Online algorithm selection method based on double-increment normal form

Abstract

The invention relates to an online algorithm selection method based on a double-increment normal form. The method comprises the following steps: (1) obtaining a feature vector of a problem; (2) obtaining an optimal algorithm label; (3) taking the feature vector of the problem and the optimal algorithm label as a pair of inputs, and training a classifier; (4) calculating the similarity by adopting the Euclidean distance between the question characteristics, finding out a plurality of questions with the highest similarity with the current question, and calculating a threshold value for similarity comparison and judgment; (5) when a new problem is faced, performing similarity comparison, if the similarity is smaller than a threshold value, recommending according to a label of the similar problem, and if the similarity is not smaller than the threshold value, using a classifier prediction algorithm; (6) performing performance evaluation on a newly added algorithm, and replacing an algorithm with poor performance in the algorithm library; and (7) updating the system. The method has the advantages of being high in prediction speed, low in computing resource consumption and high in adaptability, resource waste can be reduced to the maximum extent, and the intelligent solving capacity of the system for the optimization problem is improved.

Description

Technical Field

[0001] The present invention belongs to the field of algorithm selection, and particularly relates to an online algorithm selection method based on a double increment paradigm. Background Art

[0002] Algorithm selection refers to the process of selecting the most effective algorithm to solve a specific problem. The most direct algorithm selection method is the trial-and-error method. Researchers have tested different algorithms one by one based on intuition and experience, and then selected the best algorithm to solve the given problem. However, it is inefficient and time-consuming.

[0003] There are several limitations in existing algorithm selection research. First, many methods rely on offline algorithm selection systems, which makes it challenging for the system to adapt to the dynamic characteristics of real data. Second, they do not consider the similarity between current and past optimization problems, resulting in unnecessary resource consumption. In addition, no new algorithms are added to the system, resulting in limited improvement in system performance and reduced adaptability. To solve these problems, the present invention proposes an online algorithm selection method based on a double increment paradigm to minimize resource waste and improve system performance. Summary of the Invention

[0004] The object of the present invention is to provide an online algorithm selection method based on a double increment paradigm, which has the advantages of fast prediction speed, low computational resource consumption, and high adaptability, minimizes resource waste and improves system performance, and has high application value.

[0005] To achieve the above object, the technical solution adopted by the present invention is: an online algorithm selection method based on a double increment paradigm, comprising the following steps:

[0006] (1) For a selected problem set for testing, use the Adaptive Local Landform Feature Vector (ALLFV) method to extract problem features and obtain the ALLFV feature vector of the problem;

[0007] (2) Use the Friedman test to measure the performance of different algorithms in the algorithm library on the problem set, compare the advantages and disadvantages of different algorithms, and store the algorithm with the best performance as the optimal algorithm label corresponding to the problem;

[0008] (3) Take the ALLFV feature vector of the problem obtained in step (1) and the optimal algorithm label obtained in step (2) as a pair of inputs and input them into a classifier for training;

[0009] (4) Calculate the similarity using the Euclidean distance between problem features, find multiple problems with the highest similarity to the current problem, and calculate the adaptive threshold for similarity comparison and judgment;

[0010] (5) When encountering a new problem, extract the feature vector of the new problem and compare its similarity with the feature vectors of existing problems according to step (4); if the similarity is less than the adaptive threshold, recommend based on the optimal algorithm label corresponding to the similar problem; otherwise, use the trained classifier to predict the applicable algorithm, and the classifier will be updated in real time according to this new problem and the corresponding predicted algorithm label;

[0011] (6) For a new algorithm to be added to the algorithm library, use the Friedman test to evaluate its performance. If the performance of the new algorithm is better than that of the existing algorithms in the algorithm library, use the new algorithm to replace the poorly performing existing algorithms in the algorithm library;

[0012] (7) After updating the algorithm library, update the optimal algorithm label corresponding to the problem at the same time, and then retrain and update the classifier according to the problem and the new label.

[0013] Specifically, the problem set is CEC2013, CEC2014 or CEC2017.

[0014] Specifically, in step (1), the method for obtaining the ALLFV feature vector is as follows: Assume that each problem has multiple search points, and each search point is x i ∈x (i = 1, 2,..., λ), x i：j (1 ≤ j ≤ M) represents the j-th nearest neighbor of x i , where x i,1 is the point closest to x i ; the fitness value of x i is expressed as f(x i ), and f(x i:j ) refers to the fitness value of the j-th neighbor closest to x i ; the average fitness of all search points is expressed as f mean ; then the relevant parameter ε i of each search point is calculated according to the following formula:

[0015]

[0016] For each search point x i , according to the difference in the fitness values of x i and its nearest neighbor x i：j , calculate the value of the set δ(ε i , x i , x i:j ), and its judgment basis is as follows:

[0017]

[0018] Then, calculate U(δ i ) and P(xi ) and add the P(x i ) values with the same search points to the corresponding histograms to obtain UP(X). Finally, the adaptive local terrain feature vector ALLFV(X) is calculated from UP(X), and the calculation formula is as follows:

[0019]

[0020] Specifically, the classifier in step (3) is a KNN classifier.

[0021] Specifically, the similarity calculation formula in step (4) is as follows:

[0022]

[0023] Among them, X i represents the feature vector of the current problem, X j represents the feature vector of other problems, l represents the dimension of the feature vector, 1 ≤ l ≤ m; d(X i , X j ) represents the Euclidean distance between X i and X j , X i，l represents the l-th dimensional eigenvalue of X i , X j，l represents the l-th dimensional eigenvalue of X j ;

[0024] Select k distance values with the smallest distance from X i from all the calculated Euclidean distances to form a distance set d s (X i ) as follows: d s (X i ) = {d1, d2,....d k} ≠ ∅, where d1, d2,..., d k are the k smallest distance values;

[0025] The adaptive threshold is β, and its calculation is as follows:

[0026]

[0027] The beneficial effect of the present invention is that the online algorithm selection method of the present invention synchronously improves the coverage breadth and decision confidence of the algorithm selection system by using the dual-increment learning paradigm of adding new problems and new algorithms, and thus can more flexibly cope with complex and changeable optimization requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0029] Figure 1 It is the overall flowchart of the online algorithm selection method described in the embodiments of the present invention. Detailed implementation manners

[0030] The following will further elaborate on the present invention in conjunction with the drawings and embodiments, but it is not used as a basis for any limitation to the invention.

[0031] An online algorithm selection method based on the double-increment paradigm. The online algorithm selection framework relied on by this method has Figure 1 The following modules: (1) Database module, including a problem library and an algorithm library. The problem set adopted by the problem library is CEC2013, CEC2014 or CEC2017, or other problem sets. The algorithm library includes various algorithms, such as differential evolution algorithm, particle swarm optimization algorithm, etc.;

[0032] (2) Topographic feature module, used for data sampling and calculating topographic features, and outputting topographic feature vectors;

[0033] (3) Algorithm performance module, used for evaluating the algorithms in the algorithm library for the problems in the problem library to obtain the optimal algorithm labels for each problem;

[0034] (4) Offline learning module, using the topographic feature vector and the optimal algorithm label as a pair of inputs to train a basic classifier;

[0035] (5) Incremental learning module, including the addition of new problems and the addition of new algorithms;

[0036] (6) Online algorithm selection module, for new problems, measure their similarity with the problems in the problem library, and recommend the algorithms of known problems similar to them or use the classifier to predict algorithms according to the comparison results, and update the classifier in real time; after the new algorithm is evaluated, replace the original algorithm with poor performance with the new algorithm, and update the classifier according to the problem and the corresponding new optimal algorithm label.

[0037] The following will further elaborate on the online algorithm selection method of the present invention. Among them, the topographic feature vector adopts the ALLFV feature vector, and the algorithm performance evaluation adopts the Friedman method. [[ID=--]]

[0038] It includes the following steps:

[0039] Step 1: For the selected problem set for testing, use the Adaptive Local Topographic Feature Vector (ALLFV) method to extract problem features and obtain the ALLFV feature vector of the problem. The specific method is as follows:

[0040] Assume that each problem has multiple search points, and each search point is \(x_i\) (\(i = 1, 2, \ldots, \lambda\)), and \(x_{ij}\) (\(1\leq j\leq M\)) represents the \(j\)-th nearest neighbor of \(x_i\), where \(x_{i1}\) is the point closest to \(x_i\); the fitness value of \(x_i\) is denoted as \(f(x_{ij})\), and \(f(x_{ij})\) refers to the fitness value of the \(j\)-th neighbor closest to \(x_i\); add up and average the fitness values of all search points to obtain the average fitness \(\overline{f}\); i \(x_i\in X(i = 1, 2, \ldots, \lambda)\), \(x_{ij}\) i：j (1\leq j\leq M) represents the \(j\)-th nearest neighbor of \(x_i\), where \(x_{i1}\) i is the point closest to \(x_i\); the fitness value of \(x_i\) is denoted as \(f(x_{ij})\), and \(f(x_{ij})\) refers to the fitness value of the \(j\)-th neighbor closest to \(x_i\); add up and average the fitness values of all search points to obtain the average fitness \(\overline{f}\); i,1 is the point closest to \(x_i\); the fitness value of \(x_i\) is denoted as \(f(x_{ij})\), and \(f(x_{ij})\) refers to the fitness value of the \(j\)-th neighbor closest to \(x_i\); add up and average the fitness values of all search points to obtain the average fitness \(\overline{f}\); i is the point closest to \(x_i\); the fitness value of \(x_i\) is denoted as \(f(x_{ij})\), and \(f(x_{ij})\) refers to the fitness value of the \(j\)-th neighbor closest to \(x_i\); add up and average the fitness values of all search points to obtain the average fitness \(\overline{f}\); i The fitness value of \(x_i\) is denoted as \(f(x_{ij})\), and \(f(x_{ij})\) refers to the fitness value of the \(j\)-th neighbor closest to \(x_i\); add up and average the fitness values of all search points to obtain the average fitness \(\overline{f}\); i ), \(f(x_{ij}\) i:j ) is the fitness value of the \(j\)-th neighbor closest to \(x_i\); add up and average the fitness values of all search points to obtain the average fitness \(\overline{f}\); i ) is the fitness value of the \(j\)-th neighbor closest to \(x_i\); add up and average the fitness values of all search points to obtain the average fitness \(\overline{f}\); mean ;

[0041] Then calculate the relevant parameter \(\varepsilon_i\) for each search point. First, calculate an index \(k_i\) that measures the difference between the neighbor fitness value and the average fitness, and then multiply the fitness value \(f(x_{ij})\) of the search point \(x_i\) by \(k_i\) to obtain the parameter \(\varepsilon_i\). The calculation formula is as follows: i First, calculate an index \(k_i\) that measures the difference between the neighbor fitness value and the average fitness, and then multiply the fitness value \(f(x_{ij})\) of the search point \(x_i\) by \(k_i\) to obtain the parameter \(\varepsilon_i\). The calculation formula is as follows: i First, calculate an index \(k_i\) that measures the difference between the neighbor fitness value and the average fitness, and then multiply the fitness value \(f(x_{ij})\) of the search point \(x_i\) by \(k_i\) to obtain the parameter \(\varepsilon_i\). The calculation formula is as follows: i First, calculate an index \(k_i\) that measures the difference between the neighbor fitness value and the average fitness, and then multiply the fitness value \(f(x_{ij})\) of the search point \(x_i\) by \(k_i\) to obtain the parameter \(\varepsilon_i\). The calculation formula is as follows: i ) multiplied by \(k_i\) i to obtain the parameter \(\varepsilon_i\) i ; The calculation formula is as follows:

[0042] \(\varepsilon_i\) i = f(x_{ij})\cdot k_i i )·k_i i To obtain the ALLFV feature vector, first make pairwise comparisons of the set \(\delta(\varepsilon_i, x_i, x_{ij})\). For each search point \(x_i\), put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i To obtain the ALLFV feature vector, first make pairwise comparisons of the set \(\delta(\varepsilon_i, x_i, x_{ij})\). For each search point \(x_i\), put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i To obtain the ALLFV feature vector, first make pairwise comparisons of the set \(\delta(\varepsilon_i, x_i, x_{ij})\). For each search point \(x_i\), put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i:j To obtain the ALLFV feature vector, first make pairwise comparisons of the set \(\delta(\varepsilon_i, x_i, x_{ij})\). For each search point \(x_i\), put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i For each search point \(x_i\), put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i,j put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i：j put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i:j put \(\varepsilon_i\), \(x_i\), and its neighbor \(x_{ij}\) together, and calculate the value of the set \(\delta(\varepsilon_i, x_i, x_{ij})\) according to the difference in the fitness values of \(x_i\) and its neighbor \(x_{ij}\), as follows: i

[0043]

[0044] Then, first calculate \(U(\delta(\varepsilon_i, x_i, x_{ij}))\), and then calculate \(P(x_i)\) for each search point i )), and then calculate \(P(x_i)\) for each search pointi ) and add the P(x i ) values with the same search points to the corresponding histograms to obtain UP(X). Divide UP(X) by the total number of search points λ to get the normalized histogram, which is the final adaptive local terrain feature vector ALLFV(X). The calculation formula is as follows:

[0045]

[0046] Among them, U(δ i ) is the calculation result of δi. It is obtained by summing the absolute values of the differences between adjacent values of δ(ε i , x i , x i:j )(j ranges from 1 to M). Intuitively, it measures the degree of change of the calculation result δ based on the fitness value comparison within the M nearest neighborhoods of the search point x i , and reflects a change amount of the local terrain feature.

[0047] P(x i ) represents a feature value of the search point x i .

[0048] UP(X) is a vector, which is an integer within the range [-M + 1, -M + 2,..., M + 1], representing the bin index in the histogram.

[0049] This step focuses on extracting problem features from the original data to obtain the adaptive local terrain feature vector ALLFV, providing an organized and feature - characterized data basis for subsequent algorithm processing.

[0050] Step 2: Algorithm performance evaluation. Measure the performance of the algorithm on a given dataset (CEC2013, CEC2014, or CEC2017), and compare the advantages and disadvantages of different algorithms. Specifically, use the Friedman test to evaluate each algorithm running on the above - mentioned test problems after feature extraction, so as to determine the algorithm ranked first, and store the algorithm ranked first as the optimal algorithm label.

[0051] Step 3: Classifier training. Use the ALLFV feature vector of the problem and the optimal algorithm label as a pair of inputs to train the classifier. The KNN classifier is used in this experiment.

[0052] Step 4: Similarity calculation. Use the Euclidean distance between problem features as the similarity calculation method. The specific calculation process is as follows:

[0053] (1) Similarity calculation, and its calculation formula is as follows:

[0054] Among them, Xi The feature vector representing the current problem, X j The feature vector representing other problems, l represents the dimension of the feature vector, 1 ≤ l ≤ m; d(X i， X j ) represents X i and X j The Euclidean distance between them, X i，l represents X i The l-th dimensional eigenvalue of X j，l represents X j The l-th dimensional eigenvalue of X; By calculating the distance to measure the difference degree of the feature vectors of two problems, the smaller the distance, the more similar the two problems are;

[0055] (2) Formation of the distance set:

[0056] Select the k distance values with the smallest distance from X i from all the calculated Euclidean distances to form the distance set d s (X i ) as follows: d s (X i ) = {d1, d2,....d k}, where d1, d2,..., d k are the k smallest distance values;

[0057] (3) Calculate the adaptive threshold for similarity comparison

[0058] The adaptive threshold β is calculated as follows:

[0059] Set the adaptive threshold for subsequent judgment on whether the similarity between other problems and the current problem meets the requirements.

[0060] Step 5: Based on the previous steps, give an algorithm selection strategy for new problems. When encountering a new problem: First, through the calculation method in Step 4, compare the similarity between the feature vector of the new problem and the feature vectors of existing problems; if the similarity between the new problem and the existing problems is less than the adaptive threshold calculated in Step 4, it means that the new problem is very similar to some existing problems, then recommend the algorithm according to the optimal algorithm labels corresponding to these similar problems, that is, directly adopt the optimal algorithm for dealing with similar problems; if the similarity is not less than the adaptive threshold, it means that the new problem is quite different from the existing problems. At this time, rely on the KNN classifier trained in Step 3, input the feature vector of the new problem, and let the classifier predict the algorithm suitable for this new problem.

[0061] Step 6: Mainly evaluate and screen the newly added algorithms. When a new algorithm needs to be added, the Friedman test is used to evaluate the performance of this new algorithm. By evaluating to judge the performance of the new algorithm, if it is found that the performance of some existing algorithms in the algorithm library is inferior to that of the new algorithm, then the new algorithm is used to replace these poorly performing algorithms, so as to ensure that the algorithms in the algorithm library always maintain good performance for better solving various problems in the future.

[0062] Step 7: System update.

[0063] It should be noted that: Each feature vector in Step 1 and the optimal algorithm label in Step 2 will form the corresponding relationship of the optimization problem - optimal algorithm. For example, the CEC2013 dataset has 28 problems, the CEC2014 dataset has 30 problems, and the CEC2017 dataset has 30 problems. There will be a total of 88 corresponding relationships of the optimization problem - optimal algorithm. These known multiple corresponding relationships are the basic data for training the classifier. When encountering a new problem with a similarity not less than the threshold, it proves that the feature vector of this problem is not in the existing basic data. Therefore, the corresponding relationship of the optimization problem - optimal algorithm of this optimization problem will be added to the known basic data. Thus, after encountering a new problem in Step 5, the new problem and the corresponding algorithm label will be added to the database, and the classifier will retrain and update the system according to the database.

[0064] It should be noted that: After evaluating the performance of the new algorithm in Step 6, if the poorly performing algorithms in the algorithm library are replaced, this will change the corresponding relationship of the "optimization problem - optimal algorithm". In order to enable the KNN classifier to adapt to the new situation and more accurately predict the algorithm for new problems, it is necessary to retrain it or adjust relevant parameters using the updated set of the optimization problem - optimal algorithm to complete the system update and improve the adaptability of the system to the matching of problems and algorithms.

[0065] The above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Those of ordinary skill in the art should understand that the specific implementation manners of the present invention can be modified or equivalently replaced by referring to the above embodiments. Any modification or equivalent replacement that does not depart from the spirit and scope of the present invention is within the scope of the claims pending for approval.

Claims

1. An online algorithm selection method based on the double-increment paradigm, characterized in that It includes the following steps: (1) For a selected set of problems for testing, use the Adaptive Local Landform Feature Vector (ALLFV) method to extract problem features and obtain the ALLFV feature vector of the problems; (2) Use the Friedman test to measure the performance of different algorithms in the algorithm library on the problem set, compare the advantages and disadvantages of different algorithms, and store the algorithm with the best performance as the optimal algorithm label corresponding to the problem; (3) Take the ALLFV feature vector of the problem obtained in step (1) and the optimal algorithm label obtained in step (2) as a pair of inputs and input them into the classifier for training; (4) Calculate the similarity using the Euclidean distance between problem features, find multiple problems with the highest similarity to the current problem, and calculate the adaptive threshold for similarity comparison and judgment; (5) When encountering a new problem, extract the feature vector of the new problem and compare its similarity with the feature vectors of existing problems according to step (4); If the similarity is less than the threshold, make a recommendation based on the optimal algorithm label corresponding to the similar problem; otherwise, use the trained classifier to predict the applicable algorithm, and the classifier is updated in real time according to this new problem and the predicted corresponding algorithm label; (6) For a new algorithm to be added to the algorithm library, use the Friedman test to evaluate its performance. If the performance of the new algorithm is better than the existing algorithms in the algorithm library, use the new algorithm to replace the poorly performing existing algorithms in the algorithm library; (7) After updating the algorithm library, update the optimal algorithm label corresponding to the problem at the same time, and then retrain and update the classifier according to the problem and the new label.

2. The online algorithm selection method according to claim 1, wherein The problem set mentioned above is CEC2013, CEC2014 or CEC2017.

3. The online algorithm selection method according to claim 1, characterized in that In step (1), the method for obtaining the ALLFV feature vector is as follows: Assume that each problem has multiple search points, and each search point is x i ∈x (i = 1, 2,..., λ), x i：j (1 ≤ j ≤ M) represents the j-th nearest neighbor of x i , where x i,1 is the point closest to x i ; the fitness value of x i is denoted as f(x i ), and f(x i:j ) refers to the fitness value of the j-th nearest neighbor to x i ; the average fitness of all search points is denoted as f mean ; then the relevant parameter ε i of each search point is calculated according to the following formula: ε i = f(x i )·k i For each search point x i , based on x i and the difference in fitness values between it and its nearest neighbor x i：j , calculate the value of the set δ(ε i , x i , x i:j ), and the judgment basis is as follows: Then, calculate U(δ i ) and P(x i ), and add the P(x i ) values that are the same for all search points to the corresponding histogram to obtain UP(X). Finally, calculate the adaptive local terrain feature vector ALLFV(X) from UP(X), and the calculation formula is as follows: UP(X) = [UP(-M + 1), UP(-M + 2),..., UP(M + 1)] 4. The online algorithm selection method according to claim 1, characterized in that The classifier in step (3) is a KNN classifier.

5. The online algorithm selection method according to claim 1, characterized in that The similarity calculation formula in step (4) is as follows: Among them, X i represents the feature vector of the current problem, and X j represents the feature vector of other problems. l represents the dimension of the feature vector, where 1 ≤ l ≤ m; d(X i， X j ) represents the Euclidean distance between X i and X j . X i，l represents the l-th dimensional eigenvalue of X i , and X j，l represents the l-th dimensional eigenvalue of X j . Select the k distance values with the smallest distances from all the Euclidean distances calculated for X i to form the distance set d s (X i ) is as follows: d s (X i ) = {d1, d2,.... d k}, where d1, d2,..., d k are the k smallest distance values; The threshold is the adaptive threshold β, and its calculation formula is as follows:

Images