Scientific Data Ranking Methods

Theory and Applications

,

• Publisher: Elsevier Science
• Year: 2008
• Print ISBN: 9780444530202
• eBook ISBN: 9780080931937
• Topic: Physical Sciences
• Subtopic: Applied Mathematics

This volume presents the basic mathematics of ranking methods through a didactic approach and the integration of relevant applications. Ranking methods can be applied in several different fields, including decision support, toxicology, environmental problems, proteomics and genomics, analytical chemistry, food chemistry, and QSAR. - Covers a wide range of applications, from the environment and toxicology to DNA sequencing - Incorporates contributions from renowned experts in the field - Meets the increasing demand for literature concerned with ranking methods and their applications

Chapter 1 Introduction to Ranking Methods

L.A. Sarabia, M.S. Sánchez, M.C. Ortiz

Publisher Summary

The statistical analysis based on the distribution of the ranks (order of the experimental values) has had an increasing development. Outcomes associated with an experiment may be numerical in nature, such as quantity in an analytical sample. The types of measurements are usually called “measurement scales” and are, from the weakest to the strongest, nominal, ordinal, interval and ratio scale. This chapter describes procedures that can be used with data in nominal scale. It presents statistical methods, which is the most powerful for data in ordinal scale—they are the test of ranks. The tests of ranks are valid for data with continuous, discrete or both continuous and discrete distributions. The chapter discusses order in graphs and optimization problems. Graphs are highly versatile models for analyzing many practical problems in which points and connections between them have some physical or conceptual meaning. Optimization refers to finding one or more feasible solutions that correspond to extreme values of one or more objectives or criteria. When an optimization problem involves only one objective, the task of finding the optimal solution is called “single-objective optimization,” whereas if the problem involves more than one objective, it is known as “multi-objective optimization.”

1 Definition of Order Relations

From the mathematical point of view, an ordering (order relation) R in a set E is a binary relation among the elements in E that verifies:

(i) u R u, for each u in E (reflexive);

(ii) if u R v and v R u, then u = v (antisymmetric);

(iii) if u R v and v R w, then u R w (transitive).

A total ordering (or linear ordering or simple ordering) is, in addition, connected (complete), that is, every two members of the set are comparable (either u R v or v R u, for all u, v in E), and thus enables every member to be ordered relative to every other, and that generates a unique “linear” chain.

If this is not so, R is called a partial ordering (that is, it is transitive and antisymmetric but not necessarily connected) and thus generates possibly different chains of comparable elements; members of distinct chains may be incomparable.

Typical examples are the relation “less than or equal to” in the real numbers which is a total order whereas the set inclusion is a partial order. Hence, the real numbers form a unique chain of comparable elements (which is usually represented by the real line), whereas, for instance, the set of even natural numbers and the set of odd natural numbers are two incomparable sets; neither the set of odd numbers is included in the set of even numbers nor vice versa, so that they will be in different chains of comparable elements.

Let us consider an order and denote it as ≤, for simplicity. There are some special elements within such an order. One of the most important is the least element of a (sub)set S, which is an element u such that u ≤ v, for all elements v ε S. A value that is less than or equal to all elements of a set of given values is called a lower bound. The infimum or greatest lower bound is the unique largest member of the set of lower bounds for some given set, and it is equal to its minimum if the given set has a least element.

Analogously, the greatest element of a subset S of a partially ordered set (poset) is an element of S, which is greater than or equal to any other element of S, that is, u, such that v ≤ u, for all elements v of S; an upper bound is a value greater than or equal to all of a set of given values. The unique smallest member of the set of upper bounds for a given set is the supremum or least upper bound, and it is equal to its maximum if the given set has a greatest member.

In that respect, note the difference between minimum (resp. maximum) and minimal (resp. maximal) element. An element in an ordered set is minimal (resp. maximal) when there is no element smaller (resp. greater) than it, that is, it is the least (resp. greatest) element of a chain. If we do not need to specify, we use the generic term optimal.

Least and greatest elements may fail to exist but, if they e...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright Page
• Contributors
• Preface
• Chapter 1 Introduction to Ranking Methods
• Chapter 2 Total-Order Ranking Methods
• Chapter 3 Partial Ordering and Hasse Diagrams
• Chapter 4 Partial Ordering and Prioritising Polluted Sites
• Chapter 5 Similarity/Diversity Measure for Sequential Data Based on Hasse Matrices
• Chapter 6 The Interplay between Partial-Order Ranking and Quantitative Structure–Activity Relationships
• Chapter 7 Semi-Subordination Sequences in Multi-Measure Prioritization Problems
• Chapter 8 Multi-Criteria Decision-Making Methods
• Chapter 9 The DART (Decision Analysis by Ranking Techniques) Software
• Index