Multi-indicator Systems and Modelling in Partial Order

Many modern applications need evaluation of statements, the truth values “true” and “false” alone may not suffice, a statement can be neither true nor false, it may be true (or false) “in a certain sense.” They also need modeling of linguistic expressions and of fuzzy situations. “Binary thinking” does not suffice in many cases. Moreover, the choice of methods might better be problem driven, depending, for example, if we better use a pessimistic or an optimistic reasoning. Here is a brief introduction of how we can choose tools that are appropriate for mathematically modeling this kind of problems.

Ideal formulation of a multi-indicator system (MIS) would be to define, design, and acquire the entire construct with complete consensus among all concerned. However, such would be an extreme rarity in actuality. Experts have differing views. Factors may not express monotonically, as when either extreme is unfavorable. The entirety cannot be assessed and must be sampled. Empirical experience to validate expectations is inadequate. Consequently, exploratory examination of any available datasets collected for collateral purposes can augment insights relative to suitable surrogates for ideal indicators, with particular attention to ordering relations for subsets of quantifiers and ensembles of entities (objects, cases, instances, etc.).

Multivariate datasets are comprised of several quantifiers (variates or variables) as columns recorded for multiple entities as rows. The data matrix thus realized is not necessarily directly useful nor fully informative for analytically inferring order among entities. In this chapter, some approaches are discussed which may be helpful in extracting insights on ordering properties that are embodied in multivariate datasets and applicable in configuring suites of indicators. These procedures may be particularly helpful in finding suitable surrogates and applying partial order theory when expediency is essential. We consider orientation, crispness of data, and culling of candidates according to importance in respect of some desirable criteria.

Usually, there are only two stages of comparability between two objects: they are comparable or incomparable (see, for instance, the theory of partially ordered sets). The same holds with respect to equality/inequality. In this publication, measures of incomparability u _ij and of inequality v _ij between two objects g _i and g _j with m attributes with respect to the relation ≤ are introduced. Based on these definitions the (non-metric) distance measure \( {a}_{ij}=\frac{1}{2}\left({u}_{ij}+{v}_{ij}\right) \) with maximal possible values \( m+1+\left[\frac{m}{2}\right]\cdot \left(m-\left[\frac{m}{2}\right]\right) \) is proposed. The distance matrix A = (a _ij ) will be used for clustering starting from the corresponding complete graph 〈g〉 (g – number of objects), whose edges g _i –g _j are valued by a _ij . The result of the classification consists of a set of complete subgraphs, where, for instance, the objective function of compactness of a cluster is based on all pairwise distances of its members. The same edge-valued graph is used to construct a transitive-directed tournament. Thus, a unique seriation of the objects can be obtained which can also be used for further interpretation of the data. For illustrative purposes, an application to environmental chemistry with only a small data set is considered.

We discuss two complexity indicators reported in the literature for partially ordered sets (posets), the first one based on linear extensions and the second one on incomparabilities. Later, we introduce a novel indicator that combines comparabilities and incomparabilities with a Shannon’s entropy approach. The possible values the novel complexity indicator can take are related to the partitions of the number of order relationships through Young diagrams. Upper and lower bounds of the novel indicator are determined and analysed to yield a normalised complexity indicator. As an example of application, the complexity is calculated for the ordering of countries based on their performance in chemical research. Finally, another complexity indicator is outlined, which is based on comparabilities, incomparabilities, and equivalences.

In many decision-making situations, ranking of objects with related tasks is a fundamentally important issue. In these situations, a number of objects are ranked on the basis of measurements on a set of several indicators. A prevalent approach is to form a composite index from these several measurements using weights of relative importance for the selected indicators determined by experts and/or stakeholders. An entirely different approach for ranking uses the theory of partially ordered sets (posets). In classical poset ranking derived by average ranks (AR) method, unequal indicator weights of any kind do not play any part in the computation of ranking based on a given data matrix. Here we present a novel method of poset ranking that involves stochastic order of weighted indicator cumulative rank frequency (CRF) distributions. We then investigate how this data-validated evidence-based ranking can be used to construct a composite index reproducing an identical ranking. We further seek reconciliation between databased weighted poset CRF ranking and ranking induced by an arbitrary subjective composite index. This investigation acquires particular importance today in view of issues of trade-offs among indicators, implicit in the apparent advocacy involved in the choice of weights of the composite index. This chapter is based on research conducted in the spirit of start small even for big data. The concept of databased weighted poset ranking introduced here may open doors to still other ways of weighting schemes and other reconciliation approaches for comparative knowledge discovery using partial orders and composite indicators. Meaningful ability to deal with big data is an urgent need of comparative knowledge discovery with partial orders and composite indicators in this infometrical computer science and software engineering age of statistical information science and technology. This chapter is prepared in the spirit of a concept paper for digital age infometrics and comparative knowledge discovery critical in several fields, such as document discovery, drug discovery, gene discovery, chemical discovery, criminal discovery, geospatial critical area discovery, etc. The ranking, prioritization, and selection of objects and indicators carrying a variety of names in a variety of contemporary issues of societal and scientific importance based on relevant evidence embodied in data matrices provide insightful leads in these substantive investigations involving variously big data.

Environmental aspects often are in conflict with the criteria for a best/optimal behavior under technical aspects only. In these cases, common methods to compare different options and to come to a decision show methodological disadvantages. In this situation, this chapter intends to demonstrate the situation firstly by giving a typical example, secondly to show, how a software platform might support the decision, thirdly, to provide different methods for comparison to demonstrate the effects of the method chosen, and finally to sensitize the users for the interdependencies between the comparison method and the resulting ranking. The example will deal with the decision to find a new car according to individually scalable ratios. General data on different cars mainly are in conflict with the ratio of CO₂ expressing the environmental aspects of the cars to select. The chapter proposes a software platform that allows dealing with these conflicting parameters by individually weighting and a flexible interface for comparison.

In this “position paper” we discuss the potential role of partial order theory in socio-economic statistics and social indicators construction. We maintain that the use of concepts and tools from poset theory is needed and urgent to improve currently adopted methodologies, which often prove ineffective for exploiting ordinal data. We also point out that the difficulties in spreading partial order tools are cultural in nature, and that some open-mindedness is needed among social scientists. We address these issues introducing some examples of open questions in socio-economic data analysis: (i) the problem of multidimensional poverty evaluation, (ii) the problem of assessing inequality and societal polarization, and (iii) the problem of clustering in multidimensional ordinal datasets.

This work identifies hazardous chemicals that cause chemical accidents in plants using simple and available properties. Particular attention is given to reactive chemicals and the relation between corrosion and accidents frequencies. The identification of hazardous chemicals is done by categorized ranking using the Hasse diagram technique. Hasse diagrams are the most promising method due to simplicity, wide use, and nonparametric advantage. To achieve our goal, the large number of isolated objects in Hasse diagrams was reduced/eliminated using a heuristic approach that presents strategies to speed up the ranking task. The basis is to collect suitable indicators and to define the suitable ranking objective. The ranking is presented using a case study of 22 chemicals with 8 simple hazardous indicators. Results show that the reduction of isolated objects is essential before evaluating the hazardous results. Also, simple and readily available data were used successfully as indicators for identifying chemicals causing accidents.

Lichens and mosses have extensively been used in multielemental large-scale biomonitoring studies of atmospheric metal pollution. Despite its high importance in the assessment of cumulative risk and the communication with risk managers, the presentation and interpretation of biomonitoring results have only been partially the center of interest for a standardized methodology and for the harmonization of the techniques. Here we attempt to expand and improve the up-to-date formal presentation of biomonitoring results, combining the Hasse diagram technique with GIS techniques. The implementation using real data has demonstrated that such an expansion and improvement, in the direction of cumulative risk assessment and management, is feasible and it is suggested for incorporation in biomonitoring studies.

Contaminated site cleanup is a multibillion dollar issue in Canada. Practical decision support tools are needed to help prioritise contaminated site cleanup funding across a portfolio of many sites. This chapter illustrates the concept of prioritising contaminated site management decisions using environmental risk, societal perception and environmental liability as the measures. Using previously published information on 20 contaminated sites located in Canada, these three aspects are combined by applying partial order concepts to prioritise sites. The authors show how societal perception and environmental liability influence site prioritisation compared to prioritisation based on environmental risk alone. They recommended additional research in quantifying the societal perception of contaminated sites prior to practical application of the concepts explained in the chapter.

Multi-indicator matrices represent a set of objects or alternatives characterized simultaneously by several criteria or attributes. In many situations, a decision-maker is interested in assessing each object, by considering simultaneously all criteria, and defining a ranking able to synthesize the global characteristic of each object, for example, from best to worst. However the assessment could be influenced by uncertain factors. For example, the cost of a project could be affected by variations in the interest rate. The effects of such variations could affect the initial or base rank. In this chapter, the robustness of the base rank is analyzed. The first part analyzes how the uncertainty in the numerical value of the criteria associated with each objects affects its rank. Additionally, it proposes some ideas for assessing the rank robustness. The second part proposes the use of global sensitivity analysis to assess the importance of each uncertain factor on, for example, the base rank. An example related to a real portfolio management, using three techniques that do not require additional preference parameters, is presented.

This chapter deals with the successive application of self-organizing map (SOM) classification and Hasse diagram technique (HDT) as chemometric tools for assessment of river water and sediment quality. Both studies are carried out by using long-term water quality monitoring data from the Struma River catchment, Bulgaria and lake sediment samples from Mar Menor lagoon in Spain. The advantages of the SOM algorithm for advanced visualization and classification of large datasets are used for proper selection of chemical parameters being most effective in quality assessment combined with some state directives for surface water quality parameters in the river water study and as preprocessing procedure of the initial sediment data matrix. The simultaneous application of the SOM methodology or legislation norms with Hasse diagram technique allows to visualize the spatial and temporal evolution of water quality parameters or to reveal specific sediment pollution patterns.

The paper introduces PARSEC, a new software package implementing basic partial order tools for multidimensional poverty evaluation with ordinal variables. The package has been developed in the R environment and is freely available from the authors. Its main goal is to provide socio-economic scholars with an integrated set of elementary functions for multidimensional poverty evaluation, based on ordinal information. The package is organized in four main parts. The first two comprise functions for data management and basic partial order analysis; the third and the fourth are devoted to evaluation and implement both the poset-based approach and a more classical counting procedure. The paper briefly sketches the two evaluation methodologies, illustrates the structure and the main functionalities of PARSEC, and provides some examples of its use.

We extend exploration and application of a compound ranking regime for multi-indicator systems based on partial order theory that coordinates principal down-set, up-set, and comparability. We use this “balance of normalized definitive status (BONDS)” ranking in conjunction with augmented hierarchical clustering for comparing clusters as graded groups to obtain information relevant to number of clusters, group-wise ordering, and interaction of indicators. A case study is conducted with localities as instances (objects) and percentages for kinds of land cover as indicators. Hierarchical clustering is accomplished with the hclust facility of R software in conjunction with customized computational support in R.

In this chapter we evaluate the data of 18 Organochlorine pesticides (OCPs) found in breast milk samples from 44 mothers in the Taurus Mountains in Turkey. In this approach the association of concentration levels in breast milk samples with the two confounding factors: smoking habit and habit of taking medication is the goal. For all data evaluation approaches, we applied the Hasse diagram technique and its software package, namely the PyHasse software. Special emphasis was laid on the software features “similarity” and “Local Partial Order Model” to draw further conclusions out of the data. The data analyses resulted in differences between the smoking women and those who did not smoke as well as between the medication and non-medication breast milk samples. Little differences were found comparing hormone taking mothers and mothers taking other medication.

Simple elements of partial order theory appear helpful for a causal analysis in the context of ranking. The Hasse diagrams may seem as a confusing system of lines and a high number of incomparabilities. Thus, they indicate that metric information may be lost, but, on the other side partial order tools offer a wide variety of additional information about the interplay between the objects of interest and indicators. In this chapter a series of tools are presented to reveal such information.

As an illustrative example the so-called Failed State Index (FSI) is used. FSI is a composite indicator based on 12 individual indicators by simply summarizing the single values. The FSI comprises 177 states, which are the objects of our study.

A selection of appropriate partial order tools are applied to reveal specific information about the interplay between the states and the 12 indicators, such as A: sensitivity analysis, where the indicators are ordered relatively to their impact on the structure of the partially ordered set, B: a “vertical,” i.e., chain analysis that is directed towards the comparabilities within a Hasse diagram, and C: a “horizontal,” i.e., antichain analysis focusing on incomparabilities, including also the use of tripartite graphs as well as a derivation of an ordinary graph.

Partial order does not necessarily constitute as a Multicriteria Method solving all inherent problems. However, this chapter discloses that a detailed analysis by partial order tools prior to a possible derivation of a ranking index apparently is highly attractive.

The software PyHasse is an elaborated “experimental” software for ordinal analysis of data matrices. PyHasse is based on the interpreter programming language Python. A brief introduction to the programming language Python is given and the general principles behind PyHasse are outlined. An actual overview about PyHasse (status, April 2013) is provided. Today PyHasse comprises 91 modules covering 9 different categories, such as basic Partial Order Analysis, i.e., the drawing Hasse diagrams and the calculation of some important quantities. A selection of newer or rarely used modules are discussed in detail in order to explain some principles of PyHasse. As a leading example the pollution by Lead, Cadmium, and Zinc of regions of south-western Germany is discussed.

An outlook is given, where future projects are discussed. Such projects comprise among others, Internet access to some of the more important modules, inclusion of the Formal Concept Analysis tools, and of tools derived from POSAC and the variance-based sensitivity.