Optimal choice of prototypes for ceramic typology Uzy Smilansky (WIS) Avshalom Karasik (WIS) M. Bietak (Vienna) V. Mueller(Vienna) Acknowledge support.

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

4

6

8

a.

cm

cm

0 0.2 0.4 0.6 0.8 1-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3b.

normalized arc length

x(s

)

0 0.2 0.4 0.6 0.8 1-1

0

1

2

3

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

0.4

0.6

0.8

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

50

100

150

200

250

300

350

400

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

0.6

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 !

-100 0 100 200 300

50

100

150

200

250

300

350

400

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

50

100

150

200

250

300

350

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

50

100

150

200

250

300

350

-100 0 100 200 3000

50

100

150

200

250

300

350

400

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