JP5603827B2 - Generation of regression equation for control factor identification - Google Patents

Description

  The present invention relates to a method for generating a regression equation for specifying a control factor that adapts a control result to a desired range in the control of various devices such as a vehicle.

  In the control of various devices including a vehicle, the control factor is appropriately adjusted according to various situations, and the control result (evaluation index) falls within a predetermined allowable range. In performing such control, it is common to obtain in advance an expression representing a correspondence relationship between a related control factor and a control result (evaluation index). Specifically, a regression equation representing the correspondence between these control factors and control results (evaluation indicators) is derived in advance from a plurality of sample data consisting of combinations of control factors and control results (evaluation indicators). Then, a control factor necessary for obtaining a control result (evaluation index) within an allowable range in the control of various devices is calculated using the regression equation. In this way, various requirements regarding the characteristics of various devices are met.

  Incidentally, a prediction model (regression equation) is created using learning data (a plurality of sample data consisting of combinations of control factors and control results (evaluation indicators)), and a prediction value (control result) is created using the created prediction model. An apparatus and a method for calculating the value are disclosed in, for example, Japanese Unexamined Patent Application Publication No. 2005-135287 (Patent Document 3) and Japanese Unexamined Patent Application Publication No. 2006-338123 (Patent Document 4).

  On the other hand, a technique called a support vector machine (Support Vector Machine: hereinafter referred to as “SVM”) or a support vector regression method (Support Vector Regression: hereinafter also referred to as “SVR”) has attracted attention in the field of pattern recognition and regression estimation. Yes. These methods are also studied as a method for deriving a regression equation from sample data as described above. A technique for setting a control target value of a vehicle using such SVM or SVR is disclosed in, for example, Japanese Patent Application Laid-Open No. 2010-2010. No. 78084 (Patent Document 1) and Japanese Patent Application Laid-Open No. 2008-45475 (Patent Document 2).

  SVR has the advantage of being able to generate regression equations with good prediction performance. However, depending on the formulation, it may be necessary to solve the quadratic programming problem without using a solution (for example, an algorithm such as the SMO method) that can reduce the calculation time of the regression equation. In such a case, the solution time of the quadratic programming problem is not limited to the SVR, and is generally the order of the cube of the number of sample data, and the storage area required when the quadratic programming problem is solved by the computer. It is known that the capacity increases in the order of the square of the number of sample data. Accordingly, when the number of sample data is large, the calculation time for generating the regression equation is very long.

  Note that “sequential learning method” is known as one of the means for speeding up the generation of regression equations in SVR. Various methods are known as “sequential learning methods”. However, as a main method, sample data is divided into a plurality of groups, SVR is performed on the first group, and a sample that becomes a support vector at that time There is a method of repeating a cycle in which only data is left, and SVR is further performed on a sample data group in which these remaining sample data are combined with sample data of the next group.

  The “sequential learning method” has an advantage that the generation of a regression equation in SVR can be speeded up. However, the method of leaving the sample data that became the support vector in the SVR for the sample data of the previous group as the sample data also in the SVR executed in the next cycle is disadvantageous in that it is easily affected by the abnormal value mixing. Have Further, when the number of sample data serving as support vectors increases, the number of sample data to be processed in one SVR increases as a result, and the advantage of speeding up the generation of regression equations is diminished.

JP 2010-078084 A JP 2008-045475 A JP 2005-135287 A JP 2006-338123 A

  The present invention was created to solve the above-described problems. That is, the present invention does not significantly increase the calculation time required to generate the regression equation even when the number of sample data is enormous, and is not easily affected by outliers in the sample data, An object of the present invention is to provide a method for generating a regression equation. More specifically, in the present invention, for example, when generating a regression equation used in the control of various devices such as vehicles, even when a large amount of sample data is used, the regression is compared with the conventional regression method. It is an object of the present invention to provide a method for generating a regression equation for specifying a control factor, which can shorten the time required for generating an equation and can maintain accuracy even when an abnormal value is mixed.

The above object of the present invention is to
A method of generating a regression equation representing a relationship between a control factor of a device to be controlled and an evaluation index corresponding to the control factor,
A sample data dividing step for dividing a plurality of sample data including information on control factors and evaluation indices into a plurality of groups;
A regression equation generating step for generating a regression equation based on sample data belonging to one of a plurality of groups divided in the sample data dividing step,
If the execution of this step is the first time, a regression equation representing the relationship between the control factor and the evaluation index included in the sample data belonging to the group is obtained,
When the execution of the step is not the first time, the result obtained from the control factor included in the sample data belonging to the group based on the regression equation obtained by the previous execution of the step and the sample data belonging to the group To calculate an error with the evaluation index included in and to obtain a regression equation representing the relationship between the error and the control factor,
Regression equation generation step,
A regression equation addition step for generating a new regression equation by additively overlapping the regression equation obtained in the regression equation generation step with an existing regression equation,
When the execution of this step is the first time, the regression equation obtained in the regression equation generation step is a new regression equation,
If the execution of the step is not the first time, the regression equation obtained in the regression equation generation step and the existing regression equation obtained by the previous execution of the step are additively superimposed to obtain a new regression equation Generate
Regression equation addition step,
Achieved by a method comprising:

  According to the present invention, even when the number of sample data is enormous, the calculation time required to generate the regression equation does not increase remarkably, and it is difficult to be affected by the abnormal value mixing in the sample data. A method for generating a regression equation is provided. More specifically, according to the present invention, even when a large amount of sample data is used for generating regression equations used in the control of various devices such as vehicles, for example, compared with the conventional regression method. There is provided a method for generating a regression equation for specifying a control factor that can reduce the time required for generating a regression equation and can maintain accuracy even when an abnormal value is mixed.

It is a graph showing the outline of the regression type production | generation method which concerns on one embodiment of this invention. It is a flowchart showing the processing flow in the regression type production | generation method which concerns on one embodiment of this invention. The extent of the influence on the measurement time by mixing of an abnormal value and measurement noise about the regression equation generation method, the conventional method (sequential learning method), and the normal method (collective regression) according to one embodiment of the present invention is shown. It is a graph. Regression formula generation method according to one embodiment of the present invention, conventional method (sequential learning method), and normal method (collective regression) to the regression accuracy (correlation coefficient) due to mixing of abnormal values and measurement noise It is a graph which shows the grade of influence. In the regression equation generation method according to one embodiment of the present invention, the relationship between the prediction error and the width R of the Gaussian function used as the basis function of the regression equation and all sample data are obtained. It is a graph compared with the case where it asked about. In the regression equation generation method according to one embodiment of the present invention, the prediction accuracy of the regression equation (generalized performance estimation) by the combination of the Gaussian function width R of the first regression equation and the Gaussian function width R of the second regression equation It is a graph showing a change of (value). In the regression equation generation method according to one embodiment of the present invention, the prediction accuracy of the regression equation (generalized performance estimation) by the combination of the Gaussian function width R of the first regression equation and the Gaussian function width R of the second regression equation It is a contour diagram that represents a change in (value) two-dimensionally.

  As described above, the present invention, for example, in the generation of regression equations used in the control of various devices such as vehicles, even when using a huge number of sample data, compared with the conventional regression method, An object of the present invention is to provide a method for generating a regression equation for specifying a control factor, which can shorten the time required for generating a regression equation and can maintain accuracy even when an abnormal value is mixed.

That is, the method according to the first aspect of the present invention is:
A method of generating a regression equation representing a relationship between a control factor of a device to be controlled and an evaluation index corresponding to the control factor,
A sample data dividing step for dividing a plurality of sample data including information on control factors and evaluation indices into a plurality of groups;
A regression equation generating step for generating a regression equation based on sample data belonging to one of a plurality of groups divided in the sample data dividing step,
If the execution of this step is the first time, a regression equation representing the relationship between the control factor and the evaluation index included in the sample data belonging to the group is obtained,
When the execution of the step is not the first time, the result obtained from the control factor included in the sample data belonging to the group based on the regression equation obtained by the previous execution of the step and the sample data belonging to the group To calculate an error with the evaluation index included in and to obtain a regression equation representing the relationship between the error and the control factor,
Regression equation generation step,
A regression equation addition step for generating a new regression equation by additively overlapping the regression equation obtained in the regression equation generation step with an existing regression equation,
When the execution of this step is the first time, the regression equation obtained in the regression equation generation step is a new regression equation,
If the execution of the step is not the first time, the regression equation obtained in the regression equation generation step and the existing regression equation obtained by the previous execution of the step are additively superimposed to obtain a new regression equation Generate
Regression equation addition step,
It is a method including.

  Here, although the apparatus used as the said control object contains moving bodies, such as a vehicle and a ship, for example, these apparatuses are not limited to a moving body. That is, the present invention calculates a target value of a control factor suitable for obtaining a control result within an allowable range using a regression equation, and controls the control factor to the target value thus obtained. Therefore, the present invention can be applied to any device as long as it is intended to obtain a desired control result.

  In the case where the device is a vehicle, the vehicle here means, for example, a train, a train, an automobile, or the like, a vehicle that transports passengers / cargo, etc. Examples include, but are not limited to, engines and electric motors. Moreover, the control in a vehicle refers to satisfying various requirements regarding various characteristics of the vehicle. More specifically, for example, fluctuations in engine torque in an internal combustion engine, the amount of NOx (nitrogen oxide) generated in the combustion chamber, the amount of unburned HC (unburned hydrocarbon) discharged from the combustion chamber, AT ( Various characteristics such as a shift shock in an automatic transmission) and compatibility between vehicle attitude maintenance during driving and steering stability are used as evaluation indexes, and adjustment so that these evaluation indexes fall within a desired range is referred to as vehicle control. However, it is not limited to these specific examples.

In order to adjust the evaluation index, the corresponding control factor is appropriately adjusted according to the state of the device to be controlled (for example, the driving situation of the vehicle), and the control result as the evaluation index is in a desired range. Need to fit in. When the device to be controlled is a vehicle, the control factors include, for example, the opening degree of the throttle valve, the ignition timing by the ignition plug, the opening / closing valve characteristics of the intake valve or the exhaust valve, the fuel injection amount from the fuel injection valve , timing of transmission, various parameters such as the rotation angle of the rotary valve of Abu source over bar (control factor).

  That is, when trying to control the state of the device (for example, the driving situation of the vehicle, etc.), the control factor is appropriately adjusted according to various situations, and the control result (evaluation index) is within a predetermined allowable range. Can be stored. In performing such control, it is common to obtain in advance an expression representing a correspondence relationship between a related control factor and a control result (evaluation index). Specifically, from multiple sample data (experimental values) consisting of combinations of control factors and control results (evaluation indicators), a regression equation representing the correspondence between these control factors and control results (evaluation indicators) is pre- The target value of the control factor necessary for obtaining the control result (evaluation index) within the allowable range in the control of the device to be controlled is calculated using the above regression equation. By adjusting the control factor to the target value thus calculated, various requirements regarding various characteristics of the device are satisfied.

  Hereinafter, an outline of one embodiment of the present invention will be described with reference to the drawings. FIG. 1 is a graph for explaining a difference between a normal regression equation generation method and a regression equation generation method according to one embodiment of the present invention.

  FIG. 1A shows a method for generating a regression equation by a normal method. This method is a method of regressing the relationship between factors and evaluation indices for all sample data at once. As described above, if the regression equation is obtained at once for all the sample data in this way, there is a disadvantage that the calculation time becomes very long to generate the regression equation when the number of sample data is large.

  On the other hand, FIGS. 1B to 1D illustrate an outline of a data processing procedure in the regression equation generation method according to one embodiment of the present invention.

  First, FIG. 1B divides the same sample data as in FIG. 1A into two groups (corresponding to the “sample data division step”), and belongs to one of the groups (first group). The figure shows the step of obtaining a regression equation for sample data only (corresponding to the first execution of the “regression equation generation step”). Next, in FIG. 1C, a step of obtaining a regression equation representing an error from sample data belonging to the other group (second group) with respect to the regression equation obtained in FIG. This corresponds to the case where the “generation step” is not executed for the first time). Further, in FIG. 1 (d), the regression equation obtained in FIG. 1 (b) and the regression equation (with respect to error) obtained in FIG. This corresponds to the case where the “regression formula addition step” is not executed for the first time). By this step, regression equations corresponding to all sample data belonging to the first group and the second group (that is, all sample data) can be obtained.

  In the above description, the case where the sample data is divided into two groups has been described. However, in the case where the sample data is divided into three or more groups, depending on the number of groups, FIGS. ) Is repeatedly executed (corresponding to the case where the “regression equation generation step” is not executed for the first time), the regression equations corresponding to the sample data belonging to all groups (that is, all sample data) are obtained.

  By the way, the method for generating the regression equation in the method according to the present invention is not particularly limited, and various regression methods can be applied. As an example of a preferable regression method, as described above, there is a method called a support vector regression method (Support Vector Regression: hereinafter also referred to as “SVR”).

  Therefore, the method according to the second aspect of the present invention is the method according to the first aspect, wherein the regression equation is obtained by support vector regression in the regression equation generation step.

  Next, referring to FIG. 2, there is shown a flowchart 10 representing a processing flow in the regression equation generation method according to one embodiment of the present invention. In this example, the sample data group is divided into a plurality of groups in steps 11 and 12 (corresponding to the “sample data dividing step”). When dividing the sample data group into multiple groups, each sample data must be divided so that sample data existing in a specific range in the control factor (input) space does not exist in a specific group. It is desirable to distribute evenly to groups.

  Therefore, the method according to the third aspect of the present invention is the method according to the first or second aspect, wherein in the sample data dividing step, sample data existing in a specific range on the control factor space is stored. In this method, sample data is uniformly distributed to each group so as not to be biased to a specific group.

  An example of a procedure for uniformly grouping the sample data groups as described above will be described in detail below. First, sample data (experiment point) closest to the center (center of gravity) in the factor (input) space is selected for the sample data group, and the order is determined to be “confirmed” and “confirmed order = 1”. (Step 11 in FIG. 2).

  Next, for each piece of sample data that has not yet been “order fixed”, the distance in the factor (input) space with all the sample data that has already been “order fixed” is obtained. For each sample data, the minimum distance (minimum distance) is specified from the distances thus determined. The sample data having the maximum minimum distance specified in this way is specified, this sample data is set to “order fixed”, and the fixed order is counted up by one and given to this sample data. This process is repeated until all the sample data is “order fixed” and a fixed order is given. Thereafter, the sample data is sorted in ascending order of the confirmation order, and the sample data as many as necessary for each group are selected and grouped (step 12 in FIG. 2).

  In other words, the order is determined starting from the sample data closest to the center (center of gravity) in the factor (input) space, and the least sparse (the minimum distance is the maximum) for all the sample data for which the order has already been determined. The order is determined one after another for the sample data, and the sample data is sorted according to the order thus determined, and as many sample data as necessary for each group are sorted.

  The above procedure is merely an example of a procedure for dividing a sample data group into a plurality of groups, and sample data existing in a specific range in the factor (input) space is biased to a specific group. As long as the sample data can be uniformly distributed to each group so that there is nothing, there is no limitation to the above procedure.

  After dividing the sample data group into a plurality of groups as described above, the processing flow enters the regression number loop (steps 20 to 70 in FIG. 2). First, there is a conditional branch as to whether or not it is the first regression (step 30 in FIG. 2), and in the case of the first regression, a regression equation representing the relationship between the factor and the evaluation index is obtained (step 41 in FIG. 2). (This corresponds to the case of the first execution of the “regression formula generation step”).

  Thereafter, the process flow proceeds to step 60 corresponding to the first execution of the “regression formula addition step”. That is, in the case of the first execution, since only one regression equation has been generated yet, the regression equation is not added in the step 60, the process proceeds to step 70, the process returns to step 20, and the next group of sample data Proceed to the process.

  In the second and subsequent regressions, the process flow proceeds to step 51 by a conditional branch (step 30 in FIG. 2). In this step, instead of obtaining a regression equation that represents the relationship between the control factor and the evaluation index as in the first regression, the error from the previously obtained regression equation is calculated as individual sample data (control factor and evaluation index 2) (step 51 in FIG. 2), and a regression equation representing the relationship between the calculated error and the control factor is obtained (step 52 in FIG. 2, when the “regression equation generation step” is not the first execution). Correspondence).

  Thereafter, the processing flow proceeds to step 60 corresponding to the case where the “regression formula addition step” is not executed for the first time. That is, when it is not the first execution, that is, in the second and subsequent regressions, the regression equation generated by the previous processing has already been generated. Therefore, the existing regression equation and the regression equation obtained in step 52 are obtained. Are added to form a new regression equation. Thereafter, the processing flow between step 20 and step 70 is repeated until the processing for all groups of sample data is completed.

  As described above, when the processing for all the groups of sample data is completed, the processing flow ends. The regression equation at this point is a regression equation corresponding to all sample data in which the regression equation obtained in the first regression and all of the regression equations obtained thereafter are additively superimposed. ing.

  Incidentally, as described above, in the flowchart shown in FIG. 2, the process of adding the regression formula is executed for each cycle of the processing loop for generating the regression formula. There is no need to execute at the above timing. For example, a regression equation addition step that adds a regression equation generated in the previous processing loop to an existing regression equation to form a new regression equation may be executed immediately before step 51 (however, in this case, After the processing of the last group of sample data is completed, a separate process of adding the regression equation corresponding to the group (regression equation of error from the previous regression equation) to the existing regression equation is required).

  As described above, the present invention should not be applied only to the control of a specific device, but the target value of a control factor suitable for obtaining a control result within an allowable range is expressed by a regression equation. The present invention is applicable to any apparatus as long as it is intended to obtain a desired control result by calculating using it and controlling the control factor to the target value thus obtained. Therefore, the method according to any one of the first to third aspects, wherein the device to be controlled is a vehicle, is also an aspect of the present invention. Included in the range. That is, the method according to the fourth aspect of the present invention is the method according to any one of the first to third aspects, wherein the device to be controlled is a vehicle.

  By the way, as described above, by dividing the sample data into a plurality of groups and performing regression on each group, the quadratic programming problem can be solved without using the SMO method or the like that can shorten the calculation time of the regression equation. However, even when the number of sample data is large, the time required for generating the regression equation can be shortened. As described above, the solution time of the quadratic programming problem is the cube of the number of sample data. This is because the capacity of the storage area required when the quadratic programming problem is solved by a computer in the order of 2 is increased by the order of the square of the number of sample data.

More specifically, there are N pieces of sample data, and when these sample data are collectively regressed, assuming that the calculation time of the regression equation is T ₀ , these sample data are divided into k groups. The number of sample data included in each group is N / k, and the calculation time of the regression equation in each group is T ₀ × (1 / k) ³ . Therefore, the calculation time of all groups is k × T ₀ × (1 / k) ³ , and when this is grouped with sample data or when the second and subsequent groups are processed, an error from the existing regression equation is added. The total processing time is obtained by adding the time required for the calculation or adding the regression equations obtained for each group. However, the time required for these incidental processes is sufficiently small compared with the calculation time of the regression equation. Therefore, for example, when the sample data is divided into 10 groups, the calculation time is reduced to about 1/100 compared with the case where the sample data is processed in a lump.

  As described above, according to the method according to the present invention, the sample data is divided into a plurality of groups, and the regression equation is calculated for each group. Therefore, when the number of sample data is large, the calculation time becomes very long. Can be resolved.

  Further, as described above, since the capacity of the storage area required when solving the quadratic programming problem with a computer increases in the order of the square of the number of sample data, according to the method according to the present invention, Since it is possible to reduce the number of sample data when calculating the regression equation for the group, it is possible to reduce the memory capacity of the computer required to obtain the regression equation, so even in a computer with a relatively small storage capacity The regression equation can be calculated using a lot of sample data.

  By the way, according to the method according to the present invention, the sample data is divided into a plurality of groups and the regression equation is calculated for each group. The accuracy tends to decrease as the sample data is divided into more groups (in other words, as the number of sample data belonging to each group decreases). That is, the calculation speed and the regression accuracy are in a trade-off relationship. Therefore, by appropriately adjusting the number (size) of groups of sample data according to the accuracy required for controlling the target evaluation index (control result), the calculation speed and the regression accuracy can be appropriately balanced. it can.

  As described above, the regression accuracy in the method according to the present invention tends to decrease as the sample data is divided into more groups, but this tendency is extremely small. This is because, in the method according to the present invention, in the second and subsequent regressions, the error with respect to the regression equation obtained in the immediately preceding stage is approximated. Since the error that becomes positive and the error that becomes negative are distributed almost evenly, a regression equation for these errors is obtained, and this regression equation is added to the reference regression equation to form a new regression equation. In this case, it is considered that the error in each part acts in the direction to be canceled and the error is hardly increased.

  Further, in the method according to the present invention, when noise or abnormal value is included in the sample data, the regression equation for the group including the sample data is affected by the noise or abnormal value. It does not affect the regression equation. Therefore, as a result of additively superimposing the regression equations obtained for each group to obtain a new regression equation, the regression equation affected by noise and anomalous values and the regression equation not affected by this are compatible. Averaged. As a result, in the method according to the present invention, the influence of noise and abnormal values is reduced.

  As described above, according to the present invention, even when the number of sample data is enormous, the calculation time required for generating the regression equation is remarkably increased as compared with the conventional regression method. Provided is a method for generating a regression equation for specifying a control factor, which can avoid a decrease in accuracy due to the influence of an abnormal value in sample data.

  The regression equation for specifying the control factor generated by the method according to the present invention is, for example, a signal input from a detection device such as a CPU, a storage device (for example, a volatile memory, a non-volatile memory, a hard disk, etc.) or various sensors. Control unit (for example, ECU in a vehicle equipped with an internal combustion engine) including an actuator, an actuator such as a servo motor, a signal output unit to other devices, etc. When it is performed, it is read out by a control program or the like, and is used for processing for calculating a target value of a control factor for obtaining a desired control result under a constraint condition according to the situation of the apparatus at that time.

  By the way, in the regression equation generation method according to the present invention, a regression equation employing a so-called “Gaussian function” as a basis function is also assumed. The Gaussian function is represented by the following general formula (A), for example.

Therefore, a regression equation using a Gaussian function as a basis function is represented by the following general formula (B) (wherein β ₁ is a coefficient).

  As described above, when a Gaussian function is adopted as the basis function of the regression equation, it is necessary to separately determine the width of the Gaussian function (R in the general formula (B)) using an appropriate method. As a method for determining the width R of the Gaussian function, any of various methods generally performed in the technical field may be used. As an example, for example, several candidates for the width R of the Gaussian function are listed, the prediction error of the regression equation using each width R is obtained, and the width R used in the regression equation that gives the smallest prediction error is adopted. The method of doing is mentioned.

  Whichever method is adopted, when a Gaussian function is adopted as the basis function of the regression equation, determining the width R of the Gaussian function is an unavoidable procedure. Therefore, even in such a case, the present invention reduces the calculation time required to generate the regression equation even when the number of sample data is enormous, and maintains high accuracy even if there are abnormal values mixed in. In order to achieve the above objective, a method capable of determining the width R of the Gaussian function that gives a highly accurate regression equation with a smaller amount of processing (amount of work, calculation amount) is required.

  Therefore, as a result of earnest research, the present inventor, when adopting a regression equation using a Gaussian function as a basis function, first determines the width R of the Gaussian function as described above at the first execution of the regression equation generation step, At the second and subsequent executions of the regression formula generation step, the width R of the Gaussian function is set to a value smaller than the width R at the first execution, so that the second and subsequent processings can be performed with a smaller amount of processing (amount of work and calculation) It has been found that the width R at the time of regression execution can be determined, and the accuracy of the regression equation thus generated is also high.

That is, the method according to the fifth aspect of the present invention is:
A method according to any one of the first to fourth aspects,
The regression equation uses a Gaussian function as a basis function, and in the regression equation generation step, compared to the width of the Gaussian function when the regression equation generation step is executed for the first time, the regression equation generation step The width of the Gaussian function is smaller when the execution is not the first time,
It is the method characterized by this.

  As described above, the present embodiment assumes a case where a regression equation using a Gaussian function as a basis function is employed. In the present embodiment, as described above, in the regression formula generation step, the regression formula generation step is executed in comparison with the width (R) of the Gaussian function when the regression formula generation step is executed for the first time. The width (R) of the Gaussian function when it is not the first time is made smaller.

  More specifically, in the present embodiment, at the second and subsequent executions of the regression equation generation step, only a value smaller than the Gaussian function width (R) obtained at the first execution is used as the Gaussian function width (R). ) As a candidate. In other words, when the regression equation generation step is executed for the second and subsequent times, a value larger than the Gaussian function width (R) obtained at the first execution may be excluded from the Gaussian function width (R) candidates. it can. Thereby, the amount of calculation for determining the width (R) of the Gaussian function can be reduced. In some cases, when the regression equation generation step is executed for the second time or later, an optimal value is not selected from a plurality of candidates, but a value sufficiently smaller than the width (R) of the Gaussian function obtained at the first execution ( For example, 1) may be set as the width (R) of the Gaussian function.

  In this embodiment, the above procedure is repeated at the third and subsequent executions of the regression equation generation step, and a smaller Gaussian function width (R) is adopted each time the regression equation generation step is executed. May be. Alternatively, when the regression equation generation step is executed for the third time or later, the above procedure is omitted, and the width (R) of the Gaussian function obtained during the second execution of the regression equation generation step is calculated as the third or subsequent time of the regression equation generation step. You may use it as it is even when executing.

  As described above, in the regression equation generation method according to the present invention, sample data is divided into a plurality of groups, and regression is performed for each group. Accordingly, when a Gaussian function is used as the basis function of the regression equation, the width (R) of the Gaussian function is obtained based on only sample data belonging to one group among a plurality of groups at the first execution of the regression equation generation step. It will be. Thus, it can be considered that the validity (gauss function width) of the Gaussian function obtained based only on a part of sample data is concerned.

  However, in practice, the width (R) of the Gaussian function obtained based on only a part of the sample data is compared with the width (R) of the Gaussian function obtained based on a larger number of sample data. Although there is a difference in the magnitude of the prediction error (generalization performance estimate), there is no difference in the optimal Gaussian function width (R) value that gives the smallest prediction error (generalization performance estimate). I can't. That is, even if the Gaussian function width (R) is obtained based on only a part of sample data as in the regression equation generation method according to the present invention, the optimum Gaussian function width (R) can be determined ( Details will be described in Examples).

  By the way, as described above, the optimum value of the width R of the Gaussian function, for example, lists several candidates for the width R of the Gaussian function, obtains the prediction error of the regression equation using each width R, and obtains the smallest prediction. The width R used in the regression equation giving the error can be adopted as the optimum value. A preferable specific example of such a procedure is, for example, a so-called “cross validation method” (Cross Validation).

That is, the method according to the sixth aspect of the present invention is:
A method according to the fifth aspect,
In the regression equation generation step, at least a part of the width of the Gaussian function when the execution of the regression equation generation step is the first time and the width of the Gaussian function when the execution of the regression equation generation step is not the first time are: Determined by the cross-validation method,
It is the method characterized by this.

  As described above, the intersection confirmation method may be applied only at the first execution of the regression equation generation step, or may be applied at any one of the second and subsequent executions of the regression equation generation step. Alternatively, the intersection confirmation method may be applied multiple times during the first and second and subsequent executions of the regression formula generation step, or at the first and second and subsequent executions of the regression formula generation step. You may apply.

  In the intersection confirmation method, for example, the original data is first divided into n blocks, the model is constructed and the accuracy is calculated using the first block as test data and the other blocks as training data. This is a method that builds a model using the second block as test data and the other blocks as training data, repeats this procedure n times, and uses the average accuracy calculated at each time as the estimated accuracy of the model. Even if it exists, it is characterized by a small estimation error.

  A plurality of types of specific execution procedures of such a cross-confirmation method are known. For example, “a cross-confirmation method for taking out only one” or “one-out method (LOO method)” (Leave-one-out) Cross Validation), “10-fold Cross Validation”, “2-fold Cross Validation”, and the like.

  The present invention will be described in more detail with reference to the following examples, but the technical scope of the present invention is not limited to these examples.

1. Verification of calculation time and regression accuracy of method according to one aspect of the present invention (1) Artificial data The effectiveness of the method according to one aspect of the present invention was verified using the following artificial data.
a) The input x _i is sampled at 1300 points randomly from {0, 3000}.
b) The measurement noise η _i is generated with a normal distribution N (0, σ ² ) (when noise is added in this embodiment, σ ² = 0.15).
c) The values θ _i to be added to five abnormal values are ± 0.0, ± 3.0, and ± 5.0.
d) All five abnormal values are included in the learning data.
e) The output y _i is defined by the following formula (I).

(2) Evaluation of calculation time of regression equation It took 300 points randomly extracted from the above-mentioned artificial data 1300 points as learning data, and it was necessary to learn by solving a hard margin problem and to obtain a regression equation. The calculation time was measured for each regression method listed below.
i) Method according to one aspect of the invention
ii) Conventional method (sequential learning method)
iii) Ordinary method (batch regression of all test data without using sequential learning method)
In any of the above regression methods i) to iii), the regression equation was obtained by SVR using a Gaussian kernel. In the regression methods i) and ii), the learning data was divided into six groups to obtain a regression equation, and the time required for the calculation was measured. The obtained results will be described below with reference to FIG.

As described above, FIG. 3 illustrates the calculation of the regression equation by mixing out abnormal values in the regression equation generation method, the conventional method (sequential learning method), and the normal method (collective regression) according to one embodiment of the present invention. It is a graph which shows the grade of the influence on time. In FIG. 3, (a) shows the influence on the calculation time of the regression equation due to mixing of abnormal values when measurement noise is not added and (b) when measurement noise is added. Incidentally, “noise 0.15” in (b) indicates that σ ² = 0.15 in the measurement noise generated by the normal distribution N (0, σ ² ) as described above. Yes.

  As shown in FIG. 3, the regression equation generation method according to one aspect of the present invention performs a regression compared with the normal method (collective regression) regardless of the presence of abnormal values and the presence or absence of measurement noise. The calculation time of the equation is extremely short, and the fluctuation of the calculation time of the regression equation accompanying the mixing of abnormal values into the learning data is also extremely small. In addition, the conventional method (sequential learning method) also showed a tendency similar to that of the regression equation generation method according to one aspect of the present invention. Fluctuations in the calculation time of the regression equation due to the addition of abnormal values to measurement data and the addition of measurement noise are large. As described above, the regression equation generation method according to one aspect of the present invention has an extremely short calculation time for the regression equation as compared with the normal method (collective regression), and the conventional method (sequential learning method) and the normal method ( Compared to (batch regression), it was confirmed that the fluctuation in the calculation time of the regression equation due to the inclusion of abnormal values and measurement noise in the learning data was also small.

(3) Evaluation of accuracy of regression equation Using the remaining 1000 points of data as test data, for each regression method (learning method) of i) to iii) above, the true value and the regression value of the test data Correlation coefficient was obtained.

  The evaluations in (2) and (3) above are based on the values of the additional value θi (± 0.0, ± 3.0, and ± 5.0) to the abnormal value and noise for each regression method. The measurement was performed 15 times for each combination with and without, and the average values of the correlation coefficient and the calculation time were obtained. The obtained results will be described below with reference to FIG.

As described above, FIG. 4 illustrates the regression accuracy (correlation) due to mixing of abnormal values in the regression equation generation method, the conventional method (sequential learning method), and the normal method (collective regression) according to one embodiment of the present invention. It is a graph which shows the grade of the influence on number). In FIG. 4, (a) shows the influence on the correlation coefficient due to mixing of abnormal values when no measurement noise is added, and (b) when measurement noise is added. Incidentally, “noise 0.15” in (b) indicates that σ ² = 0.15 in the measurement noise generated by the normal distribution N (0, σ ² ) as described above. Yes.

  As shown in FIG. 4, the regression equation generation method according to one aspect of the present invention has an abnormal value in the learning data as compared with the conventional method (sequential learning method) and the normal method (collective regression). The decrease in correlation coefficient due to the mixing of is extremely small. In other words, the regression equation generation method according to one aspect of the present invention has extremely high robustness against the inclusion of abnormal values in the learning data, and has high generalization (accuracy) even when abnormal values are mixed. Can be maintained. Further, it is confirmed that the regression equation generation method according to one aspect of the present invention has a smaller variation in the correlation coefficient due to the presence or absence of measurement noise compared to the conventional method (sequential learning method) and the normal method (collective regression). It was done.

(4) Summary As described above, the regression equation generation method according to the present invention has an extremely short calculation time for the regression equation as compared with the ordinary method (collective regression), and the conventional method (sequential learning method) and the ordinary method. Compared to the method (batch regression), it is confirmed that the robustness of the regression accuracy is extremely high (the decrease in the correlation coefficient is extremely small) against the addition of abnormal values to the learning data and the addition of measurement noise. It was. Therefore, according to the method according to the present invention, the regression equation representing the relationship between the control factor and the control result (evaluation index) in the control of various devices can be calculated in a very short time compared to the conventional technique, Even if an abnormal value or measurement noise is mixed, a highly generalized regression equation can be generated, so that highly accurate control can be performed.

2. In this embodiment, the relationship between the control factors of the vehicle and the corresponding control results (evaluation indices) is sampled from actual data at 3,584 locations. A regression equation was generated using 588 points of sample data randomly selected from as learning data. At this time, the size (number of data points) of the group created by dividing the learning data of 588 points is changed as shown in Table 1, and the regression equation is calculated by the method according to one aspect of the present invention. The change of time was investigated.

  Next, the accuracy (average error) of each regression equation was also investigated using the remaining 2996 points. The details of the regression method were the same as in 1 (2) above. The “average error” in Table 1 is an average of the absolute value of the difference between the calculated value based on the regression equation and the actually measured value over all data.

  As is clear from Table 1, according to the method according to the present invention, the sample data is divided into a plurality of groups, and a regression equation is calculated for each group, so that all sample data are compared with the usual method for performing a collective regression. The calculation time for the regression equation was greatly reduced. However, as the sample data was divided into a larger number of groups (in other words, the smaller the number of sample data belonging to each group), the regression accuracy tended to decrease (average error increased).

  Therefore, as described above, the degree of decrease in regression accuracy is negligible compared to the degree of reduction in calculation speed associated with grouping of sample data, but the required accuracy is required when extremely high accuracy is required. Accordingly, by adjusting the number (size) of groups of sample data, it is possible to appropriately balance the calculation speed and the regression accuracy in a trade-off relationship.

  In the present embodiment, the control of the vehicle has been verified, but as described above, the present invention uses a regression equation to obtain a target value of a control factor suitable for obtaining a control result within an allowable range. Any device can be applied as long as the desired control result is obtained by controlling the control factor to the target value thus obtained.

1. Determination of the width R of the Gaussian function of the first regression equation In the method of generating a regression equation according to the present invention, when a Gaussian function is used as the basis function of the regression equation, a series of processing is basically executed according to the flowchart shown in FIG. The However, since a Gaussian function is used as the basis function of the regression equation, a process for determining the width R of the Gaussian function is further required. Specifically, the Gaussian function width R is determined at the time of the first regression execution, and the Gaussian function width R is determined at the time of the second and subsequent regression executions. The method for determining the width R of the Gaussian function in these steps can be selected from various methods as described above, and an example is an intersection confirmation method.

  By the way, when the Gaussian function is used as the basis function of the regression formula in the regression formula generation method according to the present invention, as described above, when the regression formula generation step is executed for the first time, it belongs to one group among a plurality of groups. The width (R) of the Gaussian function is obtained based only on the sample data. However, in practice, the optimum Gaussian function width (R) can be determined by obtaining the Gaussian function width (R) based on only a part of the sample data. This point will be described in detail below with reference to FIG.

  As described above, FIG. 5 shows the relationship between the prediction error and the width R of the Gaussian function used as the basis function of the regression equation in the regression equation generation method according to one embodiment of the present invention. It is a graph compared with the case where it calculates | requires about and the case where it asks about all the sample data. In this example, 500 points of sample data are prepared, of which 329 points are training data and 171 points are generalization performance confirmation data. As a method for determining the optimum value of the width R, the aforementioned “10-fold cross validation method” (10-fold Cross Validation) was adopted. Further, five values of 1, 2, 3, 4, and 5 were prepared as candidates for the width R.

  First, as an example of a regression equation generation method according to one embodiment of the present invention, an initial regression equation is obtained using only 100 points of the training data 329 points, and generalization performance confirmation data 171 points are used. For each candidate value of width R, a prediction error (generalized performance estimation value) was obtained (black square plot in the graph). On the other hand, as a reference example, using all the training data 329 points, the prediction error (generalized performance estimation value) was obtained in the same manner as described above for each candidate value of the width R (the black circle in the graph). plot).

  As is clear from the graph shown in FIG. 5, in the example of the regression equation generation method according to one embodiment of the present invention in which only 100 points of training data are used, the reference using all 329 points of training data is used. Compared to the example, the prediction error value itself is large in all the candidate values of the width R. However, the optimal value of the width R that minimizes the prediction error is the same (3) whether only 100 points of training data are used or 329 points of all training data are used.

  As described above, in the regression equation generation method according to one embodiment of the present invention, even if the width (R) of the Gaussian function is obtained based only on a part of sample data, a smaller amount of processing (amount of work, calculation) It has been confirmed that the optimum width (R) of the Gaussian function can be determined.

2.2 Determining the width R of the Gaussian function of the regression equation after the first time (1)
As described above, in the regression formula generation step of the regression formula generation method according to one embodiment of the present invention, the regression formula generation step is compared with the width of the Gaussian function when the regression formula generation step is executed for the first time. The width of the Gaussian function is smaller when execution of is not the first time. That is, a value smaller than the width R of the Gaussian function of the first regression equation is used as the width R of the Gaussian function of the regression equation for the second and subsequent times in the regression equation generation method according to one embodiment of the present invention. Thereby, the width R at the time of the second or subsequent regression execution can be determined with a smaller amount of processing (work amount, calculation amount), and the accuracy of the regression equation thus generated is also improved. This point will be described in detail below with reference to FIG.

As described above, FIG. 6 shows a regression based on a combination of the Gaussian function width R of the first regression equation and the Gaussian function width R of the second regression equation in the regression equation generation method according to one embodiment of the present invention. It is a graph showing the change of the prediction precision (generalization performance estimated value) of a type | formula. More specifically, in the graph of FIG. 6, (a) represents the first regression equation F (solid line) obtained using sample point 1 (0, 1) and sample point 2 (3, -1). Here, the coordinates of the sample points represent (position, value). That is, in the first regression shown in (a), two sample points with positions 0 and 3 and values +1 and −1 were used. As shown in (a), the width R of the Gaussian function of the first regression equation is 3. Here, F1 represents a regression equation for only sample point 1, and F2 represents a regression equation for only sample point 2. That is, the first regression equation F is an overlay of F1 (dotted line) and F2 (dashed line).

  Next, in the second regression, two sample points 3 and 4 whose positions are 1 and 2 and whose values are the residuals from the first regression equation F are +0.2 and −0.2, respectively. It was. Note that (b) and (c) in FIG. 6 both represent the second regression equation, the first regression equation F is represented by a thin solid line, and the second regression equation F is represented by a thick solid line. Yes. Also, res1 and res2 represent regression equations for the residuals of the sample points 3 and 4 and the first regression equation F, respectively, and the second regression equation F is the first regression equation F and the res1 and res2 It is a superposition of.

  However, in (b) and (c), the width R of the Gaussian function is set to a value different from 1 and 3, respectively. That is, in (b), since the width R of the Gaussian function of the second regression equation is smaller than the width R of the Gaussian function of the first regression equation, it corresponds to the regression equation generation method according to one embodiment of the present invention. In (c), since the width R of the Gaussian function of the second regression equation is equal to the width R of the Gaussian function of the first regression equation, this corresponds to the regression equation generation method according to the comparative example.

  As is clear from the comparison between (b) and (c) in FIG. 6, the second regression, which is an overlay of the first regression equation F (thin solid line) and res1 (dotted line) and res2 (dashed line). The accuracy of Formula F (thick solid line) for the second sample points 3 and 4 is better in (b) than in (c). This is due to the difference in the degree of influence on the surroundings of res1 and res2. In other words, in (c) using the same value as the width R at the time of the first regression, the influence of one sample point extends to the position of the other sample point, which reduces the accuracy of the regression equation. ing. On the other hand, in (b) using a smaller value than the first regression execution as the width R, the influence from one sample point to the other sample point is small, and as a result, the sample of the second regression equation The deviation from the point is small (the accuracy of the regression equation is high).

  As described above, in the regression equation generation step of the regression equation generation method according to one embodiment of the present invention, as the width R of the Gauss function of the regression equation for the second and subsequent times, from the width R of the Gauss function of the first regression equation It was confirmed that the accuracy of the regression equation after the second time is improved by using a small value. Further, since the value R larger than the width R of the Gaussian function of the first regression equation as the width R of the Gaussian function of the regression equation after the second time does not need to be raised as a candidate, a smaller amount of processing (work load, calculation) Quantities) can determine the width R for the second and subsequent regression runs.

3. Determination of the width R of the Gaussian function of the regression equation after the second time (2)
As described above, by setting the width R of the Gaussian function of the regression equation for the second and subsequent times to be smaller than the width R of the Gaussian function of the initial regression equation, the processing amount (work amount, calculation amount) can be reduced to 2 Not only can the width R at the time of the regression execution after the first time be determined, but also the accuracy of the regression equations after the second time can be improved. Accordingly, the accuracy of the resulting regression equation was evaluated for various combinations of the width R of the Gaussian function of the first regression equation and the width R of the Gaussian function of the regression equation for the second and subsequent regression equations. The evaluation result will be described in detail below with reference to FIG.

  As described above, FIG. 7 shows a combination of the width R of the Gaussian function of the first regression equation and the width R of the Gaussian function of the regression equation for the second and subsequent times in the regression equation generation method according to one embodiment of the present invention. It is a contour figure which represents the change of the prediction accuracy (generalization performance estimated value) of a regression equation two-dimensionally. The white solid line shown in FIG. 7 represents a state in which the width R of the Gaussian function of the first regression equation and the width R of the Gaussian function of the second and subsequent regression equations are in a one-to-one relationship (that is, equal). As shown in FIG. 7, in the region below the white solid line (that is, the region where the second and subsequent widths R are smaller than the initial width R), the accuracy of the resulting regression equation is high. (Average error is small). On the other hand, in the region above the white solid line (that is, the region in which the width R after the first time is larger than the first width R), the accuracy of the resulting regression equation is low (average error is smaller). large).

  Here, the white alternate long and short dash line in FIG. 7 represents a state where the width R of the Gaussian function of the first regression equation is 4. That is, on the one-dot chain line, the width R of the Gaussian function of the first regression equation is fixed at 4, and the width R of the Gaussian function of the regression equation for the second and subsequent times represents various values. Table 2 below shows changes in the average error when the width R after the second time is changed to various values on the one-dot chain line.

  As is clear from the results shown in Table 2, the regression equation obtained as a result is obtained by setting the width R of the Gaussian function of the regression equation for the second and subsequent times smaller than the width R of the Gaussian function of the first regression equation. Accuracy can be improved. Further, since the value R larger than the width R of the Gaussian function of the first regression equation as the width R of the Gaussian function of the regression equation after the second time does not need to be raised as a candidate, a smaller amount of processing (work load, calculation) The amount R) of the width R at the second and subsequent regression executions can be determined.

  As described above, according to the present invention, even when the number of sample data is enormous, the calculation time required to generate the regression equation does not increase remarkably, and an abnormal value in the sample data is obtained. Provided is a method for generating a regression equation that is not easily affected by contamination. Therefore, according to the present invention, compared with the conventional regression method, even when a large amount of sample data is used, the calculation time is significantly shortened and high accuracy is maintained even if abnormal values are mixed. Thus, a regression equation can be generated. The method according to the present invention is useful in generating a regression equation for specifying a control factor in controlling various devices such as a vehicle.

  Furthermore, according to the present invention, even when a Gaussian function is used as a basis function of a regression equation, a Gaussian in a case where the regression equation generation step is not executed for the first time (that is, after the second time) in the regression equation generation step. By setting the function width (R) to a value smaller than the width (R) of the Gaussian function when the regression equation generation step is executed for the first time, the processing amount (work amount, calculation amount) is reduced to 2 Not only can the width (R) for the regression execution after the first time be determined, but also the accuracy of the resulting regression equation can be improved.

10 Process Flow of Regression Formula Generation Method According to One Embodiment of the Present Invention 11 Step of Selecting Center Sample Data on Factor Space Step 12 Sorting and Grouping Sample Data Step 20 Regression Loop Inlet 30 Initial Regression Conditional branch 41 depending on whether or not a step 41 A step of obtaining a regression equation representing the relationship between the factor and the evaluation index 51 A step of calculating an error from the existing regression equation for each sample data 52 A step of calculating the error and the control factor Step 60 for obtaining a regression equation representing the relationship Step 70: adding the regression equation obtained in Step 52 to the existing regression equation to obtain a new regression equation Step 70 Regression loop exit

Claims (6)

A method of generating a regression equation representing a relationship between a control factor of a device to be controlled and an evaluation index corresponding to the control factor,
A sample data dividing step for dividing a plurality of sample data including information on control factors and evaluation indices into a plurality of groups;
A regression equation generating step for generating a regression equation based on sample data belonging to one of a plurality of groups divided in the sample data dividing step,
If the execution of this step is the first time, a regression equation representing the relationship between the control factor and the evaluation index included in the sample data belonging to the group is obtained,
When the execution of the step is not the first time, the result obtained from the control factor included in the sample data belonging to the group based on the regression equation obtained by the previous execution of the step and the sample data belonging to the group To calculate an error with the evaluation index included in and to obtain a regression equation representing the relationship between the error and the control factor,
Regression equation generation step ,
The regression equation determined in those said step when executed is a first-time before Symbol regression equation generating step as a new regression equation, if the execution of the step is not the first time obtained in the previous SL regression equation generating step and existing regression equation determined by a previous execution of the regression equation and those said step were additively superimposed to generate a new regression equation,
Regression equation addition step,
Only including,
A method of repeating the regression equation generation step and the regression equation addition step until execution of the regression equation generation step and the regression equation addition step is completed for all groups divided in the sample data division step .   2. The method according to claim 1, wherein a regression equation is obtained by support vector regression in the regression equation generation step.   3. The method according to claim 1, wherein in the sample data dividing step, the sample data is divided into groups so that sample data existing in a specific range on the control factor space does not exist in a specific group. A method characterized by sorting evenly.   4. The method according to claim 1, wherein the device to be controlled is a vehicle. A method according to any one of claims 1 to 4, wherein
The regression equation uses a Gaussian function as a basis function, and in the regression equation generation step, compared to the width of the Gaussian function when the regression equation generation step is executed for the first time, the regression equation generation step The width of the Gaussian function is smaller when the execution is not the first time,
A method characterized by. 6. A method according to claim 5, wherein
In the regression equation generation step, at least a part of the width of the Gaussian function when the execution of the regression equation generation step is the first time and the width of the Gaussian function when the execution of the regression equation generation step is not the first time are: Determined by the cross-validation method,
A method characterized by.