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Estimating subjectivity of typologists and typological
classification with fuzzy logic
1. Classification
A basic concern of prehistorians is the delineation of cultural configurations, or patterns, from the assemblages uncovered (Trigger 1968,15). In order to achieve this goal, artifacts are analyzed by defining a system of ordered units (as typological groups), used for a comparison between assemblages. The primary unit of comparison is the component, defined as a single occupation in the history of a given site (Willey, Phillips 1958, 21). For a component to be meaningful, the time segment that is isolated must delimit a static cultural situation, without any significant change. Thus, the cultural material associated with this component should represent the people who inhabited the site, at a given point in their cultural development.
Intuitively, a major component of prehistoric material culture, flint tools, are defined as: artifacts with intentional retouch (a series of removals), intended either for the preparation of a hafting or a working edge. The system of ordered units, our typology, follows the rules of taxonomy:
it is based on the identification of certain forms, of certain characteristics.
the chosen characteristics have to be exclusive, present at each case of type.
each case can be classified in one and only one type.
Two essential properties are characteristic of types: identity, or identifiability, and a meaning, relevant to some purpose. One of the major weaknesses of typological research is the nature itself of types being defined partly intuitively and partly rationally, partly essential and partly instrumental, most typological lists are polythetic, so there are no fixed criteria of typehood.
Archaeological types are usually distinguished by norms or central tendencies, rather than distinct boundaries (Adams, Adams 1991, 20). The main essence of types is whether they serve a specific (scientific) purpose, or not. Every type necessarily has to have a diagnostic attribute, or a cluster of attributes, that sets it apart from all other types.
A valid classification approach is the one often called lumping (see Dunnell 1971; Everitt 1974), in which the emphasis is on external isolation of types (see Adams, Adams 1991), rather than searching for minor variations, which can be the result of internal cohesion among similar entities (splitting). A basic assumption is that although types are defined a posteriori, as a result of criteria used by typologists, they also existed a priori, since they can be identified and described.
Several approaches to typological studies were adopted since archaeology raised as a science (for a good summary, see Whittaker et al. 1998). Following is a survey of main ideas adopted through years in the typological research, and their application.
One approach to the concept of fossile directeur (supposedly defining components of a given culture, usually particular tooltypes) was formulated by Gallus (1977), who described (prehistoric) culture as an assemblage of objects and observable manifestations or traitswhich according to form and expression (observable externalization) can be regarded as different (disparate) from any other such assemblages of traitsthe objects and traits, which document a culture, are the synchronic end resultsof a typological series in a particular time period (Gallus 1977, 143). In this sense, the traditional cultural historic view of types, defining cultural fossils directeurs, is adopted in its larger sense, including other aspects of lithic industries. Thus, types with potential cultural identifiabilty were used to classify assemblages in a timescale of developing cultures.
The possibility of using typological studies to explore cognitive capabilities of past humans was expressed since the sixties in works of Chang (1967, 228 cognitive system of the makers) or Cowgill (1967, 236 alien system of cognition). These concepts were adopted by Gallus, for his definition of an organic typology, where the concept of a tooltype, as formulated by the prehistory researcher in his mind, should match as closely as possible the idea formed by the prehistoric artisan himself (Gallus 1977, 134). The organic typology is based therefore on a comparison (matching) between the ancient mental template (of the knapper) and the modern one (of the prehistorian). The use of typology is based on the assumption that human brain, when confronted with a variety of perceptions in his environment, notices repeated occurrences of identical or similar occurrences, forming cognitive units, or engrammes (a complex of neural networks, which reflect an idea, or a concept) (Margenau 1950, 58). Thus, the aim of typological analysis is to establish tooltypes that reflect the concept of a prehistoric artisan, externalized in the form of an actual tool, in order to serve a need, to perform a task (Gallus 1977, 134).
This view contrasts Tixiers, who declared that tools are artifacts that prehistorians define as such, and not what prehistoric people might have regarded as such (Tixier 1967, 815). This view still dominates the scientific prehistoric research, even though it was criticized that it dehumanizes prehistoryweare in danger of arriving at historically anachronistic results (Gallus 1977, 135).
Already in the 60s, LeroiGourhan pointed to the problematic of the classic (of those days) typologically oriented prehistoric research, proposing a research framework, the chane opratoire, aimed to produce a biology of techniques a structural and functional awareness of techniques; he formulated the means of action on matter, which included terms as percussion, abrasion, etc. He concluded that 'techniques are both gestures and tools, organized in a true syntax (LeroiGourhan 1964, 164). The act of fabrication is a dialogue between the knapper and the worked material (LeroiGourhan 1965, 132) implicating the natural determination of the tendance and the cultural idiosyncrasy of the fait (LeroiGourhan 1943, 2343).
Since the beginning of this century, and especially during the fifties, under the influence of Franois Bordes, typological research was regarded as the main objective tool in establishing chronological frameworks and comparisons between sites. This view is best reflected in Bordess work: la typologie est la science qui permet reconnatre, de dfinir, et de classer les diffrentes varits doutils se rencontrant dans les gisements (Bordes 1961). Selected flint tool types were regarded as fossile directeurs, which could be used for dating archaeological horizons, in a similar manner to fossils, used for dating geological layers.
The recognition of repetitive morphological forms enabled researchers to group them into types, defined by their characteristic retouch. A special vocabulary was needed for the definition of types, the terms being taken from ethnographic analogies, mode of preparation or supposed function (e.g. burin, scraper, etc.). For a more detailed classification of tools, subtypes were defined, according to specific morphological features (convergent scraper), modes of preparation (dihedral burin) or toponymical appurtenance to a particular site (Kebara point).
At this stage of research, most of the effort was directed towards the definition of chronological sequences, based on the fossile directeur concept that was supposed to define cultural entities. Following a rapid development in the prehistoric research, a need for standardization of the terminological vocabulary was needed (Brezillon 1968). Several symposiums were held, for the definition of universal typological list, suited for all regions, given a specific time period (see Hours 1974).
A further development of the prehistoric research followed the observations of Bordes, which concluded that fossiles directeurs are meaningless in their singular forms, and a method of percentage comparison between assemblages should be developed (Bordes 1950, 1961). The idea was to define a representative typological list, a complete inventory of tools characteristic of a given region/period and compare it (graphically) with inventories of other sites. Parallelisms and differences between the (cumulative) graphs were used for definition of cultures and cultural regions.
During this period, following intensification of statistical methods, alternative typological approaches were proposed, as the analytical typology of Laplace (1966) or the descriptive morphology of LeroiGourhan (1966). The main assumption underlying the Laplacian approach is that cultural changes can be explained by a process of predetermination the analysis of flint tools should be based on an hierarchy of complete characteristics, build on discrete and continuous variables (Laplace 1966). Thus, each item is not only nominated (scraper, burin, etc.), but also morphologically described, by a series of codes. Thus, the subjective aspect of the typological definition is eliminated, each item being classified using mathematical aid of factorial analysis and automatic classification.
This somehow promising approach, to eliminate the subjectivity from typological studies, was forwarded by intensive researches in the application of various statistical methods of identifying, defining and classifying past objects into meaningful types. This research was oriented in several directions, including elaborate dataextraction through its visualization in elaborated graphic displays (e.g. Hovers, Raveh 2000), evaluation, using Bayesian methods, of subjectivity errors of typologists (e.g. Whittaker et al. 1998; Read 1989; Gnaden, Holdaway 2000) or the application of various statistical methods to eliminate the subjective side of the types nature (Bailey 1994; Read, Russell 1996; Shott 2000, Djindjian 2001).
Despite these efforts (see also Bisson 2000), it seems that any tentative to separate between subjectivity and typology may have limited results, given the nature itself of typological research: le bon typologiste peroit avec lexprience aprs avoir analyse quelques milliers de pieces, spending a good deal of time au course de longs ttette avec les outillages lithiques (Demars, Laurent 1992, 20).
2. The subjective nature of objective classification
Apparently, a part of the classification process is an inherently subjective one, intuition having a larger or lesser influence, which depends on the researcher and on the classification method (the typological list) applied. Thus, objective methods are largely subjective, consisting upon the measurement of certain parameters, often arbitrarily defined, allowing in many cases a great degree of freedom, having a loose definition. For example, given some characteristic C, a classification based on C requires measuring a set of physical parameters p1, p2 , pn and verifying if
fC (p1, p2 , pn) = 1 or not.
fC being some predefined function that completely describes the characteristic C.
In most cases, the definition of the classification function will be very simple, for instance it may require simply that pi lies in a given interval Pi, so that in this case
EMBED Equation.3
Ik being the characteristic function of the interval Pi, that is
It is simple to verify that any objective definition of tools may be easily translated in this way: for instance, the definition of scraper (the characteristic C of our case), according to the Laplace method, is an item with a continuous, simple or scaled retouch. Thus, two parameters defines C:
p1 = delineation of retouch, with the value of continuous, described as a retouch that draws a continuous line, straight, concave, convex or sinuous along the edge of the item)
p2 = mode of retouch, with two potential values: simple (along the edge of a thin flake, maintains the cutting edge, the retouch being a series of more or less elongated scars, sometimes en caille, which form with the striking plane of the item a narrow angle; surleve (along the edge and the face of a thick flake, conserving or not the cutting edge, the retouch being irregular, wide and scalariforme, with subparallel bladelet scars.
Therefore C, the scraper in our example, is a tool on a thin or thick flake, with a retouch varying from simple to scalariforme, covering some or all the cutting edge, modifying it or not. The ambiguity of this definition is selfevident, especially when confronted with definitions of other tooltypes, as retouched flakes or truncations (see also Tixier 1967).
This process gives a complete illusion of objectivity being based, as it is, on the absoluteness of mathematical formulae, and as far as mathematics is applicable, it is so. However, the application of such a system requires some operations that are outside the realm of mathematics, and here the objectivity goes awry.
First of all, who decided that fC characterizes the class C? Did prehistoric industries have a quality control that discarded tools not obeying to this production function, which was thereafter communicated to modern archaeologists so that they could check the product characteristics and infer the product name, type and purpose? Often, as noticed before, it is a modern statistical analysis that groups together different artifacts recognizing (possibly after having applied statistical analysis) common characters that may also correspond to some hypothized use rule (e. g. from wear) and abstracting these results to obtain a general rule, which is then applied to other assemblages. Further deductions are then inferred by this classification and the process continues, basing repeatedly on simplifications, approximations, errors and inference. So the process that leads from artifacts to theories does not only rely on numbers and computations, but has to cope with decisions: if the prescribed length for a class is 4 cm, should we accept something 4.1 cm long? And something 4.2 cm long? And when should we stop accepting and start rejecting?
The fact is that items to be classified may be grouped in three heaps: what certainly does belong to that class, what for sure does not and what is uncertain.
The only validation that can be applied is the acceptance by the scientific community: every scholar may, in principle, check the deductions of another researcher, both from what he or she communicated and even if with greater difficulty on the real thing, the objects that were accurately collected and documented; verify by subjective judgment if the results are convincing; argue (publicly or privately) in favor or against; then the conclusions are accepted or rejected. In this chain, statistical and mathematical considerations have to be convincing, not decisive, even if they play a substantial role in deciding, classifying and concluding. So, any contribution to the transparency of computations supports the scientific correctness of the process.
What usually happens is that uncertain objects are assigned to some class and then deductions are made basing on the count, percentage or some other global index of the number of objects belonging to different classes. Thus, wrong assignments may lead to erroneous results. Still worst conditions may derive by the use of a computer, which leaves no space to uncertainty and hence forces the researcher to assign objects to one determinate class, by ticking some box and inserting the value into a database: with the automatism of its deductions and the impossibility of computational errors, a computer adds to the false sense of reliability of the final result.
The above considerations do not mean that all the theory of classification based on statistical analysis is to be rejected, but only that extreme caution must be exerted when basing deductions only on statistical computations; they moreover suggest that the method that will be introduced later may help to make visible the uncertainty that underlies any such deduction and help to evaluate some numerical reliability index.
3. Fuzzy set theory
Fuzzy sets, first introduced by Zadeh in 1965 (Zadeh 1965), generalizes the familiar concept of sets extending the indicator function. As stated above, the indicator function of a set is a function that values 1 within the set, 0 outside. Since a set is its indicator function, a fuzzy set is defined as a function with values in the interval [0, 1], namely given some (traditional) set U, a fuzzy subset of U is a function
EMBED Equation.3
If the only possible values are 0 and 1, this reduces to the classical definition of a subset.
As for classical set theory, this traduces into logic, the truth function of a predicate being the fuzzy set corresponding to its truth values.
Fuzzy set theory has been thoroughly studied (see, for instance Yager, Filev 1994 and its extensive bibliography, and Li, Yen 1995) and has many applications in engineering. Usually, the goal is the defuzzification, that is the reduction of control problems to the nonfuzzy case. Our case is exactly the opposite.
Fuzzy logic has seen also many applications in linguistic and other science, being able to manage situations in which binary logic is inapplicable.
As far as archaeology applications are concerned, fuzzy set theory and fuzzy logic has been applied to databases, creating and implementing a RDBMS model with fuzzy functions and using it to study an Etruscan cemetery, and to GIS: spatial relations are a good example of fuzzy statements as in the statement this place is near that one. For both applications, see Niccolucci et al. (2001) and Crescioli et al. (2000) and the references quoted there.
For the present problem we can use the fuzzy set model defining the grade of belonging of the item i to class A as a number ai, with EMBED Equation.3 , which expresses the subjective degree of belief that the assertion item i is a member of class A is true.
This has some kinship with probability theory, but is not identical to it: the main difference is that no normalization condition of the type EMBED Equation.3 is imposed. The numbers ai are merely a way of communicating how reliable the assignment is from the researchers viewpoint. They help them to do a better overall analysis and help other researchers to evaluate the authors results. As it will be seen in what follows, the use of fuzzy set theory may also solve conflicts of attribution, that is situations in which different researchers classify differently the same assemblage of objects, resolving in different ways the classification of uncertain artifacts.
4. Using fuzzy set theory for the classification of artifacts.
The fuzzy method may be easily applied to flint tool classification as a generalization of the usual methodology. So far, the result of classification consists of a list of numbered items (the finds) accompanied by a label (the tool class assigned to it). Then statistical computations may be applied, as simple counts or percentage evaluation, clustering or whatever the researcher estimates as more significant for the investigation to be carried on.
With the fuzzy method, the starting point of this statistical processing is, instead of a onedimensional list, a matrix A having all the items as rows and all possible tool classes as columns. Each element aij of the matrix is a number in [0, 1] expressing the degree of possibility the evaluator assigns to item i belonging to tool type j. If for each row there is only one aij = 1 and all the others are 0, this method reduces to the classical one.
Table 1 gives an example of fuzzy classification for an assemblage of 50 flint tools. Starting from this table, various statistical analysis may be performed; for instance the relative presence of different types requires computing the column totals and evaluating the relative weight of each type as a fraction or percentage of the total, as shown in the last line of the table.
To evaluate the fuzzy coefficients, the researcher has several alternatives. Some of these methods are outlined in Li, Yen (1995) as far as subjective evaluation is concerned; when measurements are involved, a reliability function as the following may be used. Let us suppose that some type assignment is based on a numeric parameter x, such that the item belongs to the type under consideration if x > b, b being a numeric threshold established in previous research. Let us also assume that it is unquestioned that if x < a the item does not belong to this type (if no such value exists, we may assume a = 0). A simple reliability function f may then be created as follows:
This is equivalent to use a linear function as shown by the following graph.
Of course the latter method may result rather cumbersome to use manually, but it may be easily implemented when using a computer. This may occur when finds are recorded automatically or semiautomatically using, for instance, a 3Dscanner and measurements are done by the software on the resulting digital 3D model. The method may be adapted to take into account systematic errors of the automatic method, due to inaccuracy in scanning, roundoff, and so on.
The above method may be used also when more than one numeric parameter is involved. If, for instance, two measurable features have to be used, after evaluating each reliability coefficient f1 and f2, to compute the global reliability coefficient f one may adopt a pessimistic viewpoint
f = min(f1, f2)
(the global reliability is the worst of the two), which is equivalent to require an AND condition between the fulfillment of the two requirements; or an optimistic viewpoint
f = max(f1, f2)
(the global reliability is the best of the two), which corresponds to an OR condition between the two distinct requirements in the Boolean traditional framework. Also intermediate values have been proposed and they may be used to give a greater weight to the principal parameter and a lesser importance to a secondary one.
In the experiment shown in the following paragraph, we adopted a much simpler method: ask the expert who is classifying the artifacts to express a subjective mark for each type, basing on his or her experience. Subjective as it may be, this method generalizes what is done in most cases: evaluate tools by observation and experience. It simply adds to common practice to express numerically the referees confidence in the typology just assigned and to list other possible types, each one with its own confidence index.
5. The case study
To verify the impact of the fuzzy classification method we chose an assemblage of 50 flint tools from a protohistoric site in Southern Israel. The material originates from a single occupation phase, therefore, intrusions were basically absent.
This set of artifacts was given to 5 different experts and they were invited to classify them according to 12 classes and a residual (Varia) one. The result is expressed by Table 1. As it can be seen, the agreement on classification was not complete even if the researchers came from the same school, had similar training and experience and had a good expertise on the typology from which the sample derived. Of course it was assumed that no mistake was done.
This posed a series of problems: who should be the trusted expert? How could uncertainty be managed? And then, how scientifically reliable could classification be considered, since it was not replicable even in so simple laboratory conditions?
The fact is that as previously discussed classification does not lead to a precise, crispy result, but carries some amount of uncertainty that cannot be completely eliminated. In most cases, this has no consequence on the final result, but in some it may make the difference. Thus, the fuzzy method is proposed as a possible solution.
The experts were asked to extend their classification expressing any possible typology assignment by means of a numeric coefficient in the range [0, 1], 0 meaning that they did not consider such an assignment as possible, 1 meaning that they considered the related typology as most reliable. No constraint was set on the number of 1s, so that multiple most reliable assignment were allowed: this is no contradiction, since it may happen that after excluding a number of classes it is almost nondecidable which one of the remaining must be chosen, all appearing as equally reasonable.
This led to the compilation of 5 tables, one for each referee, showing different assignments accompanied by the reliability coefficients assigned by each expert, which could then be used to obtain a joint classification table showing the compound reliability coefficients.
The final assignments and coefficients may be calculated by means of a weighted average of each evaluators coefficients. This was done with equal weights, obtaining the values shown in Table 3.
6. The impact of the fuzzy method
It might be argued that the differences between the usual classification method and the fuzzy one are not so relevant, and that they do not justify the waste of time.
The first reason of using the fuzzy method is to make comparable different assignments, as came out by our experiment, without having to choose between different referees.
Second, there are differences, which make one typology prevail in the modal ranking only if the fuzzy method is used. This happens for typologies that are always a second choice in the referees classification, with different first choices: if this is taken into account, as the fuzzy method allows, they add up a global score which makes them relevant in the global analysis. This result is possible only with the fuzzy method, since with the usual one second choices are always discarded.
The fuzzy classification method helps also to identify difficult items. For this we can define a reliability index of each item which takes jointly into account the spread of possibilities assigned to that item (i.e. the number of different typology choices considered as possible for it) and the value of the most reliable assignment.
The value we propose for this index is given by the following formula:
EMBED Equation.3
relating the reliability index R of a set of fuzzy coefficients x1, x2, xm to the maximum value of the xs and inversely to their sum. R is defined as zero if all the x are zero. The reliability index is a number in the interval [0, 1] with some interesting properties that make it a good candidate for this role.
R is 1 only if the set of xs is a crispy set, namely if the assignment has no fuzziness: one and only one of the x is 1, all the others are zero. As a function of the coefficients, R is a continuous function, in particular when all x approach zero. Moreover, R decreases if the number of nonzero fuzzy coefficients increases, thus increasing the fuzziness of the classification; R increases, as expected, if the maximum value of the fuzzy coefficients is greater, reducing the fuzziness by expressing more confidence in the preferred typology. Thus R may be used to evaluate the global reliability of type assignments for an item and hence the difficulty of that specific item to be classified. A global R for an assemblage can be easily defined by averaging R over the items: the result is an average index of the reliability of the assemblage as a whole.
7. Conclusions and further work
It has been shown that applying concepts of fuzzy logic to typological classification of artifacts may aid in various aspects to a better interpretation of past relics. Future research will focus on two main aspects: evaluation of R coefficient of fuzzy types in main periods/cultures and subsequently, evaluation of R at assemblage level, identifying characteristic Rs for various assemblages representing different cultures and periods. Moreover, it is intended to explore the fuzziness of not only tools, but waste products as well, hoping to achieve a simple level of elaboration of R in a way that synthesis of research and accuracy of explanations will be easily achieved.
Sorin Hermon
BenGurion University of the Negev, Israel
Franco Niccolucci
University of Florence, Italy
Abstract
It is well known that interpretation always conveys a certain degree of subjectivity, which disappears as soon as interpreted data are stored in a computer database. This may lead to dangerous approximation and possibly to fallacious conclusions. To avoid this oversimplification, it has been suggested to use fuzzy databases, in which attributes may have a fuzzy nature and be indexed by a numerical coefficient, the fuzzy coefficient, which can be interpreted as the degree of confidence the researcher has in each possible assigned value. This technique has been successfully applied to gender and age assignment for deceased in a cemetery investigation: in this case anthropological data offered statistical parameters that could be used to compute the fuzzy coefficient.
Lithics classification is another field in which fuzzy databases have a potential usefulness, but in this case, no previous statistics may help in determining the fuzzy coefficient. We decided to perform an experiment during a standard typological classification of a flint tool assemblage from Israel. It concerned the classification of 50 tools, by different researchers. Each one was asked to note, besides the typology of each item, an evaluation of the degree of sureness, or the possibility of an item to belong to a particular type, in other words his or her guessed estimate of the fuzzy coefficient.
This paper reports the results of this experiment, in order to evaluate the difference between researchers when performing a classification of tools, to recognize problematic types or items (which mostly differed between the typological lists presented) and eventually to compute a fuzzy coefficient for each type assignment, balancing the different evaluations of experts. Finally, data are set into a fuzzy database, performing examples of fuzzy queries.
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Table 1. Results of typological classification of an assemblage containing fifty tools, by five different researchers. Note difference between inventory lists, by refererees and after computing fuzzycoefficients.
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