Skip to main content
SpringerLink
Log in

Introduction to partial order theory exemplified by the Evaluation of Sampling Sites

  • Chapter
Partial Order in Environmental Sciences and Chemistry

Abstract

The first part of this chapter gives a detailed introduction to partial order ranking and Hasse Diagram Technique (HDT). Thus, the construction of Hasse diagrams is elucidated as is the different concepts associated with the diagrams. The analysis of Hasse diagrams is disclosed including structural analysis, dimension analysis and sensitivity analysis. Further the concept of linear extensions is introduced including ranking probability and averaged rank. The evaluation of sampling sites is, in the second part of the chapter, used as an illustrative example of the advantageous use of partial order ranking and Hasse Diagram Technique.

When a ranking of some objects (chemicals, geographical sites, river sections etc.) by a multicriteria analysis is of concern, it is often difficult to find a common scale among the criteria and therefore even the simple sorting process is performed by applying additional constraints, just to get a ranking index. However, such additional constraints, often arising from normative considerations are controversial. The theory of partially ordered sets and its graphical representation (Hasse diagrams) does not need such additional information just to sort the objects.

Here, the approach of using partially ordered sets is described by applying it to a battery of tests on sediments of the Lake Ontario. In our analysis we found: (1) the dimension analysis of partially ordered sets suggests that there is a considerable redundancy with respect to ranking. The partial ranking of the sediment sites can be visualized within a two-dimensional grid. (2) Information, obtained from the structure of the Hasse diagram: For example six classes of sediment sites have high priority, each class exhibits a different pattern of results. (3) The sensitivity analysis identifies one test as most important, namely the test for Fecal Coliforms/ Escherichia coli. This means that the ranking of samples is heavily influenced by the results of this specific test.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Atkinson MD (1989) The complexity of Orders. In: Rival, I. (Ed.) Algorithms and Order NATO ASI series. Series C Mathematical and Physical Sciences Vol. 255 Kluwer Academic Publishers, Dordrecht, pp 195–230

    Google Scholar 

  • Atkinson MD (1990) On the computing the Number of Linear Extensions of a Tree. Order 7:23–25

    Article  Google Scholar 

  • Beckenbach EF, Bellman R (1971) An Inequality of Karamata. In: Inequalities, 3 edn. Springer-Verlag, Berlin, pp 1–107

    Google Scholar 

  • Birkhoff G. 1984. Lattice theory. Providence, Rhode Island: American Mathematical Socienty, Vol XXV

    Google Scholar 

  • Borg I, Shye S (1995) Facet Theory-Form and Content. Sage Publications, Thousand Oaks, California

    Google Scholar 

  • Brans JP, Vincke PH (1985) A Preference Ranking Organisation Method (The PROMETHEE Method for Multiple Criteria Decision-Making). Management Science 31:647–656

    Google Scholar 

  • Brans JP, Vincke PH, Mareschal B (1986) How to select and how to rank projects: The PROMETHEE method. European Journ Oper Research 24:228–238

    Article  Google Scholar 

  • Brüggemann R, Bartel HG (1999) A Theoretical Concept to Rank Environmentally Significant Chemicals. J Chem Inf Comput Sci 39:211–217

    Article  Google Scholar 

  • Brüggemann R, Bücherl C, Pudenz S, Steinberg C (1999a) Application of the concept of Partial Order on Comparative Evaluation of Environmental Chemicals. Acta hydrochim hydrobiol 27:170–178

    Article  Google Scholar 

  • Brüggemann R, Halfon E (1997) Comparative Analysis of Nearshore Contaminated Sites in Lake Ontario: Ranking for Environmental Hazard. J Environ Sci Health A32(1):277–292

    Google Scholar 

  • Brüggemann R, Halfon E, Welzl G, Voigt K, Steinberg C (2001b) Applying the Concept of Partially Ordered Sets on the Ranking of Near-Shore Sediments by a Battery of Tests. J Chem Inf Comp Sc 41:918–925

    Article  CAS  Google Scholar 

  • Brüggemann R, Pudenz S, Carlsen L, Sørensen PB, Thomsen M, Mishra RK (2001a) The Use of Hasse Diagrams as a Potential Approach for Inverse QSAR. SAR and QSAR Environ Res 11:473–487

    Google Scholar 

  • Brüggemann R, Pudenz S, Voigt K, Kaune A, Kreimes K (1999b) An algebraic/graphical tool to compare ecosystems with respect to their pollution. IV: Comparative regional analysis by Boolean arithmetics. Chemosphere 38:2263–2279

    Article  Google Scholar 

  • Brüggemann R, Schwaiger J, Negele RD (1995a) Applying Hasse diagram technique for the evaluation of toxicological fish tests. Chemosphere 30(9):1767–1780

    Article  Google Scholar 

  • Brüggemann R, Simon U, Mey S (2005) Estimation of averaged ranks by extended local partial order models. Math Commun Math Comput Chem 54(3):489–518

    Google Scholar 

  • Brüggemann R, Sørensen PB, Lerche D, Carlsen L (2004) Estimation of Averaged Ranks by a Local Partial Order Model. J Chem Inf Comp Sc 44:618–625

    Article  CAS  Google Scholar 

  • Brüggemann R, Welzl G (2002) Order Theory Meets Statistics-Hassediagram technique-. In: Voigt K and Welzl G (eds) Order Theoretical Tools in Environmental Sciences-Order Theory (Hasse diagram technique) Meets Multivariate Statistics-. Shaker-Verlag, Aachen, pp 9–39

    Google Scholar 

  • Brüggemann R, Zelles L, Bai QY, Hartmann A (1995b) Use of Hasse Diagram Technique for Evaluation of Phospholipid Fatty Acids Distribution in Selected Soils. Chemosphere 30(7):1209–1228

    Article  Google Scholar 

  • Brüggemann R, Welzl G, and Voigt K (2003) Order Theoretical Tools for the Evaluation of Complex Regional Pollution Patterns. J Chem Inf Comp Sc 43:1771–1779

    Article  CAS  Google Scholar 

  • Carlsen L, Lerche DB, Sørensen PB (2002) Improving the Predicting Power of Partial Order Based QSARs through Linear Extensions. J Chem Inf Comp Sc 42:806–811

    Article  CAS  Google Scholar 

  • Carlsen L, Sørensen PB, Thomsen M (2001) Partial Order Ranking-based QSAR’s: estimation of solubilities and octanol-water partitioning. Chemosphere 43:295–302

    Article  CAS  Google Scholar 

  • Carlsen L, Sørensen PB, Thomsen M, Brüggemann R (2002) QSAR’s based on Partial Order Ranking. SAR and QSAR in Environmental Research 13:153–165

    Article  CAS  Google Scholar 

  • Colorni A, Paruccini M, Roy B (2001) A-MCD-A, Aide Multi Critere a la Decision, Multiple Criteria Decision Aiding. JRC European Commission, Ispra

    Google Scholar 

  • Davey BA, Priestley HA (1990) Introduction to Lattices and Order. Cambridge University Press, Cambridge

    Google Scholar 

  • Dutka BJ, Walsh K, Kwan KK, El Shaarawi A, Liu DL, Thompson K (1986) Priority site selection from degraded areas based on microbial and toxicant screening tests. Water Poll Res J Canada 21(2):267–282

    CAS  Google Scholar 

  • Freese R. 2004. Automated Lattice Drawing. In: Eklund P, Ed. ICFCA 2004, Berlin: Springer-Verlag, p 112–127

    Google Scholar 

  • Ganter B and Wille R (1996) Formale Begriffsanalyse Mathematische Grundlagen, Springer-Verlag, Berlin, pp 1–286

    Google Scholar 

  • Halfon E, Hodson JA and Miles K (1989) An algorithm to plot Hasse diagrams on microcomputer and Calcomp plotters. Ecol Model 47:189–197

    Article  Google Scholar 

  • Heinrich R (2001) Leitfaden Wasser-Nachhaltige Wasserwirtschaft; Ein Weg zur Entscheidungsfindung. Wasserforschung e.V., Berlin

    Google Scholar 

  • Jonsson B (1982) Arithmetic of Ordered Sets. In: Rival I (Ed) Ordered Sets. D. Reidel Publishing Company, Dordrecht, pp 3–41

    Google Scholar 

  • Lerche D, Brüggemann R, Sørensen PB, Carlsen L, Nielsen OJ (2002) A Comparison of Partial Order Technique with three Methods of Multicriteria Analysis for Ranking of Chemical Substances. J Chem Inf Comp Sc 42:1086–1098

    Article  CAS  Google Scholar 

  • Lerche D, Sørensen PB (2003) Evaluation of the ranking probabilities for partial orders based on random linear extensions. Chemosphere 53:981–992

    Article  CAS  Google Scholar 

  • Lerche D, Sørensen PB, Brüggemann R (2003) Improved Estimation of the Ranking Probabilities in Partial Orders Using Random Linear Extensions by Approximation of the Mutual Probability. J Chem Inf Comp Sc 53:1471–1480

    Article  CAS  Google Scholar 

  • Muirhead RF (1900) Inequalities relating to some Algebraic Means. Proc Edingburgh Math Soc 19:36–45

    Article  Google Scholar 

  • Muirhead RF (1906) Proofs of an Inequality. Proc Edingburgh Math Soc 24:45–50

    Google Scholar 

  • Neggers J, Kim HS (1998) Basic Posets. Singapore: World Scientific Publishing Co

    Google Scholar 

  • Patil GP, Taillie C (2004) Multiple indicators, partially ordered sets, and linear extensions: Multi-criterion ranking and prioritization. Environmental and Ecological Statistics 11:199–228

    Article  Google Scholar 

  • Pavan, M. and R. Todeschini, 2004. New indices for analysing partial ranking diagrams. Analytica Chimica Acta 515: 167–181.

    Article  CAS  Google Scholar 

  • RandiĆ M (2002) On Use of Partial Ordering in Chemical Applications. In: Voigt K and Welzl G (Ed.) Order Theoretical Tools in Environmental Sciences-Order Theory (Hasse Diagram Technique) Meets Multivariate Statistics. Shaker-Verlag, Aachen, pp 55–64

    Google Scholar 

  • Rival I (1981) (Ed.) Ordered Sets, NATO ASI Series, Series C. Mathematical and Physical Sciences Vol. 83, Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Rival I (1983) Optimal Linear Extensions by Interchanging Chains. Proc AMS 89:387–394

    Article  Google Scholar 

  • Rival I (1985a) (Ed.), Graphs and Order, NATO ASI Series, Series C. Mathematical and Physical Sciences Vol. 147, Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Rival I (1985b) The diagram. In: Rival I (1995 a) pp 103–133

    Google Scholar 

  • Rival I (1989) (Ed.) Algorithms and Order, NATO ASI Series, Series C. Mathematical and Physical Sciences Vol. 255, Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Rival I, Zaguia N (1986) Constructing Greedy Linear Extensions by Interchanging Chains. Order 3:107–121

    Article  Google Scholar 

  • Roy B (1972) Electre III: Un Algorithme de Classements fonde sur une representation floue des Preferences En Presence de Criteres Multiples. Cahiers du Centre d’Etudes de Recherche Operationelle 20:32–43

    Google Scholar 

  • Roy B (1990) The outranking approach and the foundations of the ELECTRE methods. In: Bana e Costa (Ed.) Readings in Multiple Criteria Decision Aid. Springer-Verlag, Berlin, pp 155–183

    Google Scholar 

  • Ruch E (1975) The diagram lattice as Structural Principle. Theor Chim Acta 38:167–183

    Article  CAS  Google Scholar 

  • Ruch E, Gutman I (1979) The Branching Extent of Graphs. J of Combinatorics-Inform System Sciences 4:285–295

    Google Scholar 

  • Schröder BSW (2003) Ordered Sets An Introduction. Birkhäuser, Boston

    Google Scholar 

  • Shye S (1985) Multiple Scaling-The theory and application of partial order scalogram analysis. North-Holland, Amsterdam

    Google Scholar 

  • Stanley P (1986) Enumerative Combinatorics. Volume I, Wadsworth & Brooks/Cole, Monterey, California

    Google Scholar 

  • Systat Program http://www.spss.com/software/science/SYSTAT/, SPSS Inc., 2000

    Google Scholar 

  • Trotter WT (1991) Combinatorics and Partially Ordered Sets Dimension Theory; John Hopkins Series in the Mathematical Science. The J Hopkins University Press: Baltimore

    Google Scholar 

  • Urrutia J (1989) Partial Orders and Euclidian Geometry, in Rival, I, Ed. 1989, pp 387–434

    Google Scholar 

  • Voigt K, Welzl G, Brüggemann R (2004) Data analysis of environmental air pollutant monitoring systems in Europe. Environmetrics 15:577–596

    Article  CAS  Google Scholar 

  • Wille R (1987) Bedeutungen von Begriffsverbänden: Beiträge zur Begriffsanalyse (Hrsg.: Ganter B, Wille R, Wolff KE). BI Wissenschaftsverlag Mannheim pp 161–211

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department: Ecohydrology, Leibniz-Institute of Freshwater Ecology and Inland Fisheries, Germany

    Rainer Brüggemann

  2. Awareness Center, Roskilde, Denmark

    Lars Carlsen

Authors
  1. Rainer Brüggemann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lars Carlsen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. Ecohydrology, Leibniz Institute of Freshwater Ecology and Inland Fisheries, Müggelseedamm 310, 12587, Berlin-Friedrichshagen, Germany

    Rainer Brüggemann

  2. Awareness-Center, Hyldeholm 4, 4000, Roskilde-Veddelev, Denmark

    Lars Carlsen

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Brüggemann, R., Carlsen, L. (2006). Introduction to partial order theory exemplified by the Evaluation of Sampling Sites. In: Brüggemann, R., Carlsen, L. (eds) Partial Order in Environmental Sciences and Chemistry. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-33970-1_4

Download citation

Publish with us

Policies and ethics

Search

Navigation