Probability approach applied for
of the phenomenon of concern and (2) how to deal with the uncer-tainty associated with the necessary input parameters (descriptors) for the ranking procedure. The topic of this paper will be the design and application of a ranking procedure, leaving the uncertainty as-pect with resas-pect to the input data for other investigations. In the area of environmental management the problems will typically be rather complex where multiple factors are going to be taken into account simultaneously. So, dealing with higher complexity more than one criterion is often needed to provide a sufficient valid description and this paper deals with this type of multiple criteria ranking. The con-ventional ways for ranking of objects is to use a model, which can yield a single ranking number for each object. The outcome of the model is then an exact rank of each object in relation to all the other objects. Such a model for exact ranking can be more or less compli-cated. If the model is judged to be sufficiently valid then this method will solve the ranking problem. However, in some cases dealing with high complexity it can be difficult to suggest a model of sufficiently known validity. One could be tempted to use a more or less doubtful scoring system and add the scoring of each descriptor to form a sin-gle number. The actual validity of such a system can easily be too unclear and difficult to quantify on a scientific basis.
The problem of multi criteria ranking is illustrated in a simple exam-ple in Figure 1 where two criteria are applied in terms of two scriptor values for each object to be ranked. The four objects are de-noted X1-4 and the descriptor values are shown in the table in Fig 1. In the same table two different models for exact ranking are shown each in one column and based on respectively addition and multiplication of the two descriptors and the resulting values are shown in the table.
The resulting ranking from these two models are different because X1>X2 using addition while X1<X2 using multiplication. As an alter-native, a partial order can be applied where no model is applied to form an exact ranking. In the partial order a pair of objects is only ranked when no contradiction exists among the ranking of the single descriptors (Davey and Priestley, 1990). So, the ranking between X3 and X1 is included in the partial order ranking because both 7>2 and 7>5 is true yielding the rank X3>X1. Contrary, the ranking between X1 and X2 is not included in the set of rankings because 2<3 and 7>5, which makes the ordering of the objects X1 and X2 indefinite. The partial order is graphically mapped in the Hasse diagram, where connecting lines are drawn between objects between which there ex-ist determined orders (Hasse, 1952).
The indefiniteness in the ranking using partial order forms the basis of ranking probability. This is illustrated in Figure 1, where three possible exact rankings are seen to exist all being in agreement with the partial order (Hasse diagram). Each of these exact rankings is de-noted a linear extension and any exact ranking model using all the descriptors will make a rank similar to one of the linear extensions. In this way the exact ranking model using addition is seen to reproduce one of the linear extensions while the model based on multiplication identifies another linear extension. A third linear extension is seen to exist, which is identified neither by the simple addition nor by the simple multiplication of the descriptors but other exact ranking
mod-Figure 1. A simple example for illustration of the relationship between the partial order and models for exact ranking, here in terms of addition and multiplication of the descriptors.
In this paper a situation is considered where no model for exact ranking can be identified and where no ranking between two objects a priori can be claimed to be more relevant (important) than another ranking. So, in this way in Figure 1 none of the rankings X3>X2, X3>X1
nor X3>X4 will be considered a priori as more relevant than the others.
But, the fact that: X3>X2 and X2>X4 in the partial order yields a higher resulting weight to the ranking of X3>X4 compared to the ranking X3>X2. Under these circumstances, none of the linear extensions can be said to be most likely and random selections of linear extensions can thus create a probability space for a specific object to be ranked at a specific level. In this way it can be seen for object X2 that the most likely rank is at level 3 (next highest) as this is the rank in two out of 3 linear extensions. So, more precisely the object X2 is ranked at level 3 by a probability of 2/3 and ranked at level 2 by a probability of 1/3.
This principle is described by e.g. Winkler, 1982, and Trotter, 1992.
A more general illustration is seen in Figure 2 for at series of N ob-jects where each has I different descriptors. In such cases the partial order theory can be a better alternative for solving the ranking prob-lem as illustrated in Figure 1 (Lerche et al., 2002). But, there is a price Objects Descriptors Addition Multiplication
X1 2 7 9 14
X2 3 5 8 15
X3 7 7 14 49
X4 3 4 7 12
X3
X2 X1
X4
X3
X2
X1
X4
X3
X1
X2
X4
X3
X4
X2
X1
All possible exact rankings (Linear extension) The partial order
(Hasse diagram)
applying the partial order where the total ranking level for an object comes out in form of a probability distribution for a series of possible ranking levels as illustrated in Figure 2. The ranking probability makes the result more fuzzy and thus less useful compared to an ex-act rank. However, the probability distribution yields a direct meas-ure for the uncertainty related to the missing quantified interrelation between the different descriptors.
The ranking probability formed by the partial ordering yields a prob-ability distribution for all possible ranking values of each object. The probability distribution thus represents in Bayesian terms the a priori knowledge related to the ordering using the descriptors alone with-out knowing a valid model for exact ranking. The use of the ranking probability may also be regarded as a maximum entropy (maximal variability) estimate of the ranking variability purely as a result of the selected descriptors where no exact ranking model is assumed. When it is possible to make a useful conclusion based on this ranking prob-ability the validity is relatively strong because the uncertainty related to any assumed inter-relationship between the different descriptors is avoided.
The concept of entropy in partial order theory has been discussed by several references (Dhar, 1978, Dhar, 1980 and Brightwell et al., 1996).
The findings in these references lead to the consideration of partial orders as a real gas, where the phase transition is considered as a transition from an unordered state having a higher degree of indefi-niteness in the ranking towards partial orders having a higher degree of determined rankings. The phase transition of partial order is now an item of intensive mathematical research (Proemel, personal com-munication).
For discrete systems an equation for entropy (E) for each object can be defined (Berger, 1985) as
∑
=⋅
−
= n
i
i
i p
p E
1
)
log( (1)
where n is the number of possible alternatives and pi is the probabil-ity for alternative i to be true. The alternatives in our context are the different rankings so n is equal to the number of objects. If no order-ing is realised in a partial order then the probability for a given object to be placed at a given position in the linear extension is
n
1 and Eq. 1 becomes:
( )
nn E n
n
i
no 1 log
1 log
1
inf =
⋅
−
=
∑
=
(2)
This relation sets up the upper limit of entropy equivalent to “no in-formation about ranking”. The lowest possible entropy value is zero and comes out when an exact rank exists for an object as the p value in this case will be unity for the true rank and all other p values will be zero. So the interval of E is closed between zero and log(n).
Figure 2. The principle of ranking using either a ranking model or a partial order.
In recent years new methods have been developed to generate rank-ing probabilities for prioritisation of up to a few thousand objects depending on the computer power available and this opens up for a wide use of the method of partial ordering (Sørensen et al., 2001 and Lerche and Sørensen, 2003). The concept of ranking probability is the fundament in the paradigm in this paper and we will introduce the use of this concept for practical purposes in cases where the ordering can be formulated as an event space and used for a high number of objects in GIS.
Descriptor Object
x⋅ ,1 x⋅ ,2 - - x⋅ ,i - - x⋅ ,I
X1 x1,1 x1,2 - - x1,i - - x1,I X2 x2,1 x2,2 - - x2,i - - x2,I
X3 x3,1 x3,2 - - x3,i - - x3,I
- - - - - - - -
-- - - -
-Xn xn,1 xn,2 - - xn,i - - xn,I
- - -
-- - -
-XN xN,1 xN,2 - - xN,i - - xN,I
Exact ranking model Calculation of a unique ranking number as a function of the descriptors
Rank of Xn
Partial order
Calculation of a ranking probability as a function of the descriptors
Rank of Xn
Probability Two possible ways of
ranking Additional information
about the inter-relations between the descriptors.
lll llm
llh
lml lmm
lmh
lhl lhm
lhh
mll mlm
mlh
mml mmm
mmh
mhl mhm
mhh
hll hlm
hlh
hml hmm
hmh
hhl hhm
hhh