Classifications

• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B17/00—Systems involving the use of models or simulators of said systems
  • G05B17/02—Systems involving the use of models or simulators of said systems electric
• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
  • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
  • G05B13/0265—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion

Definitions

• the present invention relates to a method for determining the value to be given to a set of so-called system-specific parameters from the values of a set of so-called system measurement parameters.
• a method can be used to control various systems such as a character recognition system, a fault diagnosis system for electrical components, a transport cost evaluation system, etc.
• FIG. 1 represents an example of a system using such a method.
• a car 1 tries to follow a truck 2, the car and the truck being scale models moving on a flat surface without obstacles.
• the car is equipped with a camera 3 and an autonomous control system.
• the control system has the function of defining at regular intervals, for example every 100 ms, the direction and the speed which the car must take 1, as well as the inclination of the camera so that the truck remains permanently in the field of view of the camera.
• the control system estimates where the truck will be 100 ms later, at a time t ⁇ . In the event that the truck actually moves as planned, it will be at the estimated position Pe (t ⁇ ), in the center of image 4 that the camera will take at time t] _. However, in general, the truck will be in a measured position P (t ⁇ ) distinct from Pe (t ⁇ ).
• the position Pm (t ⁇ ) is offset in x and y with respect to Pe (t ] _) according to a horizontal offset c x and a vertical offset Cy.
• the horizontal offset c x indicates whether the truck has moved further to the left or to the right.
• the vertical offset c v indicates whether the truck has accelerated or slowed down.
• FIG. 2A is a side view of the car 1 moving at speed v provided with the camera 3 and represents the angle of vertical inclination cty of the axis of the camera.
• FIG. 2B is a top view of the car 1, and represents the horizontal angle of inclination ⁇ ⁇ of the axis of the camera, as well as the angle of rotation ⁇ ro ⁇ of the wheels of the car 1.
• the FIG. 3 represents a possible configuration of the car 1 and of the truck 2, at 1 'instant t] _, the truck 2 being in the position Pm (t ⁇ ) of FIG. 1. It would therefore seem that the truck is going to the right ( hypothesis a). The truck can also go straight and accelerate (hypothesis b).
• the truck can also leave to the left (hypothesis c).
• hypothesis c We will consider below "that the truck has only these three possibilities and is only likely to go to one of the three estimated positions Pe (t 2 ) a, Pe (t 2 ) b and Pe (t 2 ) c.
• the control system of the car must then decide if the car should take the direction dj_ allowing to join the truck Pe (t 2 ) c or if it will take the direction d allowing to join the truck in position Pe (t 2 ) a or Pe (t 2 ) c.
• the three possibilities Pe (t 2 ) a, Pe (t 2 ) b, Pe (t 2 ) c being a priori eq iprobable, the choice of the direction d 2 seems to be the most judicious because it covers a greater number of possibilities , and we will consider below that d 2 has been chosen.
• the control system must then choose to increase or decrease the speed v of the car 1. Analysis of the image suggests that the truck has accelerated because it's on the top of the image. A first decision could be to request an acceleration of the car. However, the control system chose to go in the direction d 2 and it is possible that the truck turns slowly to the right to find itself in the position of the truck Pe (t 2 ) a. Knowing this probability, it then seems wise not to accelerate too much so as not to collide with the truck.
• the control system likewise defines the angles ⁇ and ⁇ n to be given to the camera as a function of the decisions previously taken and of the estimate of the future movement of the truck.
• the control system of car 1 must be able to define the new values of the parameters v, ct ro t-, coy- and ⁇ n , with a computer and a memory of small sizes and low costs. In addition, it is necessary for the control system to take decisions quickly, every 100 ms.
• the control system must also process a large number of measurement parameters c x , c v and specific or more specifically control parameters, v, ⁇ ro t, oty, ⁇ n in order to be able to make effective decisions which make it possible to follow the truck regardless of its path.
• the realization of a control system requires to define beforehand the interdependencies between the parameters.
• the choice of the horizontal tilt angle ⁇ n and the rotation angle d Q t depends on the horizontal offset c x measured; the choice of the vertical tilt angle ⁇ - ⁇ and the choice of the speed v depend on the vertical offset c v measured.
• the angle ⁇ n and the angle ⁇ 0 t- are dependent on each other otherwise the truck may leave the field of vision of the camera, the car can no longer follow the truck.
• the speed v and the angle cty are dependent on each other, the angle o having to be adapted to the speed and vice versa.
• a model of the joint probability distribution of the set of system parameters is defined from a set of independent probability distributions defined for each of the previously identified interdependencies.
• the probability p (v, ⁇ ro t, cty, ⁇ ⁇ , c x , Cy) that a given combination of values of all the parameters is possible and clever, can be defined for the example described above. with the following formula:
• conditional probabilities such as p ( ⁇ ] / c x ) can be defined by a family of analytical functions, which can for example be Gaussians centered on the value c x .
• Simple probabilities, p (c x ), or conditional probabilities, p ( ⁇ j1 / c x ), can also be calculated from a database listing the number of times a horizontal shift c x or a couple of values ( ⁇ , c x ) has been observed for example during a learning phase.
• the model of the joint probability distribution, the analytical functions and the databases are stored.
• the control system After taking an image at an instant t ⁇ , the control system must decide the value to be given to each parameter of command v, ct j -ot, oty, cc ⁇ from the values recorded for each measurement parameter at the instant t ⁇ , c x j_ and Cyj_.
• the system first chooses a couple of values . Only the choice of a couple ( ⁇ ro t, v) will be described, the choice of a couple (cty, () being carried out identically.
• the probability p (ct ro t, v / c x j_, Cyj for that choosing a couple
• a representation of the probability distribution known as the "Gaussian mixture” consists of modeling the distribution by a set of Gaussians, each Gaussian being defined from a maximum probability value.
• an optimization method is used to identify the pairs ( ⁇ ro t, v) of maximum probabilities.
• the couples are then divided into groups containing more or less couples depending on whether or not the couples have values ⁇ ro t and v close to the identified maximums.
• a probability value is calculated for each group, the probability values forming gaussians around the maximum probabilities.
• the choice of a couple is then made by random draw or search for the maximum probability couple. This method allows a more or less precise representation depending on the available memory space. However, an increase in the precision of the representation requires redoing the distribution of the couples and the calculation of the probabilities of each group. In addition, the modeling of the probability distribution by a set of gaussians is not adequate for all systems.
• An object of the present invention is to provide a method for determining the values to be given to one or more specific parameters of a system knowing one or more measurement parameters, in the case where the number of parameters is very large and / or in the case where some of the parameters can take a large number of values.
• Another object of the present invention is to provide a method for determining the values to be given to all the specific parameters of a system in a time interval which can be very short.
• Another object of the present invention is to provide such a determination method using a memory of variable size and possibly very small.
• Another object of the present invention is to provide such a determination method using a simple calculation device.
• the present invention provides a method for determining the value to be given to a set of specific parameters of a system from the values of a set of measurement parameters of this system, each of the parameters being able to take a finite number of values, the system being associated with a means for providing a probability value for any combination of values of the specific parameters, said probability value being all the higher the more the choice of the combination considered being relevant knowing the value of the measurement parameters, the method comprising the following steps: - reading the value of each measurement parameter; construct a tree-shaped representation of the probability distribution of all possible combinations of values of specific parameters corresponding to the values noted, the set of combinations, constituting a first branch, being divided into several subsets of combinations, constituting second branches, each subset grouping combinations having values of similar specific parameters, each second branch being able to divide into several third branches in a similar manner and so on, a value of probability being assigned to each branch , this probability value being that obtained for one of the combinations of the branch considered or for one of the combinations of one of the branches
• the branches resulting from the division of the same branch are two in number and contain the same number of combinations
• the first branch dividing into two second branches, each second branch can be divided into two third branches and so on.
• the division of a branch into two branches comprises the following steps: a) choosing a different combination from the combinations which have already served to define the probability value of the existing branches and calculating the probability of this chosen combination; b) divide the so-called "mother” branch containing the chosen combination into two so-called "daughter”branches; and if the combination chosen and the "mother” combination used to define the probability value of the mother branch belong to the same daughter branch, assign the two daughter branches the probability value of the mother branch and divide the branch daughter containing the combination chosen by repeating the process in step b), this daughter branch becoming the parent branch, and in the case where the chosen combination and the mother combination do not belong to the same daughter branch, assign the probability value of the chosen combination to the daughter branch containing the chosen combination and assign the
• the selection criterion consists in choosing one of the combinations having the maximum probability.
• the selection of a combination consists in implementing the recursive method comprising the following steps: a) randomly choosing a number p between 0 and 1; b) calculate the sum of the probability values assigned to the two so-called daughter branches resulting from the division of the first branch, and calculate for each daughter branch, a new probability value equal to the ratio between the probability value assigned to this daughter branch and the calculated amount; c) define two contiguous probability intervals between 0 and 1, the first interval being associated with a first daughter branch, the second interval being associated with the second daughter branch, the first interval ranging from 0 to the probability value of the first branch daughter included and the second interval going from this probability value to 1; d) identify in which interval the number p is located and select the daughter branch associated with this interval, and in the case where the selected daughter branch branches out into other branches, repeat the recursive process in step a
• the selection criterion consists in choosing one of the combinations having a predetermined probability value or between two given probability values.
• the probability values assigned to each branch are not normalized and can be greater than one.
• a weighting is assigned to each branch, the weighting of the branches of the last ramifications being equal to the product of the probability value assigned to this branch and the number of combinations of this branch , the weighting of the other branches being equal to the sum of the weightings of the branches coming from the branch considered and being on the next branching level.
• the probability value assigned to each branch can be normalized, the normalized probability value of a branch being obtained by dividing the probability value of this branch by the weight assigned to the first branch of the tree.
• the choice of a combination is made by implementing a method producing combinations having high probability values.
• the representation of the probability distribution of all the combinations is stored and can be refined later by the creation of additional branches, or can be simplified by the deletion of certain branches.
• the number of values that can be taken by a parameter is artificially increased, the probability value of a combination of values of control parameters including at least a value of one of the parameters corresponds to an added value is zero.
• FIG. 1 represents a car trying to follow a truck
• Figure 2A illustrates a side view of the car of Figure 1
• Figure 2B illustrates a top view of the car of Figure 1
• FIG. 3 illustrates a possible configuration of the car and the truck as well as three possible future positions of the truck
• FIGS. 4A to 4G illustrate steps of a method according to the present invention for constructing a representation of a probability distribution
• FIG. 5 illustrates in the form of a branched tree the steps of FIGS. 4B to 4G
• FIG. 6A to 6D illustrate two possible cutting modes of the set of couples (cty, ⁇ ⁇ ) according to the method of the present invention
• FIG. 7 illustrates an application of the method of the present invention to diagnosing a failure of a set of electrical components
• FIG. 8 illustrates an application of the method of the present invention to the recognition of figures.
• the method of the present invention applies to any system defined according to the criteria set out below. We must decide on the value to give to n specific parameters XS ⁇ to XS n of the system, knowing the values of n measurement parameters XM ⁇ to Mm corresponding to a determined state of the system. Each of the parameters can take a finite number of values. The values of the specific parameters form a continuous series of integers.
• the measurement parameters can be symbolic variables, the possible values being yellow, blue, green and red.
• a model of the joint probability distribution of the set of system parameters is known and the probability p (XS ⁇ , ..., XS n , XM, ..., XMrr ⁇ ) can be calculated for a given combination of all parameters (Si, ..., XS n , XM 1 , ... / Mrn) are relevant.
• the analytical functions and databases used by the system's probability distribution model are known.
• an inference consisting in defining the probability distribution of combinations of values of all or part of the specific parameters, for example (XS ⁇ , XS 2 XS n ), knowing the values of all or part of the measurement parameters, for example (XM ⁇ , M3).
• the probability p (XS-L, XS XSn / XMi, XM3) for the choice of a combination
• the present invention provides a method of constructing a representation of the probability distribution of the combinations of values of the k specific parameters chosen, obtained for the values read.
• a combination of values of the k parameters chosen will hereinafter be called "a combination”.
• the set of combinations can be represented by a set of points defined in a space E to k dimensions.
• the probability distribution is then represented in a space with k + 1 dimensions.
• the construction method of the present invention aims to divide the space E into several sets of points and to assign an identical probability value to all the points of the same set in order to obtain a representation of the probability distribution of the combinations.
• the choice of a combination is made according to one of several selection criteria.
• the method of constructing a representation of the probability distribution of combinations consists in successively choosing different combinations from the set of possible combinations and in calculating their respective probability values. After each choice of a combination, the set of points in space E containing the chosen combination is divided into several sets of points. The set of points containing the chosen combination takes the probability value of this combination. The other sets of points keep the probability value they had before the division. Initially, the space E is not divided and all the points of the space E take the value of probability p ⁇ of the first chosen combination C ⁇ .
• the choice of a second combination C 2 results in a division of the space E into several sets of points, the set of points containing the second chosen combination C 2 taking the value of probability p 2 of the. second chosen combination C 2 , the other sets of points taking the probability value Pi of the first chosen combination C ⁇ .
• the choice of a third combination C3, different from the first and second combinations chosen C ⁇ and C 2 involves the division of the set of points "father” containing the third combination chosen C3 into several sets of points "sons", l set of "child” points containing the third chosen combination C3 taking the probability value P3 of the third chosen combination, the other sets of "child” points taking the probability value of the set of points "father", p ] _ or p.
• the construction method of the present invention can be executed for a period of time. variable, the choice of execution time can be adapted to each system.
• the successively chosen combinations can be obtained according to a pseudo-random process producing combinations distributed uniformly over space E or according to an optimized process producing combinations having high probability values.
• each set of "child” points, resulting from the division of a set of "father” points comprises an identical number of combinations.
• the present invention provides for keeping track of the construction of the probability distribution via a construction tree.
• the first branch of the construction tree represents space E and takes the value of probability ⁇ -
• the first branch branches into second branches each representing one of the sets of points resulting from the division of space E.
• Each second branch takes the probability value of the points of the second branch considered.
• each second branch is likely to branch into several third branches according to the new combinations chosen.
• Each third branch can branch into several fourth branches and so on.
• the construction tree is memorized as it is built.
• the terminal branches of the tree give the final division of all the combinations.
• the final representation of the probability distribution will be by the sequence used to choose one of the combinations as will be described in the second part.
• the creation of a construction tree has several advantages, as will be specified below, in particular for obtaining standardized probability values and for selecting a combination by random drawing according to a random drawing process of the present invention. 1.3. Illustration for car / truck system
• the car's control system determines the values to the control parameters ( ⁇ ro t, v, cty () ⁇ - es one after the other in the case of a. choice of a speed, the probability p (v) so that the choice of a speed v at a time tj_ is relevant knowing the offset values c x j_ and Cyj_ noted at time tj_, can be calculated as follows:
• FIG. 4A represents a probability distribution 10 of the velocity values obtained for the offset values C XQ and Cy 0 recorded at time to- It is this distribution for which we seek to obtain an approximation, in a simple, rapid and minimizing the means of calculation and storage used.
• the speed v can take an integer value between 0 and 15 km / hour inclusive.
• the speed is represented on the abscissa, the probability p (v) is represented on the ordinate.
• the probability distribution 10 of the velocity values is a continuous function which is worth 0 when the speed is zero or greater than 14 and which has two maximums for speeds v equal to 4 km / h and 10 km / h.
• the set of FIGS. 4B to 4G illustrates the construction of a representation of the probability distribution of FIG. 4A.
• the sets of speed values, or branches, are represented by a two-way arrow positioned under the speed values forming part of the branch.
• a horizontal line intersecting the probability distribution 10, and placed above a two-way arrow, represents the probability value associated with the speed values of the branch represented by the two-way arrow.
• FIG. 4B illustrates a first stage of construction linked to the choice of a first speed value] _ equal to 4 km / h of probability p ⁇ , the set of speed values, forming a first branch B, takes the value of probability p ⁇ _.
• FIG. 4C illustrates a second stage of construction linked to the choice of a second speed value v 2 equal to 12 km / h, of probability p.
• the set of speed values is divided into two sets of speed values, each forming two second branches B ⁇ and B 2 .
• the branch B] _ groups together the lowest speed values ranging from 0 to 7 km / h.
• the branch B 2 groups together the highest speed values ranging from 8 to 15 km / h.
• the two branches B ⁇ and B 2 combine the same number of speed values.
• Branch B contains the second speed value chosen v 2 , so it takes the probability value p.
• the branch B ⁇ keeps the probability value p ⁇ .
• the branching of a branch leads to the creation of two branches comprising the same number of speed values, one of the branches comprising the lowest speed values, the other branch comprising the highest speed values.
• Figures 4D and 4E illustrate two phases of a third stage of construction linked to the choice of a third speed value V 3 equal to 6 km / h of probability value P 3 .
• the branch B ⁇ containing the third selected speed value V 3 branches into two branches B ] _] _ and i 2 as it appears in FIG. 4D.
• the branch B ⁇ 1 gathers the speed values going from 0 to 3 km / h
• the branch B ⁇ _ 2 gathers the speed values going from 4 to 7 km / h.
• the first and third chosen speed values v ⁇ and V 3 belong to the same branch B] _ 2 . In this case, the branches
• the branching of a "mother" branch containing the last speed value chosen continues until a "daughter" branch contains only the newly selected speed value and no other previously selected speed value.
• the intermediate "daughter" branches take the probability value of the "mother” branch.
• Figure 4E illustrates the second phase of the third step.
• the branch B ⁇ 2 containing the third speed value chosen V 3 branches into two branches B ⁇ 2 ] _ and B ⁇ 1 2 comprising the speed values 4, 5 and 6, 7 km / h respectively.
• the branch B] _ 2 2 contains only the third speed value chosen V 3 and no other speed value chosen. We therefore assign the probability value P 3 to the branch B ⁇ 2 2 •
• the branch Bi 2 , 1 keeps the probability value p ⁇ of the branch B ] _ 2 from which it comes.
• FIG. 4F illustrates a fourth stage of construction linked to the choice of a fourth speed value V 4 equal to 10 km / h of probability value p 4 .
• the branch B containing the fourth speed value chosen 4 branches into two branches B 2 ⁇ and B 2 , the branches grouping together the speed values 8 to 11 and 12 to 15 km / h respectively.
• the fourth speed value V4 belongs to the branch B 2 1 and no other speed value chosen belongs to this branch.
• P4 is then assigned probability value to the branch B 2 1 • T he branch B 2 2 Cover the probability value p.
• FIG. 4G illustrates a fifth stage of construction linked to the choice of a fifth value of speed V5 equal to 1 km / h, of probability, p ⁇ .
• Branch B] _ 1 containing the fifth selected V5 speed value branches into two branches B] _ 1 1 and B l 12> ⁇ - es branches respectively gathering speed values 0, 1 and 2, 3 km / h.
• the fifth speed value chosen 5 does not belong to the branch B ⁇ 1,1 and no other speed value chosen belongs to this branch.
• the branch B ⁇ 1 2 keeps the value of probability l. Note that at this last stage, we obtained in the form of segments a good approximation of the probability distribution 10 of FIG. 4A.
• FIGS. 6A to 6D represent the two-dimensional space E of the set of pairs of values of horizontal and vertical tilt angles ( ⁇ ⁇ , cty).
• a couple of values of horizontal and vertical tilt angles ((, () will hereinafter be called a .couple.
• the horizontal tilt angle (is shown on the abscissa.
• the vertical tilt angle cty is shown on the ordinate.
• the probability p ( ⁇ , cty / c x; j_, Cyj so that the choice of a couple ((, cty) is relevant knowing the offset values c x ⁇ and Cyj_ can be calculated according to the following formula:
• the embodiment of the method of the present invention for this example takes up the first and second aspects described above.
• the branches from a branch are two in number and contain the same number of pairs.
• the branching of a branch continues until the last couple chosen is the only chosen couple of one of the branches.
• the horizontal tilt angle (can take six values between 0 ° and 5 °
• the vertical tilt angle cty can take four values between 0 ° and 3 °.
• the number of values that a parameter can take is increased to the power of two immediately higher.
• the probability of couples for which one of the parameters corresponds to an added value (not initially planned) is zero.
• the horizontal tilt angle cc ⁇ can initially take six values. We therefore artificially increase the number of values to 8 (2 3 ), the possible values are now 0 ° to 7 °. No increase in the number of values is made for the vertical tilt angle cty for which 4 (2 2 ) values are possible.
• FIG. 6A represents the set of couples ( ⁇ n , cty) constituting the first branch B.
• the branch B branches into two branches Bi and B 2 according to a vertical limit 12 passing between the values 3 ° and 4 ° of angle of horizontal inclination.
• Branch B groups together couples with a horizontal angle of inclination strictly less than 4 (2 2 ).
• the branch B 2 groups together the couples having a horizontal angle of inclination greater than 4 (2 2 ).
• the branching of a "mother" branch leads to the creation of two branches along a vertical limit. In the case where it is impossible to define a vertical limit, that is to say when the couples of the "mother” branch all have the same value of angle of horizontal inclination (, the division is done according to a limit horizontal passing between two vertical tilt angle values cty.
• FIGS. 6C and 6D illustrate another possible ramification of branch B, the second pair chosen C2 • ⁇ j ⁇ ⁇ l) being different from C.
• FIG. 6C illustrates a first division of branch B along the same vertical limit 12 as that previously defined.
• the branches Bi and B 2 take the probability value pi of the first couple Ci and we proceed to a new branching of branch B containing the couple C 2 t until the two selected couples C and C 2 ⁇ are in separate branches.
• FIG. 6D illustrates the branching of the branch Bi according to a horizontal limit 13 passing between the values 1 ° and 2 ° of angle of vertical inclination cty.
• the branch Bl, l groups the couples having a vertical tilt angle value greater than or equal to 2.
• the branch Bi 2 groups the pairs having a value, vertical tilt angle strictly less than 2.
• the branch B 2 takes the probability value p 2 of the second pair C 2 ⁇ and the branch B 2 2 the probability value i-
• the successive ramifications of a "mother" branch are carried out successively according to a vertical limit and a horizontal limit until the new combination chosen to be the only one of the combinations chosen to belong to a given "daughter" branch.
• XS n obtained from formula (3) are in fact defined to within a constant, the normalization constant Z.
• This constant can be calculated only when the probability values of all combinations (XS ⁇ , XS 2 , XS n ) are known and have been calculated without taking Z into account (Z taken equal to 1).
• the normalization constant Z is then equal to the sum of all the probability values.
• the present invention provides for defining an intermediate normalization constant Zi which is an estimate of the normalization constant Z.
• the normalization constant Zi is calculated during the construction of the representation of the probability distribution. It is equal to the sum of the probability values of all the combinations, the probability value of a combination being that assigned to the branch containing the combination considered.
• the present invention provides for assigning a weight to each branch.
• the weighting of the branches of the last ramifications is equal to the product of the probability value assigned to the branch considered and the number of combinations of this branch.
• the weighting of one of the other branches is equal to the sum of the weightings of the branches from the branch considered and located on the next branching level.
• the weighting of the first branch is then equal to the intermediate normalization constant Zi.
• the weights are updated during a branching of one of the terminal branches of the construction tree.
• the weights of the branches between the first branch and the branching terminal branch must be recalculated.
• the weighting of the branch B at the end of the first step, FIG. 4B is equal to 16 * p ⁇ .
• the weightings of the new branches Bi and B 2 are worth 8 * p ⁇ and 8 * p 2 respectively
• the weighting of branch B is updated and is now worth 8 * p ⁇ + 8 * p 2 .
• FIG. 4C At the end of the third step, FIG.
• the weights of branches B ⁇ # 2 1 and B ⁇ , 2.2 are worth 2 * p ⁇ and 2 * P3 respectively, the weighting of branch B ⁇ 2 is now 2 * p ⁇ + 2 * P3, the weighting of branch Bi is 4 * p + (2 * p ⁇ + 2 * p3) and the weighting of branch B is ⁇ 4 * p ⁇ + (2 * p ⁇ + 2 * P3)) + 8 * p 2 .
• An advantage of the method of the present invention is that it makes it possible to quickly know the normalized probability value of a speed since it is not necessary to calculate all the probability values. 1.5. Advantages The method of the present invention makes it possible to represent very diverse probability distributions, unlike the "Gaussian mixture" method.
• the method of the present invention makes it possible to obtain a more or less detailed representation as a function of the available memory and the calculation time allowed.
• the implementation of an optimized method for the choice of combinations makes it possible to obtain in a very short time, a representation occupying little memory and having sufficient precision for the areas of space E where the combinations have high probability values.
• the process of the present invention is thus "multi resolution" in the sense that the cutting of the space E can be very fine for certain parts and coarse for others.
• Known modes of selection of one of the combinations from a representation of the probability distribution of the combinations consist in choosing one of the combinations having * the maximum probability or in choosing a combination by random draw, the principle of which will be recalled below.
• other selection criteria can be defined such as the choice of one of the combinations having a predetermined probability value or the choice of one of the combinations having a probability value between two given probability values.
• a memory register is used to store an indication of the branch or branches having the maximum probability value.
• the register initially memorizes the first branch of the construction tree, then it is updated during the construction of the representation of the probability distribution. At each branching of a branch, it is checked whether the probability value of the last combination chosen is greater than the probability value of the branch memorized by the register and if this is the case, the register is updated and memorized the new branch containing the last chosen combination. In the case where the branch of maximum probability branches and gives one or more "daughter" branches of the same probability, the register is updated and stores all the "daughter" branches.
• the choice of a combination then consists in identifying the branch memorized by the register containing the greatest number of combinations and in then choosing one of the combinations of this branch.
• a random draw consists in choosing one of the possible combinations in such a way that a combination with a high probability is very likely to be chosen and a combination with a low probability is unlikely to be chosen. Following a large number of random draws, the probability distribution of the "drawn" combinations is identical to the distribution of initial combination probability on which the random draw process is based.
• the random drawing of a combination is carried out according to a recursive selection method using
• each branch branches into two branches
• the method of the present invention comprises several steps described below.
• a number p between 0 and 1 inclusive is chosen at random.
• a second step we calculate the sum S of the probability values assigned to the 2 "daughter" branches from the branching of the first branch. Then, for each "daughter" branch, a new probability value equal to the ratio between the probability value assigned to the branch considered and the sum S calculated is calculated.
• a third step two contiguous probability intervals are defined between 0 and 1, the first interval being associated with a first daughter branch, the second interval being associated with the second daughter branch. The first interval goes from 0 to the probability value of the first child branch included and the second interval goes from this probability value to 1.
• a fourth step we identify in which interval the number p is located and we select the "daughter" branch associated with this interval.
• the recursive process is repeated in the first step, the first branch being replaced by the "daughter” branch selected.
• the first number p chosen may be used again.
• the recursive process stops and one of the combinations of the selected “daughter” branch is chosen.
• the aforementioned recursive selection method is used in the example of the car / truck system to choose a speed or a couple of angles of inclination (cty, and] -,).
• An advantage of the method of the present invention is that the random drawing method is simple and easy to implement.
• the accuracy of the representation of the probability distribution can be reduced by removing more or less terminal branches from the construction tree.
• FIG. 7 represents an electrical device which comprises several components between an input I and an output 0, each component being capable of passing a part of the input current K when it is operating, no current flowing when the component is broken down.
• a component A is placed between the input I and a first intermediate point 100.
• the current KA at the output of component A is equal to 100% of the current K when the component is operating (and equal to 0% of the current K when the component is Out of order) .
• Components B and C are placed in parallel between the first intermediate point 100 and a second intermediate point 101.
• the output current KB of the component B is at most equal to 40% of the current K when the component B is operating.
• Component C consists of two components Ci and
• Each component Ci and C 2 can pass up to 30% of the current K.
• the output current KC of the component C can therefore be equal to 0%, 30%, or 60% of the current K, depending on whether both components have failed, only one component has failed, or both components are functioning.
• a component D is placed between point 101 and the output O.
• Component D consists of eight components Di to Dg in parallel. Each component Di to Dg can pass up to 15% of the current K when it is operating. In addition, it is necessary that at least six of the components Di to Dg function for the component D to function.
• KBC current can be 0%, 30% (only Ci or C works), 40% (only B works), 60% (Ci and C 2 work), 70% (Ci or C 2 and B work) or 100% (Ci, C 2 and B work) of current K.
• the output current KD of component D can be equal to 0%, 30%, 40%, 60%, 70% (KBC is respectively equal to 0%, 30%, 40%, 60%, 70% of current K and at least six of the components Di to D 8 work), 90% (KBC is equal to 100% of the current K and 6 components Di to Dg work) or 100% (KBC is equal to 100% of the current K and at least 7 components Di to Dg operate) of current K.
• Each component A, B, C, C 2 and Di to Dg can be considered as a parameter of the device which can take the value 0 when the component is faulty and 1 when the component is operating.
• the output currents KA, KB, KC, KBC and KD are also considered to be parameters of the device which can take more or less values.
• KA and KB can take the value 0 or 1 according to whether the current is worth respectively 0% or 100% of the current K.
• KC can take the values 0, 1 or 2 according to whether the current is worth respectively 0%, 30% or 60% of the current K.
• KBC can take the values 0, 1, 2, 3, 4 or 5 depending on whether the current is respectively 0, 30, 40, 60, 70 or 100% of the current K.
• KD can take the values 0 to 7 depending on whether the current is respectively 0, 30, 40, 60, 70, 90 or 100% of the current K.
• p (KA / A) is the probability that KA has a given value knowing the value of A
• p (KB / B, KA) is the probability that KB has a given value knowing the value of B and KA
• p (KC / C, KA) is the probability that KC has a given value knowing the value of C and KA
• p ( KBC / KB, KC) is the probability that KBC has a given value knowing the value of KB and KC
• p (KD / KBC, D) is the probability that KD has a given value knowing the value of KBC and D.
• the simple probabilities p (A) to p (KD) can be defined from databases recording the cases of failure that appeared during tests carried out by the company manufacturing the device.
• the current can be measured at a point on the device (KA, KB, KC, KBC or
• the parameters A, B, Ci, C 2 , Di to Dg, KA, KB, KC, KBC and KC can be either specific parameters whose value is to be determined, or measurement parameters because a measurement of. current or operating state is achieved, i.e. parameters not retained because we do not want to determine their value and no measurement is made of their value.
• the measurement parameters are one or more currents, for example KBC
• the specific parameters are one or more components, in this example the components A, B, Ci and C located upstream of the KBC current.
• the probability p (A, B, C ⁇ , C 2 / KBC) for a combination (A, B, C, C 2 ) to represent the state of the device knowing the value of KBC can be calculated as before by summing of all probabilities p (A, B, C ⁇ , C 2 , D ⁇ , .., D 8 , KA, KB, KC, KBC, KE) for all the values of the parameters not retained in this inference (Di to Dg, KA, KB, KC, KD).
• control parameter will be the current that one wishes to know, for example KBC
• the measurement parameters can be one or more of the others settings.
• a representation of the probability distribution of the possible values of the KBC current we will construct, according to the method of the present invention, a representation of the probability distribution of the possible values of the KBC current. The value of the KBC current having the maximum probability is then selected.
• FIG. 8 represents an image 200 of a handwritten figure which one wishes to identify.
• the image is broken down into 64 squares or pixels of identical size. For each pixel, a gray level representing the surface occupied by the lines of the digit in the pixel considered is measured on a scale from 0 to 16.
• the system comprises a single parameter to be determined (or specific parameter) NUMBER which can take the values 0 to 9 and 64 measurement parameters Pix [i], i being an integer between 1 and 64, which can each take the values 0 to 15.
• p (NUMBER) is the probability d 'have a given number
• p (Pix [i] / NUMBER) is the probability of having a given gray level for the pixel Pix [i] knowing the number.
• p (NUMBER) is equal to 1/10.
• the conditional probabilities p (Pix [i] / NUMBER) are defined at the end of a learning phase consisting in measuring the gray levels of each pixel for different models of handwritten numbers.
• Each probability p (Pi [i] / NUMBER) can be defined by a histogram (with 16 columns) normalized according to Laplace's law.
• the recognition of a figure comprises a first phase of measuring the gray level of each pixel.
• a second phase we build from formula (5) and method of the present invention a representation of the probability distribution of the figures knowing the gray levels of each pixel.
• the choice of a figure consists in identifying the figure presenting the maximum probability.
• a shipping company transports cargo from a European port to another port.
• the transported goods can be very diverse: foodstuffs, medicines, electrical appliances or clothing.
• Different kinds of containers are used for the storage of goods in the cargo ship and in the port of arrival.
• the containers can be refrigerated and more or less large.
• the shipping company wants to quickly establish, during a telephone conversation for example, transport cost quotes knowing which ports of departure (PortDep) and arrival (PortArr), the type of goods transported (Tue) , the container used (Cont), the customer (Cl) and the month (M) during which the transport will take place. These parameters, informed prior to the establishment of the estimate, constitute the measurement parameters of the transport system.
• the company has a whole range of information concerning, in particular, the time for preparing containers in the European port of departure (TdP), the time for maritime transport (TdTM) (outward or return, the containers are full by '' and empty on return), the waiting time for the container in the port of arrival (TdA), the time for unloading the container at the customer (TdDC), the time for reconditioning the containers on their return to Europe ( TdRE), times being counted in days.
• information is available on the daily cost of renting a container (CdLJ), the daily cost of immobilizing a container for its repackaging (CdU), the cost of maritime transport (CdTM), the cost of repairing a container (CoR).
• the model of the joint probability distribution of all the previously defined parameters is constructed from the independent probability distributions defined for each time parameter (TdP, TdTM, TdA, TdDC and TdRE), and for each cost parameter (CdLT, CdIT , CdTM, CdE, CdR, CT).
• Probability values are obtained from a set of data acquired over the life of the business.
• the probability distributions of container preparation times in the port of departure (TdP) knowing the nature of the goods (Mar) are a family of Gaussians.
• the probability distributions of maritime transport costs (CdTM) knowing the type of container (Cont), the port of departure (PortDep), the port of arrival (PortArr), and the goods transported (Mar) are functions of Dirac .
• TC total cost of transport
• TC total cost
• TC total cost
• the seller first wants to know what is the maximum cost, for example 2000 euros. He then makes an initial quote, possibly taking a 10% margin from the maximum cost and then offers 2200 euros. In the event that the customer does not accept price, the seller can then estimate what is the average cost or what is the range of costs containing for example 90% of the possible cost values. The average cost can be easily calculated by dividing the intermediate normalization constant Zi by the number of possible total costs. The seller will then make a second quote, taking a possibly lower margin compared to the average cost or compared to one of the costs in the cost range.
• the established estimate can also detail all the costs, in this case the system measurement control parameters are (CdLT, CdIT, CdTM, CdE, CdR).
• the total cost is then calculated from the chosen cost combination.
• This application example shows that from a representation, one can easily determine the maximum probability combination, calculate the mean value and the standard deviation of the values of a parameter and select one of the combinations having a given probability.
• the method of the present invention therefore makes it possible to easily implement several selection criteria.
• the present invention is susceptible to various variants and modifications which will appear to those skilled in the art.
• those skilled in the art will be able to define the branching process of a branch that is most suited to the system studied.
• those skilled in the art will be able to define new criteria for selecting a combination from the tree-like representation of the probability distribution.

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• 2002
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