STATISTICA & APPLICAZIONI - 2011 - Special issue. Partial orders in applied sciences

Applied Statistics is certainly devoted to extract information from any kind of data. Data which characterize objects of interest. Ranking, i.e. finding a complete order among objects, belongs to the tasks of Applied Statistics. Not anticipating a historical research, it seems, however, as if the task of ranking in terms of the theory of partially ordered sets does not play that role in statistics which it should do, taking into account the enormous multitude, diversity and popularity of ranking studies. The idea is simple, not to say trivial. Let objects x1, …, xn be characterized by m data, attributes, say qj(j = l, …, m), then xi1 < xi2 if and only if qj (xi1) = qj (xi2) for all j, and with at least one qj* with a strict inequality. It can be easily seen that this definition (being the basis of the ‘‘Hasse diagram technique’’ (HDT)) does not guarantee that every object is in a <-relation with every other object; objects can in fact be incomparable with others. Hence in general we arrive at a partially ordered set starting from a data matrix qj (xi). Partially ordered sets can be visualized by directed acyclic graphs. Considered as ordinary graphs, they are triangle free due to the axiom of transitivity of order relations. Drawn in a special manner, this kind of directed graph is known as a Hasse diagram, which is an extremely useful tool to analyze partially ordered sets. Examples and some lines of interpretation will be found everywhere in this Special Issue.

Poverty is a fuzzy and complex phenomenon which is intrinsically multidimensional. First attempts of tackling poverty with multidimensional measures trace back to the seventies with the conceptual writings on income poverty by Amartya Sen (1976). Since then much research has been devoted to answer questions of the type: (i) Who is poor? (ii) How poor is a poor? The measure of poverty and social exclusion is certainly a key point in poverty description. While much effort has been put in the last decades to the measurement of poverty, less attention has been paid to find relations among different poverty aspects. In this paper, we start from a classical definition of the population of the poor and we employ Fuzzy Multi-Criteria Analysis to provide an attempt to relate poverty aspects to one another, which we call a ‘structural representation of poverty’. Our focus is on the pattern of implications existing among different descriptors characterizing poverty aspects. We show how fuzzy relation theory and partially ordered set techniques are effective in representing complex relational structures and provide new insights into multidimensional poverty. As simple test cases the method is applied to data concerning two Italian regions based on EU-SILC database 2004.
Keywords: Multidimensional Poverty, Multi-criteria Analysis, Poverty Structure, Ordinal Variables, Posets, Fuzzy Quasi-order Relations.

The evaluation of material deprivation, quality of life and well-being very often requires to deal with multidimensional systems of ordinal variables, rather than with classical numerical datasets. This poses new statistical and methodological challenges, since classical evaluation tools are not designed to deal with this kind of data. The mainstream evaluation methodologies generally follow a counting approach, as in a recent proposal by Alkire and Foster pertaining to the evaluation of multidimensional poverty. Counting procedures are inspired by the composite indicator approach and share similar drawbacks with it, computing aggregated indicators that may be poorly reliable. A recent and alternative proposal is to address the ordinal evaluation problem through partial order theory which provides tools that prove more consistent with the discrete nature of the data. The goal of the present paper is thus to introduce the two proposals, showing how the evaluation methodology based on partial order theory can be integrated in the counting approach of Alkire and Foster.
Keywords: Partial Order theory, Counting Approach, Evaluation, Material Deprivation, Quality of Life

The paper outlines an approach, applicable to both the problem of clustering and to (‘‘optimum’’) ordering, which starts from a formulation of the objective function and the constraints, equivalent to a binary mathematical programming problem. This formulation, for both ordering and clustering, represents a number of very positive features, like possibility of dealing with incomplete and inconsistent data, while posing essential numerical difficulties. For clustering, it implies a globally optimal solution in that both cluster content and cluster number are obtained. We reformulate this problem by parameterising it and show that, under certain additional assumptions, an effective algorithm can be deduced for both clustering and ordering, which suboptimises the objective function. In the case of clustering, the algorithm is an analogue of the classical hierarchical merger procedures, while in the case of ordering it relies on iterations, in which just one object is moved. Some essential properties are given, along with a simple illustration. In spite of the analogy, the properties of the approach and the respective algorithms are different for the two cases considered, i.e. clustering and ordering.
Keywords: Clustering, Ordering, Mathematical Programming, Parameterisation, Suboptimisation Algorithms, Objective Functions.

In the paper presented here, we use Formal Concept Analysis (FCA) to solve a problem that arises when working with partially ordered sets (posets). In detail, the task here is to look for incomparable subsets which are related to a given poset. A way to solve this problem is to use FCA based on a context which can be derived in some steps from the _-matrix of the (simple directed) graph corresponding to the given poset. The requested incomparable subsets result from the set of concepts obtained from this context. For illustrative purposes, small toy data sets are presented. At the end, a real data application to environmental chemistry is given in detail. The data consist of ten chemicals found in the German river Main. As the result a set of twelve incomparable pairs of subsets are figured out.
Keywords: Bipartite Graph, Adjacency Matrix, Formal Concept Analysis, Partially Ordered Set, Incomparability of Sets.

This paper is concerned with introducing a modified Copeland method as a relative and categorized ranking tool. Using the concept of partially ordered sets and the social choice theory, the Copeland score ranking methodology is applied outside its usual political voting environment to rank objects in the scientific field. The ranking methodology was assessed using 45 data sets with different number of objects and indicators and compared with other methods. Results show that the Copeland method appears as a good and stable tool for ranking objects giving results comparable to the Dominance and the Simple Additive Ranking methods with the advantage of lower sensitivity and CPU time. Also, it solves the problem of isolated objects found in some Hasse diagrams.
Keywords: Copeland Method, Hasse Diagram, Categorized Ranking, Relative Ranking, Sensitivity.

Importance Measures (IMs) are valuable tools that have been used to quantify and rank the components of a system with respect to their contribution to a considered measure of performance. For example, IMs have been used for characterizing the importance of element failures with respect to the overall system reliability. In general, different IMs based on different definitions may lead to different importance rankings of the components within a system. This fact could affect a decision-maker for achieving, for example, a better global performance level.
In this paper we propose the use of the Hasse Diagram Technique (HDT) to make a preliminary assessment for detecting possible conflicts among IM and selecting, if required, a convenient combination or aggregation of IMs, based on parametric or non-parametric techniques, such as Ordered Weighted Average (OWA) or Copeland Score (CS). Numerical examples illustrate the assessment.
Keywords: Copeland Score, Hasse Diagram Technique, Importance Measures, Multi-Criteria Decision, Ordered Weighted Average (OWA).

The DPSIR (Driving forces, Pressures, State, Impacts, Responses) framework takes into account a chain of past and present situations as well as suggests future activities as responses aiming at improving the environmental health. Thus, DPSIR constitutes an advantageous directive for integrated environmental assessments. The driving forces are centered on economic sectors and human activities, i.e. activities in the society that directly or indirectly are causing the pressures on the environment. The pressures on the environment develop from the human activities that are associated with the above mentioned ‘needs’ (driving forces). The state refers to the environmental and human health as a result of the pressures. The impacts refer to environmental and economic factors changing the physical, chemical or biological states of the environment as well as impacts on human health. The responses comprise a priori the reactions by authorities, regulators or society in general to the changes induced through the other element in the DPSIR chain. The paper will discuss the qualification of the DPSIR approach by applying partial order ranking (POR) to the single stages of the assessment, eventually applying the hierarchical partial ranking (HPOR) methodology in order to select the more appropriate responses. Further the paper will describe the possible involvement of expert groups in the assessment process applying a partial order based DPSIR approach.
Keywords: DPSIR, Integrated Environmental Assessment, Partial Order Ranking, Hasse Diagrams.

Assessments of the behavior and impact of chemicals in the environment typically require a multicriteria approach as a multitude of parameters has to be taken into account in order to disclose the full picture. In the present paper we demonstrate how selected partial order ranking tools can be applied as decision support. As an illustrative example a series of 14 chemicals, belonging to 4 different classes of chemicals, found in the river Main (Germany) have been assessed based on 3 parameters, i.e. volatilization, sedimentation and advection, determinative for their exposure behavior. In addition to ordinary partial order ranking more advanced tools as average ranks and dominance analysis have been applied leading to conclusions as to which class of chemicals should receive primary attention. Further, the analyses suggest directions for risk management such as pointing to specific sources for the most problematic pollutants.
Keywords: Partial Order Ranking, Hasse Diagram Technique, Average Rank, Dominance Analysis, Environmental Impact Assessment, EIA, River Main.

Jar-test is a useful tool for chemicals selection for physical–chemical wastewater treatment. The results show the treatment efficiency in terms of suspended matter and organic matter removal. However, in spite of having all these results, coagulant selection is not an easy task because one coagulant can remove efficiently the suspended solids but at the same time increase the conductivity. In this paper, the use of Partial Order Scaling Analysis (POSA) is proposed to help on the selection of the coagulant and its concentration in a sequencing batch reactor (SBR). An evaluation of two commonly used coagulation-flocculation aids was conducted and based on jar tests and POSA model, Ferric Chloride (100 ppm) was the best choice.
Keywords: Coagulation, Jar Test, Partial Order Scaling Analysis, Treatment Selection.

It has been evident for decades that many environmental chemicals pose an enormous risk to the environment as well as to humans. There is increasing pressure to intensify the research and to more efficiently evaluate the data on persistent and bioaccumulative chemicals in the environment as well as in human bodies. An appropriate data analysis method is based on the theory of partially ordered sets. The program PyHasse, developed by the third author, provides several features which are useful for gaining information out of the data and drawing conclusions concerning the impact of those chemicals and their prevention. In our data analysis approach we investigated data sets of breast milk samples of women in Denmark and Finland which contained measurable levels of 32 persistent organic pollutants (POPs). Three important features of the PyHasse program are used: The Sensitivity Analysis, the Similarity Analysis and the Separability Analysis. The aim of this discrete mathematical approach is to find differences in the chemicals’ contamination between the healthy boys and those boys who were suffering from congenital malformations (cryptorchidism).
Keywords: Environmental Health Data, Persistent Organic Pollutants (POPs), Cryptorchidism, Partial Order, PyHasse Program.