Abstract—Taking motivation from ɛ-insensitive twin support vector regression (ɛ-TSVR) and the projection idea, this paper proposes a novel ɛ-twin projection support vector regression models, called ɛ-TPSVR. The proposed ɛ-TPSVR, which is based on ɛ-TSVR, determines the regression function through a pair of nonparallel hyperplanes solved by two smaller sized quadratic programming problems. Different from ɛ-TSVR, a projection axis is sought for each optimization problem of ɛ-TPSVR such that the variance of the projected points is minimized. Therefore, the empirical correlation coefficient between each hyperplane and the projected inputs can be optimized. The experimental results indicate that the proposed ɛ-TPSVR obtains the better prediction performance than TSVR and ɛ-TSVR methods that were widely adopted.
Index Terms—Support vector machine (SVM), Regression analysis, Projection algorithms, Benchmark testing, ɛ-twin support vector regression (ɛ-TSVR)
I. INTRODUCTION
upport vector machine (SVM) [1-3] has become a powerful tool in pattern classification and regression field for its excellent generalization performance, and has been successfully applied to various real-world problems [4-6]. As for support vector regression model (SVR), which is a standard tool in regression tasks, there are some classical methods, such as ε-support vector regression (ε-SVR) [2], the sequential minimal optimization (SMO) [7], smooth SVR [8], the parametric insensitive SVR [9], PSO-SVR [10], and v-support vector classification (v-SVC) [11].
One of the main challenges for the SVR is the high cost of training, i.e. , where m is the number of training samples. Motivated by the researches on twin support vector machine (TSVM) [12-15], Peng [16] proposed a twin support vector regression (TSVR), in which the complexity of algorithm is reduced to 2 /2 , that means the TSVR is four times faster than the usual SVR in theory.
Different from SVR, the TSVR method generates two
Manuscript received June 4, 2018; revised November 3. This work was supported by the National Natural Science Foundation of China under Grants 61304002 and 61403177.
Xinyu Oyang is with School of International Finance & Banking, University of Science and Technology Liaoning, Anshan, CO 114051 China.
(corresponding author: e-mail: ouyangxinyu@ ustl.edu.cn).
Nannan Zhao is with School of Electronics and Information Engineering, University of Science and Technology Liaoning, Anshan, CO 114051 China.
(e-mail: [email protected]).
Chuang Gao is with School of Electronics and Information Engineering, University of Science and Technology Liaoning, Anshan, CO 114051 China.
(e-mail: [email protected]).
Lidong Wang is with School of Electronics and Information Engineering, University of Science and Technology Liaoning, Anshan, CO 114051 China.
(e-mail: [email protected]).
nonparallel functions by solving a pair of smaller sized quadratic programming problems (QPPs) instead of a large size one, so the algorithm TSVR is much faster than a usual SVR. Consequently, many kinds of twin-type SVR have been studied extensively. For example, Shao et al. [17] proposed an ε-insensitive twin support vector regression (ε-TSVR), in which the structural risk minimization principle is implemented by introducing the regularization term in the primal problems. R. Rastogi et al. [18] presented another twin model for regression termed as v-TSVR, which extended the ε-TSVR techniques by automatically adjusting the value ε1 and ε2 via user defined parameters v1 and v2, so that the fraction of errors and support vectors can be controlled.
However, the previous algorithms and models are very suitable for the assumption that the noise level is uniform on training data, or at least. For the heteroscedastic noise structure, in which the amount of noise depends on location or input, the assumption of a uniform noise level is not satisfied. Recently, motivated by the par-v-SVR [19], Peng [20] presented a novel parametric- insensitive SVR model for data regression, termed as twin parametric insensitive SVR (TPISVR). Although both par-v-SVR and TPISVR are more suitable for the case that the noise is more heteroscedastic than the classical SVR, the difference between them is that the strategy TPISVR is to solve two smaller sized QPPs rather than one large QPP, which makes it learn faster than par-v-SVR and SVR. Later, another efficient twin projection support vector regression (TPSVR) algorithm proposed by Peng [21] improved TPISVR through seeking a suitable projection axis such that the empirical variance of projected points is minimized.
In addition, it is interesting to note that, in the ε-TSVR model, the regularization term /2, 1,2 is added to the primal optimization problems because of the structural risk minimization principle. The regularization terms in the TPSVR model, however, is /2, 1,2.
Because minimizing the projection zone of input points is unrelated to the offset , has to be removed from the regularization term in order to getting the solutions of optimization problems correctly. This implies that the method of solving has to be deduced separately, which will increase the complexity of the model. In fact, what we expect is that the augmented vector is directly solved, not directly obtained . To this end, inspired by ε-TSVR and the idea of projection, an ε-twin projection support vector machine for regression is proposed in this paper, which is termed as ε-TPSVR. Some of the major features of the proposed ε-TPSVR are as follows:
An Efficient Twin Projection Support Vector Machine for Regression
Xinyu Ouyang, Nannan Zhao, Chuang Gao, and Lidong Wang
S
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(1) The proposed ε-TPSVR model aims to seek a suitable projection axis such that the empirical variance of projected points is minimized. It is more suitable for the assumption that the noise level is heteroscedastic. In addition, because the propose method is based on the ε-TSVR model, the structural information of data is then embedded into the regression model.
(2) The Introduction of matrixes J and K makes it possible to solve the augmented vector . It also reduces computational complexity. For the definition of matrixes J and K, see section III for details.
(3) In terms of generalization performance, the results of the experiment have indicated that the performance obtained by the proposed ε-TPSVR is superior to that of other classical twin-type SVR algorithms for the artificial dataset with the heteroscedastic error structure and UCI datasets.
The rest texts of paper are organized as follows. Section II introduces notations used in this paper and briefly describes ε-TSVR and the projection axis. Section III proposes the linear ε-TPSVR model and its extension for the non-linear case. Numerical experiments have been done on both synthetic and real-world datasets and their results have been compared with TSVR, ε-TSVR and v-TSVR in Section IV, and section V contains concluding remarks.
II. BACKGROUND
In this section, we first give a brief description of ε-twin support vector regression (ε-TSVR) [17] and then introduce the concept of projection axis [21]. Without loss of generality, given a training set , , 1,2, … , ， where
∈ and ∈ . Also let ; ; … ; be the input training sample, and ; ; … ; be the response of the training samples, where is a matrix, the i-th row represents the i-th training sample, Y is a 1 vector, and represents the i-th response.
A. ε-Twin Support Vector Regression
Following the idea of TWSVM and TSVR, Shao et al. [15]
proposed an approach termed as ε-twin support vector regression (ε-TSVR). Just like any other twin-type SVR, it also finds two insensitive proximal linear function:
w , 1,2. By introducing the regularization terms /2 and the slack variables , ∗, and ∗ , the primal problems can be expressed as
, , ,∗
1 2
1
2 ∗ ∗ ,
. . ∗,
, 0,
1
and
, , , ∗
1 2
1
2 ∗ ∗ ,
. . ∗,
, 0,
2
where , , , , and are positive parameters, e is column vector of 'ones' of appropriate dimension. The main difference between TSVR and ε-TSVR is an extra regularization term /2, 1,2, in (1) and (2), thus the structural risk minimization principle is implemented [17].
In order to get the solutions of problems (1) and (2), their dual problems need to be derived. By introducing the
Lagrangian multiplies and , the dual problems of the ε-TSVR are given by
1
2 ,
. . 0 , 3
and
1
2 ,
. . 0 ,
4
where . Once the QPPs (3) and (4) are solved, , and the final regressor function can be obtained as follows
, 5 , 6 1
2
1 2
1
2 . 7
B. Projection Axis
In the twin-type SVR, there is usually a pair of insensitive
bound function w , 1,2 , i.e. the
insensitive up- and down-bound functions by solving a pair of smaller sized QPPs. In order to obtain a suitable , the variance of are supposed to be as small as possible. In the other word, in each QPP, it finds a projection axis, denoted as 1 , 1,2, such that each points will be projected on the projection axis , and the projected zone is expected to be minimized . It's important to note here that does not contribute to the variance of
because is a fixed variable. The projected zone for each bound function is minimized as
1
2 1 1
1
2 1 1
1 2
1
2 , 1,2, 8
where Σ , Σ are the empirical covariance matrices of inputs, and responses, and Σ is the empirical covariance matric of the inputs and responses. They are defined respectively as
1 1
̅ ̅ ,
1 1
,
1 ̅ , ,
where ̅ and are the centroid points of inputs and responses, respectively.
III. ε-TWIN PROJECTIONSUPPORT VECTORREGRESSION In this section, a novel ε-twin projection support vector machine for regression, termed as ε-TPSVR is presented, which is motivated by the ε-TSVR and the idea of projection.
Using (8), the optimization problems of (1) and (2) can be rewritten as the following formulations,
, , ，
∗,
1 2
1 2 ∗ ∗
1
2 ,
. . ∗,
, 0,
9
and
, , ,∗,
1 2
1 2 ∗ ∗
1
2 ,
. . ∗,
, 0.
10
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To solve (9), the Lagrangian function is given by
,
(11)
where , , … , and , , … , are the vectors of Lagrange multipliers. The K.T.T necessary and sufficient optimality conditions for (9) are given by 0, 12
0 , (13)
0 , (14)
, 0, (15)
0, 0, (16)
0, 0 . (17)
Since 0, from (14), we have 0 . Next, combing (12) and (13) leads to 0 0 0 0 0, (18)
By defining , , 0 00, 0 , (19)
equation (18) can be rewritten as 0, (20)
Then the augmented vector is given by . (21)
Then putting (21) into (11) and using the above K.T.T. conditions, the dual problem of the problem (9) is obtained as ax 1 . . 02 , , 22 where G G and to simplify our expression. In the same way, the dual of the problem (10) can be given as ax 1 2 λ . . 0 , , 23 where λ and . The augmented vector is given by , (24) Once the solutions , , and are obtained, the estimated regressor can be constructed as (7). For the case of nonlinear regressors, an appropriately chosen kernel function K should be introduced, and the more detailed derivation is similar to the references [17, 21].
IV. EXPERIMENTS
To check the performance of the proposed ε-TPSVR, we compare it with the popular TSVR, ε-TSVR and v-TSVR in several artificial and benchmark [22] datasets. All the computations are carried out on Windows 7 OS Intel Core i5-4210U CPU(2.4GHz) with 4GB RAM and MATLAB R2014a environment. In order to decrease the computational complexity of parameter selection, we set , , and . And Gaussian kernel function defined by , exp ‖ ‖ ⁄ is used to process nonlinear data, where vectors , ∈ , and the parameter 0 . Some commonly-used evaluation criterions [16] shown in Table I are introduced before evaluating the performance of these methods.
To test the regression performance of our proposed
ε-TPSVR, sin ⁄ , ∈ 4 , 4 and
| ⁄ |, ∈ 4 , 4 are introduced to generate all artificial datasets. The training data's observed values are polluted by
the form , where the noise
0.5 | | 8⁄ depends on input. Variable is the form of , or , , where , represents the uniformly random variable in , and , represents the Gaussian random variable with means and variance . It implies that these examples have the heteroscedastic error structure.
TABLE I
PERFORMANCE METRICS AND THEIR CALCULATION
Metric Calculation
SSE ∑
SSR ∑
SST ∑
SSE/SST ∑ ⁄∑
SSR/SST ∑ ⁄∑
Fig. 1 shows the one run results of ε-TPSVR and our ε-TPSVR on with Gaussian noise with mean zero and standard deviation 0.2, and Fig. 2 makes the same comparison with . In Fig. 1 and Fig. 2, the solid line is the noiseless test data, and heteroscedastic noise is added to the test data as training data, which is the cross in the figures.
It can be seen from the figures that the heteroscedastic noise different from usual additive noise has the characteristic of depending on location or input, so it's harder to deal with. 260 noisy data are selected as training data, 500 data are selected as test data without noise, and the final regression function is trained by ε-TSVR and our ε-TPSVR. Horizontal coordinate represents the number of training points. In particular, in order to see the regression effect more clearly, the horizontal coordinates only show the range from 50 to 450 in Fig. 1, and the horizontal coordinates only show the range from 200 to 300 in Fig. 2. One of the dotted lines is the regression result of the proposed ε-TPSVR method, and the other dotted line is the regression result of the ε-TSVR method. The SSE values of ε-TSVR and ε-TPSVR algorithms in Fig. 1 are 0.4276 and 0.3674 in order, and the SSE values of them in Fig. 2 are 1.5449 and 1.0983 in order. It can be clearly observed from the figures that the result of ε-TPSVR is closer to the black solid line than the result of ε-TSVR. It is because that some points in the heteroscedastic zone are discarded, whereas ε-TSVR cannot filter out the possible noise points. It is also for this reason that data structure information is embedded in our ε-TPSVR learning process.
Next, the effectiveness of the proposed ε-TPSVR is further verified by comparing it with TSVR, ε-TSVR and v-TSVR.
To fairly compare with the performance of TSVR, ε-TSVR, v-TSVR and ε-TPSVR, 20 independent groups’ data on the two functions with different types of noise are generated randomly using Matlab toolbox, including 260 samples during training and 500 samples during testing for each function. Besides, testing data points are uniformly sampled from the objective function without any noise. The measure results are listed in Table Ⅱ and Table Ⅲ. It is easy to see that our method obtains the smaller SSE values and SSE/SST values than the other methods on these two problems, which indicates that the proposed ε-TPSVR gets better performance
Engineering Letters, 27:1, EL_27_1_13
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th th
br str al te va Ta ou
han the other a hat the noise is
Fig. 1 Predic
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PERFORMANCE C DataSets
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ABLE II ARTIFICIAL DATA
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0.01
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V. CONC a novel ɛ-tw ls (ɛ-TPSVR) VR is based e the term of p pirical correla nd targets an VR derives be
m. The expe I datasets sho better gener and v-TSVR.
ameters and h problems an
TABLE MANCE COMPARIS gressor SSE TSVR 6.236
4.724 TSVR 6.630
5.340 TSVR 6.630
5.340 TPSVR 5.554
4.130 TSVR 4.284
1.543 TSVR 4.014
1.635 TSVR 4.017
1.63 TPSVR 3.918
1.576 TSVR 1.875
0.667 TSVR 1.652
0.647 TSVR 1.652
0.647 TPSVR 1.595
0.61
REFERE stical Learning Th
2.
tutorial on sup ta Min. Knowl. D
EⅢ
RTIFICIAL DATASE SE SSE/SS 499±
252
0.0023±
0.0004 029±
64
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CLUSION
win projection ) is proposed d on ɛ-TSV projection axis ation coeffici nd the projec etter approxim erimental resu ow that our pr
ralization per Moreover, ho how to determ nd should be
E IV
SON ON UCI DAT E SSE/SST 64±
43 0.2959±
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09
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12
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NANNAN ZH China, in 1975 from Northeas Science and Te and 2003 resp degree in intelli Northeastern U She is curren Communication Science and Te rs. Her research i
image processing CHUANG GA China, in 1982 from Warwic London, UK in
He is curren Science and T authored over and EI. His r system contro control.
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n Liaoning Provin S. and M.S. degr and University ning, China in 19 she received Ph
expert system fr 6.
te Professor in with University ng. She has autho
pattern recogniti
Liaoning Provin S. and M.S. degr nd King's Coll respectively.
date in University ning, China. He pers indexed by S
s include nonlin ning and intellig
n Liaoning provin S. and M.S. degr Technology a echnology Liaoni espectively, and Gyeongsang Natio
with University aoning, and he r in Communicat y of Science a
s authored over ecognition, mach
nce, rees of 998 h.D.
rom the of ored ion,
nce, rees ege y of has SCI near gent
nce, rees and ing, he onal of is tion
and 30 hine
