Créer une présentation
Télécharger la présentation

Download

Download Presentation

Representation of Symbolic Objects According to the description structure

117 Vues
Download Presentation

Télécharger la présentation
## Representation of Symbolic Objects According to the description structure

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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**• 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)**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)**• 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 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 assertion in the original space**where the descriptors are interval variables**Example of tranformation of an interval 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)**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**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)**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)**1 2 3 Red Green Blue**Transformation multinominal variable(3)**Multicategorial modal variable with weights at interval Red Green Blue**Example of representation of a symbolic assertion in the**original description space(An interval variable combined with a Multinominal Variable) (1)**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**Blue Green Red**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)**• 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)**Choice of the most suitable SO visualization (2)**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**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**• 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**• 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