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Representation of Symbolic Objects According to the description structure. Antonio Irpino*, N.Carlo Lauro**, Rosanna Verde* * Second University of Naples, Italy ** University of Naples Federico II, Italy. Problem

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Representation of symbolic objects according to the description structure

Representation of Symbolic Objects According to the description structure

Antonio Irpino*, N.Carlo Lauro**, Rosanna Verde*

* Second University of Naples, Italy

** University of Naples Federico II, Italy


Representation of symbolic objects according to the description structure

  • Problem description structure

  • The results of the factorial analysis as well as the symbolic objects visualization depend on the nature and structure of symbolic descriptors

  • Aims

  • To transform symbolic data in input in order to:

  • homogenize the nature of the different descriptors;

  • study (non-linear) relationships among descriptors;

  • Find the most suitable kind of visualizatiob of the symbolic objects (by connected shapes)



Representation of symbolic objects according to the description structure

GCA analysis allows to perform a reduction of dimensionality of the original descriptors space.

The different kind of descriptors in input are transformed in multi-categorial descriptors with associated weights


Coherent transformation of descriptors homeomorphism
Coherent Transformation of descriptors of the original descriptors space.(Homeomorphism)

  • From a geometrical point of view – let give a topology on a space, the topology defined on the tranformed space have to be an homeomorphism.

  • The transformation function have to be biettive and bicontinue.

  • From a statistical point of view - the transformation do not allow to lose information related to the internal variability of the objects.

  • Furthermore we have to take into account the metric of spaces.

  • For istance, the classical categorization of a real variable is not a homeomorphic transformation


Transformation of a single numerical variable into a multicategorial modal one
Transformation of a of the original descriptors space.single numerical variable into a multicategorial modal one

k

R

Euclidean metric

Y1 = k

L

M

1

Metric : c2

1

F(k)=(L(k), M(k), H(k))

Lk+Mk+Hk=1

Lk,Mk,Hk[0,1]

0

1

H

f:(S,m)(S’,m’) biettive and bicontinue (f,f -1are continue)

where: S=[min, Max]  R , min < MAX

S’=(L, M, H) R3 L,M,H[0,1] L+M+H=1; m=c2


Representation of symbolic objects according to the description structure
Representation of symbolic assertion in the original space where the descriptors are interval variables


Example of tranformation of an interval in a multi categorial with associated weights
Example of tranformation of an where the descriptors are interval variablesinterval ina multi-categorial with associated weights

R

u

l

Euclidean metric

Min<l<k<u<Max

L

M

L(k), M(k), H(k)

L(k)+M(k)+H(k) =1

X2 metric

H

It is worth to note that it needs to codify with respect to others points than the min and Max in order to keep an homeomorfism:

(0,1,(0)


Representation of symbolic objects according to the description structure
Example of tranformation of an interval in a multi-categorial with associated weights using semilinear B-spline functions

Euclidean metric

R

u

l

Min<l<k<u<Max

L

M

L(k), M(k), H(k)

L(k)+M(k)+H(k) =1

X2 metric

H


Representation of symbolic objects according to the description structure
Example of tranformation of an interval in a multicategorial weighted value using semilinear B-spline function(2)

Euclidean metric

R

u

l

Min<l<k<u<Max

ML

L(k), ML(k), MH(k),H(k)

k+ML+MH+H=1

H

c2 metric

L

MH


Transformation of multinominal variable 1
Transformation of multinominal variable(1) weighted value using semilinear B-spline function(2)

What is the topology which take into accout the variability?

{Red, Green, Blue} is not a metric space

Red

Red

Green

1

Green

1

1

1

c2 metric

Two Points

A segment

Blue

Blue

0

0

1

1


Transforming multinominal variable 2
Transforming multinominal variable(2) weighted value using semilinear B-spline function(2)

1

2

3

Red

Green

Blue


Transformation multinominal variable 3
Transformation multinominal variable(3) weighted value using semilinear B-spline function(2)

Multicategorial modal variable with weights at interval

Red

Green

Blue


Representation of symbolic objects according to the description structure

Example of representation of a symbolic assertion in the original description space(An interval variable combined with a Multinominal Variable) (1)


Representation of symbolic objects according to the description structure

Example of representation of Symbolic Assertion in the original description space(An interval variable conbined with a Multinominal Variable) (2)

The representation is no convex as MCAR but connected


Example of representation of a symbolic assertion
Example of representation of original description spacea symbolic assertion

Blue

Green

Red


Effects of the descriptors transformation on the visualization on factorial planes
Effects of the descriptors transformation on the visualization on factorial planes

  • In factorial analysis are performed orthogonal projections onto factorial subspaces.

  • A factorial subspace is a linear combination of the symbolic descriptors

  • This means that: the space spanned by factorial variables is an affine transformation of the originary space

  • An affine trasformation is invariant with respect to linear properties of the originary space i.e.: a linear projection of a convex shape is a convex shape and so on.


Choice of the most suitable so visualization 1
Choice of the most suitable SO visualization(1) visualization on factorial planes

  • A linear projection of a convex shape is a convex shape

MCAR

Originary space

Factorial plane

Convex hull of the vertices is the best visualization shape

(i.e. no-overfitting)


Representation of symbolic objects according to the description structure

Choice of the most suitable SO visualization (2) visualization on factorial planes

A linear projection of a connected shape is a connected shape

Convex hull of the vertices is better than MCAR

but it presents an overfitting

CH

MCAR

Originary space

Factorial plane


Descriptors space with rules
Descriptors’ space with rules visualization on factorial planes

When there are dependences rules, then the descriptor space loses the convexity properties

R: If y1>h then Y2<k

Y2

Incoherent sub space induced by the dependence rule

k

Y1

h


Analysis of non linear relationships between variables
Analysis of non-linear relationships between variables visualization on factorial planes

  • By tranforming interval variables into categorical modal ones - by means of semilinear B-splines functions - it is possible to study non linear relationships among variables.

  • It allows to study relationships between categories of two or more variables which represent an High Medium or Low level of them.


Open problems
Open problems visualization on factorial planes

  • A new kind of symbolic variable needs

  • Multicategorial modal having weights at interval

  • What is the topological structure and the properties of this kind of geometrical space?

  • New visualization shapes need in order to solve overfitting problem.

  • Symbolic interpretation of the shapes of SO representation