Ordered Sets for Data Analysis

TL;DR

This book explores the mathematical foundations and algorithmic challenges of data analysis based on order theory, emphasizing scalability and complexity in handling structured and big data across various domains.

Contribution

It introduces a formal framework using order theory for data analysis and discusses the computational complexity of related algorithms, with applications across multiple fields.

Findings

Analysis of algorithmic complexity for data analysis tools

Application of order-based methods in diverse domains

Scalability considerations for large data sets

Abstract

This book dwells on mathematical and algorithmic issues of data analysis based on generality order of descriptions and respective precision. To speak of these topics correctly, we have to go some way getting acquainted with the important notions of relation and order theory. On the one hand, data often have a complex structure with natural order on it. On the other hand, many symbolic methods of data analysis and machine learning allow to compare the obtained classifiers w.r.t. their generality, which is also an order relation. Efficient algorithms are very important in data analysis, especially when one deals with big data, so scalability is a real issue. That is why we analyze the computational complexity of algorithms and problems of data analysis. We start from the basic definitions and facts of algorithmic complexity theory and analyze the complexity of various tools of data analysis we consider. The tools and methods of data analysis, like computing taxonomies, groups of similar objects (concepts and n-clusters), dependencies in data, classification, etc., are illustrated with applications in particular subject domains, from chemoinformatics to text mining and natural language processing.