Comparison-based centrality measures

Recently, learning only from ordinal information of the type “item x is closer to item y than to item z” has received increasing attention in the machine learning community. Such triplet comparisons are particularly well suited for learning from crowdsourced human intelligence tasks, in which workers make statements about the relative distances in a triplet of items. In this paper, we systematically investigate comparison-based centrality measures on triplets and theoretically analyze their underlying Euclidean notion of centrality. Two such measures already appear in the literature under opposing approaches, and we propose a third measure, which is a natural compromise between these two. We further discuss their relation to statistical depth functions, which comprise desirable properties for centrality measures, and conclude with experiments on real and synthetic datasets for medoid estimation and outlier detection.

Functional Symmetry and Statistical Depth for the Analysis of Movement Patterns in Alzheimer’s Patients

Black-box techniques have been applied with outstanding results to classify, in a supervised manner, the movement patterns of Alzheimer’s patients according to their stage of the disease. However, these techniques do not provide information on the difference of the patterns among the stages. We make use of functional data analysis to provide insight on the nature of these differences. In particular, we calculate the center of symmetry of the underlying distribution at each stage and use it to compute the functional depth of the movements of each patient. This results in an ordering of the data to which we apply nonparametric permutation tests to check on the differences in the distribution, median and deviance from the median. We consistently obtain that the movement pattern at each stage is significantly different to that of the prior and posterior stage in terms of the deviance from the median applied to the depth. The approach is validated by simulation.

Depth classification based on affine-invariant, weighted and kernel-based spatial depth functions

Several multivariate depth functions have been proposed in the literature, of which some satisfy all the conditions for statistical depth functions while some do not. Spatial depth is known to be invariant to spherical and shift transformations. In this paper, the possibility of using different versions of spatial depth in classification is considered. The covariance-adjusted, weighted, and kernel-based versions of spatial depth functions are presented to classify multivariate outcomes. We extend the maximal depth classification notions for the covariance-adjusted, weighted, and kernel-based spatial depth versions. The classifiers' performance is considered and compared with some existing classification methods using simulated and real datasets.

Robust multivariate estimation based on statistical depth filters

In the classical contamination models, such as the gross-error (Huber and Tukey contamination model or case-wise contamination), observations are considered as the units to be identified as outliers or not. This model is very useful when the number of considered variables is moderately small. Alqallaf et al. (Ann Stat 37(1):311–331, 2009) show the limits of this approach for a larger number of variables and introduced the independent contamination model (cell-wise contamination) where now the cells are the units to be identified as outliers or not. One approach to deal, at the same time, with both type of contamination is filter out the contaminated cells from the data set and then apply a robust procedure able to handle case-wise outliers and missing values. Here, we develop a general framework to build filters in any dimension based on statistical data depth functions. We show that previous approaches, e.g., Agostinelli et al. (TEST 24(3):441–461, 2015b) and Leung et al. (Comput Stat Data Anal 111:59–76, 2017), are special cases. We illustrate our method by using the half-space depth.

Breast Cancer Diagnosis Using Relational Discriminant Analysis of Malignancy-Associated Changes

The chapter is devoted to description of a novel method of breast cancer diagnostics based on the analysis of the distribution of the DNA concentration in interphase nuclei of epitheliocytes of buccal epithelium with the aid of novel algorithms of statistical machine learning, namely: novel proximity measure between multivariate samples, novel algorithm of construction of tolerance ellipsoids, novel statistical depth and novel method of multivariate ordering. In contrast to common methods of diagnostics used in oncology, this method is a non-invasive and offers a high rate of accuracy and sensitivity.