In the southern cemetery at SBa-72 Megathura ornaments were the most common type of artifact found associated with burials. Burials with Megathura

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1 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

2 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 to My talk will be on an analysis of the Megathura crenulata ornaments from ten burials in the cemetery of SBa-72S. SBa-72S

3 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

4 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 to My talk will be on an analysis of the Megathura crenulata ornaments from ten burials in the cemetery of SBa-72S. SBa-72S

5 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).

6 A.D SBa-72 Southern Cemetery. David Banks Rogers (1926) Unit and Burial Numbers. Early Late Middle Late Middle Early Late Burials in my analysis

7 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).

8 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.

9 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.

10 M5C1 chipped ends VEN-27 (Pitas Point Site) M5c-L1a Megathura crenulata ornaments M5C2-L1a one end perforated M5C2-L1a wing-shaped

11

12 SBa-46 SCrI-83 VEN-27 VEN-110 LAn-264 SCrI-100 End-Chipped Megathura crenulata ornament blanks from Catalina Island.

13 Santa Barbara Topanga Canyon SBa-46 SCrI-83 VEN-27 VEN-110 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.

14 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).

15 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: ).

16 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).

17 Hypotheses being explored by my analysis (H1) There is an emic-based conceptually structured dependency between orifice size and specific Megathura crenulata ornament types.

18 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.

19 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.

20 Sequence of Analysis Step 1 Collection of Giant Keyhole Limpet ornaments from burial.

21 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.

22 Step 3 Sequence of Analysis 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.

23 Cluster Analysis Sub-step 3b Giant Keyhole Limpet shell size Emic selection of shell size For making an ornament. Orifice size ( max L and max W) 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.

24 Raw Data Orifice max L max W Cluster Analysis Orifice Size Class #1 #2 #3 #4 #5 X X X X X X X X X X X Qualitative Data chipped ends X X X X X X X X X one end ground one chipped X X Cross Table of Burial 2 Trench 8 Section B.

25 Dendrogram resulting from weighted average clustering algorithm using Euclidean distance as the dissimilarity measure. The cluster analysis here serves to convert numerical data into discrete classes that can be entered into a binary matrix.

26

27 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).

28 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.

29 Attributes p q r s t Obj 1 X X Obj 2 X X Obj 3 X X Obj 4 X X Obj 5 X X Obj 6 X X 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.

30 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.

31 ( { }, {p, q, r, s, t}) object set is empty attribute set is full

32 ({2}, {p, r}) ({3}, {p, s}) ({1}, {p, t}) ({4, 5}, {q, s}) ({6}, {q, t}) ({ }, {p, q, r, s, t})

33 ({1, 2, 3}, {p}) ({2}, {p, r}) ({3}, {p, s}) ({1}, {p, t}) ({4, 5}, {q, s}) ({6}, {q, t}) ({ }, {p, q, r, s, t})

34 ({1, 2, 3}, {p}) ({2}, {p, r}) ({3}, {p, s}) ({1}, {p, t}) ({3, 4, 5}, {s)) ({4, 5}, {q, s}) ({6}, {q, t}) ({ }, {p, q, r, s, t})

35 ({1, 2, 3}, {p}) ({2}, {p, r}) ({3}, {p, s}) ({3, 4, 5}, {s)) ({1, 6}, {t}) ({1}, {p, t}) ({4, 5}, {q, s}) ({6}, {q, t}) ({ }, {p, q, r, s, t})

36 ({2}, {p, r}) ({1, 2, 3}, {p}) ({3}, {p, s}) ({4, 5, 6}, {q}) ({3, 4, 5}, {s)) ({1, 6}, {t}) ({6}, {q, t}) ({1}, {p, t}) ({4, 5}, {q, s}) ({ }, {p, q, r, s, t})

37 object set is full attribute set is empty ({1, 2, 3, 4, 5, 6}, { }) ({2}, {p, r}) ({1, 2, 3}, {p}) ({4, 5, 6}, {q}) ({3, 4, 5}, {s)) ({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

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

39 There is one absolute implication in the lattice which is: r implies p object set is full ({1, 2, 3, 4, 5, 6}, { }) attribute set is empty ({2}, {p, r}) ({1, 2, 3}, {p}) > ({3}, {p, s}) ({3, 4, 5}, {s)) ({4, 5, 6}, {q}) ({1, 6}, {t}) ({6}, {q, t}) ({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

40 Additive line type of lattice diagram. Drawn with chain decomposition algorithm. Burial 2 Trench 8B The initial diagram produced by the algorithm usually requires manual adjustment by the analyst.

41 16/20=0.8 Luxenburger Basis 5/6=0.83 Orifice Size Class #1 implies chipped ends Orifice Size Class #2 implies chipped ends Green refers to One end ground one chipped implies absolute implications Orifice Size Class #3 and red to partial Orifice Size Class #3 implies chipped ends implications [80% of the time] Orifice Size Class #4 implies chipped ends [89% of the time] Threshold for Luxenburger basis set at greater than or equal to 80% Burial 1 Trench 5A

42 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).

43 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

44 > >< < > > < Burial 4 Trench 5 or 4D

45 > > > Burial 7 Trench 6A

46 <> < < > > < < > < > > 6/7=0.86 4/5=0.8 4/5=0.8 Orifice Size Class #3 implies wing-shaped square and vice versa Burial 3 Trench 8B

47 entire margin ground Burial 5 Trench 6B 5/6=0.83

48 Burial 3 Trench 5D ext.

49 Burial 1 Trench 5B

50 Burial 4 Trench 5B 5/6=0.83

51 9/10=0.9 7/8=0.88 Burial 1 Trench 8C Orifice Size Class #5 implies thin oval ring thin oval ring implies Orifice Size Class #5

52 B1T5A

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

54 (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.

55 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.

56 ground end ground end

57 Transitional burial

58 Transitional burial wing-shaped ornaments

59

60

61

62 thin oval ring ornaments

63 My Analysis Provides Good Support for H1 and H2 Ring Wing-shaped To End-ground refine ornaments this Megathura analysis square structure and a better crenulata the rectangular largest typology ornaments orifice Megathura of the size appear wing-shaped classes crenulata to (e.g. Class ornaments contribute #5 in the both provide is needed most B4T5 the structure to or most discriminate 4D and structure to the B1T8C). middle ground to the and middle from large chipped and orifice large orifice forms size classes. size well classes to precisely in the later identify (L1a) burials subtypes in the based sample. on shape variation in the outline (e.g. compute outline curvature profiles using an Elliptical Fourier Function approach).

64 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.

65 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.

66 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.

67 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 graphtheoretic 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.

68 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 graphtheoretic 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).

69 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 graphtheoretic 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 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.

70 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.

71 Level in Lattice Node Degree Burial B2T8B B1T5A B5T6B B3T5Dext B4T5or4D B7T6A B1T5B B3T8B B4T5B B1T8C Node degree frequency matrix by lattice level for the lattices of each of the ten burials.

72 Burials B2T8 B B1T5 A B5T6 B B3T5D ext B4T5o r4d B7T6 A B1T5 B B3T8 B B4T5 B B1T8 C B2T8B B1T5A B5T6B B3T5D Ext. B4T5o r4d B7T6A B1T5B B3T8B B4T5B B1T8C Correlation matrix computed from the above table. Provides a measure of structural similarity for each pair of lattices.

73 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).

74 Burials B2T8 B B1T5 A B5T6 B B3T5 Dext B4T5o r4d B7T6 A B1T5 B B3T8 B B4T5 B B1T8 C B2T8B B1T5A B5T6B B3T5D ext B4T5o r4d B7T6A B1T5B B3T8B B4T5B B1T8C Adjacency matrix determined from the correlation coefficient matrix using a cutoff of 0.82 or greater for the correlation coefficient.

75 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. Lattices with a very heterogeneous structure. Clique 1 Clique 2 Lattice Structural Similarity Classification Lattices with a moderate to very heterogeneous structure.

76 contrast between Galois lattice and PCA (principal components analysis)

77 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.

78 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% sister lighthearted PC2 vacillating self younger sister 47.4% PC1 Attributes: Eigenvectors Objects (= Persons): Component Scores (= Left Singular Vectors times Square Root of Eigenvalues)

79 The biplot suggests that SELF is depressive and vacillating which is not true.

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

81 PCA uses a numeric metric scale which results in the misleading In the lattice sister and younger sister share the same node in complete impression in the biplot that sister and younger sister have different agreement with the raw data. degrees of affinities with respect to attributes. Examining the lattice chains it is clear that SELF does not have any paths leading to either depressive or vacillating.

82 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.

83 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.

84 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;