A computational geometry approach for linear and nonlinear discriminant analysis (Q1584187)

The aim of discriminant analysis is the construction of a rule allowing to allocate individuals to predetermined classes. The allocation rules proposed in literature can be divided into parametric and distribution free procedures. As for the parametric ones, the probability density function is described as a finite mixture of the group-conditional density functions (gcdf) in the class. Parametric procedures have some drawbacks: they deal with a priori distribution assumptions, often the multinormal distribution and, in addition, the projective method and the Fisher-type discriminant functions deal with linearly separable groups. On the other hand, several non parametric approaches have been proposed. These are based on the kernel method for the density estimation of the gcdf, on the nearest neighbour density estimation, or on the tree-structured allocation rule. However, methods based on non parametric density estimation do not deal satisfactorily with the tails of the underlying distributions. The aim of this paper is to present a completely data driven discriminant function and an allocation rule which do not rely on any distributional assumption and allow to deal as well with non linear or non convex population structures. For this purpose the author proposes a geometric procedure based on the Voronoi tessellation (or Voronoi diagram). The proposed discriminant analysis induces a space partition, that allows to deal efficiently with non linearly separable or non convex population structures. The computational cost of the proposed procedure and the topological conditions concerning the group-conditional density functions that optimize the procedure performance. Because of its geometric properties, the method can be also usefully applied in statistical pattern recognition is analyzed.