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By Michael Merrill

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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By Michael Merrill

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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. 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

  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. 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

  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. 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

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

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

  12. 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.

  13. “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).

  14. “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).

  15. 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).

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

  17. 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.

  18. 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.

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

  20. 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.

  21. 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.

  22. 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.

  23. Cluster Analysis Raw Data Qualitative Data Cross Table of Burial 2 Trench 8 Section B.

  24. 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.

  25. 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).

  26. 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.

  27. 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.

  28. 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.

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

  30. ({2}, {p, r}) ({3}, {p, s}) ({1}, {p, t}) ({4, 5}, {q, s}) ({6}, {q, t}) ({ }, {p, q, r, s, t})

  31. ({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})

  32. ({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})

  33. ({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})

  34. ({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})

  35. 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

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

  37. 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

  38. 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

  39. 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

  40. 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).

  41. 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

  42. > > < > > < < Burial 4 Trench 5 or 4D

  43. > > > Burial 7 Trench 6A

  44. < > < > < > > < < < > > 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

  45. entire margin ground 5/6=0.83 Burial 5 Trench 6B

  46. Burial 3 Trench 5D ext.

  47. Burial 1 Trench 5B

  48. 5/6=0.83 Burial 4 Trench 5B

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