Ranking of pesticides on the basis of multi indicator systems. How can knowledge of stakeholders assist the ranking process.

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

29. Example continued

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