![Idea
• Let Wi be the ith weight interval (located within the
span of [0,1])
• U = f(Wi) i = 1,…,m ; m number of indicators (1)
• Equation (1) by far too detailled
• Hence: Introduction of an artificial parameter:
Vr = (realized wi,max – realized wi,min)
for all (realized wi,max – realized wi,min ) 0](https://image.slidesharecdn.com/florincomp26anim-150421103156-conversion-gate01/85/Flor-incomp26-anim-14-320.jpg)
Flor incomp26 anim
- 1. Incomparable, what now ? IV: An (unexpected) modelling challenge R. Bruggemann(1) and L. Carlsen(2) (1): Leibniz-Institute of Freshwater Ecology and Inland Fisheries, Berlin, Germany (2): Awareness Center, Roskilde, Denmark Flor_incomp25_anim.ppt 27.2.2015 – 4.4.2015
- 2. Ranking I • Most often there is no measure for the ranking aim • Hence a multi-indicatorsystem (MIS) is needed as a proxy for the ranking aim. Examples: Poverty Sustainability Child-well being
- 3. Ranking I (cont‘d) • MCDA-methods, so far a ranking is intended, are constructing a ranking index CI. • The order due to CI is by construction a weak (or better a linear) order. • Hence, there are no incomparabilities. Everything seems to be on its best way! Really??
- 4. Consequences • By construction a linear order is intended: – No ambiguity – Most often no ties • A metric is available • Modelling of the knowledge of stakeholders/decision makers
- 5. However…. • Conflicts are hidden from the very beginning • There may be robustness problems • Problems to get the needed parameters (for example the weights) of the MCDA-method
- 6. Ranking II • HDT-equation: x y: qi(x) qi(y) for all qi of MIS • The HDT-equation is very strict: Incomparabilities are arising: – Minute numerical differences i: = abs(qi(x) – qi(y)) – No care how many indicator pairs taken from MIS are contributing to x || y – No care, whether or not an incomparability is induced by contextual similar qi of the MIS • Regarding knowledge to eliminate some incomparabilities: How to model within the framework of HDT?
- 7. Our talk here in Florence: 1. An empirical data set (taken from environmental chemistry) 2. Weight intervals 3. Towards a controlling law 4. Results 5. Discussion, Future tasks
- 8. Pesticides in the environment DDT Aldrin (ALD) … and nine other pesticides. How do they affect the environment? There is no single measure. Hence a multi-indicator System: Persistence (Pers), Bioaccumulation (BioA) and Toxicity (Tox) Hexachlorobenzene (HCB)
- 9. Hasse diagram (HD) of 12 Pesticides There are many incomparabilities (such as DDD || HCL), so there is a need for modelling. Pesticides: DDT ALD CHL DDE DDD HCL HCB MEC DIE PCN PCP LIN Characterized by a MIS {Pers, BioA, Tox} U = |{(xi,xj)XxX, xi||xj, with i<j}| = 31
- 10. Modelling by weight intervals 1. Basic paper: Match – Commun. Math. Comput. Chem. 2013, 69, 413-432 2. Idea: Let (x) = wi qi(x) the value of a composite indicator , one of the simplest constructions to get a linear or weak order.
- 11. Modelling by weight intervals (cont‘d) 1. The problem is the selection of weights, which causes – subjectivity and – hides incomparabilities expressing major conflicts in the data 2. Hence: A relaxation by weight-intervals
- 12. U = 5 0 1 Range of weights Persistence Bioaccumulation Toxicity In our example: Many weak orders Concatenation of these orders Algebraically: Intersection of these orders. CI1 CI2 …
- 13. Questions 1. What can be said about stability? Could it happen that another MC – run changes the result? 2. How can we judge the role of weight- intervals? – Effect of lengths of the intervals – Effect of location of the intervals
- 14. Idea • Let Wi be the ith weight interval (located within the span of [0,1]) • U = f(Wi) i = 1,…,m ; m number of indicators (1) • Equation (1) by far too detailled • Hence: Introduction of an artificial parameter: Vr = (realized wi,max – realized wi,min) for all (realized wi,max – realized wi,min ) 0
- 15. Expectation U, incomparability Vr = 0 Certainty about weights, i.e. exactly one weight for each indicator: U = 0 0 < Vr < 1 Some uncertainty Hence intervals with Lengths < 1 0 < U < Umax Vr = 1 Complete weight‘s span. No knowledge. Original poset U = Umax 0
- 16. U = f(Vr) • Having m indicators, m* may be the number of indicators, where Vr 0. Then: • U = f(Vr, m*) • Focus on the length of the intervals • Disregarding the position of the intervals
- 17. Hypothesis • U = Umax* Vr s, s = 1/m*, • Umax = U(Vr= 1), • Vr = 1: all intervals have length 1, maximal uncertainty, original poset (without any weights) • The values U(Vr) may be considered as describing a kind of normal behaviour. • Realistic assumption?
- 18. Results: Pesticides data 0 5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 1.2 Urealmin Urealmax U Vr 0 5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 1.2 U(V) U(V) Umax = 31
- 19. Results/Summary • Modelling stakeholders knowledge within the framework of partial order theory: weight intervals • Need of a function to get an overview U = f(Vr). • A power law seems to be a suitable approach • Other modelling concepts (not shown here): power law seems to describe pretty well U = f(p), i.e. U = Umax *ps, p methods parameter.
- 20. Check of the ideas with another dataset • Polluted sites in South-Westgermany • Four indicators, 59 regions • Power law seems to fail!
- 21. Results: Pollution in a South-Western region of Germany 0 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 0.8 1 1.2 Ucalc Ureal The values obtained by the MC simulations seem to be larger then the values obtained by the power law U = 1386 * V(1/4) The problem: In contrast to the pesticides data, the single indicators induces orders with large equivalence classes. I.e. the degree of degeneracy is high. Umax = 1386
- 22. Pollution data Playing with ideas….. K(qi) describes the degeneracy with respect to indicator qi. K = Ni*(Ni – 1) , Ni = |ith equivalence class| The degree of degeneracy k(qi) related to each indicator: )1( )1( :)( nn NN qk ii i 1/m is valid, only if no degeneracy appears, because then each indicator contributes its own linear order to the poset. Idea: (1/m) eff(ectiv) = f(m, k(qi)) n: number of objects
- 23. Approach: ))(1( 1 : 1 ,..,1 mi i eff qk mm Pesticide data : k(qi) = 0 for all i. No correction needed Pollution data: k(q1) = 0.074 , k(q2) = 0.036, k(q3) = 0.037, k(q4) = 0.016 (1/m)eff = 0.21. UcalcP is calculated following (1/m)eff: 0 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 0.8 1 1.2 Ureal UcalcP U Vr Eq. 2 -100 -50 0 50 100 150 200 250 300 350 400 0 0.2 0.4 0.6 0.8 1 1.2 delta1 delta2 Vr
- 24. Future Work • Explain, why a power law is the correct law! • Find out, whether Vr is a good selection • Find methods for a correct interpretation • Are there other more reasonable controlling parameter? • Is s generally well described as a funktion of 1/m and k(qi)? Could we do it better?
- 25. Thank you for attention!!
- 26. Background
- 27. Remarks 1) „Differential“ view for effect of V for CI‘s 2) Linear orders w.r.t. to the indicators or any CI imply equal distance to the original poset 3) In linear orders the number of 1 in the -matrix equals (n*(n-1)/2) and is independent of the special CI All 1 in for the original poset are realized in all (CI). Therefore the distances of all CI to the original poset are the same, namely (in the case of the pesticides) 31 4) If not a linear but weak orders with nontrivial equivalence classes appear then irregularities appear as in case of the second data set nji otherwise jiif ji ,...,1, 0 1 ,
- 28. Example wi (i=1,2,3 and wi = 1) w3 w2 w1 w*(1) w*(2) w*(3) Three tuples w* selected
- 30. Pollution data: 59 objects , 4 indicators) orig Pb cd zn S CIrelat orig 0 1513 1448 1449 1414 1386 Pb 1513 0 1851 1884 1455 1359 cd 1448 1851 0 1183 1922 742 zn 1449 1884 1183 0 1703 1185 S 1414 1455 1922 1703 0 1534 CIrelat 1386 1359 742 1185 1534 0 Orig (poset) Max distance: 1513, • The attributes Pb, Cd, Zn and S induce weak orders with • Pretty large nontrivial equivalence classes. Degeneracy (K) : CI 0; Pb 254; Cd 124; Zn 126; S 56 • Assumption s = ¼ may not a good one, because some indicators are insufficient in differentiating the order.
- 31. Check of the exponent s in U = Umax*Vs Pollution data 0 200 400 600 800 1000 1200 1400 1600 0 0.2 0.4 0.6 0.8 1 1.2 Ureal Ucalc3 Ucalc4 Ucalc3: s = 0.1 Ucalc4: s = 0.2 Hypothesis: s = f(m*,K) U V
- 32. Discussion • Selection of interval length: – Seems to describe crudely the behavior of U – Position of intervals in dependence of any indicator is considered as „fine tuning“ – There is no need to find a law which exactly meets the points (U,Vr). • Main effect at the very beginning of the Vr-scale: Vr = 0 Vr = – Sensitivity? – Distances (orders due to single indicators)
- 33. V 0 •V = 0 means, we start from a weak order , •V = means, there is an influence from other indicators Orig (partial order) Pers Tox BioA 58 16 50 31 31 31 46 16 12 31 A composite indicator due to equal weights
- 34. The behaviour U = f(Vr) could be as follows: Range a) due to mc-runs and b) how different weight intervals are associated with the indicators Vr U Umax U = 0 Vr = 0 Vr = 1
- 35. Constructed test data set 7 objects, three indicators. By construction a high degree of degeneracy: K(q1) = 8, K(q2) = 8, K(q3) = 14. i.e. k(q1) = 0.19, k(q2) = 0.19, k(q3) = 0.333 0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 1.2 U Ucalc Ucalcwithoutdeg Calculated individually Calculated, by eq. 2 Calculated Without regarding degeneracy
- 36. Balance equation a, b d, e, f c V = |{(c,a),(c,b),(c,d),(c,e),(c,f),(a,b),(b,a),(d,e),(e,d),(d,f),(f,d),(e,f),(f,e)}| U = |{(a,d),(a,e),(a,f),(b,d),(b,e),(b,f)}| K = = |{(a,b),(b,a)} {(d,e),(e,d),(d,f),(f,d),(e,f),(f,,e)}| 2*U + 2*V – K = n*(n-1)
- 37. Discussion • Assume U = f(p) can be established, then • The control is concerned with stability, not… • …at which values of p the „best“ partial order will be found (see with respect to fuzzy modelling: Annoni et al., 2008, De Baets et al., 2011) • The distance graph will help us to decide the effect of mixing the indicators due to their weight intervals • As to how far Vinput can be seen as a better leading quantity is a task for the future