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Galois Lattices and the Formal Analysis of Orifice Size in Relation to Emic Type Selection, Valuation, and Temporal Change of Giant Keyhole Limpet (Megathura crenulata) Ornaments in Late Middle to Early Late Period Coastal Chumash Society. By Michael Merrill.

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slide1

Galois Lattices and the Formal Analysis of Orifice Size in Relation to Emic Type Selection, Valuation, and Temporal Change of Giant Keyhole Limpet (Megathura crenulata) Ornaments in Late Middle to Early Late Period Coastal Chumash Society

By Michael Merrill

slide2

CA-SBa-72 (hereafter SBa-72) is located on a mesa overlooking the Pacific Ocean, next to Tecolote Canyon several kilometers north of UC Santa Barbara. The southern site at SBa-72 (hereafter SBa-72S) was inhabited from about A.D. 1050 to 1250. My talk will be on an analysis of the Megathura crenulata ornaments from ten burials in the cemetery of SBa-72S.

SBa-72S

slide3

An abrupt change in the size and shape accompanied by a dramatic change in the diversity of Megathura crenulata ornaments began at the time SBa-72S was first occupied, which suggests major changes had taken place in one or more of the social subsystems within Chumash society over a very short period of time.

SBa-72S

slide4

CA-SBa-72 (hereafter SBa-72) is located on a mesa overlooking the Pacific Ocean, next to Tecolote Canyon several kilometers north of UC Santa Barbara. The southern site at SBa-72 (hereafter SBa-72S) was inhabited from about A.D. 1050 to 1250. My talk will be on an analysis of the Megathura crenulata ornaments from ten burials in the cemetery of SBa-72S.

SBa-72S

slide5

“In the southern cemetery at SBa-72…Megathura ornaments were the most common type of artifact found associated with burials. Burials with Megathura ornaments were found in all areas of the cemetery though they tended to be concentrated in the western half of the cemetery “(King 1990: 147).

slide6

A.D. 1050-1250

Early Late Middle

SBa-72 Southern Cemetery. David Banks Rogers’ (1926) Unit and Burial Numbers.

Late Middle

Early Late

Burials in my analysis

slide7

Changes in Giant Keyhole Limpet Ornaments as Indicators of a Changing Cultural System

“The size of the ornaments gradually increased until the end of the Middle period. I interpret this change, along with the data indicating less exclusive use by the political elite as indicating a decrease in their use in the political economic subsystem and an increase in their use as ornaments worn and exchanged by many members of Santa Barbara Channel society” (King 1990: 148).

slide8

Changes in Giant Keyhole Limpet Ornaments as Indicators of a Changing Cultural System

The L1a types were probably used as money as were the M1-M5b callus ring types. The L1a Megathura ornaments were probably being used as money by people at all levels of society.

slide9

Changes in Giant Keyhole Limpet Ornaments as Indicators of a Changing Cultural System

The M1-M5b ring ornaments are believed to have been used in an economic system limited to the social elite. The orifice and shell size in these ornaments tends to be larger than in the post M5b types.

slide10

VEN-27 (Pitas Point Site) M5c-L1a Megathura crenulata ornaments

M5C1 chipped ends

M5C2-L1a wing-shaped

M5C2-L1a one end perforated

slide12

VEN-27

SBa-46

VEN-110

SCrI-83

LAn-264

SCrI-100

End-Chipped Megathura crenulata ornament blanks from Catalina Island.

slide13

Santa Barbara

Topanga Canyon

SBa-46

VEN-110

VEN-27

SCrI-83

LAn-264

Los Angeles

SCrI-100

White’s Landing

Sites with M5c-L1a Megathura crenulata ornaments.

The above ornament is an end-chipped type believed to have been traded from Catalina Island to the indicated sites.

slide14

“Megathura crenulata ornaments were apparently being used by more of the people than during the early Middle period when they were concentrated in the areas containing the greatest amount of wealth and symbols of power” (King 1990: 147).

slide15

“The large M5c-L1a Megathura crenulata ornaments often retain traces of red ochre paint and several with painted designs have been recovered. Burial A9 from SCrI-100 had 11 Megathura crenulata ornaments which were at the front of a necklace of Olivella biplicata and mussel disc beads. Five of these were painted with white spots on an orange-red background “(King 1990: 145-146).

slide16

Original Impetus for my Undertaking this Analysis

“It is probable that during the period of use of the SBa-72 south cemetery there were changes in the types of Megathura ornaments used. Further analysis is necessary to determine the temporal sequence of the ornaments” (King 1990: 39).

slide17

Hypotheses being explored by my analysis

(H1) There is an emic-based conceptually structured dependency between orifice size and specific Megathura crenulata ornament types.

slide18

Hypotheses being explored by my analysis

(H2) Megathura crenulata ornament value is positively correlated with shell size. Also, grinding requires a greater expenditure of time and energy than chipping, which leads to the expectation that larger shells were more often chosen for making ornaments with ground ends, edges, and/or surfaces as well as for ring ornaments which require a maximum removal of shell to make.

slide19

Hypotheses being explored by my analysis

(H3) Changes in the types of Megathura crenulata ornaments in the cemetery in SBa-72S provide detailed conceptual information about the re-structuring of specific subsystems in the rapidly changing socio-cultural system of the Chumash at the end of the Middle period.

slide20

Sequence of Analysis

Step 1 Collection of Giant Keyhole Limpet ornaments from burial.

slide21

Sequence of Analysis

Step 2 Classification of ornaments into types based on qualitative attributes (e.g. end-chipped) that were recognizable to the makers and users of these ornaments.

slide22

Sequence of Analysis

Step 3

Sub-step 3a Measure the orifice length and width for each limpet ornament recovered from a burial.

Sub-step 3b Group ornaments into orifice size classes using a cluster analysis.

slide23

Cluster Analysis

Sub-step 3b

Orifice size

( max L and max W)

Giant Keyhole Limpet shell size

Emic selection of shell size

For making an ornament.

Class (Ideational Domain)

Inference

Similarity Group

Based on orifice

length and width

Cluster analysis of orifice maximum length and maximum width. Here a minimum number of measurements related to orifice size are made in order to maximize the chance that the results of the analysis will converge on the underlying emic structure.

slide24

Cluster Analysis

Raw Data

Qualitative Data

Cross Table of Burial 2 Trench 8 Section B.

slide25

The cluster analysis here serves to convert numerical data into discrete classes that can be entered into a binary matrix.

Dendrogram resulting from weighted average clustering algorithm using Euclidean distance as the dissimilarity measure.

slide27

Using a cluster analysis on two variables does not reduce the dimensionality of the data since both the data space and analysis space have the same dimension, which is 2. Therefore there is no concern about erroneous classes resulting from projecting a high dimensional data space onto a two-dimensional analysis space. This increases our confidence that our analytical classes are close to or are the same as the actual emic orifice size classes (Read 2007).

slide28

Step 4. Construct the formal context by organizing the ornaments (objects) and ornament size classes and ornament types (attributes) into a “cross table”. This is the raw data for Step 5.

slide29

Attributes

“Cross Table” where X’s represent 1’s and blank cells represent 0’s. Also called an Incidence Matrix. For example, Object 1 in this table has attributes p and t since there is an X in the cells of the table corresponding with the intersection of the row belonging to Object 1 and the columns belonging to attributes p and t.

slide30

Step 5. Draw and label the concept lattice (isomorphically equivalent to a Galois lattice) with a computer. The structure of the context unfolded in the partial order of the concept lattice is required for Step 6.

slide31

( { }, {p, q, r, s, t})

object set is empty

attribute set is full

slide32

({2}, {p, r})

({3}, {p, s})

({1}, {p, t})

({4, 5}, {q, s})

({6}, {q, t})

({ }, {p, q, r, s, t})

slide33

({1, 2, 3}, {p})

({2}, {p, r})

({1}, {p, t})

({4, 5}, {q, s})

({6}, {q, t})

({3}, {p, s})

({ }, {p, q, r, s, t})

slide34

({1, 2, 3}, {p})

({3, 4, 5}, {s))

({2}, {p, r})

({3}, {p, s})

({4, 5}, {q, s})

({6}, {q, t})

({1}, {p, t})

({ }, {p, q, r, s, t})

slide35

({1, 2, 3}, {p})

({3, 4, 5}, {s))

({1, 6}, {t})

({2}, {p, r})

({3}, {p, s})

({6}, {q, t})

({1}, {p, t})

({4, 5}, {q, s})

({ }, {p, q, r, s, t})

slide36

({1, 2, 3}, {p})

({4, 5, 6}, {q})

({3, 4, 5}, {s))

({1, 6}, {t})

({2}, {p, r})

({6}, {q, t})

({3}, {p, s})

({1}, {p, t})

({4, 5}, {q, s})

({ }, {p, q, r, s, t})

slide37

object set is full

attribute set is empty

({1, 2, 3, 4, 5, 6}, { })

({1, 2, 3}, {p})

({4, 5, 6}, {q})

({3, 4, 5}, {s))

({2}, {p, r})

({1, 6}, {t})

({6}, {q, t})

({3}, {p, s})

({1}, {p, t})

({4, 5}, {q, s})

({ }, {p, q, r, s, t})

object set is empty

attribute set is full

slide38

Step 6. Determine the Luxenburger basis of partial (and absolute) implications using a computer implemented algorithm.

slide39

There is one absolute

implication in the lattice

which is: r implies p

object set is full

attribute set is empty

({1, 2, 3, 4, 5, 6}, { })

({1, 2, 3}, {p})

({4, 5, 6}, {q})

({3, 4, 5}, {s))

>

({2}, {p, r})

({1, 6}, {t})

({6}, {q, t})

({3}, {p, s})

({1}, {p, t})

({4, 5}, {q, s})

Luxenburger Basis

r implies p

({ }, {p, q, r, s, t})

object set is empty

attribute set is full

slide40

Additive line type of lattice diagram.

Drawn with chain decomposition algorithm.

The initial diagram produced by the algorithm usually requires manual adjustment by the analyst.

Burial 2 Trench 8B

slide41

16/20=0.8

5/6=0.83

Luxenburger Basis

Orifice Size Class #1 implies chipped ends

Orifice Size Class #2 implies chipped ends

One end ground one chipped implies

Orifice Size Class #3

Orifice Size Class #3 implies chipped ends

[80% of the time]

Orifice Size Class #4 implies chipped ends

[89% of the time]

Green refers to absolute implications and red to partial implications

Threshold for Luxenburger basis set at greater than or equal

to 80%

Burial 1 Trench 5A

slide42

Step 7. Determine the percentage of each of the ornament types in the Luxenburger basis in relation to the total sample size of ornaments. Then make a histogram of percentage verses ornament type (in Luxenburger basis). In this analysis this provides a visual means for inferring the temporal order of the burials and for refining the temporal sequence of Megathura crenulata ornament types in SBa-72S (Merrill 2007).

slide43

The blue upper half circle means that an attribute is attached to a node (=concept) and a black lower half circle identifies that one or more objects are attached to this concept.

Burial 2 Trench 8B

slide44

>

>

<

>

>

<

<

Burial 4 Trench 5 or 4D

slide45

>

>

>

Burial 7 Trench 6A

slide46

<

>

<

>

<

>

>

<

<

<

>

>

4/5=0.8

6/7=0.86

4/5=0.8

Orifice Size Class #3 implies

wing-shaped square and vice versa

Burial 3 Trench 8B

slide47

entire margin ground

5/6=0.83

Burial 5 Trench 6B

slide50

5/6=0.83

Burial 4 Trench 5B

slide51

9/10=0.9

7/8=0.88

Orifice Size Class #5 implies thin oval ring

thin oval ring implies Orifice Size Class #5

Burial 1 Trench 8C

slide53

(H1) There is an emic-based conceptually structured dependency between orifice size and specific Megathura crenulata ornament types.

slide54

(H2) Megathura crenulata ornament value is positively correlated with shell size. Also, grinding requires a greater expenditure of time and energy than chipping, which leads to the expectation that larger shells were more often chosen for making ornaments with ground ends, edges, and/or surfaces as well as for ring ornaments which require a maximum removal of shell to make.

slide55

New hypothesis (H4). End-chipped ornaments made from larger Megathura shells on Catalina Island were traded as “blanks” that were then made into more refined types such as end-ground by people in the sites they were traded to.

slide56

ground end

ground end

slide58

“Transitional burial”

wing-shaped ornaments

slide63

My Analysis Provides Good Support for H1 and H2

Ring ornaments structure the largest orifice size classes (e.g. Class #5 in both B4T5 or 4D and B1T8C).

Wing-shaped square and rectangular Megathura crenulata ornaments provide the most structure to the middle and large orifice size classes in the later (L1a) burials in the sample.

To refine this analysis a better typology of the wing-shaped ornaments is needed to discriminate ground from chipped forms as well as to precisely identify subtypes based on shape variation in the outline (e.g. compute outline curvature profiles using an Elliptical Fourier Function approach).

End-ground Megathura crenulata ornaments appear to contribute the most structure to the middle and large orifice size classes.

slide64

Percentage of 5 types of Megathura crenulata ornaments in each of the burials in the sample that belong to the Luxenburger basis for a given burial.

slide65

Results of my analysis that refine our understanding of the temporal sequence of Megathura ornaments in SBa-72S.

Phase M5c can be subdivided into two sub-sub-phases C1 and C2 based on the following.

M5C1 is identified with respect to Megathura crenulata ornaments by a predominance of chipped end ornaments followed by a relative abundance of ground end ornaments and a very low to zero frequency of wing-shaped ornaments.

slide66

Results of my analysis that refine our understanding of the temporal sequence of Megathura ornaments in SBa-72S.

Phase L1a is characterized by an extreme predominance of wing-shaped rectangular types and the emergence of very low frequencies of thin oval ring Megathura ornaments toward the end of L1a.

slide67

VEN-27 excavation Area 1 North (King 1970), interpreted as an outdoor activity area (Gamble 1983, Merrill 2005). Galois lattice of typed artifacts (objects) and cliques (attributes) resulting from a correlation-based graph-theoretic method of analysis (Merrill 2005) of artifact frequencies in several successive 10-centimeter excavation levels.

Megathura ornament and

Olivella split punch bead

manufacturing “tool kit”

Small end-battered stones have previously been interpreted in VEN-27 (Gamble1983) as flaking hammers used in flaked stone tool manufacture. My formal analysis provides a very different interpretation.

slide68

VEN-27 excavation Area 1 North (King 1970), interpreted as an outdoor activity area (Gamble 1983, Merrill 2005). Galois lattice of typed artifacts (objects) and cliques (attributes) resulting from a correlation-based graph-theoretic method of analysis (Merrill 2005) of artifact frequencies in several successive 10-centimeter excavation levels.

Megathura ornament and

Olivella split punch bead

manufacturing “tool kit”

Small end-battered stones have been found in association with end-chipped Megathura crenulata ornaments in various stages of manufacture on Catalina Island (Decker early 1970’s UCLA Archaeological Survey).

slide69

VEN-27 excavation Area 1 North (King 1970), interpreted as an outdoor activity area (Gamble 1983, Merrill 2005). Galois lattice of typed artifacts (objects) and cliques (attributes) resulting from a correlation-based graph-theoretic method of analysis (Merrill 2005) of artifact frequencies in several successive 10-centimeter excavation levels.

The results of this analysis of VEN-27 (a coastal Chumash village site occupied contemporaneously with SBa-72S) provides independent and analytical support for the observed spatial association of small end-battered stones and end-chipped Megathura crenulata ornaments on Catalina Island.

Megathura ornament and

Olivella split punch bead

manufacturing “tool kit”

slide70

Suggested future work to refine the method and expand its analytical reach.

Develop and implement a procedure to classify the lattices produced by the analysis based on structural similarity.

Then relate the lattice classification to burial attributes (e.g. flexure) and associations (types and quantities of grave goods) in order to better understand the social dimensions of the mortuary population being examined.

slide71

Node degree frequency matrix by lattice level for the lattices of each of the ten burials.

slide72

Correlation matrix computed from the above table. Provides a measure of structural similarity for each pair of lattices.

slide73

Frequency distribution of the correlation coefficients. The threshold correlation coefficient 0.82 acts as a “noise filter” (here the chosen cut-off is greater than or equal to 0.82).

slide74

Adjacency matrix determined from the correlation coefficient matrix using a cut-off of 0.82 or greater for the correlation coefficient.

slide75

Linked pair of lattices. One has a slightly, the other a moderately homogeneous structure (determined by the percentage of nodes with the same degree).

Linked pair of lattices with a

moderately heterogeneous

structure.

Clique 2

Lattices with a very heterogeneous

structure.

Clique 1

Lattices with a moderate to

very heterogeneous structure.

Lattice Structural Similarity Classification

slide77

In contrast to a PCA biplot (and many other methods of multivariate analysis) the line diagram of a Galois lattice achieves the aim of an injective representation, which permits the reproduction of patterning in the raw data exactly.

slide78

PCA Biplot from grid (scores 1-6) using 8 attribute pairs of a 24 year-old female patient suffering from bulimia nervosa (Spangenberg, N. and K.E. Wolff 1991)

31.3%

light-

hearted

sister

younger

sister

vacillating

PC2

47.4%

self

PC1

Attributes: Eigenvectors

Objects (= Persons): Component Scores (= Left Singular Vectors times Square Root of Eigenvalues)

slide80

This erroneous information in the biplot results from “reduction of dimensionality” by projecting a multi-dimensional space onto 2-space in the biplot.

slide81

PCA uses a numeric metric scale which results in the misleading impression in the biplot that sister and younger sister have different degrees of affinities with respect to attributes.

In the lattice sister and younger sister share the same node in complete agreement with the raw data.

Examining the lattice chains it is clear that SELF does not have any paths leading to either depressive or vacillating.

slide82

PCA uses a numeric metric scale which results in the misleading impression in the biplot that sister and younger sister have different degrees of affinities with respect to attributes.

This erroneous information in the biplot results from “reduction of dimensionality” by projecting a multi-dimensional space onto 2-space in the biplot.

slide83

In the lattice sister and younger sister share the same node in complete agreement with the raw data.

This erroneous information in the biplot results from “reduction of dimensionality” by projecting a multi-dimensional space onto 2-space in the biplot.

slide84

Luxenburger basis for the “Bulimia” Repertory Grid Lattice

1 < 5 > creative =[100%]=> < 5 > resolute;

2 < 6 > resolute =[83%]=> < 5 > creative;

3 < 4 > uncompromising resolute =[100%]=> < 4 > light-hearted open-hearted;

4 < 4 > uncompromising open-hearted =[100%]=> < 4 > light-hearted resolute;

5 < 4 > light-hearted =[100%]=> < 4 > uncompromising resolute open-hearted;

6 < 4 > soft =[100%]=> < 4 > open-hearted;

7 < 4 > resolute open-hearted =[100%]=> < 4 > uncompromising light-hearted;

8 < 5 > open-hearted =[80%]=> < 4 > uncompromising light-hearted resolute;

9 < 5 > open-hearted =[80%]=> < 4 > soft;

10 < 5 > uncompromising =[80%]=> < 4 > light-hearted resolute open-hearted;

11 < 3 > uncompromising light-hearted soft resolute open-hearted =[100%]=> < 3 > creative;

12 < 3 > uncompromising light-hearted creative resolute open-hearted =[100%]=> < 3 > soft;

13 < 2 > aggressive typically male =[100%]=> < 2 > creative resolute;

14 < 2 > aggressive resolute =[100%]=> < 2 > typically male creative;

15 < 2 > peaceful =[100%]=> < 2 > uncompromising light-hearted resolute open-hearted;

16 < 2 > vacillating =[100%]=> < 2 > helpless undecided;

17 < 2 > uncompromising enjoying =[100%]=> < 2 > light-hearted soft creative resolute open-hearted;

18 < 2 > performance oriented =[100%]=> < 2 > resolute;

19 < 2 > enjoying resolute =[100%]=> < 2 > uncompromising light-hearted soft creative open-hearted;

20 < 2 > enjoying open-hearted =[100%]=> < 2 > uncompromising light-hearted soft creative resolute;

21 < 2 > typically male resolute =[100%]=> < 2 > aggressive creative;

22 < 2 > helpless =[100%]=> < 2 > vacillating undecided;

23 < 2 > undecided =[100%]=> < 2 > vacillating helpless;

24 < 1 > aggressive uncompromising =[100%]=> < 1 > inhibited;

25 < 1 > aggressive depressive =[100%]=> < 1 > performance oriented typically male creative resolute inhibited;

26 < 1 > vacillating enjoying helpless undecided =[100%]=> < 1 > typically male inhibited;

27 < 1 > vacillating depressive helpless undecided =[100%]=> < 1 > soft open-hearted;

28 < 1 > vacillating typically male helpless undecided =[100%]=> < 1 > enjoying inhibited;

29 < 1 > vacillating helpless undecided inhibited =[100%]=> < 1 > enjoying typically male;

30 < 1 > vacillating helpless undecided open-hearted =[100%]=> < 1 > depressive soft;

31 < 1 > uncompromising performance oriented light-hearted resolute open-hearted =[100%]=> < 1 > peaceful;

32 < 1 > uncompromising inhibited =[100%]=> < 1 > aggressive;

33 < 1 > performance oriented creative resolute =[100%]=> < 1 > aggressive depressive typically male inhibited;

34 < 1 > enjoying typically male =[100%]=> < 1 > vacillating helpless undecided inhibited;

35 < 1 > enjoying inhibited =[100%]=> < 1 > vacillating typically male helpless undecided;

36 < 1 > depressive typically male =[100%]=> < 1 > aggressive performance oriented creative resolute inhibited;

37 < 1 > depressive resolute =[100%]=> < 1 > aggressive performance oriented typically male creative inhibited;

38 < 1 > depressive inhibited =[100%]=> < 1 > aggressive performance oriented typically male creative resolute;

39 < 1 > depressive open-hearted =[100%]=> < 1 > vacillating soft helpless undecided;

40 < 1 > resolute inhibited =[100%]=> < 1 > aggressive performance oriented depressive typically male creative;