EP3274929A1

EP3274929A1 - Method of explaining a score

Info

Publication number: EP3274929A1
Application number: EP16714822.0A
Authority: EP
Inventors: Hélia Pouyllau; Christophe Labreuche; Bénédicte GOUJON
Original assignee: Thales SA
Current assignee: Thales SA
Priority date: 2015-03-26
Filing date: 2016-03-25
Publication date: 2018-01-31
Also published as: US20180060746A1; SG11201707927RA; IL254658B; US11195110B2; CA2980875A1; FR3034230A1; CN107660288A; CN107660288B; WO2016151145A1; IL254658A0

Abstract

There is proposed a method of explaining a score comprising at least steps of : a1) providing a first score associated with a first vector containing first values of the parameters, b) generating a first set of lists, each list comprising a second number of indicators, c) generating, on the basis of a list, at least one third vector in which each parameter exhibits a third value, the third value being equal to the corresponding first value when the list does not comprise an indicator of the corresponding parameter, and different from the corresponding first value in the converse case, d) calculating the score of at least one third vector, e) evaluating, on the basis of the scores of the third vectors, an indicator of importance of each parameter, f) devising an explanation of the first score.

Description

Method of explaining a score

The present invention relates to a method of explaining a score associated with a vector by a function. The invention also relates to an associated computer program product and to an information carrier on which is encoded such a computer program product. The invention also relates to a system for explaining a score associated with a vector by a function.

In many areas, decision support systems are used to evaluate choices available to decision makers. For example, a banker evaluates a client's repayment capacity based on information such as monthly salary, the existence of previous loans, or the capital contributed by the client. These data are grouped together by the decision support system, which calculates a score from these data. The banker then decides, based on the score obtained by the client, whether to grant him a loan and, if so, at what rate.

The decision support system uses, to calculate the score, a mathematical model drawn from the expert knowledge of the field of application concerned. Thus, such a decision support system makes it possible to quickly make a decision that can be complex, especially in the case where each alternative comprises a large number of parameters. Indeed, instead of having to consider all the parameters, some of which may be favorable and others unfavorable, the decision maker has to consider only the calculated score. In addition, such mathematical models are usually developed from the expertise of several experts.

However, the mathematical model used may not apply in the same way to all situations and its complexity may make the overall score difficult to understand for the decision maker. This is all the more true as the number of parameters considered is high. It may therefore be important for the decision-maker to know which parameters played the most important role in calculating the score, so that the decision can be adapted accordingly.

There are therefore methods for generating an explanation of a score. However, the implementation of such methods involves a large number of calculations, and the time required to develop the explanation may be too high for some application areas. If the calculation time is reduced, the process loses in precision.

In addition, the explanation generated is not always relevant.

In addition, such methods are generally related to the decision support system used: the explanations are generated from the mathematical model used. It is therefore not easy to adapt existing processes to systems using other models mathematics. Thus, the existing methods of explanation assume the knowledge of the mathematical model with which it will be used.

There is therefore a need for a method of explaining a score, obtained with the aid of a known or unknown function, presenting better performances, particularly in terms of the relevance of the explanation.

For this purpose, there is provided a method of explaining a score associated with a vector of a first number of parameters by a function, each parameter being ordered with an indicator varying from one to the first number, the first number being a integer strictly greater than one, the method comprising at least steps of:

a1) providing a first score associated with a first vector containing first values of the parameters,

b) generating a first set of lists, each list comprising a second number of indicators, the second number being an integer strictly less than the first number and strictly greater than zero,

c) generating, from a list, at least a third vector in which each parameter has a third value,

the third value being equal to the first corresponding value when the list does not include an indicator of the corresponding parameter, and different from the first corresponding value in the opposite case,

d) calculating the score of at least one third vector,

e) evaluating, from the scores of the third calculated vectors, an indicator of importance of each parameter,

f) developing, based on the assessed importance indicators, an explanation of the first score.

According to a particular embodiment, the method comprises one or more of the following characteristics, taken separately or in any technically possible combination:

the importance indicators calculated during step e are either values of

Shapley, or Sobol clues;

the first set comprises a total number of lists, the total number satisfying the mathematical inequality:

Nt≤Co * 2 ^m

wherein Co is a predetermined coefficient; step b comprises generating at least part of the lists of the first set according to a probability law relating to either the second number or the indicators contained in the list;

step b comprises generating all the lists whose second number is less than or equal to a second predetermined integer number strictly smaller than the first number;

the method further comprises a step a2 of providing a second score associated with a second vector containing second values of the parameters, and wherein, in step c, the third value is equal to the second value; corresponding when the list includes an indicator of the corresponding parameter;

There is also provided a method of explaining a score associated with a vector by a function, the function having an overall maximum associated with a first extreme vector and an overall minimum associated with a second extreme vector, the method comprising:

g1) implementing, from the first score and the overall maximum, steps b to e of an explanation method as defined above to obtain first indicators of importance,

g2) implementing, from the first score and the overall minimum, steps b to e of an explanation method as defined above to obtain second importance indicators, and

h) generating, from the first and second importance indicators, an explanation of the first score.

There is also provided a computer program product comprising software instructions which, when executed by a computer, implement an explanation method as described above.

There is also provided an information carrier on which is stored a computer program product as described above.

There is also provided a system for explaining a function associating a score with a vector of a first number of parameters, each parameter being ordered with an indicator varying from one to the first number, the first number being a strictly greater integer to one, the system being proper to:

provide a first score associated with a first vector containing first values of the parameters, generating a first set of lists, each list comprising a second number of indicators, the second number being an integer strictly less than the first number and strictly greater than zero,

generating, from a list, at least one third vector in which each parameter has a third value,

calculate the score of at least one third vector,

to evaluate, from the scores calculated for each of the third vectors, an indicator of the importance of each parameter,

- develop, based on the assessed importance indicators, an explanation of the first score.

The features and advantages of the invention will become apparent on reading the description which follows, given solely by way of nonlimiting example, and with reference to the appended drawings, in which:

FIG. 1 is a schematic view of an exemplary system for implementing an explanation method;

FIG. 2 is a flowchart of a first example of implementation of an explanation method;

FIG. 3 is a graphical representation of vectors used and / or generated by the explanation method of FIG. 2;

FIG. 4 is a flow diagram of another example of implementation of an explanation method;

FIG. 5 is a flow chart of another exemplary implementation of an explanation method;

A system 10 and a computer program product 12 are shown in FIG.

The interaction of the computer program product 12 with the system 10 makes it possible to implement a method.

The system 10 is a computer.

More generally, the system 10 is an electronic calculator adapted to manipulate and / or transform data represented as electronic or physical quantities in the registers of the system 10 and / or memories in other similar data corresponding to physical data in the data. memories, registers or other types of display, transmission or storage devices.

The system 10 includes a data processing unit 14 comprising a processor 16, memories 18 and an information carrier reader 20. The system 10 also comprises a keyboard 22 and a display unit 24.

The computer program product 12 comprises a readable information medium 20.

A readable information medium 20 is a medium readable by the system 10, usually by the information processing unit 14. The readable information medium 20 is a medium suitable for storing electronic instructions and capable of being coupled. to a bus of a computer system.

By way of example, the readable information medium 20 is an optical disk, a CD-ROM, a magneto-optical disk, a ROM memory, a RAM memory, an EPROM memory, an EEPROM memory, a magnetic card or a memory card. optical card.

On the readable information medium 20 is stored the program comprising program instructions.

The computer program is loadable on the data processing unit 14 and is adapted to result in the implementation of a method of explaining a score associated with a vector by a function when the computer program is implemented on the processor 16.

The operation of the system 10 interacting with the computer program product 12 is now described with reference to FIG. 2 which illustrates an exemplary implementation of a method of explaining a score associated with a vector by a function .

This process will be explained below using the simplified case, given solely by way of example, of the comparison between two automobiles.

The function is noted U in the following. The U function associates a score with a vector. The function U is, for example, an integral of Choquet. The Choquet integral is an integral defined by the mathematician Gustave Choquet in 1953. The Choquet integral was initially used in the framework of statistical mechanics and potential theory before being applied to the theory of decision in the years

1980. The Choquet integral makes it possible to represent, among other things, complex decision strategies such as veto criteria, preference criteria, synergies between criteria and redundancy between criteria.

Alternatively, the function U is a generalized additive independence model.

Introduced by Fishburn in 1970, the Generated Additive Independence Model builds a function U by addition of utility functions u each relating to a subset of parameters. Generalized Additive Independence models allow the modeling of decisions where the parameters are not independent individually, but where we observe independence between subsets of parameters (not necessarily disjoint). Thus a parameter can be used by several utility functions u when the parameter is present in several subsets. The importance of a parameter in the score obtained is therefore difficult to establish.

Alternatively, the function U is a weighted sum.

Each score is noted Uv, where V is the name of the vector. Thus, for a first vector X, the first associated score is noted Ux, and for a second vector Y, the second associated score is noted Uy.

The score Ux, Uy is a unique value associated with the vector X, Y by the function U. The score Ux, Uy is preferably a real number.

Each vector X, Y is a set of values xi, yi, each value xi, yi being associated with a parameter.

Preferably, each parameter is distinct.

Each vector X, Y contains a first number N-ι of values xi, yi of parameters.

Each value xi, yi is indicated by an index i representative of the position of the value xi, yi in each vector X, Y.

For example, each vector X, Y takes the form of an N1 -uplet of values xi, yi. Each vector X, Y comprises a single value xi, yi of each parameter.

Preferably, each vector X, Y takes the form of an N1 -uplet of values xi, yi, and the value xi, yi associated with the parameter Xi occupies the same position in each vector X, Y.

The first number N1 is an integer strictly greater than one. For example, the first number N1 is greater than or equal to 3, preferably greater than or equal to 10, preferably greater than or equal to 15, preferably greater than or equal to 20, preferably greater than or equal to 50, preferably greater than or equal to 50. equal to 100.

Each parameter is a feature of an entity.

The parameters are denoted Xi, where i is the index representative of the position of the value xi, yi associated with the parameter Xi in each vector X, Y.

The index i is an indicator of the parameter Xi. This means that each index i is associated with a single parameter Xi, and vice versa.

The parameters Xi are therefore ordered by the index i. The index i is an integer. The index i is, on the one hand, greater than or equal to one, and on the other hand less than or equal to the first number N1.

For example, the entity is a physical object, a concept, an action, or a set of actions or a set of physical objects.

For example, the entity comprises five parameters which are, in order: a first parameter X1, a second parameter X2, a third parameter X3, a fourth parameter X4, and a fifth parameter X5.

For example, the entity is an automobile. In this case, the first parameter X1 is the price in Euros of the car, the second parameter X2 is the number of seats in the car, the third parameter X3 is the number of doors of the car, the fourth parameter X4 is the fuel consumption in liters per hundred kilometers traveled, and the fifth parameter X5 is the horsepower of the engine.

Each parameter X can take a plurality of values xi, yi included in a respective range Pi.

The beach Pi is, for example, continuous. For example, each value x4, y4 of the fourth parameter X4 ("consumption") is a real number strictly greater than 2 and strictly less than 15.

In another example, the beach Pi is a discrete beach. For example, each value x2, y2 of the second parameter X2 ("number of places") is an integer greater than or equal to 1 and less than or equal to 8.

The explanation method comprises a step 100 of supplying, a step 1 10 of generating a first set E1 of lists Lj, a step 120 of initialization, a step 130 of selecting a list Lj, a step 140 of generating a third vector, a calculation step 150, an evaluation step 160, a selection step 170 of a set of parameters Xi, and a step 180 of developing an explanation.

During the step 100 of supply, a first score Ux and a second score Uy are provided.

The first score Ux is associated with a first vector X by the function U. The first vector X contains the first number N1 of the first values xi of the parameters Xi.

The second score Uy is associated with a second vector Y by the function U. The first score Ux is, for example, strictly greater than the second score Uy.

The second vector Y contains the first number N1 of second values yi of the parameters Xi. In the case where each entity is an automobile, the first vector X and the second vector Y represent two automobiles A and B to be compared.

The first vector X (10,000; 5; 3; 4,5; 75) represents automobile A and the second vector Y (12,000; 6; 5; 4,8; 80) represents automobile B.

For example, the first vector X and the second vector Y are generated by an acquisition device external to the system 10, and transmitted by the acquisition device to the system 10 which calculates the first score Ux from the first vector X, and the second score Uy from the second vector Y.

In the example above, the first vector X is transmitted by a user to the system 10 via the keyboard 22.

The user first enters the first value x1, equal to 10000 and representing the price of the car A.

Then, the user types the first value x2, equal to 5 and representing the number of seats of the automobile A.

Then, the user types the first value x3 equal to 3 and represents the number of doors of automobile A.

Then, the user transmits the first value x4 equal to 4.5 and representing the fuel consumption of the automobile A.

Finally, the user types the first value x5 equal to 75 and representing the engine power of the automobile A.

As a variant, the first score Ux and the second score Uy are provided by a device external to the system 10.

At the end of the generation step 1, a first set E1 of lists is generated.

Each list is denoted Lj, where j is an index locating the position of the list Lj in the first set E1.

Each list Lj comprises at least one indicator of a parameter Xi.

Each indicator included in the list Lj is denoted ij. Indeed, each indicator ij is associated with a parameter Xi, and contained in a list Lj.

Each indicator is associated with a single parameter Xi, and vice versa.

Each list Lj is thus able to define a subset SE2j of parameters Xi. This means that a parameter Xi is in the subset SE2j associated with the list Lj if and only if the list Lj contains the indicator ij associated with the parameter Xi.

Each list L comprises a second number N2j of indicators ij of the parameters Xi _. The second number N2j is an integer strictly greater than zero. The second number N2j is strictly less than the first number N1.

For example, the first set E1 contains a first list L1, a second list L2 and a third list L3.

The first list L1 contains the indicator 1 1 of the first parameter X1, the indicator

31 of the third parameter X3, and the indicator 51 of the fifth parameter X5.

The second list L2 contains the indicator 22 of the second parameter X2, and the indicator 42 of the fourth parameter X4.

The third list L3 contains the indicator 13 of the first parameter X1, and the indicator 23 of the second parameter X2.

For example, the list Lj is a N2j-tuple of indicators ij, each indicator ij of the list Lj being an integer equal to the index i of the corresponding parameter Xi.

According to this example, the first list L1 is (1, 3,5), the second list L2 is (2,4), and the third list L3 is (1,2)

In a preferred example, each indicator ij is an integer equal to 1.

The list Lj is an N1 -uplet containing N2j indicators ij and elements ei, j. Each element ei, j is a number equal to zero.

The indicator ij associated with the parameter Xi is located in the nth position in the list Lj.

According to this preferred example, the first list L1 is the 5-tuple (1; 0; 1; 0; 1), the second list L2 is the 5-tuple (0; 1; 0; 1; 0) and the third list L3 is the 5-tuple (1; 1; 0; 0; 0).

In another example, each indicator ij is the name of the corresponding parameter Xi.

The first set E1 comprises a plurality of first subsets SE1 each comprising a third number N3 of lists Lj. Each first subset SE1 groups all the lists Lj comprising the same second number N2j of indicators ij.

The first set E1 contains a total number Nt of lists Lj.

The total number Nt of lists Lj is greater than or equal to the square of the first number N1.

For example, each list Lj generated contains at most two indicators ij.

The total number Nt of lists Lj is less than or equal to the product of a coefficient Co and the maximum number Nm of possible lists Lj.

The maximum number Nm is equal to two to the power of the first number N1. The inequality is written mathematically: Nt≤Co * 2 ^Nl

The coefficient Co is a predetermined real number. The coefficient Co is, for example, strictly less than one. Preferably, the coefficient Co is equal to one tenth.

At least part of the lists Lj of the first set E1 are generated according to a probability law P.

The probability law P relates, for example, to the indicators ij contained in each list Lj.

For example, lists Lj are generated randomly according to a Latin hypercube algorithm. This means that the third number N3 of each first subset SE1 is equal to a first predetermined number Np, the first predetermined number Np being common to each first subset SE1, and the lists Lj of each first subset SE1 are generated randomly.

This means that the indicators ij of each list are generated according to a probability law P in which each indicator ij has a probability equal to that of each other indicator ij to be contained in the list Lj.

In the case where the first predetermined number Np is equal to three, the first set E1 comprises three lists Lj containing a single indicator ij, three lists containing two indicators ij, and so on.

In another example, the probability law P relates to the second number N2j of each list. This means that the probability law P is used by the system 10 to determine the number of lists Lj of each first subset SE1, and the lists Lj of each first subset SE1 are generated randomly.

Preferably, the third number N3 of each first subset SE1 is calculated, during the step 1 10 of generating the first set E1, according to a Gauss law (also called "normal law").

The Gaussian law is a law of probability in which the probability density f (N3) is given by the equation:

in which :

• σ is the standard deviation of Gauss's law,

• μ is the expectation of Gauss's law,

• π is the ratio of the perimeter of a circle to its diameter, and

• e designates the exponential function. As a variant, the probability law P is a uniform or normal law relating to the total number of lists Nt.

Alternatively, the coefficient Co is equal to one. This means that all possible Lj lists are generated.

At the end of the initialization step 120, an importance indicator Ri of each parameter Xi is generated.

The importance indicator Ri is a magnitude representative of the importance that the parameter Xi has in calculating the first score Ux and the second score Uy. Preferably, the importance indicator Ri is representative of the importance of the parameter Xi in the difference between the first score Ux and the second score Uy.

The importance indicator Ri is an element of a totally ordered space. This means that each indicator of importance Ri is suitable to be compared with each other indicator of importance Ri.

The indicator of importance Ri is preferably a real number.

The indicator of importance Ri is, for example, a value of Shapley.

The value of Shapley is a greatness developed within the framework of game theory. As part of a cooperative game in which players must come together to achieve a goal, Shapley's value is assessed for each player from the number of winning coalitions comprising the player and becoming losers if the player withdraws from Coalition. The value of Shapley therefore makes it possible to evaluate the power of a player (that is to say his ability to influence the final result), and is, in particular, used to determine the distribution of winnings in case of victory. .

In our case, the value of Shapley Ri is evaluated for each parameter Xi and makes it possible to evaluate the importance of the parameter Xi in the difference between the first score Ux and the second score Uy.

Alternatively, the importance indicator Ri is a Sobol index.

Sobol indices are estimates of the sensitivity of a function of at least one parameter to a variation of this parameter. The first-order sensitivity index of the function U to the parameter Xi is defined from the variance V (U) of the function U by the equation:

ViEjU l Xm

V (U)

where E (U | Xi) is the expectation of the function U when the parameter Xi is fixed.

The value of each indicator of importance Ri is set equal to zero.

At the end of the step 130 of selection, a list Lj is selected. For example, if no list Lj has been selected during a previous selection step 130, the list L1 of index j equal to one is selected.

The index j of the list Lj is, for example, stored in the memory 18.

In the example given above of the comparison between two cars, the list L1 (1; 0; 1; 0; 1) is selected.

At the end of the generation step 140, a third vector Cj and at least a fourth vector Di, j are generated.

The third vector Cj comprises third values ci, j parameters Xi.

The third vector Cj is generated from the selected list Lj and at least from the first vector X.

Each third value ci, j is equal to the first value xi corresponding when the list Lj does not include an indicator ij of the corresponding parameter Xi.

In the opposite case, the third value ci, j is different from the first value xi.

Preferably, when the list Lj comprises an indicator ij of the parameter Xi, the third value ci, j corresponding is equal to the second value yi corresponding.

In the given case of the comparison between two automobiles, the third vector C1 (12,000; 5; 5; 4,5; 80) is generated from the list L1 (1; 0; 1; 0; 1), the first vector X (10,000; 5; 3; 4,5; 75); and second vector Y (12,000; 6; 5; 4,8; 80).

The first vector X, the second vector Y and the third vector C1 are represented graphically in three dimensions in FIG. 3. Only the first parameter X1, the second parameter X2 and the third parameter X3 have been represented, for the sake of clarity.

At the end of the generation step 140, a fourth vector is also generated, for each parameter Xi, from the third vector Cj.

The fourth vector is denoted Di, j, and denoted by the index j of the third vector Cj and by the index i of the parameter Xi.

Each fourth vector Di, j comprises fourth values of the parameters Xi.

Each fourth value of the fourth vector Di, j is denoted di, j, m, where m is an integer index identifying the position of the fourth value di, j, m in the fourth vector Di, j. The index m of the fourth value di, j, m is greater than or equal to one, and less than or equal to the first number N1.

Each fourth value di, j, m is equal to the third value ci, j corresponding if the index m of the fourth value di, j, m is different from the index i of the parameter Xi. In the opposite case, the fourth value di, j, m is different from the third value ci, j. Preferably, the fourth value di, j, m is equal to the first value xi if the third value ci, j is equal to the second value yi, and is equal to the second value yi if the third value ci, j is equal to at the first value xi.

In other words, the fourth vector Di, j associated with parameter Xi of index i is constructed from the third vector Cj, replacing the third value ci, j of index i, and only this one, by depending on the case, the first value xi or the second value yi corresponding so that the fourth vector Di, j is different from the third vector Cj.

In the example case of the comparison between two automobiles, five fourth vectors Di, 1 are generated from the third vector C1 (12,000; 5; 5; 4,5; 80).

The fourth vector D1, 1 (10,000; 5; 5; 4,5; 80) is generated for the first parameter X1. Only the fourth index value m, which is equal to one, is different from the corresponding third value c1, 1.

The fourth vector D2,1 (12,000; 6; 5; 4,5; 80) is generated for the second parameter X2. Only the fourth index value m, which is equal to two, is different from the corresponding third c3,2 value.

The fourth vector D3,1 (12,000; 5; 3; 4,5; 80) is generated for the third parameter X3. Only the fourth index value m, which is equal to three, is different from the third corresponding value c3.1.

The fourth vector D4.1 (12000; 5; 5; 4.8; 80) is generated for the fourth parameter X4. Only the fourth index value m, which is equal to four, is different from the third corresponding value c4.1.

The fourth vector D5,1 (12,000; 5; 5; 4,5; 75) is generated for the fifth parameter X5. Only the fourth index value m, which is equal to five, is different from the third corresponding value c5.1.

At the end of the calculation step 150, a third score Ucj is calculated from the third vector Cj and a fourth score Udi, j is calculated from each fourth vector Di, j

For example, the third score Ucj and the fourth scores Udi, j are computed by the information processing unit 14.

According to another example, the third vector Cj and the fourth vectors Di, j are transmitted to an external computing device that the information processing unit 14 is able to control. The computing device transmits back to the system 10 the third score Ucj and the fourth scores Udi, j. At the end of the evaluation step 160, the importance indicator Ri of each parameter Xi is evaluated.

Preferably, the importance indicator Ri is evaluated from the third score Ucj and the fourth scores Udi, j.

For this, the absolute value of the difference between the third score Ucj and the fourth score Udi, j associated with the parameter Xi is calculated. The value of the importance indicator Ri is then increased by a quantity Q equal to the calculated absolute value, divided by the total number Nt of lists Lj of the first set E1.

This is written mathematically

The indicator of importance Ri evaluated is then an approximation of a value of Shapley.

As a variant, during this evaluation step 160, only the parameters Xi for which the first value xi is different from the second value yi are considered. This means that, when the first value xi and the second value yi are equal, the importance indicator Ri of the corresponding parameter Xi is not evaluated during this evaluation step 160.

If all the lists Lj have not yet been selected, the step 130 of selection is repeated. The reiteration is represented in FIG. 2 by an arrow 165.

The step 130 of selecting a list Lj, the step 140 of generating a third vector Cj, the step 150 of calculating at least one score and the step 160 of evaluating at least one importance indicator Ri are then reiterated in the order presented above for another list Lj.

For example, the index j stored is increased by one, and the step 130 of selecting a list Lj, the step 140 of generating a third vector Cj, the step 150 of calculating at least one score and the step 160 of evaluating at least one indicator of importance Ri are implemented for the corresponding list Lj.

When the step 130 of selecting a list Lj, the step 140 of generating a third vector Cj, the step 150 of calculating at least one score and the step 160 of evaluating at least an indicator of importance Ri has been implemented for each of the lists Lj, the evaluation step 160 is followed by a step 170 of selecting at least one parameter Xi.

For example, when the index j of the list Lj is equal to the total number Nt of lists Lj, the evaluation step 160 is followed by the step 170 of selecting at least one parameter Xi. In the case given as an example, after each evaluation step 160 has been implemented, a first importance indicator R1, a second importance indicator R2, a third importance indicator R3, a fourth indicator of importance. importance R4 and a fifth indicator of importance R5 were evaluated.

The first importance indicator R1 is associated with the first parameter X1, the second importance indicator R2 is associated with the second parameter X2, and so on.

The first indicator of importance R1 is equal to 0.5, the second indicator of importance R2 is equal to 0.05, the third indicator of importance R3 is equal to 0.10, the fourth indicator of importance R4 is equal to 0.30, and the fifth indicator of importance R5 is equal to 0.05.

This means that the parameter Xi which has the greatest importance in the difference between the first score Ux and the second score Uy is the first parameter X1. The parameters Xi which have the smallest importance in the difference between the first score Ux and the second score Uy are the second parameter X2 and the fifth parameter X5.

At the end of step 170 of selecting at least one parameter Xi, a fourth set E4 of parameters Xi is selected.

The fourth set E4 comprises a fourth number N4 of parameters Xi. The fourth number N4 is, for example, predetermined.

As a variant, the fourth number N4 is determined, during the step 170 of selecting at least one parameter Xi, from the estimated importance indicators Ri.

The parameters Xi selected during the selection step 170 of at least one parameter Xi are the parameters Xi whose importance indicators Ri are the highest.

In the example presented above, the fourth number N4 is equal to two, and the first parameter X1 associated with the first indicator of importance R1 equal to 0.5 and the fourth parameter X4 associated with the fourth indicator of importance R4 equal at 0.30 are selected.

At the end of step 180 of elaboration, an Ex explanation is elaborated.

The explanation Ex is suitable to explain to a decision maker the first score Ux.

The explanation Ex is, for example, a relative explanation of the first score Ux with respect to the second score Uy.

This means that the Ex explanation is likely to explain to a decision maker the difference between the first Ux score and the second Uy score. In particular, the Ex explanation is own to explain to a decision maker the reasons that a Ux score, Uy is greater than another Ux score, Uy.

The Ex explanation is elaborated from the selected Xi parameters. For example, the Ex explanation includes a list of selected Xi parameters.

The Ex explanation includes, for example, a hole phrase such as: "[1] is preferred to [2] because of", where [1] is the name of the entity associated with the highest score. The hole phrase is followed by the list of selected Xi parameters.

In the case given as an example of the comparison between two automobiles A and B, the explanation Ex is therefore: "A is preferred to B because of: its price, its fuel consumption. "

The explanation is displayed on the display unit 24.

The method of explaining the score Ux does not assume that the function U is known. The explanation method can therefore be used in conjunction with an external device for calculating a score whose operating details are not known. Only the first and second scores Ux, Uy and the first and second associated vectors x, y are known.

In addition, the explanation method is adaptable since it is not relative to a particular function. No assumptions about the U function are used.

In a second exemplary embodiment, the evaluation selection step 160 is followed by the step 170 of selecting at least one parameter Xi if all the lists Lj have not yet been selected and, when the step 160 of evaluation the quantity Q is strictly lower than a predetermined threshold S.

This is written mathematically

The explanation method does not take into account some non-significant lists Lj (for which the quantity Q is less than the threshold S). The explanation process is therefore faster to implement.

A flowchart of a third exemplary implementation of an explanation method is shown in FIG. 4.

The elements identical to the first example of FIG. 2 are not described again. Only the differences are highlighted.

The function U has a global maximum Ug associated with a first extreme vector G, and a global minimum Ub associated with a second extreme vector B. The global maximum Ug and the overall minimum Ub of the function U are stored in the memory 18.

For example, the global maximum Ug and the global minimum Ub of the function U have been obtained by a mathematical analysis of the function U.

As a variant, the global maximum Ug and the overall minimum Ub of the function U are obtained by an experimental method. For example, all possible X, Y vectors are generated and the score Ux, Uy associated with each of the vectors X, Y is calculated.

The operation of the third example will now be described.

Only the first score Ux is provided at the end of step 100 of providing a first score. No second Uy score is provided.

Step 1 of generating a first set E1, step 120 of initialization, step 130 of selecting a list Lj, step 140 of generating a third vector Cj, the step 150 of calculating at least one score, the step 160 of evaluating at least one indicator of importance Ri, and the step 170 of selecting a set of parameters Xi are then implemented from the first vector x and the first extreme vector G.

This means that, during step 1 of generating a first set E1, of step 120 of initialization, of step 130 of selecting a list Lj, of step 140 of generating a third vector Cj, step 150 of calculating at least one score, step 160 of evaluating at least one indicator of importance Ri, and step 170 of selecting a set of parameters Xi, the global maximum Ug is used instead of the second score Uy, and the first extreme vector G is used in place of the second vector Y.

At the end of the evaluation step 160, a first indicator of importance Rpi is obtained for each parameter Xi.

At the end of step 170 of selecting at least one parameter Xi, a fifth set E5 of parameters Xi is selected on the basis of the first estimated importance indicators Rpi.

Next, step 1 of generating a first set E1, step 120 of initialization, step 130 of selecting a list Lj, step 140 of generating a third vector Cj, l step 150 of calculating at least one score, step 160 of evaluating at least one indicator of importance Ri, and step 170 of selecting a set of parameters Xi are implemented from of the first vector x and the second extreme vector B.

This means that, during step 1 of generating a first set E1, of step 120 of initialization, of step 130 of selecting a list Lj, of step 140 of generating a third vector Cj, of step 150 of calculating at least one score, of step 160 of evaluating at least one indicator of importance Ri, and step 170 of selecting a set of parameters Xi, the global minimum Ub is used instead of the second score Uy, and the second extreme vector B is used in place of the second vector Y.

At the end of the evaluation step 160, a second indicator of importance Rdi is obtained for each parameter Xi.

At the end of step 170 of selecting at least one parameter Xi, a sixth set E6 of parameters Xi is selected on the basis of the second evaluated importance indicators Rdi.

At the end of the step 180 of elaboration, the explanation Ex is elaborated starting from the fifth set E5 and the sixth set E6.

The explanation method then makes it possible to explain the first score Ux absolutely. This means that the method is able to explain the first score Ux without comparing it to a second score Uy.

A flowchart of a fourth example of implementation of an explanation method is shown in FIG.

The coefficients of Mobius are the elements obtained, from a first sequence, by the inversion formula of Mobius. The inversion formula of Mobius was introduced into the theory of numbers during the nineteenth century by August Ferdinand Mobius.

According to the inversion formula of Mobius, a coefficient of Mobius m (Lj) is defined, for each list Lj, by the equation: m (Lj) = Σ (- \) ^m v (T) in which:

• T is a secondary list containing indicators ij and included in the list Lj,

• | Lj \ T | is the number of indicators ij contained in the list Lj and not being contained in the secondary list T, and

• v (T) is the difference between a third score Ut associated with a third vector Cj and the first score Ux, the third vector Cj being calculated from the secondary list T.

The function U is k-additive. This means that there exists a second predetermined number k for which the Mobius coefficient m (Lj) of each list Lj is equal to zero if the second number N2j of the list Lj is strictly greater than the second predetermined number k.

The second predetermined number k is an integer strictly greater than 1 and strictly less than the first number N1.

For example, the second predetermined number k is equal to 3. Preferably, the second predetermined number k is equal to 2.

At the end of the step 1 10 of generating the first set E1, all lists Lj whose second number N2j is less than or equal to a second predetermined number k are generated.

No list whose second number N2j is strictly greater than the second predetermined number k is generated.

Only the first subsets SE1 associated with a second number N2j less than or equal to the second predetermined number k are thus generated.

For each parameter Xi, a third subset SE3i of lists Lj is also defined. The third subset SE3i each comprises lists Lj including the indicator ij of the parameter Xi. Each third subset SE3i is therefore included in the first set E1.

The initialization step 120 and the selection step 130 are not implemented. At the end of step 140 of generating a third vector Cj, the third vector associated with each list Lj is generated.

No fourth vector Di, j is generated.

Only the third score Ucj of each third vector Cj is calculated at the end of step 150 of calculation.

The evaluation step 160 comprises an initialization sub-step 162, then a sub-step 165 for calculating at least one Mobius coefficient, and then a sub-step 167 for evaluating at least one indicator. of importance Ri.

The coefficient m (Lj) is a real number equal to the difference between the third score Ucj associated with the third vector Cj generated from the list Lj and the first score Ux. This is written mathematically:

m (Lj) = Ucj - Ux

An iterator is also generated. The iterator n is set equal to zero.

At the end of the calculation sub-step 165, the iterator n is increased by one. This is written mathematically:

n: = n + l A reduced list Lrj is generated, for each list Lj, if the list Lj contains an indicator ij of the parameter Xn of index equal to the iterator n,

The reduced list Lrj is constructed, starting from the list Lj, by the deletion of the indicator ij associated with the parameter Xn of index equal to the iterator n. This means that the reduced list Lrj contains each of the indicators ij contained in the list Lj, with the exception of the indicator ij equal to the iterator n.

For example, at the end of the first calculation sub-step 165, the reduced list Lr1 (0; 0; 1; 0; 1) is generated from the list L1 (1; 0; 1; 0; 1 ).

The coefficient m (Lj) of each list is set equal to the difference between the coefficient m (Lj) associated with the list Lj, and the coefficient m (Lr, j) associated with the reduced list Lrj. This is written mathematically:

m (Lj): = m (Lj) - m (Lrj)

If the list Lj does not contain an indicator ij equal to the iterator n, the coefficient m (Lj) is not modified.

If the iterator n is strictly less than the first number N1, the first calculation sub-step 165 is repeated. The reiteration is represented in FIG. 5 by an arrow 190.

Otherwise, the first calculation sub-step 165 is followed by a second calculation sub-step 167.

At the end of the second calculation sub-step 167, the importance indicator Ri associated with each parameter Xi is evaluated.

Each indicator of importance Ri is equal to the sum, for each list Lj containing the indicator ij of the parameter Xi, of the ratio between the coefficient m (Lj) of index associated with the list Lj and the second number N2j of the list Lj.

This is written mathematically:

Ri = Σ

LjeSE3i N2j

When the second calculation sub-step 167 is completed, the step 170 of selecting at least one parameter Xi is implemented.

When the function U is k-additive, the importance indicator Ri associated with each parameter is then a value of Shapley and not an approximation of a value of Shapley.

The method of explaining a score Ux is therefore more precise. The generated Ex explanation is then more relevant. The explanation process is adaptable since it is not relative to a particular function. The explanation process only supposes that the function U is k-additive.

In a fifth exemplary explanation method, at the end of step 170 of selecting at least one parameter Xi, at least two fourth subsets SE4m of parameters are selected.

At the end of the selection step 170, a fifth number N5 of fourth subsets SE4 are generated from the importance indicators Ri received.

In the remainder of this description, each fourth subset SE4 will be identified by an index m and noted SE4m.

Each fourth subset SE4m contains at least one parameter Xi.

Preferably, the fourth subsets SE4m are generated according to a data partitioning method.

Data partitioning (or data clustering in English) is one of the statistical methods of data analysis. Data partitioning aims at dividing a set of data into different homogeneous subsets, in that the data of each subset share common characteristics, which most often correspond to proximity criteria between objects.

In the case of the fourth subsets SE4m, the proximity criterion used is the difference between the indicators of importance Ri.

The parameters Xi are classified in the fourth subsets SE4m in ascending order of the calculated importance indices Ri. This means that each of the parameters Xi of the fourth subset SE41 of index m equal to 1 is associated with an indicator of importance Ri greater than each of the indicators of importance Ri associated with the parameters Xi of the fourth subset SE42 of index m equal to 2, and so on.

At the end of the step 180 of elaboration, the explanation Ex is elaborated from the fourth subsets SE4m of generated parameters Xi.

For example, the explanation Ex takes the form of a sentence with holes of the type: "The automobile A is preferred to the automobile B first because of the parameters [1], then parameters [2], and finally because of the parameters [3] ", where [1] is a list of the parameters Xi of the fourth subset SE41, [2] is a list of the parameters Xi of the fourth subset SE42 and [3] is a list parameters Xi of the fourth subset SE43. The explanation method automatically adjusts the number of parameters Xi used in the explanation Ex to the importance indicators Ri evaluated. The explanation Ex elaborated is therefore more relevant.

Each of the above examples of the explanation method is capable of being implemented with a function U which is not precisely known.

The explanation process does not depend on the U function used. The explanation method is therefore adaptable to a large number of U functions.

In addition, the explanation Ex is elaborated from the values of Shapley associated with each parameter Xi. The Ex explanation is therefore relevant.

Finally, during the implementation of the explanation method, only a small number of lists Lj is generated. The explanation method therefore has good performance, in particular in terms of calculation time.

Claims

1 .- Method for explaining a score (Ux) associated with a vector (x) of a first number (Ni) of parameters (X,) by a function (U), each parameter (X,) being ordered with an indicator (ij) varying from one to the first number (Ni), the first number (Ni) being an integer strictly greater than one, the method comprising at least steps of:

a1) providing a first score (Ux) associated with a first vector (x) containing first values (x,) of the parameters (X,),

b) generating a first set (E1) of lists (L _j ), each list (L _j ) comprising a second number (N _2j ) of indicators (ij), the second number (N _2j ) being an integer strictly less than the first number (N ^ and strictly greater than zero,

c) generating, from a list (L _j ), at least a third vector (Cj) in which each parameter (X,) has a third value (c,

the third value (QJ) being equal to the corresponding first value (x,) when the list (L _j ) does not include an indicator (ij) of the corresponding parameter (X,) and different from the first value (x,) corresponding in the opposite case,

d) calculating the score (UCj) of at least one third vector (Cj),

e) evaluating, from the scores (UCj) of the third vectors (Cj) calculated, an importance indicator (R,) of each parameter (X,),

f) drawing, based on the evaluated importance indicators (R), an explanation (E) of the first score (Ux).

2. The process according to claim 1, wherein the indicators of importance

(Ri) calculated in step e are either Shapley values or Sobol indices.

3. - Method according to claim 1 or 2, wherein the first set (E1) comprises a total number (N _t ) of lists (L _j ), the total number (N _t ) satisfying the mathematical inequality:

Nt≤Co * 2 ^m

wherein Co is a predetermined coefficient.

4. - Method according to any one of claims 1 to 3, wherein step b comprises generating at least part of the lists (L _j ) of the first set (E1) according to a probability law (P) bearing either the second number (N2j) or the indicators ij contained in the list (Lj).

5. - Method according to any one of claims 1 to 3, wherein step b comprises the generation of all lists (L _j ) whose second number (N _¾ ) is less than or equal to a second predetermined whole number (k) strictly less than the first number (Ni).

6. - Method according to any one of claims 1 to 5, wherein the method further comprises a step a2 of providing a second score (Uy) associated with a second vector (Y) containing second values ( y,) parameters and wherein, in step c, the third value (c _ii ) is equal to the corresponding second value (y,) when the list (L _j ) comprises an indicator (ij) of the parameter ( Xi) corresponding,

7. - Method for explaining a score (Ux) associated with a vector (X) by a function (U), the function (U) having a global maximum (Ug) associated with a first extreme vector (G) and an overall minimum (Ub) associated with a second extreme vector (B), the method comprising:

g1) implementing, from the first score (Ux) and the global maximum (Ug), steps b to e of an explanation method according to claim 6 to obtain first indicators of importance (Rg, )

g2) implementing, from the first score (Ux) and the global minimum (Ub), steps b to e of an explanation method according to claim 6 to obtain second importance indicator (Rb, ), and

h) generating, from the first (Rg) and second (Rb) indicators of importance, an explanation (E) of the first score (Ux).

8. A computer program product comprising software instructions which, when executed by a computer, implement a method according to any one of claims 1 to 7.

9. An information carrier on which is stored a computer program product according to claim 8.

10.- System (10) for explaining a function (U) associating a score (Ux) with a vector (x) of a first number (Ni) of parameters (X,), each parameter (X,) being ordered with an indicator (ij) varying from one to the first number (Ni), the first number (Ni) being an integer strictly greater than one, the system (10) being suitable for:

- providing a first score (Ux) associated with a first vector (X) containing first values (x,) of the parameters (X,),

generating a first set (E1) of lists (L _j ), each list (L _j ) comprising a second number (Ν _¾ ) of indicators (ij), the second number (N ₂ ) being a strictly lower integer at the first number (N ^ and above strictly zero,

generating, from a list (L _j ), at least one third vector (Cj) in which each parameter (X,) has a third value (c,

calculating the score (UCj) of at least one third vector (Cj),

evaluating, from the scores (UCj) calculated for each of the third vectors (Cj), an importance indicator (R,) of each parameter (X,),

- Develop, based on the assessed importance indicators (Ri), an explanation (E) of the first score (Ux).