The Depth Problem: Identifying the Most Representative Units in a Data Group

Abstract

This paper presents a solution to the problem of how to identify the units in groups or clusters that have the greatest degree of centrality and best characterize each group. This problem frequently arises in the classification of data such as types of tumor, gene expression profiles or general biomedical data. It is particularly important in the common context that many units do not properly belong to any cluster. Furthermore, in gene expression data classification, good identification of the most central units in a cluster enables recognition of the most important samples in a particular pathological process. We propose a new depth function that allows us to identify central units. As our approach is based on a measure of distance or dissimilarity between any pair of units, it can be applied to any kind of multivariate data (continuous, binary or multiattribute data). Therefore, it is very valuable in many biomedical applications, which usually involve noncontinuous data, such as clinical, pathological, or biological data sources. We validate the approach using artificial examples and apply it to empirical data. The results show the good performance of our statistical approach.

Towards Application of One-Class Classification Methods to Medical Data

“…Group T formed by 107 samples from tumors on liver cancer patients, group NT formed by 76 samples from nontumor tissues of liver cancer patients and group N formed by 30 samples from normal livers. In [42] it was shown that there exists a high degree of confusion between groups, so bad one-class classification results are expected. Taking groups N, NT, and T as target classes, the typicality approach obtained AUC average values and standard deviation values 86.04 ± 3.95, 80.86 ± 3.07, and 55.62 ± 3.76, respectively.…”

“…Given the real-valued coordinates Z = Ψ( Y ), it is possible to apply any standard multivariate technique. Such an approach was used by different authors [35–42]. In this context a general measure of dispersion of Y , the geometric variability V δ of C , with respect to δ can be defined by $\begin{array}{c}{V}_{\delta }\left(C\right)=\frac{\mathrm{1}}{\mathrm{2}}\underset{S\times S}{\overset{}{{\displaystyle false\int }}}{\delta }^{\mathrm{2}}({\mathrm{boldy}}_i,{\mathrm{boldy}}_j)f\left({\mathrm{boldy}}_i\right)f\left({\mathrm{boldy}}_j\right)\lambda \left(d{\mathrm{boldy}}_i\right)\lambda \left(d{\mathrm{boldy}}_j\right)\end{array}$ which is a variant of Rao's diversity coefficient [43].…”

Estatistika metodoak distantzietan oinarrituriko ikuspegitik

“…Era asko daude Z zorizko p-bektoreak adierazten duen C populazioarekiko z aleak duen sakonera neurtzeko. Bereziki, distantzietan oinarrituriko sakonera [22] honela definitzen da:…”

“…Sakonera erlatiboa delako kontzeptua [22] ere definitu da. C r , r = 1, …, k klaseak harturik, C r klaseari dagokion alearen sakonera erlatiboa, klase horrekiko dagokion sakoneraren eta gainontzeko klaseekiko duen sakonera handienaren arteko diferentzia da.…”

Weighted Distance Based Discriminant Analysis: The R Package WeDiBaDis

“…There are many possibilities how to define the depth of the data (Liu, 1990;Vardi and Zhang, 2000;Zuo and Serfling, 2000;Serfling, 2002), nevertheless the computation of the most popular depth functions is very slow, in particular, for high-dimensional data the time needed for classification grows rapidly. A new less-computer intensive depth function I (Irigoien et al, 2013a) was developed, but the authors did not study its use in relation to the classification problem.…”