Merrill and Read 2010

Society for American Archaeology

A NEW METHOD USING GRAPH AND LATTICE THEORY TO DISCOVER SPATIALLY COHESIVESETS OF ARTIFACTS AND AREAS OF ORGANIZED ACTIVITY IN ARCHAEOLOGICAL SITESAuthor(s): Michael Merrill and Dwight ReadSource: American Antiquity, Vol. 75, No. 3 (July 2010), pp. 419-451Published by: Society for American ArchaeologyStable URL: http://www.jstor.org/stable/25766210 .

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ARTICLES

A NEW METHOD USING GRAPH AND LATTICE THEORY TO DISCOVER SPATIALLY COHESIVE SETS OF ARTIFACTS AND AREAS

OF ORGANIZED ACTIVITY IN ARCHAEOLOGICAL SITES

Michael Merrill and Dwight Read

We propose a new method to analyze spatially cohesive sets of artifacts and relate them to intrasite activity areas using the

spatial similarity and cohesion of artifact types. This method can handle heterogeneous data and is able to reveal and decon struct overlapping areas into their constituent elements. Application to a Mousterian habitation site from the Levant as a

case study enables us to distinguish three spatially cohesive yet overlapping sets ("tool kits") of artifact and ecofact types within the boundaries of an excavated 50 m2 sub-area of the site.

Proponemos un metodo nuevo para el analizar de conjuntos cohesivos de artefactos, para relacionarlos a zonas de actividades

por el medir de la similitud espacial y indice de coherencia de tipos de artefactos. Puede manejar tal metodo datos heterogeneos y puede separar zonas de actividades superpuesto hasta sus elementos particulares. Una prueba del metodo con un sitio

domestico Musteriense del Proximo Oriente nos permite distinguir tres conjuntos diferentes de herramientas y 'ecofactos' dentro de una area excavada del sitio de 50 m2.

One of the primary goals of an intrasite spatial analysis is to discover patterning in the dis tribution of artifacts within an archaeolog

ical site that relates to past human activity. This goal depends on analytical methods that enable us "to make apparent a structure that is otherwise not easily observed" (Read and Russell 1996:2). Here we introduce a new method for identifying spatially cohesive sets of artifacts within a site, and illustrate the utility of the method with a case study from Bir

Tarfawi, a Middle Paleolithic site in the Levant.

Finding a satisfactory method for identifying activity areas has been problematic. From a statis tical viewpoint, the majority of these methods pre sume that the data sets exhibit homogeneity, normality, and unimodality. From an archaeologi cal viewpoint, most of the commonly used tech

niques also assume that activity areas are spatially non-overlapping. In contrast, the method we intro duce here does not require these assumptions.

Briefly, our method begins by calculating a

probabilistic measure of spatial similarity for each

pair of artifact types. This probability measure is based on the average of the pairwise Euclidean arti fact geometric distances among the constituent arti facts that make up that particular artifact type. The

probabilistic measure is then converted into a sym metric matrix showing the pattern of spatial simi

larity and dissimilarity among the types. In this

matrix, the artifact types are compared, and the

spatial relationships are shown in the individual cells by entries of either Os or Is. The value of one

represents a statistically significant measure of spa

tial similarity for a pair of artifact types; the value of zero represents a statistically nonsignificant mea sure of spatial similarity. In this way, the matrix values show the comparison of every artifact type with every other type.

This matrix of ones and zeros may then be rep resented as a graph in which each artifact type is a

node in the graph. When the matrix value is 1 for

the two artifact types corresponding to two nodes, the two nodes are connected with a line. In this

way, a line in the graph connecting two nodes rep

Michael L. Merrill School of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287

(Michael.L.Merrill @ asu.edu)

Dwight W. Read Department of Anthropology and Department of Statistics, UCLA, Los Angeles, CA 90095

(dread @ anthro.ucla.edu)

American Antiquity 75(3), 2010, pp. 419^151

Copyright ?2010 by the Society for American Archaeology

419



420 AMERICAN ANTIQUITY [Vol. 75, No. 3,2010

resents two types for which their measures of spatial similarity show no statistical difference.

With this graph representation, the matrix of

similarity and dissimilarity among the artifact types becomes an adjaceny matrix. In graph-theoretic terms, an adjacency matrix identifies whether one

node is connected to another node by a line. One

advantage of using a graph is that it provides a

visual way to determine patterning in the adjacency matrix of spatial similarity. Another is that by expressing the spatial similarity information for artifact types in the form of a graph, we can then

analyze the artifact type spatial relationships using concepts from graph theory.

We begin the analysis of the spatial relationships by first distinguishing all possible cliques of artifact

types. A clique (see Appendix 2 for more details) is a maximal set such that each pair of types in the set has a value of 1 in the adjacency matrix. By

maximal set, we mean that in these sets, no artifact

type can be added to the set and still have a value of 1 when compared with all other artifact types in the set. A clique based on the adjacency matrix will thus consist of all artifact types with statistically similar spatial distributions. As a result, the artifact

types in the clique co-occur in the same location,

indicating that they were consistently used in that

place in the performance of the same activity. If a

given artifact type is used in more than one activity, then some of the artifact types in one clique can also be types in another clique. A clique can there fore be interpreted fairly intuitively as a set of arti fact types that are used in a particular set of activities in at least one location within the site.

The next step in the spatial analysis creates a Galois lattice representation of the cliques. The Galois lattice depicts how a clique C is built up from its constituent artifact types. The lowest level of the lattice begins with artifact type(s) that co occur in the greatest number of cliques. For all other levels in the lattice, the artifact types occurring at a higher level in the lattice are in progressively fewer cliques. This stepwise process results in chains of artifact types, beginning with the artifact

type(s) in C that co-occur with the greatest number of other cliques and ending with artifact type(s) that may be unique to C. When viewed jointly, the lattice chains show the degree to which each artifact

type in C is found in the other cliques. In the next

step of the analysis, we examine the density plots

of the chains associated with the cliques. Lastly, we interpret these results using Carr's (1985) mod els for the organization of artifact types.

We illustrate our proposed method for decom

posing a spatial distribution into its constituent ele ments by analyzing a data set (Hietala and Larson

1980) from Bir Tarfawi, a Middle Paleolithic site in the Levant. The results of our analysis also add to the ongoing debate about the function of Mous terian points: whether they were manufactured pri

marily for use as spear points or as scrapers. Our results show that Mousterian points from Bir Tar fawi should be classified as converging sidescrapers (Debenath and Dibble 1994; Holdaway 1989); that

is, Mousterian points were most likely multipur pose tools that were functionally similar to con

verging sidescrapers. In conjunction with our delineation of overlapping activity areas, this result

suggests that the inhabitants may have been using a mobile foraging strategy, thus making Bir Tarfawi a seasonally occupied camp, rather than a long term habitation site. This interpretation demon strates the utility of our approach. Previously used

methods of intrasite spatial analyses would not have been feasible for this detailed scale of analysis.

Limitations of Previous Methods for Finding Patterning in Artifact Spatial Distributions

Previous attempts to analyze intrasite artifact dis tributions have used multivariate statistical methods such as factor analysis (e.g., Craig et al. 2006; Downer 1977; Hill 1966), principal components analysis (e.g., Ciolek-Torrello 1984; Whitridge 2002; Yvorra 2003), cluster analysis (e.g., Craig et al. 2006; Krasinski 2005; Moyes 2001, 2002; Simek 1984; Yvorra 2003), and more recently, cor

respondence analysis (Kuijt and Goodale 2009). A fundamental problem in using these and similar

analytical methods is that they assume homoge neous data (Read 1985,1992,2007). However, dif ferent activity areas may share tool types and the tools may have been used differently in each of these areas. In addition, activity areas may be over

lapping. Clustering methods can reduce the het

erogeneity of the data when each activity gives rise to an isolated activity area, but attempts to make the clustering more sensitive to overlapping activity areas by including variables other than artifact spa tial location may make an unwarranted conver



Merrill and Read] A NEW METHOD USING GRAPH AND LATTICE THEORY 421

gence assumption (Read 1992:1, 2007:Appendix A).

Other widely used methods for analyzing spatial pattern such as nearest neighbor analysis do not resolve the problem of spatial heterogeneity as

these methods focus on finding characteristics of

already determined artifact clusters. The summary statistic computed in first-order nearest neighbor analysis (Clark and Evans 1954) measures depar ture from randomness (no structure) in the direction of either aggregation or regular spacing. A first order nearest neighbor analysis (comparison of a

point to its spatially closest point) is effective in

evaluating the relative density of artifact clusters but not the grain of such clusters (Carr 1984:160). It is appropriate for heterogeneous spatial data only when the data are structured by activities into dis

joint clusters of points on a line, in a plane, or higher dimensional space. Often, though, we expect dif

ferent activity areas to be spatially overlapping, or

there may be hierarchical spatial patterning within the spatial distribution of artifacts. First-order near

est neighbor analysis does not decompose overlap

ping or hierarchical activity areas into their

constituent components. Though first-order nearest

neighbor analysis is effective for characterizing

degree of clustering in spatially isolated sets of arti

facts, it is not helpful in disambiguating the hetero

geneity of spatially isolated sets of artifacts into

their constituent components.

Analytical Goal: Decomposition of Activity Areas into Constituent Elements

The spatial layout of a site can be divided into activ

ity areas defined by Flannery and Winter as:

A spatially restricted area where a specific task or set of related tasks have been carried on,

which is generally characterized by a scatter

of tools, waste products, and/or raw materials;

a feature, or set of features, may also be present

[Flannery and Winter 1976:34].

Carr elaborates on this definition:

Activity areas vary greatly in size, shape, arti

fact densities, and artifact compositions. To

these characteristics may be added the follow

ing. Activity areas are not necessarily high

density clusters of artifacts in a background of

lower densities of artifacts; they may be areas

of low-artifact density surrounded by zones of

higher artifact density. Activity areas may vary in the degree to which they are internally homogeneous in their artifact composition.

The borders of activity areas may vary in their

crispness [1984:125-126].

These observations imply that the spatial locus for a particular organized activity?during the time the

activity was taking place?should contain artifact

types characteristic of that activity when artifacts are left where they were used. Though the area

where that activity took place may contain artifacts from other activities, we expect the artifacts from a single activity to be spatially structured in a similar manner by the spatial boundaries involved when

conducting that activity. Because a spatial location

may be relevant for more than one activity, the overall spatial patterning of artifacts in a location

represents the cumulative effect of all the activities with overlapping spatial locations.

The internal, spatial organization of an archae

ological site, if decomposed into its constituent

activity specific activity areas, can provide, as noted

by Carr, information regarding the spatial context for the behavior of the former occupants of the site:

For example the kinds, frequencies, and spatial organization of activities that occurred in a site can be used to infer its seasons of occupation (Binford 1978), site functions (Styles 1981),

community population (Cook and Heizer

1965; Yellen 1977) household interaction pat terns, community kinship, and social organi

zation (Brose 1968; Wiessner 1982) [Carr 1985:308].

Spatial analysis also provides valuable information about cultural and noncultural formation processes that "are critical to formulating and testing hypothe ses about the structure and dynamics of past behav

ioral systems (Binford 1977) and natural

environmental systems" (Carr 1985:308). Data

derived from paleoclimatic studies, phytolith analy sis, faunal studies, and so on, provide details of the

temporal variation and structure of the natural envi

ronmental systems present during and after the

occupation of a site.

Our analytical goal, then, is to move backward

from the spatial distribution of artifacts as they are

found in an archaeological site to identification of

the behavioral processes underlying that spatial



422 AMERICAN ANTIQUITY [Vol. 75, No. 3, 2010

Figure 1. Spatial transformation of three areas of organized activity over time into a single depositional set. Analysis pro vides a means of recovering the intrasite locations of the three areas, as well as some or all of the constituent artifact types of each of the three areas.

distribution under the assumption that the current

spatial distributions of artifacts provide signatures reflecting the various spatially structured behavioral

processes that we are trying to identify. To decom

pose the spatial distribution of artifacts in this man

ner, we need to develop and use mathematical methods sensitive to the processes giving rise to

spatial patterning and capable of disambiguating the cumulative effect of spatially overlapping sets of behavior into its constituent elements. Because noise that may obliterate patterning in the spatial distribution of cultural remains is introduced by bioturbation (Pierce 1992) and post-depositional processes (Schiffer 1987), as well as by discard

behaviors, our analytical methods need to remove

noise effects so that we can obtain "congruence between analytically arrived at groupings (here groups of artifact and ecofact types) and preexisting structure" (Read and Russell 1996:2).

Carr defines a depositional set in a manner that

captures the effect all of these processes have on the spatial distribution of artifacts:

A depositional set may be thought of as a math ematical set, the organization of which is the end product of structural transformations

(archaeological formation and disturbance

processes) operating upon a previously struc tured set (activity sets organized by human

behavior) [Carr 1984:117].

Our goal is to determine the locations of areas of

organized activity within an archaeological site (left

side of Figure 1) from the depositional set (right side of Figure 1). We posit that the analytical method introduced in this paper is capable of identifying spatially cohesive sets of artifact types embedded in depositional sets.1 Then, as we will illustrate, by examining plots for each of the spatially cohesive sets of artifact types, specific areas of deposit asso

ciated with denser concentrations of artifacts can be identified and used in further analysis.

An Analytical Method for Delineating Spatially Cohesive Sets of Artifacts

The method we introduce in this paper can distin

guish spatially cohesive sets of artifacts from spa tially overlapping artifact sets that may have been

subjected to structural transformations (see Figure 1). Figure 2 identifies the sequence of steps and

substeps comprising our method for analyzing the

spatial distribution of artifacts and artifact types. Each of these steps will be discussed in turn.

Step 0: Raw Data and Pre-analysis

The pre-analysis involves forming an artifact typol ogy based on homogenous artifact types. Dwight Read (2007; see also Read and Russell 1996) dis cusses in detail methods for the construction of an

objective artifact typology aimed at ensuring types are homogeneous. The x and y coordinates for each artifact in the artifact types are the raw data for all

subsequent steps.




StepO Collect Raw Data and Pre-analysis

Step 1 Calculate Spatial Similarity for Pairs of Artifact Types

Sub-step A MRPP Delta-value

Sub-step B MRPP Delta P-value

Step 2 Construct a Graph for Artifact Spatial Relationships

Step 3 Analysis of the Graph Structure

Sub-step A

Identify Cliques

Sub-step B Form Incidence Matrix for Artifact Types and Cliques

Sub-step C Draw Galois Lattice

Step 4 Analyze the Spatial Distribution of Artifact Types with Density Rots

Step 5 Assign Artifact Type Pairs using Carr's (1985) Models for Organization of Artifact Types Based on a Monothetic-Polythetic Dimension

End

Figure 2. Flow chart of steps in the analysis. Note that in Step 3, cliques with two or more lattice chains ascending to them should also be plotted.

Step 1: Calculate a Spatial Similarity Measure

for Pairs of Artifact Types

This step has two substeps: (A) measurement of

spatial co-occurrence and (B) computation of prob ability values for the measure of spatial co occurrence. Jointly the two substeps result in a

probability-based measure of spatial similarity for each pair of artifact types.

(A) Spatial Co-occurrence. A within-group measure of spatial co-occurrence is computed for each of the two artifact types whose spatial distri bution is being compared. The spatial structure is measured by a multiresponse permutation proce dure (MRPP) (Mielke et al. 1976), a non-parametric statistical method recommended over twenty years ago for analyzing the intrasite spatial analysis of artifacts (Berry et al. 1980, 1983, 1984) since

MRPPs make no assumption of "interval-level

measurement, homogeneity, or normality" (Berry

et al. 1980:55). An MRPP can use small and

unequal sample sizes and can incorporate all of the data available for analysis such as the distribution

pattern for an artifact type over an entire site instead of a random sample, with the latter subject to edge effect and sampling errors (Reich et al. 1991).

The MRPP measure of spatial co-occurrence with a summary statistic called delta (8) that (in this application) measures the degree of separation of pairs of artifact types within an archaeological site. Smaller values of delta correspond to greater spatial segregation for the artifact types making up a pair of artifact types. For any pair of artifact types (call them Type A and Type B), define delta to be the pairwise, weighted average Euclidean distance,

?, of the artifacts comprising Type A and Type B:

N A W B




Figure 3. Distribution of two hypothetical artifact types with inter-artifact Euclidean distances, indicated by lines con

necting pairs of artifacts within each artifact type.

where nA is the number of artifacts of Type A, nB is the number of artifacts of Type B, and nA + nB =

N. Note that the MRPP delta can be computed for artifacts distributed in one, two, or three dimensional space. Here we are only considering artifacts distributed in two-dimensional space.

To illustrate the computation of delta, consider the spatial distribution shown in Figure 3 (modified from Figure 2 of Zimmerman et al. 1985) with three artifacts of Type A and four artifacts of Type B. The Euclidean distance for each pair of artifacts in Type A or in Type B is indicated by the symbol

where 1 < i < 6, 2 < j < 7 and i ? j. Visually, the spatial distributions of artifact Types A and B in Figure 3 have little overlap, though the artifacts in one type appear to be close to the artifacts in the other type.

The MRPP delta values for artifacts Type A and B in Figure 3 are computed as follows from the x

y coordinates for each of the artifacts:

Ai,2 +

Ai,3 + A:

3

L2,3

)

V5+3 + V2 3

? 2.217

A4,5 + A4,6 + A4,7 + A5,6 + A5,7 + A(

6

L6,7

6 * 3.911

J

8 = 77^+^=|(2-217)+T(3-911^3-185

We provide additional computational details for the MRPP delta in Appendix 1.

(B) Probability Values. A MRPP delta proba bility value (P-value) is the probability of obtaining a delta of equal or lesser magnitude than the observed delta value by chance. A delta P-value is "derived through a permutation argument; thus there are no distributional requirements on the data"

(Zimmerman et al. 1985:608). The delta P-value

equals the observed delta's rank position in an ordered list of deltas computed for each of the M

possible permutation configurations of artifact

Types A and B among the artifact locations divided

by M, where

M = nA \xnB\j

Under the null hypothesis, all M permutations are

equally likely. For artifact Types A and B in our

example (Figure 3), there are

71 2*3*4*5*6*7 oe M-=- 35

(3!)(4!) (2*3)(2*3*4)

possible configurations. The MRPP delta P-value for the configuration shown in Figure 4 equals 2/35




Figure 4. The MRPP delta is approximately 2.629 and ranks second from the smallest of the thirty-five possible permu tations of artifact Types A and B. Therefore the delta P-value equals 2/35 as .057.

= .057 since 8 = 2.629 is the second largest of the 35 delta values obtained from the permutations. The two groups in Figure 4 are well separated and do not overlap. The delta P-value suggests that the

spatial distributions for artifact Types A and B are

significantly separated. This strongly contrasts with the configuration shown in Figure 5, where the dis tributions of the two groups overlap extensively and have a very high MRPP delta P-value of .943.

Step 2: Determine Sets of Spatially Cohesive Artifact Types

For this step we also have two substeps: (A) con struction of a symmetric adjacency matrix and (B) construction of a graph of artifact spatial relation

ships. (A) Adjacency Matrix. We begin by forming an

adjacency matrix for the artifact types in which we enter the value 0 if each artifact type in a pair of artifact types has dissimilar spatial distributions and the value 1 if they have similar spatial distrib utions (for a more detailed discussion of adjacency matrices, see Appendix 2).

Although the delta values are a measure of spa tial co-occurrence, we base the adjacency matrix

on the delta P-values because the delta P- values are

probabilities that take into account sampling error.

In addition, different pairs of artifact types can have

dramatically different MRPP delta P-values and identical MRPP delta values. An example of this

situation, using actual data from the data set ana

lyzed below, is shown in Figure 6. For Figure 6, the small delta P-value of .039 for Figure 6, graph A

matches the visual impression that cores and con

verging sidescrapers are spatially well segregated and so most random reassignments of cores and

converging sidescrapers to the data point locations will not yield a pattern similar to the pattern shown in Figure 6, graph A. The much larger delta P-value of .25 for Mousterian points and converging sidescrapers in the same sampling area (Figure 6,

graph B) matches the visual impression that these two artifact types have overlapping distributions.

Once the MRPP delta P-values have been com

puted for all possible pairs of artifact types, they are assembled pairwise into an adjacency matrix as follows. First, the diagonal values of this matrix will be set to 0 since only the intrasite distribution of pairs of artifact types is being compared. Second, we decide on a cut-off value for the delta P-values in order to convert delta P-values into Is and 0s. Since the delta P-values are probabilities, a typical statistically based cut-off value would be .05 or

.01. Delta P-values are replaced by 1 if they are




Figure 5. The MRPP delta is approximately 3.909 and is the thirty-third largest of the thirty-five possible permutations of artifact Types A and B. It follows that the delta P-value equals 33/35 = .94.

larger than the cut-off value and otherwise by 0.

Converting the delta P-values to 1 or 0 provides a

way to filter out, as discussed above, noise due to

sampling error, post-depositional processes, etc.

(B) Graph of Spatial Relationships. Next, a

graph is constructed using the adjacency matrix. In the graph, each node represents an artifact type and a 1 in the adjacency matrix becomes a line con

necting the corresponding pair of nodes in the

graph. In effect, the graph is constructed using the intrasite spatial relationship for each pair of artifact

types. The graph is therefore both a statistical (since the presence/absence of a line connecting two arti fact types is determined by a probability, namely the delta P-value) and a structural model of a net

work of spatial relations among a set of artifact

types. As a structural model, the structure of the

graph is amenable to analysis using methods from

graph theory and network analysis. By analyzing the structure of the graph, we can identify spatially cohesive sets of artifacts belonging to an intrasite area where organized activity took place. More

specifically, a set of artifact types used within the

spatial locus for organized activity is expected to be a coherent and spatially cohesive group of arti fact types when the spatial location of the artifacts results from localized, patterned, and recurrent past

human activity. Each pair of types in a spatially cohesive set has a similar spatial pattern represented by a 1 in the adjacency matrix.

Step 3: Analysis of the Graph Structure

This step has three substeps: (A) identify maximal,

spatially cohesive sets of artifact types, (B) con struct a matrix of artifact types and cliques, and (C) construct and analyze a Galois Lattice based on the matrix determined in (B).

(A) Maximal Sets of Artifact Types. First we

identify maximal, spatially cohesive sets of artifact

types known in graph theory as cliques.2 A spatially cohesive set of artifact types is maximal when there is no other artifact type in the graph, not already part of the set in question, that is connected by an

edge to each of the artifact types in that set. A tri

angle is defined to be the smallest, nontrivial clique possible in a graph (for a more detailed discussion and the mathematical definition of a clique refer to

Appendix 2). Thus a clique becomes, in our

method, a group of artifacts that pairwise have the same spatial distribution as determined by the cut off values in the delta P-value matrix.

In general, some of the cliques identified in this

substep will share the same artifact type when an

artifact type was used in more than one spatially




O Core X Converging Sidescraper X Converging Sidescraper + Mousterian Point

Figure 6. (A) Observed MRPP delta equals 1.733 and the corresponding delta P-value equals .039. (B) Observed MRPP delta equals 1.733 and the corresponding delta P-value equals .25.

distinct activity. The next several substeps allow us to decompose a clique (i.e., a maximal spatially cohesive set of artifact types) into a chain showing how the clique can be built up from maximally overlapping to clique-specific artifact types. The chains are used to deterrnine the spatial location for activity sets and to infer a hierarchy of usage (from multiuse to specialized use) of the artifact

types.

(B) Matrix of Artifact Types and Cliques. Form an m x n incidence matrix in which the rows are the artifact types and the columns are the cliques identified in the above substep. A 1 in the matrix indicates that a given artifact type (row) is present in the corresponding clique (column), and a 0 indi cates the artifact type is absent from that clique. The partial and dual ordering of the artifact types and cliques used to build a lattice structure (dis cussed below) acts as another noise filter in addition to the filtering provided by constructing an adja cency matrix from the delta P-values. The lattice structure also displays the degree of overlap between categories (i.e., the overlap of artifact types in cliques in our method) and in a more global sense forms an overlapping taxonomy (Roth 2006).

(C) Construct a Galois Lattice. Use themx? incidence matrix from the previous substep to draw a Galois lattice as a line diagram. A Galois lattice

(see Appendix 3 for a formal definition) is a dually ordered graph (see Figure 7) with each node con

sisting of two sets: an object set and an attribute set

for the objects in the object set. The dual ordering refers to ordering the nodes by set inclusion and set unions for the object sets and by set containment and set intersection for the attribute sets. As one moves up the graph, the object set for a lower node connected by a line to a higher node is included within the object set for the higher node (e.g., com

pare nodes B and D in Figure 7). When a higher node is connected to two or more lower nodes, the

object set for the higher node is formed from the union of the object sets belonging to the lower nodes to which it is connected (e.g., compare the

object sets for node C versus nodes A and B).

Dually, the attribute set of a lower node contains the attribute set of a higher node to which it is con nected (e.g., compare the attribute sets for nodes B and C in Figure 7). When it is connected to two or more lower nodes, the attribute set of the higher node is formed by the intersection of the attribute sets for the lower nodes to which it is connected

(e.g., compare the attribute sets for node C versus nodes A and B in Figure 7). This means that nodes

higher in the graph have fewer attributes in the attribute sets associated with them than nodes lower in the graph since there are fewer attributes common to the objects in more inclusive object sets (e.g., compare nodes A and C in Figure 7). The converse is true when moving down the graph: object sets are formed by set containment and set intersection, attribute sets are formed by set inclusion and set

union, and attribute sets are more inclusive.




An X in the table (below) indicates that an

object in the same row as the X has the attribute in the same column as the X.

For example, object a has attributes

1 and 3, and object d has attributes 2 and 4.

G = (a,b,c,d)

= the object set.

M = (1,2,3,4)

= the attribute set.

(a, b, c, d, 0)

(a, b, c,l)

(a,c, 1,3)

(b,d,2A)

(b, 1,2 A)

(G, A/, I) is often called the formal context which is explicitly depicted by the above table, where IC G X M is a binary relation consisting of all the

object-attribute pairs depicted in the table.

Note that both Galois lattice diagrams are

isomorphic. However, the bottom lattice was

drawn using the ConceptExplorer computer program, which results in the nodes not being fully labeled as is the case for the upper graph,

which was drawn manually.

(0,1,2,3,4)

The empty set.

Figure 7. An example of a formal context (Davey and Priestley 2002; Wille 1982,1984) or incidence matrix unfolded as a Galois lattice. Observe that each of the objects belonging to the object set (a, b, c, d), has one or more attributes from the attribute set (1,2,3,4). Also, note that the Xs in the table represent Is and the blank squares 0s. The four nodes of the lattice that have either a non-empty object or attribute set are indicated by either A, B, C, or D in the completely labeled lattice drawing (upper lattice).

Graphically, the ordering for the nodes has the

property that for every pair of nodes (e.g., nodes C and D in Figure 7), each with a non-empty object set, there is always a node lower in the ordering for the nodes in this pair of nodes (e.g., the object set for B is contained in the object sets for each of C and D and the attribute set for B contains the attribute sets for each of C and D) and there is a node higher in the ordering (e.g., the unlabeled node at the top of the diagram for the nodes C and D) such that the

object set for this node contains the object sets for each of the pair of nodes (e.g., the object set for the

topmost node contains the object sets for the C and D nodes) and the attribute set is included in the attribute sets for each of this pair of nodes (e.g., the attribute set for the topmost node is included in the attribute sets for the C and D nodes).

These ordering relations are expressed visually in Figure 7 through pathways in the graph, or Hasse

diagram, of the lattice containing nodes A, B, C, and D as defined above. For example, in Figure 7 there are two separate paths, one connecting A to C and the other connecting B to C, and a single path linking B to D. Also, the single, bottom node of a Galois lattice has an empty object set associated with it unless there are one or more objects in the

object set that possess all the attributes in the attribute set. Similarly, the single top node in the lattice will have an empty attribute set associated

with it when there are no attributes common to all

objects. A drawing showing the lattice structure may be

constructed using a computer program designed for this purpose such as Concept Explorer




(http://conexp.sourceforge.net/). Each of the nodes in the lattice drawing represents one or more artifact

types (i.e., it can represent a set of artifact types) and/or one of the cliques. The lower an artifact type is in the lattice (lower, for example, in the sense of set inclusion), the more cliques to which it belongs.

Multipurpose artifact types are expected to be in the lower levels of the lattice since they belong to

more cliques than artifact types occupying higher levels.

By using the lattice representation of the inci dence matrix, each clique may be decomposed into a chain of artifact types (or sets of artifact types,

depending on whether a node represents a single artifact type or a set of artifact types) in which the artifact types from a clique are entered into the chain in the order determined by the number of

cliques in which that artifact type co-occurs. Since each clique is a coherent and spatially cohesive set

of artifacts that can be interpreted as corresponding to a specific set of activities and their associated

activity area(s), the chains measure the extent to

which an artifact type is part of more than one activ

ity and its associated activity area(s) and thereby enables us to decompose overlapping activity areas

into their constituent elements.

Step 4: Analyze the Spatial Distribution

of Artifact Types with Density Plots

In this step, a scatter plot is made of the artifacts for each of the artifact types comprising a lattice

chain. Then the sub-area(s) of a given scatter plot associated with a high density of the artifact types belonging to a particular lattice chain and its cliques is determined from a density plot. These sub-areas are interpreted as areas of organized activity, and are used in the next step to assign pairs of artifact

types in a sub-area to one of Carr's (1985:333,336) models of organization of artifact types.

We begin by using a spatial filter (Holloway 1958; Zurflueh 1967) on the x-y coordinates of the

artifacts in the artifact types comprising each of the

lattice chains for the cliques. We used a robust

bivariate kernel density estimator (KDE) to con

struct density plots (see Figures 14-16 for examples of density plots). Botev (2006, 2007) provides a

bivariate KDE implemented in Matlab and down

loadable as a Matlab function. Baxter and Beardah

(1997) provide archaeological applications of

KDEs; see also the discussion by Carr( 1979,1982).

The bivariate KDE plots used here provide an

objective way to visualize the distribution of sets of spatially cohesive artifact types for a range of different densities, as well as an unbiased means for estimating the boundaries of their distribution within a site. In the bivariate KDE plots, the area(s) of highest (peak) artifact density are readily iden tified by being significantly lighter or darker than the area of the plot surrounding them. Specifically, in the left KDE plots, which also show artifact posi tions, the sub-area(s) of the plot with peak artifact

density are the lightest areas of the plot, whereas the opposite is true for the right KDE plots, where sub-areas of peak artifact concentration are the darkest areas of the plot. The density peaks revealed

by the KDE density plots are interpreted as the

location(s) in a site associated with a specific set of spatially organized activities.

In addition, the artifact types belonging to one

lattice chain and its associated cliques may occur in more than one location of a site and these may spa

tially overlap, which is the case in the Bir Tarfawi

example discussed in the next section and is appar ent in its KDE plots. Further, two sets of artifact

types belonging to two distinct lattice chains may also share a subset of artifact types. These two pos sibilities enable our method to address two func tional sets of events, one in the behavioral domain, and the other in the archaeological domain. If two sets of artifact types represented by two lattice

chains were used in different functional categories of events (e.g. two different sets of activities), and if the two lattice chains share some artifact types

(using Carr's [1985:331] characterization), then the activities corresponding to these two chains would be overlapping within the behavioral domain. In the

archaeological domain, if two sets of artifact types comprising two distinct lattice chains and their asso

ciated cliques have some of the same artifact types and represent "two different functional classes of

archaeological deposits-two different sets of

deposits" (Carr 1985:331), then these sets are over

lapping within the archaeological domain.

Step 5: Assign Artifact Type Pairs using Carr's

(1985) Models for Organization of Artifact

Types based on a Monothetic-Polythetic Dimension

The monothetic-polythetic dimension of organiza tion of artifact types relates to the internal organi




Figure 8. Plot of the converging sidescrapers and Mousterian points in the sub-area of the Bir Tarfawi site associated with Lattice Chain #1 (which forms Clique #1) and the central ellipse in Figure 12 (next section).

zation of objects belonging to a set (Carr 1985:333). Carr distinguishes between monothetic and poly thetic sets as follows:

In a monothetic set, the elements of a set all share the same character states; all character

states are essential to group membership. In a

polythetic set, the elements share a large num ber of character states, but no single state is essential to group membership (Clarke 1968:37; Sneath and Sokal 1973:21) [Carr 1985:333].

Carr also characterizes depositional sets as either monothetic or polythetic:

A depositional "set" (list of artifact types char

acterizing a set of deposits) would be mono

thetically distributed among the set of deposits if all the artifact types in the depositional set were contained in each of the deposits. A depo sitional set would be polythetically distributed

among a set of deposits if the deposits held in common most of the artifact types in the depo sitional set, but no one artifact type were

required of a deposit to be a member of the

deposits [Carr 1985:333].

This can be done visually by first making separate plots for the positions of each pairwise combination

of artifact types belonging to a lattice chain. Then the spatial pattern of each artifact type pair within the discrete sub-area(s) of the site associated with the lattice chain is assigned to one of Carr's

(1985:336) models of organization of artifact types along a monothetic-polythetic continuum by iden

tifying the following characteristics of discrete

groups consisting of three or more artifact types within the sub-area.

1. Do the sets of groups in the sub-area always have both types of artifacts (globally monothetic) or does at least one group have only one artifact

type (globally polythetic)? 2. If one artifact type occurs in each of the

groups, does the other type also occur, and vice versa (symmetrical co-arrangement)? If not, what is the relative density of one artifact type compared to the other in each of the groups (magnitude of

asymmetry)? Does this vary between groups? Does the same artifact type always have the highest fre

quency in each of the groups (the same direction of asymmetry in each of the groups), or does this

vary? 3. Does one artifact type in each of the groups

always have the other artifact type for a nearest

neighbor and vice versa (monothetic set of artifact

pairs, locally monothetic)? If not, the arrangement of artifact pairs in the sub-area is locally polythetic.




Mousterian Points (1,2,3,4,5,6,7) and Converging Sidescrapers (8,9,10,11) Dissimilarity Measure = Euclidean distance, Linkage = Ward's Method

2h

1.5

Group 2 Group 1

0.51

r 10

Figure 9. Hierarchical cluster analysis of the x and y-coordinates of the converging sidescrapers and Mousterian points in the Bir Tarfawi sub-area outlined by a circle.

For example, the plot sub-area (outlined by a circle) of a plot of converging sidescrapers and Mousterian

points (Figure 8) from the example discussed in the next section appears to consist of two distinct

groups of Mousterian points and converging sidescrapers.

The two groups of Mousterian points and con

verging sidescrapers that appear to be visually obvi ous in the sub-area is supported by the result of hierarchical cluster analysis (Figure 9).

Group 1 consists of three converging sidescrap ers and one Mousterian point (relative density 3:1). The second group is composed of six Mousterian

points and a single, converging sidescraper (relative density 6:1). Since both groups contain both Mous terian points and converging sidescrapers, we con clude that the sub-area's internal organization with

respect to Mousterian points and converging sidescrapers is globally monothetic. Also, the very locally asymmetrical co-arrangement of Mouster ian points and converging sidescrapers defines a

polythetic set of item pairs (Carr 1985:339). Addi

tionally, both groups have a different magnitude for their asymmetry measured by relative densities

(the second group has twice the magnitude of asym metry as does the first). Finally, in both groups, Mousterian points have a higher frequency than

converging sidescrapers, which means that both

groups have the same direction of asymmetry. Using Carr's table of model characteristics, it is clear that the sub-area fits Carr's Model 3. Model

3 depicts a type of artifact organization that is

expected in an archaeological deposit since most formation processes are likely to be non-uniform and therefore, at the minimum, produce a locally polythetic artifact organization (Carr 1985:355).

A Detailed Application of the Method: An Excavated Area Consisting of a Small

Sample with Few Artifact Types

The method described above will now be illustrated with data from Bir Tarfawi, a Paleolithic Levant site located in the Eastern Sahara (see Figure 10). Bir Tarfawi has been interpreted as a Neanderthal habitation site (Hietala and Larson 1980:379) since the excavated component of Area C (see Figure 11) contains Mousterian points, Levallois cores and flakes as well as other lithic types diagnostic of Neanderthal material culture.

We selected the Bir Tarfawi example because it is a small, well-preserved site that appeared to have a significant amount of overlap in its organizational structure. In addition, it already had a reasonably well-defined lithic typology. We excluded the deb

itage category, though, since debitage was not included in the portion of the Hietala and Larson

analysis we are considering (see Figure 17). One drawback with using the site as a test case

is that obviously we do not have any means to cross check our results with ethnographic data. Also, the limits and utility of our proposed method need to




Figure 10. Map that includes the Bir Tarfawi site (sub-region 1). Bir Tarfawi is located near the border of Egypt and Sudan (Figure 2.1, Wendorf and Schild 1980).

be evaluated further with simulated assemblages in order to explore more comprehensively the effect of noise on being able to discern and disaggregate different sources of spatial patterning.

Hietala and Larson (1980:386-387) conclude from the results of their intrasite spatial analysis that: (1) lithic reduction activities appear to be spa

tially segregated from areas where fauna-related activities seem to have occurred, and (2) pointed tools such as Mousterian points, converging sidescrapers, and converging denticulates "tended to be more strongly associated with faunal elements than were the other artifacts studied in the analysis" (Hietala and Larson 1980:386-387).

In their analysis, though, Hietala and Larson

acknowledge that the "primary reduction area" they had identified was subjectively and arbitrarily defined by noting the visual overlap of the spatial distributions for cores and debitage consisting of Levallois flakes and blades, non-Levallois flakes and blades, and debris (1980:384). A further pos sible source of subjectivity in their approach arises from defining an artifact to occur in the proximity

of faunal elements if the artifact is within 60 cm of a faunal element. They justify the use of a 60 cm search radius, in part, because it is roughly twice the average nearest neighbor distance of the faunal elements found at the site. However, the use of a fixed search radius presupposes uniformity and

equivalence between the formation and post depositional processes that led to the observed spa tial associations of artifacts and faunal elements

throughout Area C. Our goal is to reanalyze the data from Bir Tar

fawi using the methods presented above to see if

they lead to a more refined analysis of the spatial distribution of the artifact types in the site. The data to be used consist of the spatial location for the arti facts making up the eight types of artifacts identified

by Hietala and Larson. The x and y coordinates for each artifact were measured with %i inch precision in Adobe Photoshop from a scanned image of Fig ure A8.1 in Hietala and Larson (1980:385).

Step 0: Identification of Artifact Types

Table 1 lists the six artifact and two ecofact types







Table 1. Bir Tarfawi Artifact and Ecofact Types.

from Hietala and Larson that are used in this analy sis (1980). For the purposes of our analysis, we

must assume the artifact types are homogeneous since we do not have access to artifact measure

ments. Ideally, we would first examine the artifact

types using the methods of Read to ensure that the

types are homogenous (2007).

Number Artifact/Ecofact Type

2

3

4

5

6

7

8

Core

Denticulate

Transverse Denticulate

Sidescraper

Converging Sidescraper Mousterian Point

Bone

Teeth

Step 1: Calculate Similarity Measures

for the Artifact Types

Table 2 lists the calculated MRPP deltas, delta P values and sample sizes. Table 3 shows the MRPP delta P-value matrix.

Step 2: Compute the Adjacency Matrix and Graph

Table 4 is the resulting adjacency matrix using a cut-off value of .05. Figure 14, with an ellipse enclosing Clique 1 from Figure 12, is the graph drawn using Table 4.

Step 3: Analysis of the Graph Structure

Three cliques are identified in the graph and enclosed by an ellipse (Clique 1) and two closed Bezier curves (Cliques 2 and 3) (see Figure 12). Table 5 consists of the artifact and ecofact types

Table 2. Bir Tarfawi Raw Data and MRPP Calculations.

Artifact

Type A vs.

Type B

Observed

Delta

Probability (exact) of a smaller or

equal delta Probability (Pearson Type III) of a smaller or equal delta*

Sample Size

Artifact

Type A

Sample Size

Artifact

TypeB

1 vs.

1 vs.

1 vs.

1 vs. 1 vs.

1 vs.

1 vs. 8

2 vs. 3

2 vs. 4

2 vs. 5

2 vs. 6

2 vs. 7

2 vs. 8

3 vs. 4

3 vs. 5

3 vs. 6

3 vs. 7

3 vs. 8

4 vs. 5

4 vs. 6

4 vs. 7

4 vs. 8

5 vs. 6

5 vs. 7

5 vs. 8

6 vs. 7

6 vs. 8

7 vs. 8

1.7991945

1.78318366

1.96583797

1.73282032 1.61888307

1.97076784

2.13006227

2.17565894

2.19720144

1.98810252

1.8049994

2.2479342

2.13854839

2.32866741

2.07428262

1.78952685

2.30387504

2.21052364

2.1494025

1.98193451

2.28658278 2.21306801 1.73285506

2.2278598

2.10273049

2.14309913

1.98298988

2.24597031

.56530611

.871323529

.03955246 .016565204

.671328671

.043343653

.022818614

.877192982

.904545455

.757142857

.463208801

.249807004

.280136617

.340980895

.35488902

.271219261

.434272736

.43856752

.849254069

.914388918

.060546963

.723677014

.465953168

.142472127

.171580064

.025220271

.108821571

.681689078

15

15

15

15 15

15

15

10

10

10

10

10

10

3

3

3

3

3

16

16

16

16

9

9

9

13

13

48

10

33.

16

9 13

26

48

3

16

9

13

48

26

16

9

13

48

26

9

13

48

26

13

48

26

48

26

26

*See Appendix 1 for a discussion of Pearson Type III probabilities.




Table 3. MRPP Delta P-Value Matrix for the Six Bir Tarfawi Surface Artifact and Ecofact Types (Note that the Row and Column Headings Are the Artifact/Ecofact Numbers in Table 1).

_1_2_3_4_5_6_7_8_ 1 0 .565 .871 .28 .04 .017 .355 .341

2 .565 0 .671 .271 .043 .023 .434 .439 3 .871 .671 0 .877 .905 .757 .849 .914

4 .28 .271 .877 0 .463 .061 .724 .466 5 .04 .043 .905 .463 0 .25 .142 .172

6 .017 .023 .757 .061 .25 0 .025 .109 7 .355 .434 .849 .724 .142 .025 0 .682 8 .341 .439 .914 .466 .172 .109 .682 _ 0

Table 4. Adjacency Matrix Derived from the MRPP Values in Table 3

(Note that the Row and Column Headings Are the Artifact/Ecofact Numbers in Table 1).

_1_2_3_4_5_6_7_8 10 1110 0 1 1 2 10 110 0 1 1 3 110 1111 1 4 1110 111 1 5 0 0 1 1 0 1 1 1 6 0 0 1 1 1 0 0 1 7 1111 10 0 1

8 _1 _1_1_1_1_1_1_0

comprising each of the three cliques. Table 6 is an

8-x-3 incidence (presence/absence) matrix for the

cliques and the ecofact and stone artifact types. It is used to make the Galois lattice shown in Figure 13. Table 7 lists the artifact and ecofact types for the two lattice chains in the Galois lattice ending at Cliques 1 and 3, respectively (see Figure 13).

Step 4: Density Plots

There are three sets of density plots corresponding to the three cliques. The KDE density plots for the artifact types in Chain #1 (associated with Clique 3) are shown in Figure 14. The plots for the artifact

types in Chain #2 (associated with Clique 1) are

shown in Figure 15. The KDE density plots for

Clique #2 (which has no single, associated chain) are shown in Figure 16.

Because only Lattice Chain #2 contains cores, it follows that only this chain can be interpreted as an activity set associated with site areas that include core reduction. The inclusion of bone and teeth

with cores in Lattice Chain #2 suggests that the

conclusion of Hietala and Larson (1980) that the core reduction region they identify (see Area R, center of Figure 17) was spatially distinct from fauna-related activities (Areas F-l to F-6 of Figure

17) is incorrect. Figure 18 buttresses this conclusion for it is clear that in the sub-area of Area C outlined

by an ellipse (which is entirely within the "reduc tion area" R), there is a significant amount of over

lap in the spatial distribution of cores, bone, and teeth.

Our analysis also suggests that Mousterian

points are not spatially associated with bone, but with teeth. The overall low level of spatial overlap of Mousterian points and bone and the good spatial overlap of Mousterian points and teeth in one sub area is apparent in Figure 19 (area outlined by an

ellipse). This result refines the statement in Hietala and Larson (1980:386) that Mousterian points tend to be strongly associated with faunal elements.

In addition, the significant overlap for the con

centrations of artifact types belonging to Lattice

Chains #1 and #2, and Clique #2 can be seen in

Figure 17. The inferred overlap in areas of orga nized activity in Area C of Bir Tarfawi is supported by a more recent observation at the Levantine Mousterian site, Tor Sabiha in Jordan that "the dis

tribution of artifacts suggests that activities were

overlapping and non-segregated" (Henry et al.

1996:34).

Finally, the exclusive occurrence of cores in Lat




Figure 12. Graph with cliques produced from the adjacency matrix of Table 4.

tice Chain #2 and Mousterian points in Lattice Chain #1 suggests that core reduction was spatially distinct from activities in which Mousterian points were often used. In Figure 20 it is clear that cores and Mousterian points concentrate in separate sub areas of Area C.

The artifact types appearing lower in both Lat tice Chains #1 and #2 comprise the artifact set of

Clique #2. Not surprisingly, the areas of organized activity suggested by Clique #2 overlap extensively with the areas of organized activity associated with Lattice Chains #1 and #2 (see Figure 17). The order of the Clique #2 artifact types in the Galois lattice

(see Figure 13) implies that they were multiple pur pose tools since they belong to two, or possibly all

three, of the cliques. The sub-areas associated with

Clique #2 appear, therefore, in loci where activities occurred that required multipurpose tools rather than tools with a singular or low diversity of usage. The latter appears to be the case for Mousterian

points and denticulates since they occupy the high est level of the lattice structure (Figure 13).

Step 5: Assignment of artifact type distributions to Carr's (1985) Models 1-6

Artifact type distributions in Carr's Models 1-6 follow the pattern that artifact types in Model 1 have the most constrained relationships (1985:336). These constraints are increasingly relaxed as the models progress to Model 6, where artifact types

Table 5. Artifact and Ecofact Types for the Three Cliques in Figure 12.

Clique 1_Clique 2_Clique 3_ Core Transverse Denticulate Transverse Denticulate

Denticulate Sidescraper Sidescraper Transverse Denticulate Converging Sidescraper Converging Sidescraper

Sidescraper Bone Mousterian Point

Bone Teeth Teeth

Teeth




I Converging Sidescraperl

| Mousterian Point | Denticulate Core

Teeth Sidescraper Transverse Denticulate

Figure 13. Galois lattice of the cliques and Bir Tarfawi stone tools, bone, and teeth, constructed from the Table 7

binary matrix. Lattice Chain #1 is the left chain (teeth, sidescraper, transverse denticulate -*

converging sidescraper

-> Mousterian point) and Lattice Chain # 2 is the right chain (teeth, sidescraper, transverse denticulate -

bone (denticulate and core). Note that the arrow indicates the order in which the artifacts are added to the lattice and, as used here, the order is from lower to higher in the lattice structure.

have the most variable organization. Carr's six models are also relatable (within both the behav ioral and archaeological domains) to formation

processes along the monothetic-polythetic contin uum.

Within the behavioral domain, a sample of pairs of artifact types assignable to Model 3 or 4 are

expected to be associated with tasks that always require both of these artifact types. Samples of pairs of artifact types that most closely fit Model 5 or 6 are expected to result from tasks that only require one of the types. Also, in Model 3 one artifact type may predominate in the sets of tasks in which both artifact types are used. In Model 4, pairs of artifact

types are expected to be used together in widely varying proportions, neither one predominating in all of the tasks (Carr 1985:351).

Within the archaeological domain, different

Table 7. Artifact and Ecofact Types for the Two Lattice

Chains in Figure 13.

Lattice Chain #1_ Tooth

Sidescraper Transverse Denticulate

Converging Sidescraper Mousterian Point

Lattice Chain #2

Tooth

Sidescraper Transverse Denticulate

Bone

Denticulate

Core

processes that are correlated over space are

expected to produce depositional sets organized in the manner depicted by Models 1, 2, 3, or 5. In

opposition to this, if different processes are not

mutually correlated over space then their combined effects are expected to produce depositional sets

organized in the form of Models 4 or 6 (Carr 1985:405). Within both of these domains, formation

processes are expected to result in spatial and orga nizational patterning ascribable to Models 3,4, 5, or 6, due to the non-uniform nature of these kinds of processes. Carr states, "Given that most forma tion processes tend to act disuniformly over events or deposits, one can expect that many archaeolog ical records will have artifact organizations similar to Models 3,4, 5, or 6" (Carr 1985:355).

This is precisely the pattern observed within the sub-area of the Bir Tarfawi site associated with Lattice Chain #1, where pairs of artifact types are identified as belonging to either Model 3, 4, or 6

(see Table 8). Based on the classification of pairs of tool types from Lattice Chain #1, it is clear that several pair combinations of artifacts in the sub area of the site associated with Lattice Chain #1 resulted from a similar milieu of globally mono thetic formation process (within both the behavioral and archaeological domains), that produced the

patterning associated with Models 3 and 4. For

Table 6. 8 x 3 Presence/Absence Matrix for the Galois Lattice Shown in Figure 14.

Artifact Type_Clique #1_Clique #2_Clique #3

Teeth 1 1

Sidescraper 1 1

Transverse Denticulate 1 1

Converging Sidescraper 0 1

Mousterian Point 0 0

Bone 1 1 0

Denticulate 1 0 0

Core 1 0 0




Figure 14. Bir Tarfawi bivariate kernel density estimate plots of the artifact types of Lattice Chain #1. (A) KDE plot including artifact locations, (B) KDE plot with density gradient only. In (B), the darkest area indicates where the Lattice

Chain #1 artifact types concentrate. The lower plot (C) shows the artifact types of Lattice Chain #1 and the approximate area (ellipse) where they concentrate.

example, Mousterian points and converging sidescrapers (Model 3) appear to have been used

together in one or more activities within the area

of the site associated with Lattice Chain #1. Also, the spatial organization of several of the possible pairwise combinations of teeth, sidescrapers, con

verging sidescrapers, and Mousterian points are

assigned to Model 6 (Table 8). This pairwise struc ture appears to be the outcome of a globally poly thetic suite of formation processes (both behavioral and archaeological) that operated within the sub area of the site associated with Lattice Chain #1. The suite of processes appears to be distinct and more complex than those associated with the first

group (Models 3 and 4). For example, in the pairs assigned to Model 6, teeth and sidescrapers are linked. At a broader scale, sidescrapers, converging sidescrapers, and Mousterian points are grouped together (based on overlap) into what appears to be a very cohesive activity set. As noted by Carr, within

the non-overlap-overlap dimension, processes that are responsible for overlap in the behavioral and

archaeological domains include: 1. "Multi-use tool types that are likely to be used

in multiple sets of events in concert with different tool types" (Carr 1985:350). Single-type tools with one functional edge are one such tool type.

2. Tools that contain more than one edge, "each of which may be intended for a different set of

tasks, and which may be used in multiple different sets of events with different tool types" (Carr 1985:350).

Sidescrapers and transverse denticulates are stone tools that occur in all three of the site areas

associated with Lattice Chains #1 and 2. Sidescrap ers are unifacial, often have one functional edge, and are typically multi-use tools, which therefore links them to the first set of activity-based processes, which are expected to result in overlap ping activity sets. Transverse denticulates typically




A B

-6 Figure 15. Bir Tarfawi bivariate kernel density estimate surface plots of the artifact types of Lattice Chain #2. (A) KDE

plot with artifact locations. (B) KDE plot with density gradient only. In (B), the darkest area indicates where the Lattice Chain #2 artifact types concentrate. The lower plot (C) shows the artifact types of Lattice Chain #2 and the approximate areas (ellipses) where they concentrate.

have more than one use edge, and often have dif ferent edge types, which links them to the second

type of activity formation process, which is

expected to result in overlapping activity sets.

Conclusion

In this article, we present a spatial-analytic method for identifying coherent and spatially cohesive sets of artifact types in an archaeological site. This method avoids making unverified assumptions about as a dataset's homogeneity or assumed sta tistical distribution. The method uses measurable

aspects of spatial distributions to disambiguate spa tial groups of artifacts into constituent components. The different artifacts are first grouped into artifact

types, and then the spatial distributions for the dif ferent types are compared using a measure of spatial similarity called a delta value.

These spatial distributions may be affected by

historical events during site formation and preser vation and analytically biased by nonrandom sam

pling of the data set. To help account for these

effects, our spatial analysis assigns a probability estimate to any measure of spatial similarity such as the delta values. Here, we use the MRPP proce dure to provide probability estimates, called delta P values, for the delta values. We can then use

ordinary statistical hypothesis testing to differen tiate statistically similar or dissimilar spatial distri butions from the delta values. These values are then converted into an adjacency matrix of Os and Is. In this adjacency matrix, a "0" represents dissimilar

spatial distributions, while a "1" represents similar

spatial distributions between two artifact types. This matrix is used to construct a graph in which the nodes are the types and a line connects a pair of nodes when the associated types have similar

spatial distributions (indicated by a 1 for that pair in the adjacency matrix).




Figure 16. Bir Tarfawi bivariate kernel density estimate surface plots of the artifact types of Clique #2. (A) KDE plot with artifact locations. (B) KDE plot with density gradient only. In (B), the darkest area indicates where the Clique #2 artifact

types concentrate. The lower plot (C) shows the artifact types of Clique #2 and the approximate areas where they con centrate.

In graph-theoretic terms, the adjacency matrix can be used to distinguish a clique; that is, a maximal set of nodes all of which are pairwise connected in the graph. In our case, a clique is a set of artifact

types that share similar spatial distribution measures. The clique is our mathematical expression for the intuitive notion of a spatial grouping or cluster of different tools that we might interpret as a single tool kit. In this hypothetical scenario, each tool in the tool kit has the same spatial distribution. Cliques thus visually represent these tool clusters.

A clique is built up with chains of spatially related artifact types. Some of these artifact types may be found only in one clique; others are found in several different cliques. The "embeddedness" of the artifact type expresses this degree of

overlap?that is, how pervasive that particular type is in the entire assemblage of cliques. We use the

Galois lattice as a visual representation of the extent of overlap among cliques.

This concept of different but overlapping cliques

captures the idea of a multipurpose tool type that

might be used in a variety of different tasks. The

pattern of overlap among the cliques may be made more explicit with a Galois lattice representation for the artifact types and the cliques. In this way we take advantage of both graph-theoretic and lattice-based methods to evaluate patterning in arti fact type spatial distributions. The advantage of

using graphs for visualizing data structure is that

they are inherently more intuitive. As has been

pointed out before, "graphical models are in some sense iconic; they look like what they represent" (Hage and Harary 1983:9). The actual structure of the data set can thus be visualized and also analyzed with graph-theoretic methods.

As a last step in our case study, we include an

operationalization of Carr's (1985) models of spatial organization along a monothetic-polythetic contin uum. One critique of our method might be its seem

ing complexity. However, we argue that the

complexity of our methods is simply a consequence




Table 8. Polythetic-Monothetic Organization of Pairs of Artifact Types for Lattice Chain #1 (See Lower Ellipse, Figure 16).

Group #1 Group #2 Group #3 Pair of Types_count_Count_count_Model Teeth 4 5 6 Sidescraper 2 0

Teeth 5 4 3 Converging sidescraper 1 3

Teeth 4 5 4 Mousterian Point 2 5

Teeth 4 5 3 Transverse Denticulate 1 1

Sidescraper 3 0 6 Converging sidescraper 0 3

Sidescraper 1 3 0 6 Mousterian Point 2 15

Converging sidescraper Mousterian Point

3

1

1

6

3

of the complexity of the problem we are addressing. As outlined above, no step in the method is irrelevant to the analysis of the spatial distribution of artifact

types. We recognize the possibility that noise intro duced through incidental post-depositional processes can obliterate spatial patterning, but that is a general problem we face in analyzing archaeo

logical spatial data and applies to any statistical method used for discerning patterning. If there is, unbeknownst to us, archaeologically meaningful spatial patterning in the site and the method we pre sent here is not capable of detecting the patterning, it is difficult to imagine what other analytical method

might be successful instead.

Keeping these caveats in mind, our analysis of the Bir Tarfawi data contribute to the ongoing debate about the function of Mousterian points and whether they were used as spear points. Holdaway (1989) has argued that they were spear points. If

that interpretation is true, then we should see evi dence of the bases of broken points that were

removed from the haft for replacement. This action

should be evident at habitation sites as a higher

proportion of exhausted point proximal bases than

distal tips. His statistical analysis of the Mousterian

points from two Iranian Mousterian habitation sites

did not agree with this expected pattern, however,

leading him to conclude that these tools were not

used as spear points.

We argue here that the Mousterian points may be multipurpose tools rather than used solely as

spear points. We note that the morphological criteria used to distinguish Mousterian points from con

verging sidescrapers are in fact ambiguous (Debe nath and Dibble 1994:62). Some archaeologists have classified these tools as "sharply-converging scrapers" (Holdaway 1989), consistent with Dib ble's (1987, 1988) argument that converging sidescrapers may be the last stage in a scraper

resharpening sequence. However, Read (2007:261 265) has shown to the contrary that converging scrapers (at Castanet A and Ferrassie H in France) do not appear to be the result of resharpening. Gor don (1993) has also argued that Mousterian points are a distinct type of tool. In his argument, the tool manufacturer sought to create a pointed tool, and was "secondarily making use of its long lateral

edges" for scraping and/or cutting (Gordon 1993:212). This interpretation implies that Mous

terian "points" are in fact a multipurpose tool.1 Our analysis supports this conclusion. We use

two lines of evidence to show that Mousterian

points and converging sidescrapers were function

ally similar tools. First, as shown in the activity area associated with Lattice Chain #1, Mousterian

points and converging sidescrapers are assigned to

Carr's Model 3, indicating that both artifact types were used jointly in tasks conducted in this sub



442 AMERICAN ANTIQUITY [Vol.75, No. 3, 2010

F-2

O BONE

V TOOTH CORE

E NOTCH E DENTICULATE

# CONVERGING DENTICULATE m TRANSVERSE DENTICULATE

FOLIATED PIECE

# CHOPPER

| SIDESCRAPER A CONVERGING SIDESCRAPER

m TRANSVERSE SIDESCRAPER

A LEVALLOIS POINT

A MOUSTERIAN POINT

Figure 17. Complete plot of Bir Tarfawi Area C artifacts (modified from Hietala and Larson 1980: Figure A8.1). Ellipses enclose sub-areas that contain all of the artifact types of Lattice Chains #1 and #2, and Clique #2. These approximate closely the high density areas identified by the KDE analysis (see Figures 14 -16). The lower solid ellipse identifies the area interpreted as organized activity corresponding to Lattice Chain #1. The central and left solid ellipses are inter

preted as areas of organized activity corresponding to the artifact set of Lattice Chain #2. The dashed ellipses are inter

preted as areas of organized activity that correspond to the artifact type set belonging to Clique #2.

area of the site. This use pattern makes sense if both types were functionally scraping tools. In this

case, it seems unlikely that Mousterian points were used exclusively as spear points. Secondly, strong spatial cohesiveness of converging sidescrapers with Mousterian points is also depicted in the lattice structure (Figure 13): converging sidescrapers are located immediately below Mousterian points.

Multipurpose tools relate to settlement mobility since mobility constrains the number of tools that can be transported. High mobility generally is asso ciated with selection for generalized and flexible tool designs that enhance versatility (Kuhn 1994;

Parry and Kelly 1987; Shea 1997). This pattern has

been repeatedly observed in ethnographic studies

(see, for example, Lee 1979:119) and confirmed with several metrical analyses. For example, Short

(1986) has used metric data on ethnographic lithic tool collections to show that settlement mobility varies negatively with tool diversity and positively with augmented tool versatility or flexibility. Sim

ilarly, Read (2008) used a detailed regression analy sis of tool diversity and complexity for 22

hunter-gatherer societies located in tropical to artic zones to show that tool diversity varies inversely with residential mobility. All these studies indicate that when residential mobility is high, "flexible" or

multi-functional tools appear to be a necessary




-1 i-?-1-1-1-1-1 x * .-.

+ 0 Core * * x Bone

-2 - + + x + Teeth

?x o_^

-3 - Z*)4* \ +

-H. * V ?

/

.gl-1-?-1-1-1 1 2 3 4 5 6 7

Figure 18. Distribution of one lithic and two ecofactual types in Area C of the Bir Tarfawi site. An ellipse encloses the area where the three types spatially overlap.

component of a technological strategy for maxi

mizing resource procurement success (Hayden et al. 1996:24;Kuhn 1994;Nelson 1991:73-74; Shott

1986:20). In the Bir Tarfawi sample, our finding of pre

dominately multipurpose stone tool types may therefore indicate that the people at the site were under some pressure to maximize transportability

by reducing the size of toolkits and the number of different kinds of tools that they carried. This result

suggests that flexible multipurpose tools were part of a subsistence strategy involving periodic or fre

quent mobility. If so, the site of Bir Tarfawi is best

interpreted as a seasonal or temporary camp. The prehistoric environmental conditions

around Bir Tarfawi support this inference of site

-1i-1-1-x?.-1

Mousterian point -P x xBone

9 . . + x I + Teeth_[ _

X *

x >S<Ax ** A

\ A/ + AV \ A xx

.61-1-1-1-1-1 1 2 3 4 5 6 7

Figure 19. Distribution of one lithic and two ecofactual types in Area C of the Bir Tarfawi site. A circle encloses where Mousterian Points (MPs) and teeth overlap. Note that MPs are almost entirely segregated from bone.




Figure 20. Distribution of two types of artifacts in Area C of the Bir Tarfawi site. Note the spatial segregation of the bulk of the Mousterian Points from the bulk of the cores.

function. The interior and southern regions of the Levant around Bir Tarfawi are relatively resource-poor during the Middle Paleolithic, and local food scarcity would necessitate moving to different resource areas. In comparison, Middle Paleolithic hominids in the more productive regions of the northern Levant and coastal areas could have coped with foraging deficits either

by using short-term storage or by using alterna

tive food sources in the same general region (Shea 1997:92).

Shea used a core and periphery model to relate these two regions to hominid foraging behavior, in which the more productive woodland areas of the Levant represent the "core" and the less productive southern and interior areas represent the less

optimal "periphery." He explains [referring to White 1977] that in the periphery:

3683890 t~--?^?-?--?- j

c Gy.-z/QD

EMRPP 3683888 -

'X\ delta=7.4970

/# \\ ^ f / d=0.2666 3683886 - / ,' \\ / 3683884 - ' / \\ // / |-1 / 1 '

\ \ * X / Choppers f ^1 V ' / / BScrapers 3683882 - / I

\X/ / E (%-f / T0A / 3683880 - \ /

3683878 - S>^^^^,^\M/

3683876 -I- -.-.- - -

265360 265362 265364 265366 265368 265370 265372

Figure 21. All (Euclidean distances) are indicated as edges connecting nodes in the graph, where the nodes represent hypothetical surface artifact locations A through F.




Table 9. Computation of Delta for Observed Hypothetical Distribution of Scrapers and Choppers in Figure 21.

AB; CDEF Number Artifact Pair_Euclidean Distance (meters)

1 A, B Scraper-Scraper 4

2 C, D Chopper-Chopper 6

3 C,E Chopper-Chopper 8.825531145

4 C,F Chopper-Chopper 11.43283867

5 D, E Chopper-Chopper 12.24295716

6 D, F Chopper-Chopper 13.60403617

7 E,F Chopper-Chopper 3.367758899

Sum of Chopper-Chopper Euclidean distances

55.47312204

55.47312204/6 = 9.245520339

(2/6)*4+(4/6)*(9.245520339) Delta =

7.49701355964778_

Note: The observed Euclidean distances used in the computations are shown as dashed edges in Figure 21.

lower amounts of rain and probably wide annual variation in rainfall supported scrub woodland or steppe-desert vegetation. The

steppe (the main vegetation community type in the region where Bir Tarfawi is located) con

Figure 22. Simple graph depicting hypothetical pairwise relations of artifact types with two cliques, Clique A and

Clique B.

tains fewer edible plants than the woodland, most of which are highly-seasonal, providing edible leaves, seeds, and root/tubers mainly in the winter and spring [Shea 1997:92-93].

Shea relates this environmental core-periphery model with a model of homind behavior developed by Henry (1995) to argue that: "Facing regular seasonal deficits in plant foods, hominids in the southern and interior Levant probably coped with

foraging deficits by seasonal changes in settlement location" (Shea 1997:92-93; our emphasis added).

Our interpretation of the site of Bir Tarfawi, based on the analysis of the spatial distribution of artifacts, is consistent with this proposed pattern of land use between different resource areas.

We suggest that our proposed method of spatial analysis represents an advance in spatial analysis of archaeological data; the visual displays and

graphical presentation is also fairly intuitive and

straightforward to interpret. Like other statistical

analyses, some aspects of this method, such as

deciding on the cut-off value for the MRPP delta P-values and interpreting the lattice chains, require the intuition and experience of an archaeologist to

produce meaningful results. Judgment calls like this reinforce the remark of Read and Russell that:

Quantitative methods do not replace legitimate insight that the practicing archaeologist obtains

through familiarity with the material at hand.

Instead, they expand our insight into aspects




Table 10. Delta Values and Corresponding P-Values for Each of the Fifteen Possible Two-Four Combinations of

Hypothetical Surface Artifact Locations A Through F In Figure 22.

Observed Probability (Exact) of a

Order_Combination_delta a_smaller or equal delta

1 CD; ABEF 4.625237699 1/15 = .0667 2 BF; ACDE 6.269733131 2/15 = .1333

3 EF; ABCD 6.960123351 3/15 = .2000

4* AB; CDEF 7.49701356 4/15 = .2667 5 BE; ACDF 7.673528201 5/15 = .3333 6 AF; BCDE 7.789464271 6/15 = .4000 7 AE; BCDF 7.858820482 7/15 = .4667 8 AD; BCEF 8.003970226 8/15 = .5333 9 CE; ABDF 8.145989313 9/15 = .6000

10 AC; BDEF 8.274176757 10/15 = .6667 11 BD; ACEF 8.764633501 11/15 = .7333 12 DE; ABCF 8.78498395 12/15 = .8000 13 CF; ABDE 9.159504623 13/15 = .8667

14 BC; ADEF 9.218536904 14/15 = .9333

15_DF; ABCE_9.244619921_15/15 = 1.0000

# of Permutations M=6!/(2!*4!)=15 *Actual

otherwise unseen and unobserved [Read and

Russell 1996:18].

Acknowledgments. We deeply appreciate the critical com

ments, editorial assistance, and archaeological wisdom of

Chester King and Patricia Martz. We are also grateful for the

excellent review and needed refinement of this paper pro vided by Christopher Carr and several anonymous reviewers on a previous version of this paper. We also acknowledge and

whole-heartedly appreciate the constructive comments and

excellent suggestions provided by the current set of review ers. Their efforts added considerably to the accuracy and

quality of this paper. We thank Michael Barton for help in

translating the English abstract into Spanish.

Appendix 1: Multivariate Permutation Methods

Multivariate permutation methods such as the

multi-response permutation procedure (MRPP) were developed to analyze both heterogeneous and

homogeneous data and are not based on an a priori distribution. Mielke provides the fundamental con

cept underlying permutation methods that distin

guish them from traditional parametric statistics:

The randomization assumption alone (i.e. the

random allocation of objects to experimental treatments) provides a meaningful distribution termed the permutation distribution for any sen

sibly defined statistic under the null hypothesis.

The resulting statistical methods, which depend solely on the randomization assumption, are

called permutation methods [Mielke 1991:55].

Permutation methods have been extensively used in the earth sciences (Mielke 1984, 1991 and in

ecological studies (Biondini et al. 1988; Reich et al. 1991; Zimmerman et al. 1985).

Permutation Tests

Mielke provides a brief description of permutation tests:

Permutation tests generally come in three

types: exact, resampling, and moment approx

imation tests. In an exact test, a suitable test

statistic is computed on the observed data asso

ciated with a collection of objects, and then the data are permuted over all possible arrange

ments of the objects and the test statistic is

computed for each arrangement. The null

hypothesis H0 specified by randomization

implies that each arrangement of objects is

equally likely to occur. The proportion of

arrangements with test statistic values as

extreme or more extreme than the value of the

test statistic computed on the original arrange ment of the data is the exact P-value [Mielke and Berry 2001:2].




In our application we used an exact multi-response permutation procedure (EMRPP) to compare the

spatial distributions for the artifacts in a pair of arti fact types.

Description of the EMRPP

The equation

Au=[{Xu-Xu)2+(X2I-X2JfJ defines the Euclidean distance between two distinct artifact locations / and / within the site surface area

being sampled. In order to compare the intrasite distributions of two artifact types A and B, it is nec

essary to measure separately the spatial clustering of the artifacts belonging to each of the two types. Let nA be the number of distinct locations for arti facts of type A in the sampling area of the site and

nB the number of distinct locations for artifacts of

type B within the same area. Let N=nA +nB. The

averages of the A/7 distances across the site surface for each of artifact types A or B are given by the

equations

and

I<J

respectively, where ? is the sum over all distinct site surface locations / and / for each of the two artifact types with 1 < / < J < nt, i = A or B and the

binomial coefficient (\} is the number of distances between distinct surface artifact locations within

the sampling area for artifact type i {i -A ox B). The MRPP delta, as used here, is the average of

the intra-type Euclidean distances for two artifact

types, Types A and B, weighted by their relative

group size.

So if the surface artifacts belonging to Type A clus ter together and the surface artifacts belonging to

Type B also cluster together in the same site, then

the intra-group average Euclidean distances (ijA and

?B) will be small. This contrasts with two artifact

types that are more spread out over a site surface and as a result will more likely have overlapping spatial distributions. Therefore, it is reasonable to view the MRPP delta as a measure of spatial seg regation between two artifact types in an archaeo

logical site. The P-value associated with an observed value

of 8 (say 80) is the probability under the null hypoth esis (H0) of observing a value of 8 as extreme as, or more extreme than, 80. An exact P-value is given by:

P(b <

80\H0) = (number of 5's<50)/M,

AM where M -

KnA \xnB\j

The original algorithm for computing EMRPP P-values is given in Berry (1982). Approximation methods such as Monte Carlo resampling and Pear son Type III moment approximations are needed for computation of MRPP P-values for M> 107. All of the methods we used are available in the Blossom Statistical Software (Cade and Richards

2001) available as freeware from the USGS (Mid continent Ecological Science Center, Fort Collins,

CO) and are also given as an online supplement to Mielke and Berry (2001) as un-compiled FOR TRAN programs.

For didactic purposes, Figure 21 and Tables 9 and 10 are provided to illustrate the concepts and details for computing an EMRPP value in a simple case: a two-four combination for a pair of artifact

types, with one type consisting of two artifacts and the other of four artifacts. In general, though, the EMRPP delta values and delta P-values cannot be calculated by hand. All possible permutations of six point locations for the two artifact types must

be determined (see Figure 21 and Tables 9 and 10

for more details). The permutation set is used to

calculate the EMRPP delta values that are then

used to compute the corresponding EMRPP delta

P-values for each of the permutations of the two

four combination of artifacts (see Table 10).

Appendix 2: Graph Theoretic Definitions

The following definitions are from Gross and Yellen

(1999:2, 10,48).




Definition 1. A graph G = (VG, EG) is a math

ematical structure consisting of two sets VG and

EG. The elements of VG are termed vertices (or nodes), and the elements of EG are called edges. Each edge has a set of one or two vertices associated to it, which are known as its endpoints.

Definition 2. A graph is simple if it has neither

self-loops nor multi-edges.

Definition 3. A complete graph is a simple graph such that every pair of vertices is joined by an edge.

Definition 4. A subgraph of a graph G is a graph H whose vertices and edges are all in G.

Definition 5. A subgraph Hof (VG EG) is called a clique or maximal complete subgraph of (VG EG) if every pair of vertices in H is joined by at least one edge, and no proper superset of //has this prop erty.

Definition 6. The adjacency matrix, AG, of a

graph is a square matrix whose elements atj (i j) are 1 if nodes / and j are connected by an edge and 0 otherwise.

Graph Construction

For the Bir Tarfawi artifact distributions, the Euclid ean distances between all mapped surface locations of the occurrences of an artifact type are to compute the P-values that are then used to construct an adja cency matrix. Here P-values < .05 are interpreted as identifying two artifact types whose site-wide

spatial distributions within Bir Tarfawi do not sig nificantly overlap. Therefore, in constructing this

adjacency matrix, P-values > .05 are coded 1 and P-values < .05 are coded 0. In both matrices, a 1 in the adjacency matrix represents a connection between two artifact types and a 0 an absence of a connection. The resulting adjacency matrix for Bir Tarfawi is the raw data for the graph analysis.

The UCINET 6 software (Borgatti et al. 2002) was used to identify the cliques of the network

graph. We interpret cliques as coherent and spatially cohesive sets of artifact types and in the network

graph, labeled solid circles (nodes) depict artifact

types and lines (edges) connecting nodes identify spatial co-occurrence between artifact types. We drew and labeled the Bir Tarfawi network graph using the Pajek freeware graph drawing and net work analysis package. Additional text as well as

ellipses and beziergons were added to the same

graph using Dia, a freeware diagram creation pro gram. A mathematical procedure using methods from linear algebra for detecting cliques is given in Harary and Ross (1957). The algorithm imple

mented in UCINET 6 for detecting cliques is due to Bron and Kerbosch (1973). The Bron and Ker bosch (1973) algorithm finds all Luce and Perry (1949) cliques greater than a specified size. Figure 22 provides a simple visualization of cliques in a

connected graph.

Appendix 3: Galois Lattice

A Galois lattice (Birkhoff 1967) is recommended

by Freeman and White (1994) for the analysis of two-mode social networks and discussed in depth from a mathematical standpoint by Duquenne (1991). Mohr et al. mention the two-mode property as it relates to the lattice: "A Galois lattice, however, has the special property of representing two orders of information in the same structure such that every point contains information on both logical orders

simultaneously" (Mohr et al. 2004:10). A Galois lattice can be viewed as the unfolding of the struc ture of multidimensional, two-mode binary data.

In our method, one mode is a set of n artifact

types A = {ax, a2,... an} and the other mode is the set of m cliques C = {cv c2,... cj, each of which

comprises a subset of three or more of the artifact

types in set A. A membership relation I <Ax C links these two sets. When an artifact type at belongs to a specific clique cy, it follows that (ai9 cj e /. The membership relations between artifact

types and cliques can therefore be represented by an n x m binary matrix M, where for any element

nty in M, = 1 if (a-, cj) e / and 0 otherwise. The

mathematics required to unfold and represent the

complete two-mode structure of the clique-artifact type binary matrix as a Galois lattice are complex and beyond the scope of this brief discussion. The interested reader is referred to Davey and Priestley (2002), Duquenne (1991, 1999:419-428), and

Wille (1982,1984) for the complete mathematical definition of a Galois lattice, accompanied by a detailed discussion of a variety of ways Galois lat tices can be used to analyze the structure of two

mode binary data.




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Notes

1. This assumes that post-depositional processes have not

altered artifact distributions so much that a significant amount

of the original spatial patterning in these distributions is no

longer extant.

2. The senior author is aware of the small sample sizes of

several of the artifact types in both of the data sets used to

illustrate the proposed method. For example, the Transverse

Denticulate sample in the Bir Tarfawi data set consists of three

artifacts. Even with a robust non-parametric statistical proce dure such as MRPP that is designed to analyze small and

unequal samples, small archaeological samples are expected to be strongly affected by sampling error, so analytical results

from such samples should be viewed with caution.

3. Whallon (2010) has argued that Middle Paleolithic

assemblages differ from Upper Paleolithic assemblages by both lack of formal tool types and absence of the kind of pat

terning in time and space one finds with the Upper Paleolithic

assemblages and their relatively abundant formal tool types. Whallon argues that the Mousterian assemblages are com

posed of artifact tools for which only some can be character

ized as formal tools linked to specific functions. The other

artifact tools, and in some cases the bulk of the artifact tools, are constrained in their form only in a general sense. These

artifacts, though frequently classified as debitage and

excluded from analysis (as was the case for the Bir Tarfawi

site), are tools because they have been utilized in accordance

with the suitability of an artifact for the task at hand based on

characteristics such as the sharpness of an edge, the size

and/or weight of the artifact, the length of edges, and so on, not its overall form. Our analysis of the spatial distribution of

the Bir Tarfawi Mousterian points leading to our conclusion

that these were multipurpose tools supports Whallon's argu ment. We suggest that a combination of (1) types identified

using the methods presented by Read and Russell (1996) for

utilized flakes from a Peruvian site and (2) the kind of spatial

analysis we have presented in this article could provide a

method for addressing the issues and questions Whallon has

raised regarding the nature and analysis of Mousterian assem

blages.

Submitted May 12, 2009; Revised July 25, 2009, Accepted

August 28, 2010.

