β-integrated local depth and corresponding partitioned local depth representation

A novel local depth definition, β-integrated local depth (β-ILD), is proposed as a generalization of the local depth introduced by Paindaveine and Van Bever citepaindaveine2013depth, designed to quantify the local centrality of data points. β-ILD inherits desirable properties from global data depth and remains robust across varying locality levels. A partitioning approach for β-ILD is introduced, leading to the construction of a matrix that quantifies the contribution of one point to another's local depth, providing a new interpretable measure of local centrality. These concepts are applied to classification and outlier detection tasks, demonstrating significant improvements in the performance of depth-based algorithms.