Data Visualization and its Proof by Compactness Criterion of Objects of Classes

In this paper considered the problem of reducing the dimension of the feature space using nonlinear mapping the object description on numerical axis. To reduce the dimensionality of space used by rules agglomerative hierarchical grouping of different type (nominal and quantitative) features. Groups do not intersect with each other and their number is unknown in advance. The elements of each group are mapped on the numerical axis to form a latent feature. The set of latent features would be sorted by the informativeness in the process of hierarchical grouping. A visual representation of objects obtained by this set or subset is used as a tool for extracting hidden regularities in the databases. The criterion for evaluating the compactness of the class objects is based on analyzing the structure of their connectivity. For the analysis used an algorithm partitioning into disjoint classes the representatives of the group on defining subsets of boundary objects. The execution of algorithm provides uniqueness of the number of groups and their member objects in it. The uniqueness property is used to calculate the compactness measure of the training samples. The value of compactness is measured with dimensionless quantities in the interval of [0, 1]. There is a need to apply of dimensionless quantities for estimating the structure of feature space. Such a need exists at comparing the different metrics, normalization methods and data transformation, selection and removing the noise objects.