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GEOMETRIC MORPHOMETRICS a numerical approach to analysis of shapes of morphological objects
. .
© 2011

I. Ya. Pavlinov Zoologcial Meuseum, Moscow State Univ.

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? WHAT IS THE SHAPE
" " ­ , " " (, ) "The essense of being" of a thing is its morphe, or its "first essense" (Aristotle, Metaphysics) = Forma = Morphe = Shape (Configuration)
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SHAPE ANALYSIS
Graphical approaches
. DЭrer`s painting ( XVI )

' D'Arcy Thompson painting ( X )

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SHAPE ANALYSIS
= Analytical approaches = morphometrics
­ Principal task is numerical analysis of shapes diversity
: Particular tasks: Comparison of shapes themselves Analysis of causes of shapes differences Classification of objects by their shapes

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SHAPE ANALYSIS
= Analytical approaches = morphometrics
() ­ , , GEOMETRIC MORPHOMETRICS (GM) ­ is a set of QUANTITATIVE (NUMERICAL) and based thereupon GRAPHICAL methods, making it possible to compare objects by their SHAPE, excluding differences by their size

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= Analytical approaches = morphometrics

Initial description of the objects

( , ) The rodent skull (side view, schematically) with points located
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= Analytical approaches = morphometrics

Initial description of the objects c' a b' b c d' d a'

In STANDARD MORPHOMETRICS a set of MEASUREMENTS between points

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= Analytical approaches = morphometrics

Initial description of the objects a c bg d e f

(= ) In GEOMETRIC MORPHOMETRICS a set of CARTESIAN COORDINATES OF LANDMARKS (= points)
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= Analytical approaches = morphometrics

Initial description of the objects
Standard morphometrics
LCB 39,5 40 38,5 38,2 39,2 39,6 38,9 LR 14,4 14,3 13,8 13,7 14,5 14,5 14,2 LZ 20,3 20,4 19,4 19,2 20,3 20,4 20,1 SM 23 22,6 22,1 21,4 22,6 22,2 22

Geometric morphometrics
X-coord 599.00000 527.00000 449.00000 392.00000 335.00000 278.00000 248.00000 221.00000 168.00000 226.00000 361.00000 482.00000 Y-coord 1264.00000 1280.00000 1284.00000 1300.00000 1319.00000 1323.00000 1339.00000 1309.00000 1258.00000 1372.00000 1371.00000 1357.00000

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= Analytical approaches = morphometrics
NB! SHAPE VARIABLES
/ INDICES CORRELATIONS

«» "Standard" approach
REGRESSION ANALYSIS

b/a, c/a, c/b...
PRINCIPAL COMPONENT ANALYSIS

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= Analytical approaches = morphometrics
SHAPE VARIABLES c «» "Geometric" approach NB!
( ) Shape differences are described as differences by landmark locations (mutual displacements) between objects ­ , Shape variables are any measures describing the above differences

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Objects of geometric morphometrics
, : Morphological structures differing by their shapes:

3-

2-

3- ,

2-

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Objects of geometric morphometrics
! Important limitation! : The objects are to be solid: minimum of degrees of freedom

! / GOOD!

! / BAD!

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/ What GM can't
GM analyzes just quantitative changes in configuration variation So the quantitative changes in topology are inaccesible to GM ! impossible! ! possible!

в

« »? Are "virtual landmarks" allowable to use&

? possible?

, «» 14 glirid molar crow, differences in number of secondary ridges


/ What GM can
Pairwase comparison of objects

в
Analysis of entire sample of objects

=

=
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/ What GM can
Analysis of correlations between shapes of different objects

Correlation between shapes of axial skull and mandible

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/ What GM can
Analysis of correlations between shape and size of the same object

Correlation between shape and length of the axial skull

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/ What GM can
? Phylogenetics?



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: GM: basical formalisms





­ Kendall space is defined by totality of possible mutual shape transformations
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: GM: basical formalisms
/ Shape spaces and distances


/ Shapes projections Tangent space

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2



/ Shape space
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Procrustes distances


: GM: basical formalisms
/ Trsansformartions without shape changes
are nt studied by means of GM

/translation

/rotation

/scaling

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: GM: basical formalisms
/ Trsansformartions with shape changes
are studied by means of GM

/ uniform stretching/compressing

shearing

/ nonuniform stretching/compressing
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: GM: basical formalisms
/ Trsansformartions with shape changes
are nt studied by means of GM cockling

straight horns

bending
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: GM: basical formalisms
/ Trsansformartions without shape changes
/ mirror reflections are nt studied by means of GM

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/GM: /procedure
/Object description
(=)

c a bg

d f

e


XYa, XYb, XYc...
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/GM: /procedure
( )

/Landmark placement

1.


, ( )

2.

...

...

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/GM: /procedure
( )

/Landmark placement



, When SEMILANDMARKS are used, their entire sequence makes sense, and not positions of particular semilandmarks

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/GM: /procedure
/Data
LM=21 1236 981 725 468 211 33 217 295 519 610 701 876 1039 1111 1266 1292 390 407 526 693 848 LM=21 1182 962 741 524 306 153 4 4 4 4 3 3 1 1 1 1 1 1 1 2 2 2 2 1 2 5 5 5 4 4 3 08 84 72 39 75 01 34 83 56 29 41 08 49 16 26 56 21 87 34 61 18 00 28 11 95 42 64



, . , .
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-

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/GM: /procedure
1.

2. /landmarks

3. a/reference /consensus

NB

!

NB

!

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/GM: /procedure
4. /alignment 4. 4.

4. «»:

5.
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/GM: /procedure
Comparison by superimposition
, ,

Objects

A BC A dAA B dAB dBB C dAC dBC dCC

Procrustes distances

Procrustes distances are similar to Euclidian distances by their properties, so their matrices can be manipulated with in the similar ways

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/GM: /procedure
Comparison by thin-spline analysis

, / / (bending energy)
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/GM: /procedure
Comparison by thin-spline analysis

1

() principal deformations (PD)

/reference

2
33 /PD


/GM: /procedure
Comparison by thin-spline analysis
­ /partial warps
1
1 1 1


2 2 2

2

­ , ...
34 ...


/GM: /procedure
Comparison by thin-spline analysis
/relative warps /RW
1 2

­ , ... ...
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/GM: /procedure
­ : Thin-spline analysis ­ results representation:
, Nectomys N P P N Prometheomys

Nectomys Prometheomys

Prometheomys Nectomys 36


/GM: /procedure
­ : Thin-spline analysis ­ results representation:

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/GM: /procedure
Singular values and percent explained for relative warps: No. SV % Cum % 1 1.22223 59.23% 59.23% 2 0.54621 11.83% 71.06% 3 0.47649 9.00% 80.06% Singular values and percent explained for relative warps: No. SV % Cum % 1 1.22223 59.23% 59.23% 2 0.54621 11.83% 71.06% 3 0.47649 9.00% 80.06%

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Cricetus

Ondatra

Rhombomys

Cricetulus

Myodes

Gerbillus

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:
The least squares analysis

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RW1, RW2





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RW1, RW2





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RW1, RW2




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Sorex
12 2 1 3 11 4 10 5 6 78 9

15

caecutiens
5 8 6 7

roboratus
14 13

5 7 1 8 2 13 6 10 14 4 15 3

9 11 12

10 12

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9

1 2 4 3

isodon

araneus



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..

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() .

.

5
..
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: ? AND AGAIN: WHAT IS THE SHAPE
: ? Principal question: how to separate shape and size?
, There actual situations exist when differences by size are included unremovably in the differences by shape
Shape

Allometry
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Size


: ? AND AGAIN: WHAT IS THE SHAPE
: - ? Analytical form of the question: how to decompose the entire morpho-disparity into components of shape and size?
-: ) , ) , Two components of morpho-disparity: ) a portion attributed to differences by size ) a portion attributed to differences by shape
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PRINCIPAL CONCLUSIONS
- ­ , GM is able something, namely to compare objects by their shapes GM presumes quite simple understanding of shape So GM is able to do not a good deal of things , In applying GM, its possibilities and limitations are to be understood
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THANKS FOR ATTENTION

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