Classifications

• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
  • G06N5/00—Computing arrangements using knowledge-based models
  • G06N5/04—Inference or reasoning models
  • G06N5/048—Fuzzy inferencing

Definitions

• the present invention relates to a decision-making process in the absence of clearly identifiable rules.
• the rule-based approach is very widely used in many expert systems. It allows experts to enter their knowledge as naturally as possible in the form of “business rules”. The rule-based approach allows the expert to provide his expertise directly in an explicit and perfectly clear form.
• Decision trees are widely used to model decision-making among a finite set of alternatives (“alternative” signifying in the present description one of the possibilities offered by a choice). Their big interest is to be perfectly understandable by an expert.
• a decision tree can be represented as a set of rules. The difficulty is to take into account the imprecisions and uncertainties of the expert's knowledge in these decision trees. The imprecisions and uncertainties are conventionally modeled thanks to the use of fuzzy logic. If we consider the following rule "If Ri ⁇ i and R 2 ⁇ 2 then zeC", then this amounts to saying that Ri is greater than or equal to ai, and likewise for R 2 with ⁇ 2 . Imprecisions and uncertainties are not the only phenomena that deserve to be modeled.
• An object of the invention is to be able to model the compensation phenomena.
• the difficulty is to take into account in the decision trees the imprecisions and the uncertainties, as well as the compensatory phenomena.
• the inclusion of uncertainties and inaccuracies in standard rules is conventionally done using fuzzy logic. This amounts to saying that the condition Ri greater than or equal to ai can be more or less verified (with a certain degree), and the same for R 2 and ⁇ 2 .
• a fuzzy set ⁇ which is nothing other than a function which, with a value of Ri associates a degree between 0 and 1.
• the compensatory fuzzy rule that we have described with two variables in the premises is generalized to any number of variables. This gives: “If F (V ⁇ (R ⁇ ) N 2 (R 2 ), ... Nn (R n )) is large, so zeC ".
• the number F (V ⁇ (R 1 ), ... N ⁇ (R n )) corresponds to the degree of compensation between 0 and 1. It describes with what degree the compensation takes place and therefore with what degree the rule must be triggered.
• N n (R n )) refers to the notion of bounded unipolar scale modeling a notion whose opposite does not exist and whose degree admits a maximum value, as is the case for example for satisfaction. The degree is therefore typically modeled in a scale [0,1].
• the concept of compensation is based on the concept of bipolar scale (modeling a concept and its opposite, such as, for example, attractiveness and repulsion) since, in any phenomenon of compensation, there are necessarily positive aspects which compensate for negative aspects.
• the utility functions Vi (R) must therefore correspond to such scales.
• the function F has as argument values belonging to a bipolar scale (the utilities V ⁇ (Rj)) and returns a value belonging to a bounded unipolar scale (the degree of compensation). Consequently, there must therefore exist inside F a function T making it possible to pass from a bipolar scale to a bounded unipolar scale.
• H is a function aggregation such as those used in multi-criteria decision support.
• the H function models the compensation.
• H (u ⁇ , ..., u n ) is between the smallest value among the Uj and the largest value among the Uj.
• good aspects compensate for bad ones.
• Good aspects are variables such that Vj (R ⁇ ) is large while bad aspects are variables such that Vj (Rj) is large.
• Rj ⁇ oci for the good variables
• RJ ⁇ OCJ for the bad variables.
• fuzzy decision tree exists (cf. JM Adamo "Fuzzy decision trees”, Fuzzy Sets & Systems, Vol. 4, pp. 207-219, 1980).
• the determination of fuzzy decision trees is generally done by learning techniques. These techniques do not make it possible to introduce vagueness into an already existing decision tree. In addition, they do not deal with compensatory phenomena.
• a standard fuzzy rule of the type "If U ⁇ (x) and U ⁇ (y) are large, then zeC”, the fuzzy sets U ⁇ and U ⁇ are directly palpable for an expert, so that he will be able to explicitly determine. This is no longer directly the case in the compensatory rules, since the Vi are perceived only through the function F. It is therefore very difficult, if not impossible, for an expert to directly supply the values of as well as of F. C This is why there is not really a method to do this.
• fuzzy logic There are a number of relatively conventional methods in fuzzy logic which indirectly make it possible to model compensatory phenomena.
• Each combination of ki k n provides a priori value different from C.
• the fuzzy rule approach using an aggregation function F consists in describing the compensation globally using a mathematical function, while this approach consists in describing the point-to-point compensation (i.e. for any tuple k ⁇ , ..., k n ).
• the big disadvantage of this approach is “the combinatorial explosion” since it is necessary to explain a rule for all the possible combinations of k ⁇ , mecanic., K n .
• conjunctive rules have an interpretation which does not correspond to an implication between the conditions in the premise and the conclusion, but just to the observation of something that has happened (cf. D. Dubois & H.
• the decision-making method is a method according to which decision-making rules are established comprising at least two variables for each of which at least one limit is not strict, and it is characterized in that we formally introduce a compensation condition into the rules that are not clearly identifiable, that we determine for each parameter of a compensatory condition at least one particular point belonging to a compensation boundary and linked to the parameter, that we deduce the value of parameters, apply all the rules and deduce the decision. Note that the fact that a limit is not strict means that the conditions on the corresponding thresholds can be violated.
• the compensation is binary in nature, and there is only one compensation frontier.
• the conditions in the premises are blurred by the expert, the compensation can be more or less verified, there are two compensation boundaries, the application of the rules makes it possible to calculate a degree of possibility on the set of possible alternatives, and we must interpret the final possibility distributions to deduce the decision.
• the compensation condition is written as the aggregation by a sum, which is advantageously a simple unweighted sum, of utility functions on each variable, the utility functions are refined by pieces, an expert provides the abscissa of the points delimiting the affine parts, and the parameters of the compensation condition are the ordinates of these points.
• the expert provides in relative values with respect to the extreme values the ordinates of the utility functions for all points delimiting the affine parts except the two extreme points and the threshold, the utility at the threshold is null and the parameters of the compensation condition are the ordinates of the utility functions for the extreme points.
• the utility at the threshold is zero and the parameters of the compensation condition are the ordinates of the utility functions for all points delimiting the affine parts, except the threshold.
• the particular points are such that all of their coordinates according to the variables except one are equal to one of the values delimiting the affine parts of the utility functions, we ask the expert to provide the value according to the unfixed coordinate so that the particular point is exactly on a compensation boundary, we determine a characteristic point for any variable and any value delimiting the parts affines of the utility function on this variable such that the coordinate of the characteristic point following the variable is equal to the value and such that the ordinate of this value is a parameter (that is to say is unknown), the relations which we have on the characteristic points leading to a system of equations of which the unknowns are the parameters, and we solve this system with a classical method.
• the expert determines for each variable the type of compensation to which it belongs, which provides a set of equations and inequalities to which we add the equations from the characteristic points, and we solve this system with a classical method.
• all the variables correspond to a compensation of the type for which, for each variable Ri, there is a value of Ri beyond or below which no more compensation is possible whatever the value according to the other variables, which the expert provides in relative values with respect to the extreme values the ordinates of the utility functions for all points delimiting the affine parts except the two extreme points and the threshold, that the utility at the threshold is null, that the parameters of the compensation condition are the ordinates of the utility functions for the extreme points, that the conditions in the premises are blurred by the expert, that the compensation can be more or less verified, than the points characteristics are such that the component according to a well satisfied variable corresponds to the maximum value according to this variable, that the component according to a badly satisfied variable is free, that one asks the expert to provide the value according to the free coordinate (not fixed) so that the particular point is located exactly on a compensation frontier and that all other components are set at the thresholds.
• the rule base corresponds to a decision tree.
• the rule base corresponds to a decision tree, and a single alternative cannot be perfectly possible in the distribution of final possibilities (this is the hypothesis H described below).
• the pairs of complementary conditions are highlighted in the decision tree, including the compensation conditions, the complementary conditions are treated at the same time by separating the core from their fuzzy set by a very small number.
• FIG. 1 is an example of a simplified decision tree used to explain the invention
• FIG. 2 is a diagram drawn in the plane of the two variables of the tree of FIG. 1
• FIG. 3 is a diagram repeating that of FIG. 2, into which the compensations in accordance with the invention have been introduced,
• FIGS. 4 to 6 are diagrams of fuzzy functions used by the invention.
• FIGS. 7 to 13 are diagrams in the plane of the two variables of a decision tree, into which various compensations are introduced, in accordance with the invention, and FIGS. 14 to 16 are diagrams of utility functions implemented by the invention.
• This compensation term corresponds to the complement of the initial compensation term.
• P2 Introduction of fuzziness.
• the conditions in the premises of the decision tree form a partition of the space of variables. The fact that we obtain a partition implies that each time that l 'we find a condition, we necessarily find its complement somewhere in the same decision tree. Two conditions are complementary if, whatever the value of the variables, one and only one condition among these two conditions is perfectly true. This corresponds to an hypothesis H described later. In order to introduce vagueness, the invention proposes to jointly treat each pair of complementary conditions so that hypothesis H is satisfied.
• the operators of conjunction and disjunction are transformed respectively into minimum and maximum.
• phases P2 and P3 are sufficient. It is also quite possible to introduce compensation without blurring in a decision tree. Phases P1 and P4 are then sufficient. We can indeed use the P4 process to specify the compensation found in a non-fuzzy decision tree.
• Characteristics 10 to 13 of the invention relate to the explanation of compensation in a decision tree. Characteristics 11 to 13 of the invention are more particularly concerned with a decision tree into which one wishes to introduce blurring while satisfying hypothesis H.
• Characteristic 12 of the invention corresponds to step P2 and consists of particular to treat couples of complementary conditions at the same time.
• Characteristic 13 of the invention consists in using steps P1 to P4 to introduce blurring and compensation in a decision tree.
• step P4 alone describes a process which can be considered separately.
• P4 provides a process for explaining a compensatory phenomenon in a fuzzy rule.
• P4 taken separately can therefore be used to specify compensation in a condition of the compensation type found in the premise of a fuzzy rule.
• the characteristics 1 to 9 of the invention relate to the explanation of the compensation in a single rule, that is to say the step P4 taken alone.
• the originality of the process of the invention relates firstly to the process corresponding to phase P4 (Specification of the compensatory fuzzy conditions), and secondly to the fact of introducing the fuzzy in a certain way (in considering the pairs of complementary conditions) so as to ensure that the result (the chosen alternative) satisfies certain properties.
• the fuzzy compensation is characterized by three zones: one in which one compensates perfectly, another in which one does not compensate at all, and in the middle an zone in which one compensates a little with a degree compensation between 0 and 1.
• the expert will not be able to determine the degree to which compensation is allowed for this point.
• the points for which the most information is deduced belong to the border between two zones. Indeed, it is easy to see that each border is characterized as a certain level curve of H (V (R 1 ) N 2 (R 2 ), ...
• N n (Rn) that is to say say the set of values of variables Ri R n for which H (V 1 (R ⁇ ) N 2 (R 2 ), ... N n (Rn)) is equal to a certain value, whereas the fact of belonging to a zone gives just an inequality on H (V ⁇ (R 1 ), ... N n (R n )) - which is less informative.
• the process therefore consists in asking the expert to specify a certain number of points lying on the border between the zones “we compensate perfectly” and “we compensate a little”, and on the border between the zones “we do not compensate at all "and” we compensate a little ". Knowledge of these points will allow the parameters of compensation to be determined.
• V the utility function defined on the variable Ri
• V we characterize V by a set of discrete points of R ⁇ . This set is noted%. It then suffices to determine the value Vj (Xi) of the utility function at each point Xi of this set to entirely specify Vi. For example, V can be refined between these points.
• the method of the invention consists in questioning the expert on a set of singular points of compensation belonging to the border between two zones. These singular points are points of which all the coordinates except one belong to the sets%. We then ask the expert for what value of the last variable the singular point is located exactly on the border between two areas.
• Such a singular point noted R '(Rj) is such that for all k ⁇ i its component on the variable k belongs to the set%. and its component on the variable i is worth Rj.
• the values of R '(Rj) according to the components other than i are such that when R, varies, the point R' (Ri) necessarily crosses one of the two borders.
• a point R '(Rj) is sought, satisfying the preceding hypotheses, such that its component according to the variable j is worth Xj.
• the values of the components of R '(R ⁇ ) outside of i and j can be fixed at values whose utilities are already known, while ensuring that R' (R ⁇ ) necessarily cuts one of the two boundaries for a value of Ri.
• the present invention will be described below with reference to an example of a decision tree. It is simple, but representative of the phenomena that can occur.
• the example is a passing exam.
• the exam consists of two tests. The results following the two tests are noted Ri and R 2 .
• Ri and R 2 the results following the two tests are noted Ri and R 2 .
• the decision tree shown in Figure 1 indicates the student's situation based on their results. This decision tree stipulates that Ri and R 2 must be greater than or equal to 10. If these two conditions are satisfied at the same time, the student is accepted. If not, we look to see if these two results at the same time are greater than or equal to 8. If so, the student is caught up, if not, he is refused.
• Ri is more important than the result R 2 .
• R 2 that is to say R 2 ⁇ 8
• Ri is more important than the result R 2 .
• FIG. 3 shows these two ways of doing things.
• the compensation here extends the possible values leading to the alternative "Catch-up". It is easier to explain first the alternative for which the domain is restricted. This is the "Refused” alternative. Without compensation, the condition for granting the alternative “Refused” is: “Refused” if R ⁇ ⁇ 8 OR R 2 ⁇ 8 We have just restricted the condition "R 2 ⁇ 8". We therefore add the compensation condition to the condition "R 2 ⁇ 8", as before: “Refused” if R ⁇ ⁇ 8 OR (R 2 ⁇ 8 AND Ri does not compensate R 2 ) The compensation condition here is "Ri does not compensate for R 2 ”. The term “compensation" has a positive connotation. The fact of being able to compensate must therefore lead to a positive conclusion.
• ⁇ should be less than the numerical precision on the variable Ri.
• R_i we mean the vector R deprived of its i the e component, that is to say (Ri, ..., Ru, Rj + ⁇ , ..., R n ).
• U (R) H (V ⁇ (R ⁇ ), ..., V ⁇ (R ⁇ )), where H is an aggregation function.
• the invention plans to construct U (R ⁇ > 10) as a piecewise affine function defined by a few points between R ⁇ ⁇ f and 10.
• the value of U (R ⁇ > 10) for these points is determined using a method from the theory of measurement and allowing the construction of a difference scale. Such a scale is given to a translation and a homothety close.
• the vector (if s n ) belongs to the level curve 0 of U since
• the compensation terms are combined via min and max operators with other conditions.
• an unbounded behavior that is to say with a level curve going to infinity
• the compensation can be bounded by other conditions.
• the utility function Vi is assumed to be monotonous, that is to say either increasing or decreasing.
• ⁇ i the sign of the derivative of Vi.
• ⁇ j -1 if Vi is decreasing.
• the compensation relates to the alternative C m eC.
• the compensation introduced is of the type: Ri compensates R 2
• T ⁇ ie ⁇ 1, ..., n ⁇ / 3 (A + , A _ ) el such that i A- ⁇
• T (i) ⁇ je l " / 3 (A + , A ⁇ ) el such as ieA ⁇ and jeA + ⁇
• l ⁇ (i) ⁇ jel + / 3 (A + , A ⁇ ) el such as ieA- etjeA + ⁇ .
• R1 There is a value of Ri beyond (if below (if ⁇ r-1) of which no more compensation is possible whatever the value according to the other variables.
• R2 Whatever the value of R; (even very bad), we compensate perfectly for sufficiently good values of the other variables.
• 9 ⁇ the set of i belonging to case R1.
• 3t 2 the set of i belonging to the case R2.
• 9Î 3 the set of i belonging to the case R3.
• Vi, * Vj (Rj, *).
• Vj, ** satisfies the relation V- ,
• the value Vi * satisfies the relation Vj, * + ⁇ jeA + V j * ⁇ -1 for all (A + , A ⁇ ) l such that ieA ⁇ (one of these inequalities corresponding to an equality), so that Ri, * has a challenge Clear and precise definition referring to the case of compensation situations defined by the expert.
• V (A + , A " ) e I such that ie A " , we have Vj, * + ⁇ jeA + Vj * ⁇ 0
• C1 means that we are in case 1 (that is, we determine the utility functions only at the ends) while R1 means that all the variables are supposed to belong to the framework R1 compensation.
• the method of the invention is composed of the following steps: C1 -R1-1 - Definition of reference thresholds si. .... s n :
• the reference thresholds Si s n correspond to the level which makes it possible to compensate perfectly, but just, on the variables Ri, ..., R n .
• the expert answers the following question:
• R k * for ke l ⁇ :
• V k corresponds to a ratio scale.
• ⁇ k (R) a method from the measurement theory is used. The invention provides for the MACBETH methodology (cf. C. Bana e
• Vj * - ⁇ Rij 0 ) x ⁇ (A + , A - ) el / ieA _ - (1 + ⁇ jeA + nD + Vj * ) /
• V k , * -1 - ⁇ j ⁇ k V j *
• U (R k , j _ (Rk, j "1 )) V k (R k ) + Vj * , we have: R, j ⁇ 1 is interpreted as follows:
• R j R j . If the expert does not agree with the value of R k j ⁇ 1 , we impose - (1+ Vj * ) / V k
• Vector of variables R k , j ⁇ for keA ⁇ ⁇ i ⁇ and jeA + . According to the same explanations concerning the particular points, there exists a value R k , j ° of the variable R k such that U (R k) j ⁇ (R k , j °)) 0.
• Ry 0 is interpreted as follows:
• Vj k Vj (Rj k ) for ke ⁇ 1, ..., pj ⁇ , as shown in Figure 15.
• Vi k Vi (Ri k ) for ke ⁇ -p '-1 ⁇ .
• V ⁇ (R, ⁇ - 1 ) + ⁇ JsA + V J * l -1
• Ri K * '° and Ri * 1 the remarkable points of the vector of variables Ri K *" defined previously from K * .
• V q * ] ⁇ > Vi k e [Vi , *, Vi , * + V j * ] So if k is large, then we expect V k to be close to Vi, *. In this case, the values k q which will allow the level curve -1 to be crossed will be equal to p q +1 or in any case large.
• the process of the invention is composed of the following steps:
• C2-R1 -1 - Definition of reference thresholds if s n This step is strictly identical to step C1-R1-1.
• C2-R1 -2 - Definition of Ri * values for ie T This step is strictly identical to step C1-R1 -2.
• C2-R1 -3 - Definition of R k * values for ke l ⁇ This step is strictly identical to step C1-R1-2.
• Equation (3- [C2-R1]) therefore provides a relation satisfied by Vj k for all I l + (i) ⁇ D + and all ke ⁇ 1 Pj + 1 ⁇ .
• the indices j in l + (i) ⁇ Dj + (where Di + is the value of D + when step C2-R1-6 is at the level of the reference index i) for a reference index i will all be preferred to the indices l + (i ') ⁇ D- * for another reference index i' if the expert has given i before i '.
• the union of all the sets l + (i) ⁇ Di + for all the reference indices i is equal to f. It therefore suffices now to explain how to order the different indices among l + (i) ⁇ Dj + for a fixed i.
• Rj described [R, * , Sj] intersects one of the level curves 0 or -1.
• C1-R * -2 • »Definition of Ri * values for je l * This step is strictly identical to step C1-R1-2.
• Determination of V for ie% ⁇ The steps whose number ends with a hook [1] are specific to case R1 (ie9î ⁇ ). They will be declined according to cases R2 and R3 respectively with the brackets [2] and [3]. The steps without a hook are generic and are not repeated in cases R2 and R3.
• C1 -R * -3H1 - Definition of Rk * values for ke l ⁇ This step is strictly identical to step C1-R1-2.
• C1 -R - Determination of the utility function V k between Rk * and Sk for ke l ⁇ This step is strictly identical to step C1-R1-4.
• C1-R * -6m - Determination of the reference ie9î ⁇ ⁇ D The questionnaire will be based on a particular variable among 9l ⁇ ⁇ D ⁇ . This step is identical to step C1-R1-6.
• step C1 -R1 -7m Determination of Vi * and V * for all ie. * ⁇ ) ⁇ D *: Let I l + (i) ⁇ D + . This step is identical to step C1 -R1-7. Thanks to question Q5 [C1-R1], we determine Ry 0 . As in step C1-R1-7, we have:
• V ,, * ⁇ (A + , A _ ) e , / isA _ - (1 + ⁇ jsA + nD + Vj * ) / (1 - ⁇ jeA + ⁇ i (Ri, j ° ))
• Vj * - ⁇ i (Rij °) X ⁇ ( A +
• Vs for ie9l 2 are determined as follows.
• the expert is asked the pair (A + , A ⁇ ) for which the previous inequality is satisfied.
• ⁇ j ⁇ (R ⁇ 0 ) if V
• ,. + Vj ⁇ -1, and ⁇ iJ 1 / ⁇ j (R j °) if Vj * + V j * > -1.
• Vj * - T y - ⁇ y x ( ⁇ + ⁇ keA + ⁇ D + T ik - ⁇ kSA + n D + V k * ) /
• V kj * K - ⁇ jeA + V j * We then proceed to the determination of V, for ie9î 3 :
• Vj * ⁇ ( ⁇ jsA + ⁇ D + Tjj - ⁇ jeA + ⁇ D + Vj) / (1 - ⁇ j s A + ⁇ D + Xij) and:
• Vj * ⁇ (A +, A-) el / ieA- ( ⁇ jeA + ⁇ D + Tjj - ⁇ j ⁇ A + nD + Vj) / (1 - ⁇ j s A + ⁇ D + tjj)
• V, * ⁇ (A +, A-) sl ieA- ( ⁇ jsA + ⁇ D + Tjj - ⁇ jsA + nD + Vj) / (1 - ⁇ jeA + ⁇ D + Tjj)
• Vj * + ⁇ J S A + VJ * 0.
• Vj - Ty -Tjj X ⁇ (A +, A-) sl / isA- ( ⁇ jsA + ⁇ D + Tjj - ⁇ j s A + nD + Vj) / (1 - ⁇ jeA + ⁇ D + Xij)
• * ⁇ (A +, A-) el / keA- ⁇ ⁇ jsA + Vj
• C1-R 0 - Determination of V j * for a te l * This step is strictly identical to step C1-R1-9.
• r M min ⁇ r ( Vj * , ⁇ ), v ⁇ e [V], v]], ⁇ e [ ⁇ , ⁇ ] ⁇ or:
• indices ⁇ k q ⁇ q ⁇ A + ⁇ j considered are those maximizing r (if we wish to solve the exact problem) or r M (if we are content with the approximate problem).

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