Partial Order Concepts in Applied Sciences

The paper discusses, first, the pros and cons of moving from a poset, obtained for some data set, towards a (liner) order. Then, the reasons are provided for doing so, based on three general premises: 1. the true nature of the data, used to obtain the poset (including uncertainty and imprecision, but also the ‘‘statistical’’ features of the data), 2. the existing prior knowledge (models, theories, hypotheses, convictions), and 3. the purpose of analysis. Some hints are offered on how this can be done, and illustrative examples are sketched. Finally, a real life case is shown, where such move from a sheer poset towards an order not only could be done, but was, indeed advisable.

In addition to our first publication about measures of incomparability/inequality (Bartel and Mucha, Measures of incomparability and of inequality and their applications. In: Multi-indicator systems and modeling in partial order. Springer, New York, 2014), a new weighted measure is proposed. In particular, on the basis of these proposed pairwise distance measures, partitional graph clustering techniques are applied to real datasets. In the case of the OCR dataset of handwritten digits “0” and “1”, the error rates are low, i.e., the performance of our proposed measures is very good. In an application to archaeometry, the results are quite similar to the K-means method. Concerning additional interesting archaeological interpretation, we postpone to our ongoing research that will be published soon (Bartel and Mucha, Applications of measures of incomparability and of inequality to archaeometry. Berliner Beiträge zur Archäometrie, Kunsttechnologie und Konservierungswissenschaft, 2016).

When a mutual ranking of a selection of objects is wanted, an initial step is the development of a multi-indicator system (MIS). Many MCDA concepts, e.g., members of the ELECTRE-family or of the different PROMETHEE versions, are available for obtaining rankings from an MIS. On the one side, a major disadvantage applying these models is the need for additional parameters beyond the data matrix, whereas Partial Order Theory is a methodology that allows extracting ranking information from a data matrix without additional, often subjective and consequently questionable parameters. On the other side, additional parameters help decision-making by introducing knowledge for decision makers/stakeholders beyond the data matrix. The present study focuses on the question to what extent an MIS can be modeled within the framework of partial order theory to add knowledge similarly to the MCDA approaches. Of all the possible alternatives, applying weight intervals to the indicators system is here discussed.

Classical measurements of performances are typically based on linear scales. However, in analytical chemistry one scale may be not sufficient to measure items appropriately. However, explorative statistics provide more factors, which all tell their own story about the analytical performance. Partial order methodology offers a possibility to evaluate analytical performance based on data provided, e.g., through method development and thus method optimization. Without presumptions or pretreatment of the data, the performance can be evaluated taking into account all indicators simultaneously and thus elucidating a “distance” from a reference, i.e., a result that is considered as the “best” or “optimal” possibly based on a certified value. In the present study, we elucidate the mutual ranking of the single analytical approaches, i.e., results from different analytical procedures. Initially, a simple approach for evaluating analytical performance is presented followed by more elaborate analyses. The analyses are based on various partial order tools and lead to (1) a partial ordering of the different analytical approaches (2) the “distance” to the reference value and (3) a classification due to the concept of “peculiar points” pin-pointing certain methods that do not fall into the “main-stream”. Additionally, information on the relative importance of the single indicator for the overall performance and a ranking without assuming weights for any single indicator can be obtained. In multi-rule systems incomparabilities appear, i.e., not every analytical result can be compared with another one. Even minor differences in indicator values may lead to incomparabilities. To elucidate these factors a detailed study of incomparabilities based on scanning analyses, tripartite graphs and fuzzy partial orders are presented in order to better understand strengths and weaknesses of the different analytical approaches. Thus, the analyses may lead to detailed recommendations for subsequent improvement. Eventually, the possible use of weight intervals for the single indicators is discussed.

In this paper, we propose an axiomatic theory of real functionals on frequency distributions over finite posets. The theory links the properties of the functionals to the classical theory of quasi-arithmetic means. In particular, it is shown that, given the frequency distribution, the values assumed by any “well-behaved” functional on a poset π can be expressed as a quasi-arithmetic mean of the values assumed over the linear extensions of π. This result plays a central role in view of the construction of synthetic indicators for multidimensional system of ordinal indicators, as shown through an example pertaining to multidimensional bi-polarization.

An evaluation of a set O of m objects with respect to a set A of n attributes, using, say, n parameters with real numbers as values, can be considered—after normalization of the parameter values L is a partial order, after normalization of the parameters, but more than that, L = [0, 1] ⁿ is a lattice. L-subsets over lattices have advantages which the standard Boolean subsets (over L = { 0, 1}) don’t have. We can in fact choose in a problem oriented way a suitable set theory for such sets, and a corresponding logic, so that we can decide if we want to be very strict in our argumentation or not, for example. This will be discussed briefly.

This chapter presents a combined approach for evaluating resource planning projects considering a multicriteria decision-making process. The approach is based on a multi-indicator matrix with three synthetic attributes that take into account several criteria such as (1) economic and financial elements (attribute: base ranking); (2) uncertainty propagation (attribute: probability); and (3) risk evaluation (attribute: compliance). The final evaluation is derived by using a combined approach based on a nonparametric aggregation rule using the concept of average rank for attributes 1 and 2; a simple procedure for score assignment for attribute3; and a lexicographic decision-making rule. In addition, a preliminary analysis of the alternatives is performed by using Hasse diagrams. An application to resource planning projects illustrates the proposed approach.

Poverty can be seen as a multidimensional phenomenon. A one-dimensional measure of poverty serving as a ranking index can be obtained by aggregating the different poverty aspects into a single scalar. Ranking indexes are thought of as supporting political decisions. We propose an alternative view based on simple concepts of partial order theory and illustrate the pros and cons of this approach taking as case study a multidimensional measure of poverty comprising three components—absolute poverty, relative poverty, and income—computed for the European Union regions. The analysis enables to highlight conflicts across the poverty components with some regions detected as controversial, with for example low levels of relative poverty and high levels of monetary poverty.

Poverty has been conceptualized and measured from a multidimensional perspective, generally by applying the classical composite index approach. However, this approach is far from capturing the diversity of individual’s poverty profiles and suffers from several shortcomings, notably regarding comparability, weighting, and aggregation issues. Such multidimensional indices are based on a dichotomized simplistic view of poverty in which binary category opposition prevails such as poor and non-poor, deprived and non-deprived. Furthermore, combining dichotomized threshold-based scores hides the complexity of ‘in-between poverty and prosperity’ profiles. In this chapter, we show that in comparison to the traditional composite index approach, the partial order theory allows to detect these ‘in-between’ profiles. In our study, monetary poverty, material deprivation, and well-being, measured with objective and subjective indicators, are used to analyse multidimensional poverty in Switzerland. The empirical analysis is based on the Swiss Household Panel data of 2013 and is realized by using partial order R package PARSEC.

We propose a multidimensional approach to describe the structure of social participation based on the theory of partial ordering. Social participation has been defined as the process of taking decisions concerning the life of an individual and that of the community in which he lives (Hart 1992).

The participation is characterized by:

Participating is a place of non-neutrality; it is the awareness that our being in the world is called to take an ethical line according to principles of justice and equity. Studying this concept is really important because social participation, involving vulnerable and excluded groups, should seek the empowerment of those groups, increasing their effective control over decisions that influence their health and life quality and their access to and use of health services.

Classical methods of analysis are framed to a particular aspect of this phenomenon. In this chapter, we will explore dynamic and multidimensional measures of social participation in Italy, leveraging properties of the poset theory. To represent these measures in a synthetic way, we will adopt a fuzzy approach: our purpose is not to measure social participation but to provide its structural representation in terms of profiles while complying with the shape of ordinal data. We begin by offering a working definition of social participation, then the analysis moves from the Multipurpose Survey on households: aspects of daily life, a longitudinal database that can measure change.

In the last two decades, data-driven policymaking has gained more and more importance due to the larger availability of data (and, more recently, Big Data) for designing proper and timely economic and social policies. This larger availability of data has let decision makers have a deeper insight of complex socio-economic phenomena (e.g. unemployment, deprivation, crime, social care, healthcare) but, at the same time, it has drastically increased the number of indicators that can be used to monitor these phenomena. Decision makers are now often in the condition of taking decisions with large batteries of indicators whose interpretation is not always easy or concordant. In order to simplify the decisional process, a large body of literature suggests to use synthetic indicators to produce single measures of vast, latent phenomena underlying groups of indicators. Unfortunately, although simple, this solution has a number of drawbacks (e.g. compensation between components of synthetic indicators could be undesirable; subjective weighting of the components could lead to arbitrary results; mixing information about different phenomena could make interpretation harder and decision-making opaque). Moreover, with operational decisions, it is necessary to distinguish between those situations when decisions can be embedded in automated processes, and those that require human intervention. Under certain conditions, the use of synthetic indicators may bring to a misleading interpretation of the real world and to wrong policy decisions. In order to overcome all these limitations and drawbacks of synthetic indicators, the use of multi-indicator systems is becoming more and more important to describe and characterize many phenomena in every field of science, as they keep the valuable information, inherent to each indicator, distinct (see, for a review: Bruggemann and Patil 2011).

The aim of this work is to analyse the epistemological and methodological aspects of the links between the Partial Order Set (POSET) theory and Sen’s Capability Approach (CA). CA is one of the best-known approaches to well-being and development analysis, founded by the Nobel Laureate Amartya Sen. If the theoretical bases of CA are sound, the empirical aspects have yet to be fully explored. The complexity of CA empirical verifications involves the requirement of statistical and econometric instruments to tackle: “a plurality of evaluative spaces; a plurality of dimensions and a multiplicity of indicators and scales of a quantitative or qualitative nature, and objectively or subjectively measured; a plurality of units of analysis (individuals, households, subgroups of population) and personal heterogeneities and a plurality of environmental contexts, including socio-economic, geographical, cultural and institutional variables” (Chiappero-Martinetti and Roche 2009, p. 5).

The chemical industry is one of the most well-known high-risk areas in modern society. Chemical accidents can have a great and lasting impact on the public’s perception of a chemical facility’s risk. The number of reported accidents constitutes a natural direct source of information on accidents risk. Reports provide a better understanding of two risk criteria, frequency and consequence, and help to identify hazardous chemicals, which can thereafter prevent accidents. Many published risk and accident data reports give information on the number of accidents or the numerator of the frequency calculation. However, they appear blind to information on exposure, such as the scale of the facility operation (the frequency denominator). This study will present how neglecting the denominator in frequency calculation gives misleading risk alerts. The research also proposes suitable denominators and a method to combine different frequency data based on partial order. Higher data size is also a good confidence indicator in frequency estimation. Results show that normalization with a suitable denominator is important, and the task is possible with the availability of many accident databases. Furthermore, partial order ranking using the Hasse diagram gives an overall hazard view of chemicals; it was able to estimate the percentage of chemicals performing above acceptable risk using only a small sample size.

Recent chemical applications of Formal Concept Analysis are reviewed, showing that molecular structure and activity of substances may be related through association rules, which is exemplified for mutagenicity and hepatotoxicity cases. Nuclear chemistry and nuclear medicine cases are explored, where attributes of radionuclides are related. A study of biotechnology application to uranium bioremediation is conducted and some Gram-positive bacteria are found as better uranium uptakers.

The German core sustainability indicators are applied to determine the status of sustainable development (SD) of Germany’s federal states. As such processes depend on political measures, the chapter analyses the connection of the SD performance of federal states with the governing political parties using the Partial Order methodology. Based on the resulting Hasse diagrams and on the average heights (based on the set of linear extensions of the partial orders) the federal states Bavaria, Schleswig-Holstein, Baden-Württemberg, Thuringia, Hesse, and Rhineland-Palatinate are compared. The comparison shows no unambiguous relation between the governing political parties and the development in a federal state. Other aspects explaining the respective SD performance could not be identified. To further analyze which aspects influence the sustainable development of federal states, the potential dependence on Germany’s federal government should be assessed.

Given two sets A and B, often the question arises how far objects a of A and b of B can be combined to a pair (a,b), fulfilling certain requirements. A first example is the marriage problem, another, the successful assignment of scientific projects to the needs of small or medium-sized enterprises. A third example, which motivated this study, arises from the project iBaMs–Barriere-Reduced Machines in Innovative Interaction. This project was aiming at promoting social inclusion for people with intellectual disabilities and their integration into labor markets and everyday activities. Especially, the project iBaMs “examines the preconditions and requirements for the development of control panels for computer-numerical-controlled (CNC) machines” (Wiesner-Steiner et al., Proceedings of the International Conferences Interfaces and Human Computer Interaction 2014, Game and Entertainment Technologies 2014, and Computer Graphics, Visualization, Computer Vision and Image Processing, pp 54–61, 2014). On the one hand, different control panels can be identified and characterized by a set of indicators. On the other hand, classifications of people with intellectual disabilities are available, leading to a profile of skills. The question arises on how optimal control panels based on indicators can be assigned to the profile of skills of employees. This assignment is called a matching between optimal control panels and profiles of skills. A first approach will be discussed on how this matching can be performed. It turns out that the Copeland index (Al-Sharrah J Chem Inf Model 50(5):785–791, 2010; Saari and Merlin, J Econ Theory 8:51–76, 1996) in its simplified form can be applied to answer the question.

Undoubtedly, the use of ethnobotanical and ethnopharmacological knowledge is of high priority worldwide, especially in fields such as drug development. Beginning in the 20th century, in the period of academic ethnobotany, the fields of ethnobotany and ethnopharmacology experienced a shift from the raw compilation of data to a greater methodological and conceptual reorientation. Nevertheless, a lot of ethnopharmacological knowledge is still unprocessed and there is plenty of space for handling this knowledge through several methodologies. Here we attempt to demonstrate an implementation of Partial Order Techniques, processing ethnopharmacological information, with the purpose to reveal inner structures and characteristics of raw data, which could potentially contribute in the conceptualization and management of ethnopharmacological knowledge. The results are promising, especially in the incorporation of rank order criteria in the classification of medicinal flora, in the revealing of trends in medicinal uses of closely related taxa, independently of whether the close relatedness is based on phylogeny or not, and in the comparison of ethnopharmacological data sets, revealing similarities in the order structure of the data.

Partially ordered sets are getting more important in socio-economical applications. In particular, their application in poverty evaluation (Fattore et al., New perspectives in statistical modeling and data analysis. Springer, Berlin, 2011) shows the advantages of their use in multivariate statistics on ordinal variables. A combinatory approach is necessary to apply this methodology, therefore the development of computational tools about partial orders is required. R is a widespread environment for statistical computing and graphics. The recent publication of the parsec (PARtial orders in Socio-EConomics) package on CRAN (the Comprehensive R Archive Network) is an achievement for the diffusion of computational tools devoted to the applications of partial orders in socio-economics. The package also implements functions related to composite indicators (Fattore et al., Quality of life in Italy. Springer, Berlin, 2012) in order to provide results of different approaches that can be compared. The aim of this work is to explain the functionalities of parsec, through examples and descriptions of its main functions.

The actual state of the development of PyHasse allows an ordinal analysis in the field of decision making, especially of performing rankings. The project iBaMs—Barriere-Reduced Machines in Innovative Interaction—aimed at promoting social inclusion for intellectually disabled people and their integration into labor markets and everyday activities. Within this project ranking will be a necessary step to find best panels of CNC machines, suitable for intellectually disabled people (see the chapter of Bruggemann et al.). Up to now, however, the PyHasse software package was only accessible by a direct contact with the developer R. Bruggemann. To allow a broader access it was necessary to develop a browser-based interface. This requirement had, however, far reaching consequences for the software itself, which was up to now based on Tkinter, a variant of Tcl/Tk. Experiences within the rearrangements of the software-code, as well with the testing system [Test Driven Development (TDD)] and other aspects such as documentation will be discussed. First results will be presented and we will give an outlook on further developments.