CN113420820A - Regularized optimal transmission theory-based unbalanced data classification method - Google Patents

Regularized optimal transmission theory-based unbalanced data classification method Download PDF

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CN113420820A
CN113420820A CN202110724175.7A CN202110724175A CN113420820A CN 113420820 A CN113420820 A CN 113420820A CN 202110724175 A CN202110724175 A CN 202110724175A CN 113420820 A CN113420820 A CN 113420820A
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optimal transmission
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马丽涛
文人庆
陈继强
张峰
张丽娜
付俊丰
万杰
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Harbin Institute of Technology
Hebei University of Engineering
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Abstract

The invention discloses an unbalanced data classification method based on a regularization optimal transmission theory, which comprises the following steps: firstly, the method comprises the following steps: acquiring an unbalanced training sample set and a test sample set; II, secondly: constructing a Monge type optimal transmission problem; thirdly, the method comprises the following steps: convex relaxation of a Monge type optimal transmission problem into a discrete Kantorovitch type optimal transmission problem; fourthly, the method comprises the following steps: designing a reasonable non-convex regular term, and further constructing a non-convex regular optimal transmission problem; fifthly: designing a maximum-minimum optimal transmission solving algorithm, and calculating Pre, Rec, GM and F of the algorithm model on each data set1And M, evaluating the index value, thereby realizing effective classification of the unbalanced data set. The invention constructs the optimal transmission problem with the non-convex regular term and enriches the theoretical research of the optimal transmission. Compared with a common unbalanced data classification method, the method has higher classification precision on the unbalanced data.

Description

Regularized optimal transmission theory-based unbalanced data classification method
Technical Field
The invention relates to a classification method for processing unbalanced data, in particular to a method for classifying the unbalanced data by using a maximum-minimum algorithm designed by a regularized optimal transmission theory.
Background
An unbalanced data set refers to a set of samples of a certain class that are much smaller (or much larger) in number than samples of other classes in one data set. For most machine learning algorithms, if the training set is an unbalanced data set, the performance of the classifier is degraded. In recent years, the above problems have been highlighted in various industries, and have attracted high attention from many scholars and experts.
At present, the research of scholars at home and abroad can be roughly divided into two types: one is to reconstruct the data set from the data itself to reduce the unbalance degree of the data set, thereby improving the classification performance of a few classes; and the other type is an improved strategy which aims at the traditional classification model, provides a series of pertinences from the classification idea and classification algorithm level, is biased to pay more attention to the minority class and improves the classification precision of the minority class.
The above researches effectively improve the classification performance of the machine learning model, but the methods are not robust in performance on different data source data sets. For a while, no detailed description of authority publications is available on how to construct an algorithm with a certain robustness to unbalanced data sets.
Disclosure of Invention
In order to improve the classification precision of the unbalanced data set and enable the classifier to have certain robustness on the unbalanced data set, the invention provides an unbalanced data classification method based on a regularization optimal transmission theory.
The purpose of the invention is realized by the following technical scheme:
an unbalanced data classification method based on a regularization optimal transmission theory comprises the following steps:
the method comprises the following steps: obtaining an unbalanced training sample set
Figure BDA0003137894030000021
And test sample set
Figure BDA0003137894030000022
The proportion of class imbalance (i.e., imbalance rate) in the training sample set and the test sample set is required to be close;
step two: aiming at the training sample set and the testing sample set obtained in the step one, a single type optimal transmission problem is constructed, wherein the single type optimal transmission problem is as follows:
Figure BDA0003137894030000023
wherein mu is a training sample set obedient RnV obeys R for the test sample setnThe upper probability distribution, # is a push forward operator, T is a transmission mapping, and omega0F is a cost function for training the sample space;
step three: convex relaxation of the Monge type optimal transmission problem obtained in the step two is performed into a discrete Kantorovich type optimal transmission problem, wherein the discrete Kantorovich type optimal transmission problem after convex relaxation is as follows:
Figure BDA0003137894030000024
wherein pi is a set composed of all joint probability distributions of the distributions mu and v, xi、yjThe coordinates of the samples are represented by i and j respectively representing lower subscripts with values between 0-N and 0-M, wherein N is the number of training samples, M is the number of test samples, and gamma is a transmission plan;
step four: designing a reasonable non-convex regular term for the discrete Kantorovitch type optimal transmission problem obtained in the step three, and further constructing a non-convex regular optimal transmission problem, wherein:
the non-convex regularization term is designed as follows:
Figure BDA0003137894030000031
wherein p, q are arbitrary real numbers,
Figure BDA0003137894030000032
is 1pQ-th power of norm, IcSet of indices for samples with class c of sample, γ (I)cJ) is a vector formed by data belonging to the c-th class in the j-th column of the matrix γ, and when p is 2 and q is 2, Ω (γ) is a convex regular term; when p is 1,
Figure BDA0003137894030000033
Time omega (gamma) is a non-convex regular term;
the following non-convex regular optimal transmission problem can be obtained:
Figure BDA0003137894030000034
wherein α is a non-negative real number;
step five: aiming at the characteristics of the non-convex regular optimal transmission problem obtained in the fourth step, based on the maximum-minimum thought, a maximum-minimum optimal transmission (MMROT) solving algorithm is designed, and the classification precision (Pre), the recall rate (Rec), the Geometric Mean (GM) and the F of the algorithm model on each data set are calculated1Value (F)1M) evaluating the index values, thereby achieving effective classification of the unbalanced data set, wherein:
the specific steps of the maximum-minimum optimal transmission (MMROT) solving algorithm are as follows:
step (1): calculating the maximum linear approximation term G of the non-convex regular term, i.e. for a fixed
Figure BDA0003137894030000035
Is provided with
Figure BDA0003137894030000036
Where β is a constant, the elements of matrix G are:
Figure BDA0003137894030000037
ε is the perturbation term of the data, IcSet of labels for samples with sample class c, γ (I)cJ) is a vector formed by data belonging to the c-th class in the j-th column of the matrix gamma;
step (2): constructing cost matrix C ═ (| x)i-yj||2) + α · G, solving the following optimization problem using interior point algorithm:
Figure BDA0003137894030000041
obtaining a minimum of the above-mentioned problems, i.e. an optimal transmission plan
Figure BDA0003137894030000042
(3) According to obtaining
Figure BDA0003137894030000043
By using
Figure BDA0003137894030000044
And updating the linear approximation term G to recalculate the cost matrix C until an iteration termination condition is met.
Compared with the prior art, the invention has the following advantages:
1. aiming at the classification problem of unbalanced data, the optimal transmission problem with the non-convex regular term is constructed by designing the proper non-convex regular term, and the theoretical research of the optimal transmission is enriched.
2. The invention considers the disturbance of data when calculating the maximum linear approximate term of the non-convex regular term, so that the constructed maximum-minimum optimal transmission algorithm has certain robustness on the classification of the unbalanced data set, and can meet the classification requirement of more unbalanced data in practice.
3. The invention provides a maximum-minimum optimal transmission algorithm aiming at unbalanced data classification by combining an optimal transmission theory. Compared with a common unbalanced data classification method, the method has higher classification precision on the unbalanced data.
Drawings
FIG. 1 shows the Pre values for 12 different methods in Table 3.
FIG. 2 shows the Rec values for 12 different methods in Table 3.
FIG. 3 is F of 12 different processes in Table 31And (4) the value of M.
FIG. 4 shows the GM values of 12 different methods in Table 3.
FIG. 5 shows the Pre values for 12 different methods in Table 4.
FIG. 6 shows the Rec values for 12 different methods in Table 4.
FIG. 7 is F of 12 different methods in Table 41And (4) the value of M.
FIG. 8 shows GM values for 12 different methods in Table 4.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings, but the present invention is not limited thereto, and modifications or equivalent substitutions may be made to the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention.
The invention provides a method for effectively classifying unbalanced data by a maximum-minimum optimal transmission algorithm, which comprises the following steps:
the method comprises the following steps: and acquiring an unbalanced data set, and dividing the unbalanced data set into a training sample set and a test sample set to enable the unbalanced rate of the test sample set to be close to the unbalanced rate of the training sample set.
Step two: and (4) aiming at the training sample set and the testing sample set obtained in the step one, constructing a single type optimal transmission problem.
Suppose that a training sample set obeys RnProbability distribution of [ mu ] above, test sample set obeys RnV. probability distribution ofThe space is respectively omega0And Ω1The cost function is f: omega0×Ω1→ R. The purpose of the Monge type optimal transmission problem is to find a transmission mapping T*:Ω0→Ω1The transmission cost from distribution mu to distribution v is minimized, so that an optimal transmission problem can be constructed:
Figure BDA0003137894030000051
wherein, # is a push forward operator, indicating that it is for any omega1Can measure A, v (A) mu (T)-1(A) ) solving the optimal transmission problem (1) to obtain the optimal transmission transformation T*。
Step three: and C, convexly relaxing the Monge type optimal transmission problem obtained in the step two into a scattered Kantorovitch type optimal transmission problem.
In consideration of the difficulty in solving the single-type optimal transmission problem, the solution is more difficult particularly when the continuous single-type optimal transmission problem is solved. However, the discrete Kantorovitch-type optimal transmission problem is a linear optimization problem, and a solution algorithm is numerous. And when the Monge type optimal transmission problem has a solution, the solution of the discrete Kantorovitch type optimal transmission problem is equivalent to the solution. To this end, problem (1) convex relaxation is implemented as a discrete Kantorovitch-type optimal transmission problem using convex relaxation techniques:
Figure BDA0003137894030000061
wherein pi is a set formed by the joint probability distribution of all the distribution mu and v, and pi isxRepresenting the projection mapping of a point (x, y) in the x-direction, piyRepresents a projection mapping of point (x, y) in the y direction, i.e.: pix:Ω0×Ω1→Ω0And is
Figure BDA0003137894030000062
πy:Ω0×Ω1→Ω1And is
Figure BDA0003137894030000063
The measure γ that satisfies the constraint condition in (2) is called a transmission plan.
If the distributions μ and ν are discrete, i.e.:
Figure BDA0003137894030000064
wherein, deltazAt point z ∈ RnThe Dirac measure of (a). At this time, the following discrete Kantorovitch-type optimal transmission problem needs to be considered:
Figure BDA0003137894030000065
wherein the matrix C ∈ RN×M，Cij＝f(xi,yj)，
Figure BDA0003137894030000066
Under the general condition of
Figure BDA0003137894030000067
The discrete Kantorovitch-type optimal transmission problem is as follows:
Figure BDA0003137894030000071
step four: designing a reasonable non-convex regular term for the discrete Kantorovitch type optimal transmission problem (4) obtained in the step three, and constructing the non-convex regular optimal transmission problem.
To further enhance the robustness of the method designed in step three to the unbalanced data set and fully utilize the label information of the training samples, a reasonable regular term needs to be constructed (i.e. all training samples mapped to the same test sample should have the same label):
Figure BDA0003137894030000072
wherein p, q are arbitrary real numbers,
Figure BDA0003137894030000073
is 1pThe norm to the power q. When p is 2 and q is 2, Ω (γ) is a convex regularization term; when p is 1,
Figure BDA0003137894030000074
Time Ω (γ) is a non-convex regular term. However, in practical applications, especially in supervised classification problems, the non-convex regularization term tends to have a better effect.
Therefore, the following optimal transmission problem with a non-convex regularization term (i.e., non-convex regularization optimal transmission problem) is constructed:
Figure BDA0003137894030000075
wherein, IcFor the set of labels for samples with sample class c, α is a non-negative real number, γ (I)cAnd j) is a vector formed by data belonging to the c-th class in the j-th column of the matrix gamma.
Step five: aiming at the characteristics of the non-convex regular optimal transmission problem obtained in the fourth step, based on the maximum-minimum thought, a maximum-minimum optimal transmission (MMROT) solution algorithm is designed, and the classification precision (Pre), the recall rate (Rec), the Geometric Mean (GM), the F and the like of the algorithm model on each data set are calculated1Value (F)1M), and the like.
According to the maximum-minimum thought, firstly calculating a non-convex regular term
Figure BDA0003137894030000081
For a fixed maximum linear approximation term G, i.e. for
Figure BDA0003137894030000082
Is provided with
Figure BDA0003137894030000083
Where β is a constant, the elements of matrix G are:
Figure BDA0003137894030000084
the smaller positive number epsilon is the perturbation term of the data. Second, the cost matrix C is constructed (| | x)i-yj||2) + α · G, solving the following optimization problem using interior point algorithm:
Figure BDA0003137894030000085
obtaining an optimal transmission plan gamma*. Finally, according to the obtained gamma*And (5) updating the linear approximation term G by using the formula (6) to recalculate the cost matrix C until an iteration termination condition is met. In this way, a maximum-minimum optimal transport (MMROT) solution algorithm is completed.
In order to evaluate the performance of the algorithm from multiple angles, a plurality of indexes are selected for evaluating the performance of the algorithm, and the concept of a confusion matrix is introduced firstly.
In the binary task, the confusion matrix is a 2 × 2 matrix, and table 1 is a table of examples of the confusion matrix in the binary task.
TABLE 1 confusion matrix
Figure BDA0003137894030000086
Further, the formula is calculated according to the evaluation index as follows:
Figure BDA0003137894030000087
Figure BDA0003137894030000088
Figure BDA0003137894030000089
Figure BDA00031378940300000810
Figure BDA0003137894030000091
respectively calculating the classification precision, recall rate, geometric mean and F of the maximum-minimum optimal transmission model on each data set1The value is obtained.
Examples of the applications
To verify the effectiveness of the maximum-minimum optimal transport (MMROT) algorithm proposed by the present invention, the following application examples are performed using two categories of true unbalanced data sets downloaded from the commonly used KEEL database https:// sci2 s.ugr.es/key/equivalent/unbalanced. php, divided into two cases below 9 and above 9 according to the data unbalance rate (IR).
Example 1: unbalanced data set classification problems with unbalanced ratios below 9.
First, 8 data sets with imbalance rates lower than 9 and different feature numbers and imbalance rates are downloaded from a KEEL database (the description of each data set is detailed in Table 2), and the data sets are divided into training sample sets and testing sample sets according to a 5-fold cross validation method.
TABLE 2 unbalanced data set description with unbalance rates below 9
Figure BDA0003137894030000092
Secondly, performing experiments according to the second step to the fifth step of the MMROT algorithm to obtain the classification precision (Pre), the recall rate (Rec), the Geometric Mean (GM) and the F of each data set1Value (F)1M) and the likeAnd evaluating the index value.
In order to verify the effectiveness of the MMROT algorithm provided by the invention, the MMROT algorithm is compared with 11 classification algorithms such as a common classification method of unbalanced data, Support Vector Machine (SVM), AdaBoost, AdaC2.M1, SAMME, HDDTecoc, HDDTova, IMECOC + OVA, IMECOC + sparse, MCHDDT and PIBoost, and the results are detailed in a table 2 and a table 3 and in a figure 1, a figure 2, a figure 3 and a figure 4.
For simplicity and illustrative purposes, only classification results of ecoli-0_ vs _1 and glass-0-1-2-3_ vs _4-5-6 in table 2 are listed in table 3 and table 4, respectively, and classification results of the remaining 6 data sets are not listed one by one, wherein NaN indicates that the denominator is 0 in the calculation (i.e., for 5-fold data of such data sets, each classification precision is 0). However, the classification accuracy, recall, geometric mean and F for the 8 data sets in Table 1 were determined1The results of evaluation indexes such as values are shown in FIGS. 1 to 4, respectively, in which the numbers in the abscissa correspond to the numbers of the data sets in Table 2, and the ordinate shows the classification accuracy (Pre), recall ratio (Rec), Geometric Mean (GM) and F1Value (F)1M)。
TABLE 3 Classification results of different methods on the eco-0 _ vs _1 dataset
Figure BDA0003137894030000111
TABLE 4 Classification results for different methods on glass-0-1-2-3_ vs-4-5-6 datasets
Figure BDA0003137894030000112
As can be seen from tables 3 and 4 and fig. 1, 2, 3 and 4, the MMOT algorithm proposed by the present invention has higher classification accuracy (Pre), recall rate (Rec), Geometric Mean (GM) and F compared to other algorithms1Value (F)1M). This shows that the MMOT algorithm provided by the invention has strong advantages and effects in the aspect of processing the unbalanced data set with the unbalanced rate lower than 9And (5) fruit.
Example 2: data sets with imbalance rates higher than 9.
Firstly, 9 real data sets with imbalance rates higher than 9 and different feature numbers and imbalance rates are downloaded from a KEEL database (the description of each data set is detailed in Table 5), and the data sets are divided into training sample sets and testing sample sets according to a 5-fold cross validation method.
TABLE 5 training data set description with imbalance above 9
Figure BDA0003137894030000121
Secondly, the steps of the MMROT algorithm are utilized to carry out experiments, and classification results are compared with results of 11 classification algorithms such as SVM, AdaBoost, AdaC2.M1, SAMME, HDDTecoc, HDDTova, IMECOC + OVA, IMECOC + sparse, MCHDDT, PIBoost and the like.
Table 6 lists the classification results of yeast-2_ vs _4 data sets in Table 4 under different methods, and FIGS. 5-8 show the classification accuracy, recall ratio, geometric mean and F of 9 data sets in Table 51Values, etc., wherein the numbers in the abscissa correspond to the sequence numbers of the data sets in Table 5, and the ordinate shows the classification accuracy (Pre), recall ratio (Rec), Geometric Mean (GM) and F of the data sets, respectively1Value (F)1M)。
TABLE 6 Classification results of different methods on yeast-2_ vs _4 dataset
Figure BDA0003137894030000122
Figure BDA0003137894030000131
As can be seen from table 6 and fig. 5, 6, 7, and 8, the MMOT algorithm proposed by the present invention has higher classification accuracy (Pre), recall rate (Rec), Geometric Mean (GM), and F compared to other algorithms1Value (F)1M). This shows that the MMOT algorithm provided by the invention has stronger advantages in the aspect of classification of unbalanced data sets with the unbalanced rate higher than 9.

Claims (6)

1. An unbalanced data classification method based on regularization optimal transmission theory is characterized by comprising the following steps:
the method comprises the following steps: obtaining an unbalanced training sample set
Figure FDA0003137894020000011
And test sample set
Figure FDA0003137894020000012
Step two: aiming at the training sample set and the testing sample set obtained in the step one, a single type optimal transmission problem is established;
step three: convex relaxation of the Monge type optimal transmission problem obtained in the step two into a discrete Kantorovitch type optimal transmission problem;
step four: designing a reasonable non-convex regular term for the discrete Kantorovitch type optimal transmission problem obtained in the step three, and further constructing a non-convex regular optimal transmission problem;
step five: aiming at the characteristics of the non-convex regular optimal transmission problem obtained in the step four, designing a maximum-minimum optimal transmission solving algorithm based on the maximum-minimum thought, and calculating the classification precision, the recall rate, the geometric mean and the F of the algorithm model on each data set1And evaluating the index value, thereby realizing the effective classification of the unbalanced data set.
2. The method according to claim 1, wherein in the second step, the monte-type optimal transmission problem is:
Figure FDA0003137894020000013
wherein mu is a training sample set obedient RnV obeys R for the test sample setnThe upper probability distribution, # is a push forward operator, T is a transmission mapping, and omega0To train the sample space, f is the cost function.
3. The method according to claim 1, wherein in step three, the discrete Kantorovitch-type optimal transmission problem after convex relaxation is:
Figure FDA0003137894020000021
wherein pi is a set composed of all joint probability distributions of the distributions mu and v, xiYj is a sample coordinate, i, j respectively represent a lower subscript with the value between 0-N and 0-M, N is the number of training samples, M is the number of testing samples, and gamma is a transmission plan.
4. The method according to claim 1, wherein in the fourth step, the non-convex regularization term is designed as follows:
Figure FDA0003137894020000022
wherein p, q are arbitrary real numbers,
Figure FDA0003137894020000023
is 1pQ-th power of norm, IcSet of labels for samples with sample class c, γ (I)cJ) is a vector formed by data belonging to the c-th class in the j-th column of the matrix γ, and when p is 2 and q is 2, Ω (γ) is a convex regular term; when p is 1,
Figure FDA0003137894020000024
Time Ω (γ) is a non-convex regular term.
5. The method according to claim 1, wherein in the fourth step, the non-convex regularized optimal transmission problem is:
Figure FDA0003137894020000025
where α is a non-negative real number.
6. The method for classifying unbalanced data based on the regularized optimal transmission theory according to claim 1, wherein in the step five, the specific steps of the maximum-minimum optimal transmission solving algorithm are as follows:
step (1): calculating the maximum linear approximation term G of the non-convex regular term, i.e. for a fixed
Figure FDA0003137894020000031
Is provided with
Figure FDA0003137894020000032
Where β is a constant, the elements of matrix G are:
Figure FDA0003137894020000033
ε is the perturbation term of the data, IcSet of labels for samples with sample class c, γ (I)cJ) is a vector formed by data belonging to the c-th class in the j-th column of the matrix gamma;
step (2): constructing cost matrix C ═ (| x)i-yj||2) + α · G, solving the following optimization problem using interior point algorithm:
Figure FDA0003137894020000034
obtaining a minimum of the above-mentioned problems, i.e. an optimal transmission plan
Figure FDA0003137894020000035
(3) According to obtaining
Figure FDA0003137894020000036
By using
Figure FDA0003137894020000037
And updating the linear approximation term G to recalculate the cost matrix C until an iteration termination condition is met.
CN202110724175.7A 2021-06-29 2021-06-29 Regularized optimal transmission theory-based unbalanced data classification method Pending CN113420820A (en)

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115064250A (en) * 2022-06-06 2022-09-16 大连理工大学 Method for adjusting distribution of stay in hospital and related product
CN115861666A (en) * 2022-12-22 2023-03-28 河北工程大学 3D image point cloud matching method, system, equipment and medium

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115064250A (en) * 2022-06-06 2022-09-16 大连理工大学 Method for adjusting distribution of stay in hospital and related product
CN115861666A (en) * 2022-12-22 2023-03-28 河北工程大学 3D image point cloud matching method, system, equipment and medium
CN115861666B (en) * 2022-12-22 2023-06-27 河北工程大学 3D image point cloud matching method, system, equipment and medium

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