JP2009276967A - Outlier detection method, outlier detector, and program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To reliably detect an outlier with less computation volume. <P>SOLUTION: The outlier detector includes: a storage section 1 for storing learning data; an SVR calculation section 2 for obtaining Lagrange multiplier of respective data from learning data by μ-ε-SVR; an outlier candidate selection section 3 for selecting the data having the maximum Lagrange multiplier as the candidate of the outlier; a Lagrange multiplier total sum upper limit determination section 4 for determining whether the total sum of the Lagrange multiplier is smaller than the upper limit value; an outlier removing section 5 for removing the data selected by the outlier candidate selection section 3 from the learning data as the outlier when the total sum of the Lagrange multiplier is equal to the upper limit value; and a processing control section 6 for repeatedly executing the processing to the learning data after removing the outlier until the total sum of the Lagrange multiplier becomes smaller than the upper limit value. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to an outlier detection method, an outlier detection apparatus, and a program for detecting an outlier that is apart from the tendency of other data from among a plurality of learning data.

  A soft sensor is a technique for estimating a physical quantity that is difficult to measure online by using a combination of alternative sensors, and is widely used in the field of the process industry. Normally, a soft sensor constructs a characteristic equation based on learning data measured offline. However, since measurement data includes measurement noise due to instrument measurement errors and outliers (abnormal values) due to measurement errors, an appropriate estimation model can be obtained unless a method that supports measurement noise or outliers is used. It becomes difficult to create a high-precision estimation.

  A robust estimation method is effective as a method for suppressing the influence of measurement noise, but outliers that are far from other data tendencies need to be detected and removed in advance. For detection of outliers, a method using support vector regression (hereinafter referred to as SVR), which is a robust estimation model, is known (see, for example, Non-Patent Document 1).

Hereinafter, a conventional outlier detection method using ε-SVR disclosed in Non-Patent Document 1 will be described. FIG. 7 is a flowchart for explaining a conventional outlier detection method.
The main problem of ε-SVR can be expressed as the following equation when learning data is (x _i , y _i ) (i = 1,..., l). X _i is a vector quantity.

Here, φ (x _i ) is a mapping function to the feature space, w is a hyperplane weight vector on the feature space, b is a bias on the feature space, C is a trade-off parameter, and ε is a model representing the characteristics of the learning data Data tolerances for the function estimated as ξ _i , ξ ′ _i are slack variables. The expressions after “st” indicate constraint expressions.

The main problem of ε-SVR is to minimize the average error with the objective function shown in equation (1). In ε-SVR, as shown in FIG. 8, a function 81 is obtained so that all data 80 can be contained between two hyperplanes 82 and 83 that are parallel to and equidistant from the optimal function (hyperplane) 81. The slack variables ξ _i and ξ ′ _i represent the degree of protrusion of the data that does not fit between the two hyperplanes 82 and 83 from the two hyperplanes 82 and 83, that is, the regression error. The dual problem for equation (1) can be expressed as:

In equation (2), α _i and α ′ _i are Lagrange multipliers representing the sensitivity of each data to the objective function of the corresponding constraint equation. If α _i = 0, the error of data i is smaller than the allowable error ε. If 0 <αi <C, the error of data i is equal to the allowable error ε. If α _i = C, the data i Is larger than the tolerance ε. K (x _i , x _j ) is a kernel function. Similarly to the above, x _i and x _j are vector quantities. In the dual problem, an upper limit constraint C is defined for each Lagrangian multiplier α _i , α ′ _i .

In the conventional outlier detection method, first, the maximum value of the allowable error ε is determined (step S200 in FIG. 7). Subsequently, the Lagrange multipliers α _i and α ′ _i are obtained from the learning data (x _i , y _i ) using the equations (1) and (2) (step S201). Next, the allowable error ε is changed so as to be reduced by the change amount Δε (step S202), and it is determined whether the changed allowable error ε is smaller than 0 (step S203). When the allowable error ε after change is 0 or more, the process returns to step S201, and when the allowable error ε after change is smaller than 0, the process proceeds to step S204. Thus, the processes in steps S201 to S203 are repeated a plurality of times until the allowable error ε is smaller than 0. Here, it is assumed that the processes of steps S201 to S203 are repeated m times.

When the allowable error ε is smaller than 0, α _i = C / l or α ′ _i = C / l and ξ _i > 0.5σ or ξ ′ _i > 0.5σ is determined for each data. (Step S204). Note that σ is an error standard deviation. Next, it is determined whether or not the maximum number of times counted in step S204 is larger than the predetermined number k and the m standard error of deviation is 0.01 or less (step S205).

  If at least one of the condition that the maximum number of times is larger than the predetermined number k or the condition that the error standard deviation of m times is 0.01 or less does not hold, the data of the maximum number of times counted in step S204 is included in the learning data (Step S206), and the process returns to step S200. Thus, the processes in steps S200 to S206 are repeated until the determination in step S205 is Yes. In step S205, when the maximum number is larger than the predetermined number k and the m standard error deviation is 0.01 or less, the outlier detection is finished.

As described above, in the outlier detection method using ε-SVR, the width of the allowable error ε is changed stepwise and is calculated a plurality of times (that is, the model complexity is changed). Then, using the fact that the data whose Lagrange multipliers α _i and α ′ _i have reached the upper limit value C / 1 are likely to be outliers, outlier candidates are obtained. In order to estimate a true outlier from this outlier candidate, a plurality of calculations are performed with different tolerances ε, and data with a high probability of being the upper limit is removed as an outlier. This removal of outliers is repeated until the overall error does not change (that is, until the variation in the multiple calculation of the RMS error is reduced).

  As another outlier detection method, there is a method of removing data having a large estimation error as an outlier. This outlier detection method will be described with reference to FIG. In FIG. 9, 90 represents learning data x, 91 represents an outlier included in the learning data x, and 92 represents a function f (x) estimated from the learning data x. In this outlier detection method, data having a large error from the function f (x) estimated from the learning data x is removed as an outlier.

E.M.Jordaan et al., “Robust Outlier Detection using SVM Regression”, Neural Network, 2004 Proceeding. 2004 IEEE International Joint Conference

In the outlier detection method using ε-SVR, there are many outlier candidates whose Lagrange multipliers α _i or α ′ _i have reached the upper limit value C / l. Therefore, the tolerance ε is changed to narrow the outlier. It is necessary to take multiple counts. Therefore, if one outlier is detected, the SVR is calculated a plurality of times, and if one outlier detection requires m calculation times, n is detected in order to detect n outliers. Xm times of calculation are required. As described above, the outlier detection method using ε-SVR has a problem that it is not suitable for handling a large amount of data in a multidimensional manner because the number of repeated calculations increases and it takes time.

  In addition, the outlier detection method that removes data with large estimation errors as outliers requires a small amount of computation, but in the case of a nonlinear model, the estimation function itself is affected by the outliers and cannot be detected. There was a problem that an outlier was detected. For example, in the example of FIG. 10, since the estimation function is affected by the outlier 91, the estimation function is separated from the normal data 90.

  The present invention has been made to solve the above-described problems, and an object thereof is to provide an outlier detection method, an outlier detection apparatus, and a program that can reliably detect an outlier with a small amount of calculation.

  The outlier detection method of the present invention does not define an upper limit for the Lagrangian multiplier of individual data, but uses μ-ε-SVR (support vector regression) that defines the upper limit of the sum of the Lagrange multipliers. A calculation procedure for obtaining a Lagrangian multiplier of each of the learning data, an outlier candidate selection procedure for selecting the data having the largest Lagrangian multiplier among the learning data as an outlier candidate, and the sum of the Lagrange multipliers of each learning data is a predetermined upper limit. A Lagrange multiplier sum upper limit determination procedure for determining whether or not the value is smaller than the value, and when the sum of the Lagrange multipliers is the upper limit value, the data selected in the outlier candidate selection procedure is determined as an outlier, and the data is In the outlier removal procedure for removing from a plurality of learning data and the Lagrange multiplier sum upper limit determination procedure, The calculation procedure, outlier candidate selection procedure, Lagrange multiplier sum upper limit determination procedure, and outlier removal procedure are repeatedly executed on the learning data after removing the outlier until the sum of the Lagrange multipliers becomes smaller than the upper limit. And a control procedure to be performed.

で表され、主問題に対する双対問題は、

で表される。 Further, in one configuration example of the outlier detection method of the present invention, the learning data is (x _i , y _i ) (i = 1,..., L), and the mapping function to the feature space is φ (x _i ), Hyperplane weight vector on feature space, b on feature space, b, predetermined tradeoff parameter μ, tolerance error ε, slack variables ξ, ξ ', Lagrange multiplier α _i , When α ′ _i , the main problem of μ-ε-SVR is

The dual problem for the main problem is

It is represented by

  In addition, the outlier detection apparatus of the present invention does not define an upper limit for the Lagrange multipliers of individual data, but uses μ-ε-SVR (support vector regression) that defines the upper limit of the sum of Lagrange multipliers from a plurality of learning data. Calculation means for obtaining a Lagrange multiplier for each data, outlier candidate selection means for selecting data having the largest Lagrange multiplier among each learning data as an outlier candidate, and a total sum of the Lagrange multipliers for each learning data is predetermined. A Lagrange multiplier sum upper limit determining means for determining whether or not the upper limit value is less than the upper limit value, and when the sum of the Lagrange multipliers is the upper limit value, the data selected by the outlier candidate selection means is determined as an outlier, Outlier removal means for removing data from the plurality of learning data, and Lagrange multiplier sum upper limit determination means In the determination, until the sum of the Lagrange multipliers becomes smaller than the upper limit value, the calculation means, the outlier candidate selection means, the Lagrange multiplier sum upper limit determination means, and the outlier removal means for the learning data after removing the outliers. And a control means for repeatedly executing the process.

  Further, the outlier detection program of the present invention does not define an upper limit for the Lagrangian multiplier of individual data, but uses μ-ε-SVR (support vector regression) that defines the upper limit of the sum of Lagrange multipliers from a plurality of learning data. A calculation procedure for obtaining a Lagrange multiplier for each data, an outlier candidate selection procedure for selecting data having the maximum Lagrange multiplier among each learning data as an outlier candidate, and a total sum of the Lagrange multipliers for each learning data are predetermined. If the sum of the Lagrange multipliers is determined to be smaller than the upper limit value, and the sum of the Lagrange multipliers is the upper limit value, the data selected in the outlier candidate selection procedure is determined as an outlier, and this data Outlier removal procedure from the plurality of learning data, and the Lagrange multiplier sum upper limit determination procedure In order, the calculation procedure, the outlier candidate selection procedure, the Lagrange multiplier sum upper limit determination procedure, and the outlier removal procedure for the learning data after removing the outlier until the sum of the Lagrange multipliers becomes smaller than the upper limit. A control procedure to be repeatedly executed is executed by a computer.

  According to the present invention, since the feature of ε-SVR that detects an outlier by a Lagrange multiplier is provided, it is possible to quantify that the estimation function itself is affected by the outlier, so that the outlier can be reliably determined. Since μ-ε-SVR that can detect and remove and does not define an upper limit for the Lagrangian multiplier of each data is used, outliers can be detected and removed with a small amount of calculation.

Hereinafter, embodiments of the present invention will be described with reference to the drawings. FIG. 1 is a block diagram showing a configuration of an outlier detection apparatus according to an embodiment of the present invention.
The outlier detection device includes a storage unit 1 that stores a plurality of learning data prepared in advance, an SVR calculation unit 2 that obtains a Lagrange multiplier of each data from the learning data by μ-ε-SVR, and a Lagrange of each learning data. An outlier candidate selection unit 3 that selects data having the largest multiplier as an outlier candidate; a Lagrange multiplier sum upper limit determination unit 4 that determines whether the sum of Lagrange multipliers of each learning data is smaller than a predetermined upper limit value; When the sum of Lagrange multipliers is equal to or greater than the upper limit value, the data selected by the outlier candidate selection unit 3 is determined as an outlier, and the outlier removal unit 5 removes this data from the learning data, and the Lagrange multiplier The learning data after the outliers are removed until the sum of the Lagrange multipliers becomes smaller than the upper limit in the determination by the total upper limit determination unit 4. The SVR calculation unit 2, the outlier candidate selection unit 3, the Lagrange multiplier sum upper limit determination unit 4, and the outlier removal unit 5 have a processing control unit 6 that repeatedly executes the processing.

The μ-ε-SVR is characterized by minimizing the maximum error with the objective function instead of minimizing the average error with the objective function like the ε-SVR. The main problem of μ-ε-SVR can be expressed as the following equation when learning data is (x _i , y _i ) (i = 1,..., l). Note that the input x _i is a vector quantity, and y _i is an output for the input x _i .

In Expression (3), φ (x _i ) is a mapping function to the feature space, w is a hyperplane weight vector on the feature space, b is a bias on the feature space, μ is a trade-off parameter, and ε is a characteristic of learning data Data tolerances for functions estimated as models representing ξ and ξ ′ are slack variables. w ^T represents a transposed matrix of the vector w. The expressions after “st” indicate constraint expressions.

  In μ-ε-SVR, a function (hyperplane) 21 that minimizes the maximum error among the errors of each learning data 20 is obtained as shown in FIG. In μ-ε-SVR, the slack variables ξ and ξ ′ are not values for each data, but represent the regression error of data having the largest degree of protrusion from the allowable error ε. Here, the slack variable ξ represents the degree of protrusion of data below the function 21 in FIG. 2, and the slack variable ξ ′ represents the degree of protrusion of data above the function 21. The dual problem for equation (3) can be expressed as:

In equation (4), α _i and α ′ _i are Lagrange multipliers representing the sensitivity of each data to the objective function of the corresponding constraint equation. In FIG. 2, the constraint equation for determining whether or not the data is below the function 21 is w ^T φ (x _i ) + b−y _i ≦ ε + ξ, ξ ≧ 0, i = 1,. ., L. The Lagrange multiplier corresponding to the sensitivity of this constraint equation is α _i . Similarly, in FIG. 2, the constraint equation for determining whether the data is above the function 21 is y _i −w ^T φ (x _i ) −b ≦ ε + ξ ′, ξ ′ ≧ 0, i in Equation (3). = 1,..., L. The Lagrange multiplier corresponding to the sensitivity of this constraint equation is α ′ _i . If α _i = 0, the error of data i is smaller than the maximum error among the errors of each data with respect to the estimated function. If α _i > 0, the error of data i is equal to the maximum error, and Σα _{If i} = C, the error of data i is greater than the tolerance ε. K (x _i , x _j ) is a kernel function. Similarly to the above, x _i and x _j are vector quantities. In μ-ε-SVR, a trade-off parameter μ and an allowable error ε are set in advance. These parameters are appropriately designed based on the required specifications of the soft sensor to be created.

  The difference between the conventional ε-SVR and μ-ε-SVR is that the main problem of ε-SVR is expressed by an expression obtained by adding the regularization term and the sum of errors, whereas the main problem of μ-ε-SVR. Is expressed by a formula obtained by adding the regularization term and the maximum error, and the constraint equation of the dual problem of ε-SVR restricts the upper limit of each Lagrange multiplier, whereas the dual of μ-ε-SVR The constraint equation in question limits the upper limit of the sum of Lagrange multipliers.

  Thus, in the dual problem with respect to Equation (3), an upper limit constraint is imposed on the sum of Lagrange multipliers, so that a plurality of Grange multipliers are not equal at the upper limit. As described above, the Lagrangian multiplier represents the sensitivity of the corresponding constraint equation to the objective function. Therefore, when an outlier exists in the data, it can be estimated that the data having the largest Lagrange multiplier is the outlier that is farthest from other data. Thereby, in [mu]-[epsilon] -SVR, the amount of calculation can be reduced compared to [epsilon] -SVR. The reason for this will be described later in detail.

  Furthermore, Equation (5) can be derived from Kuhn-Tucker's condition for Equation (3), and if there is an outlier in the data, it can be said that the sum of Lagrange multipliers takes the upper limit value μ.

In the present embodiment, outlier detection is performed by the following procedure using the above-described features of μ-ε-SVR. FIG. 3 is a flowchart showing the operation of the outlier detection apparatus of this embodiment.
First, the SVR calculation unit 2 uses the equations (3) and (4) to learn data (x _i , y _i ) (i = 1,..., L) stored in the storage unit 1. To obtain the Lagrange multipliers α _i and α ′ _i of each data (step S100).

Subsequently, the outlier candidate selection unit 3 selects, as the outlier candidate, data in which either one of the Lagrange multipliers α _i or α ′ _i is the largest among the learning data (step S101). Note that one of the Lagrange multipliers α _i and α ′ _{i for} the data i is always 0.
Next, the Lagrange multiplier sum upper limit determination unit 4 calculates the sum Σ (α _i + α ′ _i of Lagrange multipliers α _i and α ′ _i of each learning data from the calculation result of the SVR calculation unit 2 as shown in Expression (6). ) Is smaller than the upper limit value 2μ (step S102).

When it is determined that the sum Σ (α _i + α ′ _i ) of Lagrange multipliers α _i and α ′ _i is smaller than the upper limit value 2 μ (YES in step S102), the processing control unit 6 and the SVR calculation unit 2 and outlier candidates An end instruction signal is output to the selection unit 3, the Lagrange multiplier sum upper limit determination unit 4, and the outlier removal unit 5, and the process of FIG.

When the outlier removal unit 5 determines that the sum Σ (α _i + α ′ _i ) of the Lagrange multipliers α _i and α ′ _i is equal to the upper limit value 2μ (NO in step S102), the outlier candidate selection unit 3 Data selected as an outlier candidate is confirmed as an outlier, and this data is removed from the learning data stored in the storage unit 1 (step S103).

If it is determined that the total sum Σ (α _i + α ′ _i ) of the Lagrange multipliers α _i and α ′ _i is equal to the upper limit value 2 μ, the processing control unit 6 and the SVR calculation unit 2 and the outlier candidate after the outlier removal. A re-execution instruction signal is output to the selection unit 3, the Lagrange multiplier sum upper limit determination unit 4, and the outlier removal unit 5. When the re-execution instruction signal is output, the process returns to step S100, and the SVR calculation unit 2 recalculates Lagrange multipliers α _i and α ′ _i of each data for the learning data from which the outlier has been removed.

In this way, the processes of steps S100 to S103 are repeatedly executed until the sum Σ (α _i + α ′ _i ) of the Lagrange multipliers α _i and α ′ _i becomes smaller than the upper limit value 2μ in step S102. In this embodiment, outliers can be reliably detected and removed point by point in one calculation, and the processing of FIG. 3 ends when all outliers are removed.

The effectiveness of the present embodiment was verified with artificially created learning data. In order to create artificial data, input x _i is sampled at 100 points from a uniform distribution of [0, 1], measurement noise η _i is generated by normal distribution N (0, 0.05), and outliers are set at 3 points. The value θ _i to be added was set to ± 0.5, and the output y _i was defined as y _i = (sin 2πx _i ) ² + η _i + θ _i .
FIG. 4A to FIG. 4D show the results of detecting outliers with the outlier detection apparatus of the present embodiment using the artificially created data as learning data. 4A to 4D, reference numeral 40 denotes learning data, 41 denotes an estimation result obtained by calculating the output y _i from the input x _i using a function estimated from the learning data, and 42 to 44 denote outliers. .

  FIG. 4A shows initial learning data, an estimation result based on the learning data, and a result of detecting the outlier 42 by performing the processing of steps S100 to S103 in FIG. 3 once. FIG. 4B shows learning data after the outlier 42 is removed, an estimation result based on the learning data, and a result of detecting the outlier 43 by performing the second processing. FIG. 4C shows the learning data after the outliers 42 and 43 are removed, the estimation result based on the learning data, and the result of detecting the outlier 44 by performing the third processing. FIG. 4D shows the learning data after the outliers 42 to 44 are removed and the estimation result based on the learning data. 4A to 4D, it can be seen that the estimation accuracy is finally improved after the three outliers 42 to 44 are appropriately removed. After the third outlier removal, the sum of the Lagrangian multipliers was 2.98, and the outlier detection device processing was terminated.

  Next, verification was performed using Stackloss data in which outliers were identified in many previous studies. The Stackloss data is disclosed in the document “K. A. Brownlee,“ Statistical Theory and Methodology in Science and Engineering ”, New York, Wiley, p. 491-500, 1960”. The Stackloss data is shown in FIG. In FIG. 5, 51 indicates an outlier.

  The Stackloss data is a data set obtained by acquiring the relationship between the operating conditions in the nitric acid production plant and the ammonia loss amount. In FIG. 5, Y is the amount of ammonia loss that has not been absorbed by the absorption tower (an inverse index of plant processing efficiency, 10 times the%), X1 is the flow rate of cooling air, X2 is the cooling water temperature of the absorption tower, and X3 is Nitric acid concentration ([per 1000, minus 500]). FIG. 5 shows an operation history for 21 days in a nitric acid production plant for producing nitric acid from ammonia. By-product nitric oxide is absorbed by the absorption tower. Out of 21 days of data, the outliers pointed out in previous studies are the 4th data of 1, 3, 4 and 21st. FIG. 5 shows that it is difficult to find outliers only by visualization.

  FIG. 6A to FIG. 6D show the results of detecting outliers with such Stackloss data as learning data by the outlier detection apparatus of the present embodiment. 6A to 6D, 60 to 63 represent outliers. FIG. 6A shows that the processing of steps S100 to S103 of FIG. FIG. 6B shows that the second process is performed and the fourth data is detected as an outlier 61. FIG. 6C shows that the third process is performed and the third data is detected as an outlier 62. FIG. 6D shows that the fourth process is performed and the first data is detected as an outlier 63.

  As a result of applying this embodiment to the Stackloss data, data having the largest Lagrange multiplier can be detected as an outlier by each Lagrange multiplier value shown in FIGS. 6 (A) to 6 (D). I understand that The sum of the Lagrangian multipliers is 0.03 after the fourth removal, and is extremely small compared to the upper limit value 2μ. Therefore, it can be determined that there are no more outliers.

  In the conventional outlier detection method using ε-SVR, the Lagrange multiplier has an upper limit, and remains a qualitative determination material. For this reason, in order to perform quantitative outlier detection, it is necessary to perform fitting multiple times while changing the allowable error ε. On the other hand, in the present embodiment, the Lagrange multipliers are individually quantified, and the Lagrange multiplier values can be different, so that quantitative outlier detection can be performed with a single fitting. In this embodiment, it is only necessary to calculate n times in order to detect n outliers, and the outliers using the conventional ε-SVR, which required n × m times of calculation, are required. The amount of calculation can be reduced as compared with the detection method.

  In addition, in the conventional outlier detection method that removes data having a large estimation error as an outlier, the determination is made based only on the error. Therefore, if the estimated curved surface is distorted, quantification is advantageous for the outlier. On the other hand, in this embodiment, the accuracy of outlier detection can be improved by adding an error as a quantitative determination material.

  Note that the outlier detection apparatus of the present embodiment can be realized by a computer having a CPU, a storage device, and an external interface, and a program for controlling these hardware resources. In such a computer, an outlier detection program for realizing the outlier detection method of the present invention is provided in a state of being recorded on a recording medium such as a flexible disk, a CD-ROM, a DVD-ROM, or a memory card. . The CPU writes the program read from the recording medium into the storage device, and executes processing as described in this embodiment according to the program.

  The present invention can be applied to a technique for detecting and removing outliers from measurement data, for example, in the field of the process industry.

It is a block diagram which shows the structure of the outlier detection apparatus which concerns on embodiment of this invention. It is a figure explaining the function estimated by (micro | micron | mu) -SVR of embodiment of this invention. It is a flowchart which shows operation | movement of the outlier detection apparatus of FIG. It is a figure which shows the result of having detected the outlier by the outlier detection apparatus of FIG. 1 by using the artificially created data as learning data. It is a figure which shows the Stackloss data which is an example of learning data. It is a figure which shows the result of having detected the outlier by the outlier detection apparatus of FIG. 1 by using Stackloss data as learning data. It is a flowchart explaining the conventional outlier detection method. It is a figure explaining the function estimated by the conventional epsilon-SVR. It is a figure explaining another conventional outlier detection method. It is a figure explaining the problem of the outlier detection method of FIG.

Explanation of symbols

  DESCRIPTION OF SYMBOLS 1 ... Memory | storage part, 2 ... SVR calculation part, 3 ... Outlier candidate selection part, 4 ... Lagrange multiplier sum total upper limit determination part, 5 ... Outlier removal part, 6 ... Process control part.

Claims (5)

A calculation procedure for determining a Lagrangian multiplier of each data from a plurality of learning data by μ-ε-SVR (support vector regression) that defines an upper limit of a sum of Lagrange multipliers without specifying an upper limit for Lagrange multipliers of individual data;
Outlier candidate selection procedure for selecting data having the largest Lagrangian multiplier among each learning data as an outlier candidate;
A Lagrange multiplier sum upper limit determination procedure for determining whether or not the sum of the Lagrange multipliers of each learning data is smaller than a predetermined upper limit;
When the sum of the Lagrange multipliers is the upper limit value, the data selected in the outlier candidate selection procedure is determined as an outlier, and the outlier removal procedure for removing this data from the plurality of learning data;
Until the sum of the Lagrange multipliers becomes smaller than the upper limit value in the Lagrange multiplier sum upper limit determination procedure, the calculation procedure, the outlier candidate selection procedure, and the Lagrange multiplier sum upper limit determination procedure for the learning data after removing the outliers, An outlier detection method comprising: a control procedure for repeatedly executing an outlier removal procedure. The outlier detection method according to claim 1,
The learning data is (x _i , y _i ) (i = 1,..., L), the mapping function to the feature space is φ (x _i ), the hyperplane weight vector on the feature space is w, the feature space When the upper bias is b, the predetermined trade-off parameter is μ, the allowable error is ε, the slack variables are ξ and ξ ′, and the Lagrange multipliers are α _i and α ′ _i , the main of the μ-ε-SVR The problem is,

The dual problem for the main problem is

An outlier detection method characterized by the following: Calculation means for obtaining a Lagrange multiplier for each data from a plurality of learning data by μ-ε-SVR (support vector regression) that defines an upper limit of the sum of Lagrange multipliers without defining an upper limit for Lagrange multipliers of individual data;
Outlier candidate selection means for selecting data having the largest Lagrangian multiplier among the learning data as outlier candidates;
Lagrange multiplier total upper limit determination means for determining whether the total sum of the Lagrange multipliers of each learning data is smaller than a predetermined upper limit value;
When the sum of the Lagrange multipliers is the upper limit value, the data selected by the outlier candidate selection unit is determined as an outlier, and the outlier removal unit removes this data from the plurality of learning data;
Until the sum of the Lagrange multipliers becomes smaller than the upper limit value in the determination by the Lagrange multiplier sum upper limit determination means, the calculation means, the outlier candidate selection means, and the Lagrange multiplier sum upper limit determination for the learning data after removing the outliers An outlier detection device comprising: control means for causing the means and the outlier removal means to repeatedly execute processing. In the outlier detection device according to claim 3,
The learning data is (x _i , y _i ) (i = 1,..., L), the mapping function to the feature space is φ (x _i ), the hyperplane weight vector on the feature space is w, the feature space When the upper bias is b, the predetermined trade-off parameter is μ, the allowable error is ε, the slack variables are ξ and ξ ′, and the Lagrange multipliers are α _i and α ′ _i , the main of the μ-ε-SVR The problem is,

The dual problem for the main problem is

An outlier detection device characterized by the following: A calculation procedure for determining a Lagrangian multiplier of each data from a plurality of learning data by μ-ε-SVR (support vector regression) that defines an upper limit of a sum of Lagrange multipliers without specifying an upper limit for Lagrange multipliers of individual data;
Outlier candidate selection procedure for selecting data having the largest Lagrangian multiplier among each learning data as an outlier candidate;
A Lagrange multiplier sum upper limit determination procedure for determining whether or not the sum of the Lagrange multipliers of each learning data is smaller than a predetermined upper limit;
When the sum of the Lagrange multipliers is the upper limit value, the data selected in the outlier candidate selection procedure is determined as an outlier, and the outlier removal procedure for removing this data from the plurality of learning data;
Until the sum of the Lagrange multipliers becomes smaller than the upper limit value in the Lagrange multiplier sum upper limit determination procedure, the calculation procedure, the outlier candidate selection procedure, and the Lagrange multiplier sum upper limit determination procedure for the learning data after removing the outliers, An outlier detection program that causes a computer to execute a control procedure that repeatedly executes an outlier removal procedure.

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