Data depth functions for non-standard data by use of formal concept analysis

In this article, we introduce a notion of depth functions for data types that are not given in statistical standard data formats. Data depth functions have been intensively studied for normed vector spaces. However, a discussion on depth functions on data where one specific data structure cannot be presupposed is lacking. We call such data non-standard data. To define depth functions for non-standard data, we represent the data via formal concept analysis which leads to a unified data representation. Besides introducing these depth functions, we give a systematic basis of depth functions for non-standard using formal concept analysis by introducing structural properties. Furthermore, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article provides the mathematical formalisation of centrality and outlyingness for non-standard data. Thereby, we increase the number of spaces in which centrality can be discussed. In particular, it gives a basis to define further depth functions and statistical inference methods for non-standard data

A note on the connectedness property of union-free generic sets of partial orders

This short note describes and proves a connectedness property which was introduced in Blocher et al. [2023] in the context of data depth functions for partial orders. The connectedness property gives a structural insight into union-free generic sets. These sets, presented in Blocher et al. [2023], are defined by using a closure operator on the set of all partial orders which naturally appears within the theory of formal concept analysis. In the language of formal concept analysis, the property of connectedness can be vividly proven. However, since within Blocher et al. [2023] we did not discuss formal concept analysis, we outsourced the proof to this note

Comparing Machine Learning Algorithms by Union-Free Generic Depth

We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we provide two examples of classifier comparisons on samples of standard benchmark data sets. Our results demonstrate promisingly the wide variety of different analysis approaches based on ufg methods. Furthermore, the examples outline that our approach differs substantially from existing benchmarking approaches, and thus adds a new perspective to the vivid debate on classifier comparison.Comment: arXiv admin note: substantial text overlap with arXiv:2304.0987

Depth Functions for Partial Orders with a Descriptive Analysis of Machine Learning Algorithms

We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies of depth functions in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we analyze the distribution of different classifier performances over a sample of standard benchmark data sets. Our results promisingly demonstrate that our approach differs substantially from existing benchmarking approaches and, therefore, adds a new perspective to the vivid debate on the comparison of classifiers.Comment: Accepted to ISIPTA 2023; Forthcoming in: Proceedings of Machine Learning Researc

Robust Statistical Comparison of Random Variables with Locally Varying Scale of Measurement

Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to exploit the entire information encoded in them properly. We address this problem by considering an order based on (sets of) expectations of random variables mapping into such non-standard spaces. This order contains stochastic dominance and expectation order as extreme cases when no, or respectively perfect, cardinal structure is given. We derive a (regularized) statistical test for our proposed generalized stochastic dominance (GSD) order, operationalize it by linear optimization, and robustify it by imprecise probability models. Our findings are illustrated with data from multidimensional poverty measurement, finance, and medicine.Comment: Accepted for the 39th Conference on Uncertainty in Artificial Intelligence (UAI 2023

Statistical Models for Partial Orders Based on Data Depth and Formal Concept Analysis

In this paper, we develop statistical models for partial orders where the partially ordered character cannot be interpreted as stemming from the non-observation of data. After discussing some shortcomings of distance based models in this context, we introduce statistical models for partial orders based on the notion of data depth. Here we use the rich vocabulary of formal concept analysis to utilize the notion of data depth for the case of partial orders data. After giving a concise definition of unimodal distributions and unimodal statistical models of partial orders, we present an algorithm for efficiently sampling from unimodal models as well as from arbitrary models based on data depth

Comparing machine learning algorithms by union-free generic depth

We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we provide two examples of classifier comparisons on samples of standard benchmark data sets. Our results demonstrate promisingly the wide variety of different analysis approaches based on ufg methods. Furthermore, the examples outline that our approach differs substantially from existing benchmarking approaches, and thus adds a new perspective to the vivid debate on classifier comparison

Depth functions for partial orders with a descriptive analysis of machine learning algorithms

We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies of depth functions in linear and metric spaces, there is very little discussion on depth functions for non-standard data types such as partial orders. We introduce an adaptation of the well-known simplicial depth to the set of all partial orders, the union-free generic (ufg) depth. Moreover, we utilize our ufg depth for a comparison of machine learning algorithms based on multidimensional performance measures. Concretely, we analyze the distribution of different classifier performances over a sample of standard benchmark data sets. Our results promisingly demonstrate that our approach differs substantially from existing benchmarking approaches and, therefore, adds a new perspective to the vivid debate on the comparison of classifiers

COATING OF URANIUM DIOXIDE POWDERS WITH METALLIC TUNGSTEN FILMS

Conditions for the turegsten coating of uranium dioxide powders by hydrogen reduction of tungsten hexachloride in a fluidized bed of the powder product have been established. The coated material should contain approxi-mately 20 wt.% tungsten in order to obtain essentially complete coverage of micron-size urailum dioxide. (auth

Robust statistical comparison of random variables with locally varying scale of measurement

Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to exploit the entire information encoded in them properly. We address this problem by considering an order based on (sets of) expectations of random variables mapping into such non-standard spaces. This order contains stochastic dominance and expectation order as extreme cases when no, or respectively perfect, cardinal structure is given. We derive a (regularized) statistical test for our proposed generalized stochastic dominance (GSD) order, operationalize it by linear optimization, and robustify it by imprecise probability models. Our findings are illustrated with data from multidimensional poverty measurement, finance, and medicine