Optimal choice of prototypes for ceramic typology

Uzy Smilansky (WIS) Avshalom Karasik (WIS)M. Bietak (Vienna)V. Mueller(Vienna)

Acknowledge support from : Bikura (ISF) ,Kimmel center for archaeological studies (WIS) .

Tel – el – Daba = AbidosThe capital of the Hyksos (1800-1600 bc)

Assemblage - 190 drinking cups w/o clear stratigraphical assignment.

Typological analysis with a single index:)D. Arnold, M. Bietak (

H

DI

ds

dxs arccos)(

2

2

2

1

)(

ds

dx

ds

xd

ds

ds

-2 0 2 4 6 80

2

4

6

8

a.

cmcm

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25b.

arc length

x(s)

0 5 10 15 20-1

0

1

2

3

4

5c.

arc length

0 5 10 15 20-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2d.

arc length

k(s)(s)

t

s

x

Various equivalent ways to characterize a curve (profile)

s : arc length,Total length = L

Formally :

s 2 [0,L]

x(s) : distance from the symmetry axis.

s) : tangent angle.

s) : curvature.

-2 0 2 4 6 80

2

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a.

cm

cm

0 0.2 0.4 0.6 0.8 1-0.05

0

0.05

0.1

0.15

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0.3b.

normalized arc length

x(s

)

0 0.2 0.4 0.6 0.8 1-1

0

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4c.

normalized arc length

0 0.2 0.4 0.6 0.8 1-0.2

-0.1

0

0.1

0.2

0.3d.

normalized arc length

k(s

)

Profiles of two cups and their characteristic functions

The distance between the curves i and j can bedefined in different ways using any of the

characteristic functions

.))()((,,))()((,,))()((,1

0

21

0

21

0

2 dsssjiddsssjiddssxsxjid jijijix

Measure arc-length in units of L. Thus, s 2 [0,1].

The corresponding scaler products

xx

x

ji

ji

jijjii

ji

dssxdssx

dssxsxxC

||

|

)()(

)()()(

22,

The correlation matrices

Similarly : )(, jiC and 11 , jiCPer definition:

dsssjidsssjidssxsxji jijijix)()(|;)()(|;)()(|

)(, jiC

We consider each profile as a “vector” is a multi-dimensional space . If the profiles are “similar” - their corresponding vectors occupy only asubspace of the space of profiles.

Typology = Identification the relevant subspace and its basis vectors.

The basis vectors are the “prototypes.”

Criterion: maximum detail using a subspace spanned of minimal dimension.

An abstract approach to typology

Sorting branches (correlated groups) using the correlation matrix

0 1 2 3 4 5

x 104

-0.2

0

0.2

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0.8

-100 -50 0 50 100 150 200 250 300 3500

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Eigenvalues : 3.39 ; 2.61 ; 1.40

0.34 ; 0.21 ; 0.02 ; 0.00 ; 0.00

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5

-1

-0.5

0

0.5

1

1.5

)(, assemblagejiC

Generate the prototype correlation matrix

Projection on the two eigenvectorsCorresponding to the largest eigenvalues

8 prototypesConstructed as means ofThe 8 branches.

-200 0 200 400 600 800-0.2

0

0.2

0.4

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0.8

1

Cluster tree for the x-coordinate function after subtracting the mean of the assemblage

0 100 200 300 400 500 600 700 800 900 1000-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Eigenvalues : 3.4342 .39030. ;15690. ; 01860

Hence- a single parameter is sufficientTo characterize the assemblage !

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-100 -50 0 50 100 150 200 250 300 3500

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Cluster analysis in terms of the Correlation matrix )(, assemblageji xxC

dssxsx assemblage ))()((

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Bitak's parameter as a function of the distance of x-coordinate function,from the mean xcoordinate function

Good correlation between typology and chronology

I

Conclusions:

• The optimal mathematical characterization of the profiles depends primarily on the nature of the features of importance.

• The best set of independent prototypes is created by the eigenvectors of the prototype correlation matrix which correspond to the dominant eigenvalues.

• The chosen set of prototypes presents the best possible compromise which minimizes the number of prototypes, while maximizing the amount of preserved details.

• For further details on the method and other applications: visit- http://www.weizmann.ac.il/complex/uzy/archaeomath/research.html