 5.44 0.28 2 0.13 0.02 3 40.0 4.9 4.32 0.27 3 0.04 0.05 4 30.0 4.1 3.54 0.17 4 0.02 0.02 5 24.0 2.4 3.11 0.49 5 0.10 0.12 6 20.0 4.2 2.67 0.20 6 0.18 0.11 7 17.1 4.1 2.44 0.33 7 0.14 0.12 8 15.0 3.1 2.26 0.25 8 0.16 0.17 9 13.3 3.8 2.09 0.31 9 0.20 0.19

10 12.0 3.9 1.98 0.18 10 0.19 0.17 11 10.9 3.3 1.85 0.21 11 0.20 0.14 12 10.0 2.2 1.74 0.25 12 0.18 0.13 13 9.2 2.5 1.60 0.24 13 0.23 0.23 14 8.6 2.6 1.54 0.18 14 0.22 0.24 15 8.0 2.6 1.46 0.20 15 0.28 0.24

Linear Distribution Analysis

Results of the linear distribution analysis are presented in Tables 4 and 5 and Figures 5 and 6 in the same format as the clustered distribution results. As expected, the SSE plot shows slight clustering but is not conclusive, while the most diagnostic statistics are r2wbar and r2wstd which show a fairly high weighted average r2 and a high weighted standard deviation of r2 with a large number of clusters. An examination of the individual cluster statistics and the configuration maps shows that the analysis tends to divide the distribution into linear segments, with high r2 values, and nodal clusters (clusters of points near where the linear segments meet), with low r2 values. Again, the nearest neighbor statistic, .98, misleadingly suggests that the distribution is random.

Analyses of Uniform and Random Distributions

Analyses of uniform and random distributions have not been problematic. Diagnostic indicators of these patterns, especially the SSE plot (as well as the nearest neighbor statistic itself), have proved adequate for identifying uniform and random patterns at the level of analysis attempted thus far. The log(%SSE) plot of the uniform distribution of Figure lb is shown in Figure 4c. In this plot the actual data are far less clustered than randomizations of those data, clearly indicating uniformity. Similarly, in Figure 4b the log(%SSE) plot of the random data of Figure id shows the original data clustering to the same degree as randomizations of those data, indicating originally random data.




AMERICAN ANTIQUITY

Table 5. Linear Pattern-Cluster Size (n) Distribution.

Cluster Clustering Stage Size 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 2 1 2 1

1 1 1 2 1 1 4 4 2 1 2 3

2 1 1

1 1 1 1 1 5 6 5 4 4 3 4 4 4 4 1 1 3 5 6

Pincevent Case Study

The value of the methods obviously depends on the results that can be obtained in actual case studies. Preliminary results of analyses involving actual data-both artifact distributions on oc-

cupation surfaces and settlement data-are encouraging. Below, we briefly describe some results obtained from an analysis of the Magdalenian camp of Pincevent in France (Cook et al. 1978).

In their report the excavators (Leroi-Gourhan and Brezillon 1972) propose a model for the use of space in the camp. They suggest that activities in the camp were centered around three huts, each opening toward an associated basin hearth. On the basis of artifact concentrations (but no direct architectural evidence) they hypothesize the locations of the huts shown in Figure 7. They further propose a ring model with a clinal decrease in activity (and hence artifact density) as the distance from the hut increases.

1 118-120 115-117 112-114 109-111 106-108 103-105 100-102

97-99 94-96 91-93 88-90 85-87 82-84 79-81 76-78 73-75 70-72 67-69 64-66 61-63 58-60 55-57 52-54 49-51 46-48 43-45 40-42 37-39 34-36 31-33 28-30 25-27 22-24 19-21 16-18 13-15 10-12

7-9 4-6 1-3

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54 [Vol. 47, No. 1,1982

1




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Using the heuristic procedures described above, six analyses, each based on a separate class of remains (three tool classes and three classes of faunal remains), were carried out. In these analyses the number of objects in each class ranged from 132 to 235. The results at the three- cluster level are shown in Figures 8a and b. In each case locations of the three-cluster centroids derived by the heuristic procedure correspond closely with the locations of the three structures

55

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AMERICAN ANTIQUITY [Vol. 47, No. 1, 1982

. I

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0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 Pincevent Grid Meters

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Figure 7. Pincevent-locations of basin hearths (stippled), hypothesized huts, awls, and burins.

proposed by the excavators on intuitive grounds. By the magnitude of the RMS values (the circle radii in Figure 9), the analysis indicates another point made by the excavators, that the artifact concentration in the lower right of the figure is more dispersed than the other two. It should be ad- ded that the hearths shown in the Pincevent figures played no part in the analysis of artifacts but are plotted only for purposes of reference.

If we turn to the results at higher cluster levels, it can be seen that the clusters begin to reflect separate activity areas of the camp (see Figure 9). Some of the clusters remain closely associated with the three main basin hearths (adjacent to the proposed structures) while many others appear

56

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to be associated with peripheral hearths. It is worth noting that although only two classes of material are plotted here, this basic patterning is repeated over each of the six classes of remains.

In this analysis it was possible to replicate, on one level, the locations of the structures intuitively identified by the excavators. However, in contrast to the excavator's ring model, this analysis has begun, on another level, to identify patterns indicative of what appears to be smaller activity loci at the site. Further work on the k-means analysis of artifact distributions at Pincevent is described in a recent paper by Simek and Larick (1981).

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Figure 8b. Pincevent-three clusters with RMS radius circles drawn around centroids for analyses of scrapers (solid line), burins (dashed line), and awls (dot and dashed line).

Bushman Camp Analysis

In addition to the Pincevent study, a number of interesting k-means analyses of Yellen's (1977) Bushman camps have been completed by a group at the University of Michigan. The relatively clear clustering of material around huts was, as expected, located by the k-means analysis. Fur- thermore, after simulating a substantial disturbance and decomposition of the sites, which de- stroyed virtually all obvious patterning, the analysis still located artifact clusters around the huts. Again, the actual locations of the huts were not used in the analysis. Additional analysis of the same data using Whallon's (1979) unconstrained clustering technique produced results which

58 [Vol. 47, No. 1,1982

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very nicely complemented those produced by k-means. Results of these analyses are now being prepared for publication.

PROSPECTS

In the preceding sections the basic outline of the procedure and some of the basic heuristics we have developed were described. From this current implementation we see three general areas in which major development can occur.

First, a number of diagnostic indices which we have formulated but not tested can be imple-

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mented. In particular, we have some ideas for better determining the compactness and separation of clusters and for improving analysis of the troublesome linear patterns. We also plan to imple- ment decision-making logic to decide when enough clusters have been obtained. (Currently, the program forms a specified number of clusters and then stops.)

Second, the basic program discussed here could be put under the control of a higher-level supervisory program, so that multiple levels of analysis could be attempted. The basic program might first be applied to distributions such as those used in this paper. Then, essentially the same program could attack the distribution of points within a single cluster to determine its structure. In addition, it could examine the distribution not of the points themselves, but of the centroids of the clusters, thus seeking the pattern behind the location of the clusters. The supervisory program would direct the examination of the different levels of analysis, using information obtained at one level to direct other inquiries.

The third area for development could be the addition of contextual information and of heuristics utilizing these data in an analysis. We are now pursuing this sort of development for living floor analysis in contexts such as Pincevent and Yellen's Bushman camps.

This last area of development would require specializing the procedure for a particular kind of archaeological problem-for example, settlement pattern analysis as opposed to analysis of living floors. This specialization would permit the incorporation of more specific analytical heuristics developed by archaeologists with expertise in the problem area as well as better use of context. It is at this stage that the heuristic framework of the project would pay off most substantially. The basic information gathering procedure now implemented could be directed to examine a large number of partitions of the data but to report only the more interesting results for the consideration of the archaeologist. The program would use heuristics both to derive the partitions of the data to be examined and to evaluate the degree of success of each analysis. The program would make this evaluation on the basis of the behavior of the heuristic indices computed at each stage of clustering. The success of a given analysis would provide information for the purpose of deciding what form of partitioning should be considered next.

For example, in a settlement pattern study the sites in each category of site type might be analyzed separately. If the indices were to suggest relatively good patterning for one category of sites, the program might further subdivide the sites of this type by environmental zone and analyze each subdivision separately in order to search for more distinct patterning. Alternately, if separate analyses of two site types indicated similar patterning, analysis of the combination of the two types might be tried to see if the patterning is clarified or obscured. The more successful analyses, ones in which the indices suggest clear patterns, would be reported to the human analyst for further consideration.

In this way the influence of different substantive variables and combinations of variables can be studied. While this approach would surely be less "intelligent" than a human archaeologist's analytical path, it is clear that the consideration of even a moderate number of data partitions can be prohibitively time-consuming for a human analyst. It should also be noted that this process of partitioning, as a method of refining ideas concerning spatial patterns, is itself a heuristic which was derived by examining the analytical process pursued by human experts in spatial analysis.

Along with this development, more theoretically sophisticated models and heuristics could be included. Thus, we might wish to examine hierarchical relationships in complex settlement systems, perhaps using models from economic geography. Many of these models would have behavioral implications (Yellen 1977; Binford 1978) which our current analysis does not attempt to identify.

CONCLUSIONS

In this article we have attempted to make some general points concerning the role of complexity, context, and intuition in the solution of archaeological problems. Furthermore, we have presented an analytical technique which is designed to operate within this conception of the nature of archaeological problems and their solutions.

60 [Vol. 47, No. 1,1982




As a means of implementing a heuristic approach to spatial analysis, the k-means technique has thus far proved to be both powerful and robust. At multiple analytical levels it provides graphical and statistical information on the general character of a point distribution as well as on the spatial components of the distribution. An important feature of the technique is iterative mapping which permits the inspection of patterns at various levels of clustering. The technique is uninhibited by boundary problems which arise in nearest neighbor analysis and avoids the problems with grids associated with quadrat analysis. In its application it has provided intuitively satisfying, interpretable results which are, at the same time, objectively derived.

We would. like to stress that the heuristics that have been implemented so far are still very general and weak. We are not recommending the k-means technique as an exclusive method of analyzing spatial data, nor is it the only way in which one can pursue heuristic spatial analysis. In the long run we expect that a range of different techniques will be used in combination with one another and that individual techniques will be refined through use and experience. What we are attempting to do is more than nominate k-means spatial analysis for the roster of useful inductive statststical techniques of spatial analysis in archaeology. Thus, we have tried to place the technique and the motivation for its use in a broader theoretical framework.

We suggest that a distribution encountered in an archaeological situation is the composite out- come or result of a number of cultural and noncultural processesses. The interpretation of a pat- terned distribution is essentially an argument for a particular mapping of the observed patterns back into a system of generative processes. It should be clear that, generally, there is no straight- forward way (or, for that matter, any way at all) to specify rigorously the relationships among the processes which produced a particular distribution. A given set of well-defined processes might be combined in any number of ways to produce identical observed distributions. The choice among a set of alternative choices will be made by reference to other evidence: hence, the impor- tance of context in the interpretative process.

Taking as a starting point the inherent complexity of spatial analysis, we have found it productive to follow an approach which tries to incorporate the much more sophisticated (as opposed to simplified or highly reductive) analytical strategies which derive from direct human experience with problem solving. Crucial features of such human strategies include: the employment of the logical sophistication and effective pattern matching capabilities of the human mind; the ability to consider multiple levels of possible patterning; and the availability of a wealth (if not overabun- dance) of information and hypotheses about the data and the problems at hand. Thus, we argue that heuristics which are developed through experience and intuition and which are objectively implemented by a computer program can play an important role in spatial analysis, simply because the problems are inherently too difficult to solve in any other way. Our efforts, therefore, have been directed toward designing and implementing a procedure which simulates certain time-consuming intuitive analytical processes and has as its immediat e display, identification, and characterization of the patterns in a spatial distribution. It is further hoped that patterns identified by the procedure will lead to interpretive insights by the human analyst.

As the last paragraphs plainly indicate, this article is primarily concerned with the inductive problems of pattern recognition and hypothesis formulation, problems which have been perhaps undeservedly neglected. Of course, we are not suggesting that hypotheses inductively derived need not be adequately tested. On the other hand, we hold that a more deductive approach does not obviate the difficulties caused by the complexity of the spatial analysis problem or the need for better use of context. Our emphasis on intuitively developed heuristics and induction may foster uneasiness or distrust among some archaeologists who are particularly concerned with the maintenance of analytical rigor. It is important to stress that this approach operates objectively with an explicitly defined set of assumptions and techniques so that the results may be replicated and fully assessed. Furthermore, it is worth recalling that such an approach to the solution of complex problems has proved successful in other fields of research.

Our initial "experiments" have focused on the motivation for employing heuristic approaches and have shown how they can be implemented. The next step is to concentrate on the exploration




AMERICAN ANTIQUITY

of a specific case study or problem for which more powerful heuristics can be learned from human experts and more contextual information incorporated in the analysis.

Technical Note. The k-means program used in this analysis was adapted from the F. R. Hodson's IBM ALGOL program which was based on the work of Geoffrey Ball. Although the basic algorithm has remained the same, extensive additions have been made to the current PL/I version to produce the extensive numerical and graphical output shown in this article. The program is able to analyze distributions with a relatively large number of points using only moderate amounts of computer time. Required CPU time varies with the number of points and the specific configuration of the distribution. Division of a 344-point distribution into 15 clusters required about 15 seconds of Amdahl V/8 CPU time. Copies of the current version of the program and its documentation (Kintigh 1980) are available from the authors.

Acknowledgments. We wish to thank the many people who have commented on various drafts of this paper. James Doran, Vin Steponaitis, Patty Jo Watson, and Robert Whallon have provided many useful com- ments and suggestions as have a number of anonymous reviewers.

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