The efficiency of support vector machine in practice is closely related to the optimal selection of kernel functions and their hyper-parameters. A novel kernel, namely the arctangent kernel, is proposed in this paper. Compared with the Gaussian kernel, the new proposed kernel has a quick similarity descent in the neighborhood of the inspection sample and a moderate similarity descent toward the infinity of the inspection sample. The experimental results on two simulated data sets and some UCI data sets show that the new proposed kernel function has better effectiveness and robustness compared with the polynomial kernel, the Gaussian kernel, the exponential radial basis function, and the former proposed kernel with moderate decreasing.

#### Keywords

- Arctangent kernel
- Classification
- Generalization
- Kernel function
- Support vector machine

Support vector machine (SVM) is a classical kernel method. It has grown substantially due to its good generalization capability even in cases of low quantity training setting over the last decades. It comes from Vapnik’s theory, and develops on the structural risk minimization principle [1]. Illuminated by SVM, many kernel based methods, such as kernel principle component analysis (KPCA) [2], kernel clustering (KC) [3] kernel Fisher discriminant analysis (KFDA) [4], kernel canonical correlation analysis (KCCA) [5], and kernel linear discriminant analysis (GDA) [6] are proposed to solve various different problems in machine learning community.

In most practical problems the data samples are linear non-separable, kernel methods map these points into a high dimensional space and even infinite-dimensional space (named feature space) to gain a better linear separability. The mapping is defined implicitly by the chosen kernel function which represents the inner product of each pair of data points in the feature space without computing their mapping functions directly. Thus the linear algorithm which has a poor performance in the default space produces a very impressive performance in a suitable feature space. This is the well-known “kernel trick”, and it is the key to the success of kernel methods. The importation of kernel functions makes the SVM more robust against the curse of dimensionality. Based on this technique of “kernel trick”, kernel methods have been numerous algorithms developed due to their attractive advantages in terms of theoretical guarantees, efficient training time, and delicate high-dimensional data handle ability.

Regardless of the prosper of kernel methods, a poorly selected kernel can lead to a worse prediction performance than that of their linear counterparts in the original input space. The reason lies in the nature of data is hard to catch. So the estimation of the underlying relationship of data is now converted into the choice of kernel function and its parameters. As we all know, selecting an appropriate kernel or an optional kernel parameter, namely the appropriate feature space is a crucial target. [1, 7, 8]

There are several commonly used kernel functions, such as the linear kernel (^{n},^{2}/2^{2}),^{2}),

In the literature, there are three approaches to choose kernel function or kernel parameters. First, the kernel selection can be reduced to real-valued parameter optimization. Especially, the Gaussian kernel has shown to be an optimal choice of kernel function. Many operations are taken to select its hyperparameters for different tasks. The most generally used method is grid search technique and cross-validation [1, 7, 8]. This approach works well in practice, however it is quite time-consuming because it need train SVMs with all desired combinations of parameters and screens them according to the training accuracy. Kernel alignment, Fisher discriminant criterion, Bayes predictive method and independence criterion are proposed subsequently [9–12]. And these operations are all based on kernel matrix and do not require the information of the trained SVMs, so they can save running time compared with cross-validation strategy. However, these methods based on target function optimization have their own disadvantages such as the minimum is not global, and their preferred accuracy depending on the type of data to be processed. Some dynamic evolutionary algorithms have been adapted to optimize the SVM parameters [13] [14]. But they are still time-consuming because of the training of SVMs from the population evolved generation by generation to converge. So far, there is no uniform standard or paradigm to solve the kernel parameter optimization problem of the Gaussian kernel function.

The second approach is combing several base kernels instead of using only one specific kernel to achieve better results. In the recent years, multiple kernel learning method (MKL) are proposed to overcome the difficulty of the Gaussian kernel optimization [15–17]. MKL adopts a set of predefined based kernels instead of selecting a single kernel function. The combination function of kernel functions has several forms, such as linear function, nonlinear function, or a data-dependent combination. Sincerely, choosing a suitable combination form, and further constructing the specific combination function is a challenging work. Many approaches are proposed to pursuit the combination coefficients by a joint optimization algorithm, or by using a two-stage alternate procedure. Testing results show that using multiple kernels can improve classification accuracy in practice compared with the single kernel approach, but the computational time of MKL is higher for determining the kernel function combination form.

The third way is creating new kernel functions to evade the kernel selection in the practice setting. A. Jacot et al. prove that the evolution of an ANN during training can also be described by a kernel, and proposed a new kernel-the Neural Tangent Kernel [18]. Built on the Cauchy distribution, Basak proposed the Cauchy kernel and observed the new proposed kernel has a long-tailed property which triggers its special performance sometime [19]. N. E. Ayat et al. propose a new kernel with moderate decreasing for pattern recognition [20]. Subsequently, Zhang and Wang presented a kernel function modifying the kernel proposed above, and the new proposed kernel can obtain better generalization performance compared with the old one [21]. Recently, some sets of generalized kernels based on orthogonal functions are proposed to simplify the kernel parameter optimization [22, 23]. M. Tian et al. analyzed the construction modes and the properties of several kinds of kernels based on orthogonal polynomials, and highlighted the similarities and differences among them [24]. But the specific construction of these kernels makes the calculation of kernel function time-consuming.

A new kernel function based on the arctangent function is presented in this paper. The obtained kernel function has concentrated on a high similarity in the neighborhood of the inspection sample and a low similarity toward infinity. Meanwhile, this kernel has a quick similarity descent rate near to the inspection sample and a slow similarity descent rate far away from the inspection sample. We provide experimental results showing their good performances on synthetic and UCI benchmark data. The paper focuses on the classification tasks and considers the SVM algorithm is popular in various applications. The applicability and robustness of the proposed kernel will be investigated here.

The remaining part of this paper is organized as follows. Section 2 shows a clear description of the SVM for classification. Section 3 provides a detailed introduction of the new presented kernel. We give experimental results and discussions on performance in section 4. Section 5 concludes this article.

The support vector machine is a classical machine learning approach. In recent years, SVM has become a hot topic in pattern recognition, and SVM has been successfully applied in many practice fields. In particular, a classical SVM classifier is capable of producing a linear decision boundary between two different classes, and maximizing the margin in the feature space between them at the same time. Introductions to SVM for binary classification task are summarized in the following.

Consider a set of training labeled sample represented by _{i} ∈ ^{n}, ^{n} denotes _{i}_{i}. The SVM classifier training process which builds the separating hyperplane between these two classes can be formulated by that:
_{1},…,ξ_{l})^{T} is the vector of slack variables and

The Lagrange dual operation converts the optimization problem above into the dual formulation, which can be rewritten as:
_{i}_{i∈{1,…,l}} are Lagrangian multipliers obtained by computing the optimization problems corresponding to each separation constraint. The samples with _{i} > 0 are named as support vectors and they are integral to define the separating boundary.

To solve real-world classification problems, the dot product < _{i},x_{j} > in the former algorithm can be replaced with a kernel function _{i}, _{j}). The kinds of kernel K satisfying some defined characterization of dot product are all samples _{i} and _{j}, which can be formulated as:
_{i} and _{j} in the augmented space

In practice, some theorems have been proved and provided to verify whether a function can be a kernel function. In these theorems, the original and most common condition for a symmetric function _{i}, _{j}) to be a kernel was raised by Mercer in 1909, namely a kernel is the function must be symmetric, continue and positive definite.

Based on unimodality and asymptotic of kernel functions, we proposed a new kind of kernel function. This kernel function is formulated as:

To prove the AtanK kernel is a Mercer kernel. We give the following theorem which is helpful to prove positive definite of it.

^{n},f^{2}).

^{n},

^{∞}. Let ^{2}, and Eq. (1) can be simplified as in the following form:

If 0 < _{1} < _{2}, we have
_{1} < ξ < _{2}. So

Fig.1 shows the output of AtanK kernel function for different

N. E. Ayat et al. pointed that a good kernel function should have different decrease speed in the neighborhood and the infinity of the inspection sample. That is to say, a quick and a moderate decrease speed should behold in the neighborhood and the infinity of the inspection sample, respectively [20]. And the authors pointed that the Gaussian kernel satisfies the requirement in the neighborhood of the inspection sample but not the requirement in the infinity of the inspection sample. Based on this, the authors proposed a kernel with moderate decreasing (denoted as KMOD), whose analytic expression is as:

Where ^{x}

Note that these two kinds of kernel functions both need a pair of parameters. Obviously, optimizing these two parameters is a very time-consuming target. For simplicity, Ref. [20] set the parameters chosen from some predetermined sets and Ref. [21] set L = ^{2} and

Fig. 2 shows the kernel outputs of the Gaussian kernel, the ERBF kernel, the UKF kernel, and the AtanK kernel for the parameters 0.2, 1, and 2, respectively. In order to give a detail comparison, the curves of the Gaussian kernel, the ERBF kernel, the UKF kernel, and the AtanK kernel are all given here. We discarded the comparison between the UKF kernel and the KMOD kernel due to the reason that Ref. [21] have shown the UKF can learn more separation information compared with the KMOD kernel. For each parameter, the kernel curves near to and far away from the neighborhood of the inspection sample zero in two-dimension plane are both shown.

From Fig.2, we can see that the AtanK kernel satisfies the quick decrease in the neighborhood of the inspection sample and the moderate decrease far away from the inspection sample. When

Seen from the kernel function discussed above, we find that when describing the change of the kernel function value with respect to the inspection sample in two-dimensional space, the curve of the kernel function presents a unimodal function curve whose maximum point is exactly the inspection sample, whether the kernel function constructed based on the inner product form of <

This paper discusses the different trends of similarity change near to and far away from the inspection sample, which in fact provides intuitive guidance for the similarity measurement requirements around the inspection sample. Various similarity measurement methods provide different inner product, or similarity, between mapping vectors in different feature spaces, which makes it possible to find the optimal feature spaces with better classification performance. However, because of the diversity and particularity of the real problems, we cannot determine which kernel function is the optimal kernel function for the specific problem. Therefore, we all choose the Gaussian kernel function as the universal kernel. As we all know, in the field of mathematical statistics, the normal distribution is a key type of data distribution. The existence of the maximum likelihood probability theorem makes the normal distribution be the highest peak in the classical statistical field. However, with the development of statistical technology, people have found that some specific data distribution cannot be characterized by the normal distribution in real life. Gradually, some new distributions such as the Chi-square distribution and the student distribution, are springing up. It can be imagined that the construction of kernel function is also faced with the same problem. The universal kernel function represented by the Gaussian kernel cannot describe all the similarity measurement. It is necessary to construct new kernel functions constantly to provide more possibilities for different practical problems and diverse similarity measurements.

The effectiveness of the proposed kernel function is assessed on 2 artificial datasets and 11 UCI benchmark datasets compared with the polynomial kernel, the Gaussian kernel, the UKF kernel, and the exponential radial basis kernel [26]. For simplicity, the polynomial kernel, the Gaussian kernel and the exponential radial basis kernel, are denoted as Poly, Gauss and ERBF, respectively.

To evaluate the performance and robustness of the AtanK kernel, we have performed a simple classification task in two dimensions. The results of the AtanK kernel in relation to the four kernel functions mentioned above (Poly, Gauss, UKF and ERBF) are performed on the checkerboard dataset.

The original training population is a set of 592 points, uniformly distributed in the unit square. We divide the square into 9 grids, and set these points of adjacent grids into two classes alternately. In the experiment, we select randomly 496 training samples and the last 96 samples as testing samples.

For simplicity, we set the coefficient

Table 1 lists the chosen parameter, the training accuracy and the minimal number of support vectors corresponding to several different kernels when they obtain their highest testing accuracy. Fig.3 shows the default checkerboard data set and the best classification resulting boundaries with different kernels.

Results on the Checkerboard dataset.

Kernel | Poly | Gauss | ERBF | UKF | AtanK |
---|---|---|---|---|---|

Parameter | σ=0.5 | σ=0.7 | σ=1.1 | σ=0.7 | |

Training accuracy (%) | 67.14 | 98.18 | 97.18 | 94.15 | |

Testing accuracy (%) | 71.87 | 92.71 | 89.58 | 92.71 | |

Number of SV | 381 | 129 | 199 | 183 |

Results on the Cosexp dataset.

Kernel | Poly | Gauss | ERBF | UKF | AtanK |
---|---|---|---|---|---|

Parameter | σ=1.1 | σ=0.7 | σ=0.5 | σ=1.3 | |

Trainging accuracy (%) | 75.20 | 88.20 | 98.00 | 78.20 | |

Testing accuracy (%) | 71.00 | 71.00 | 70.00 | 72.00 | |

Number of SV | 281 | 193 | 134 | 128 |

From the results of Table 1, it can be seen that the AtanK kernel have both the highest testing accuracy and the lowest support vector numbers in spite of its ordinary training accuracy in these 5 kernel functions. As we all know the support vector ratio can be seen as an index reflecting the generalization error rate, so we take the number of SV into consideration here. On this index, the AtanK kernel is pretty much the same as the Gaussian kernel. In all, the performance of Poly is the worst. The Gaussian kernel is the best in the training accuracy, while its generalization is not so amazing. Seen from the testing accuracy performance, the Gaussian kernel and the UKF kernel have the same results, and both fall behind the AtanK kernel. And the AtanK kernel has the best classification performance in these kernels discussed above.

Seen from Fig. 3, the polynomial kernel has the worst performance that it does not find the adjacent grids structure at all. Though the ERBF kernel has the highest training accuracy, but its classification boundary is not smooth enough. Part of the reason lies in the quick decrease in the neighborhood of

We give another artifical data set-the Cosexp dataset to evaluate the performance of the new proposed kernel [25]. Let the function

The results of the AtanK kernel compared with the four kernel functions are investigated on the Cosexp dataset in Table 2. Table 2 shows the chosen parameter, the training accuracy and the minimal number of support vectors obtained with these kernels when they each obtain their highest testing accuracy. From the results of Table 2, it can be seen that the AtanK kernel and the polynomial kernel receive the better test accuracy, in spite of their naive training accuracy. And the UKF, ERBF and AtanK all have the good performance in terms of support vector numbers.

Fig. 4 shows the best classification resulting boundaries with different kernels on the Cosexp dataset. Seen from Fig. 4, we can see that the classification boundaries obtained by the ERBF kernel and the UKF kernel are more identical to the default boundary. They both have good local structure preserving ability, and it may be responsible for the result. At the same time, the difference of boundries obtained by the Gaussian kernel and the AtanK kernel shows that these two kernel functions have an obvious difference in classification, although they both have symmetry and unimodality.

Combined the results of Table 2 and Fig. 4, we found the Poly kernel has the best classification accuracy despite it fails to find the structure of default separating boundary, while the UKF kernel has the worst classification accuracy despite it catch the more vivid structure of default separating boundary. It can be seen as a reflection of the difficulty of realizing the structural risk minimization in kernel selection. From this point of view, the Atank kernel has a naive training accuracy performance, a better testing classification result, and a rougher separating boundary structure, so it is a well-balanced choice.

We evaluate the classification performance improvement of our proposed kernel over the four commonly used kernels mentioned above on 11 data sets from the UCI benchmark repository [26]. The specifications of these data sets are listed in Table 3. The regularization parameter C in SVM is chosen from the set of {2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1}, 2^{2},2^{3},2^{4}}. And the description of the hyperparameter values of kernels are summarized in Table 4.

In the experiment, the experimental methodology is set as follows: For a given data set, we throw randomly two-thirds of samples into the training set and the remaining one-third into the testing set, the stratified sampling is exceeded in this program. For single dataset, five kernel functions are adopted separately. To ensure the objectivity of experimental results, we run 10 randomly independent performances to evaluate the classification performance by the best classification accuracy rate by average. The average classification accuracies with standard deviations over 10 trials using the optimal parameter obtained by cross validation are summarized in Table 5. In Table 5, the bold font denotes the best classification performance across the kernel functions compared. The t-test results are summarized in Win-tie-loss (W-T-L), and they are attached at the bottoms of Table 5.

According to the prediction accuracy in Table 5, we note that the AtanK kernel obtains the best accuracy on 6 out of 11 datasets. The ERBF kernel and the UKF kernel fall behind, give better performance on 2 datasets, and the Gaussian kernel provides the best accuracy on only one dataset. Of these five kernel functions, the polynomial kernel has the worst behavior in the experiment. The W-T-L summarization shows that the AtanK kernel has an obvious advantage compared with the Poly kernel, the Gauss kernel, the ERBF kernel and the UKF kernel on 10 datasets, 7 datasets, 9 datasets and 7 datasets out of 11 datasets in Table 5.

The specification of data sets for experiments.

Data set | Number of features | Number of samples | Number of classes |
---|---|---|---|

Heart | 13 | 270 | 2 |

Ringnorm | 20 | 1000 | 2 |

German | 24 | 1000 | 2 |

Wdbc | 30 | 569 | 2 |

Ionosphere | 34 | 351 | 2 |

Sonar | 60 | 208 | 2 |

Splice | 60 | 1000 | 2 |

Australian | 14 | 690 | 2 |

Twonorm | 20 | 1000 | 2 |

Yeast | 8 | 1136 | 2 |

IJK | 16 | 2241 | 2 |

The values corresponding to kernel parameters in experiments.

Kernel | Parameters | Values |
---|---|---|

Poly | 1,2,3,4,5,6,7,8 | |

Gauss | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

ERBF | 2^{–4},2^{–3},2 ^{2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

AtanK | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

UKF | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

1 |

The classification accuracy of different kernels on benchmark data sets (mean(%)± std).

Data set | Poly | Gauss | ERBF | AtanK | UKF |
---|---|---|---|---|---|

Heart | 76.00±0.0457 | 78.44±0.0393 | 78.59±0.0149 | 77.22±0.0417 | |

Ringnorm | 95.09±0.0102 | 96.29±0.0049 | 97.37±0.0457 | 97.84±0.0070 | |

German | 72.72±0.0184 | 73.38±0.0103 | 73.05±0.0252 | 73.26±0.0192 | |

Wdbc | 94.74±0.0096 | 97.00±0.0145 | 97.21±0.0050 | 96.84±0.0253 | |

Ionosphere | 86.15±0.0248 | 88.37±0.0092 | 92.48±0.0184 | 91.28± 0.0212 | |

Sonar | 91.43±0.0316 | 90.43±0.0415 | 92.43±0.0234 | 91.29±0.0549 | |

Splice | 80.36±0.0401 | 82.99±0.0229 | 81.95±0.0183 | 83.83±0.0268 | |

Australian | 82.74±0.0334 | 81.57±0.0148 | 83.48± 0.0319 | 82.87±0.0340 | |

Twonorm | 96.41± 0.0101 | 96.32±0.0047 | 96.26± 0.0046 | 96.38±0.0065 | |

Yeast | 66.78± 0.0144 | 65.80±0.0091 | 67.28± 0.0270 | 66.33±0.0210 | |

IJK | 96.80± 0.0045 | 98.11±0.0091 | 98.14± 0.0087 | 98.30±0.0054 | |

W-T-L | 10-0-1 | 7-3-1 | 9-0-2 | 7-2-2 | — |

Besides comparing the testing accuracy of these kernels for each data set, we take an overall comparison on 11 UCI data sets by combing the nonparametric Friedman’s test and the Tukey’s honestly significant difference criterion. The Friedman’s test is an extension of the sign test and is better known, it is widely useful for a much larger range of treatments [27]. The overall evaluation of the classification accuracy of these kernels is displayed in Fig. 5.

Seen from Fig. 5, the performance of the polynomial kernel is outperformed by other kernel functions. The performance of the proposed AtanK has the best rank of these five kernel functions, and it has a statistically significantly accuracy on 11 datasets. It means the AtanK kernel is a better and robust kernel compared with the polynomial kernel, the ERBF kernel, the Gaussian kernel, and the UKF kernel.

Kernel selection is fundamental to kernel based methods. Driven by the similarity measure defined by the arctangent function, we described a new universal kernel in this paper. Comprehensive experimental shows the new proposed AtanK kernel can properly learn separating boundary for many classification application. Schematic diagrams show the AtanK kernel is a new set of SVM kernels that allows a quick decreasing around the sample and a moderate decreasing toward infinity.

Experiments done on the checkerboard dataset and the Cosexp dataset show the ability of the AtanK kernel in separating the patterns. Thus we find the new kernel can preserve the all data on closeness information while still penalize the far neighborhood more reliably. Moreover, a comparison is done with previously constructed SVM models based on the UCI benchmark datasets, which was obtained by using the polynomial kernel, the Gaussian kernel, the ERBF kernel, and the UKF kernel. The result revealed that the AtanK kernel can improve classification accuracy.

Further numerical experiments are required in order to assess the power of our evolved kernel. And an automatic algorithm of the AtanK kernel parameters optimization will be valuable. Many mature kernel parameter optimization tricks can extend to the new kernel easily.

#### The classification accuracy of different kernels on benchmark data sets (mean(%)± std).

Data set | Poly | Gauss | ERBF | AtanK | UKF |
---|---|---|---|---|---|

Heart | 76.00±0.0457 | 78.44±0.0393 | 78.59±0.0149 | 77.22±0.0417 | |

Ringnorm | 95.09±0.0102 | 96.29±0.0049 | 97.37±0.0457 | 97.84±0.0070 | |

German | 72.72±0.0184 | 73.38±0.0103 | 73.05±0.0252 | 73.26±0.0192 | |

Wdbc | 94.74±0.0096 | 97.00±0.0145 | 97.21±0.0050 | 96.84±0.0253 | |

Ionosphere | 86.15±0.0248 | 88.37±0.0092 | 92.48±0.0184 | 91.28± 0.0212 | |

Sonar | 91.43±0.0316 | 90.43±0.0415 | 92.43±0.0234 | 91.29±0.0549 | |

Splice | 80.36±0.0401 | 82.99±0.0229 | 81.95±0.0183 | 83.83±0.0268 | |

Australian | 82.74±0.0334 | 81.57±0.0148 | 83.48± 0.0319 | 82.87±0.0340 | |

Twonorm | 96.41± 0.0101 | 96.32±0.0047 | 96.26± 0.0046 | 96.38±0.0065 | |

Yeast | 66.78± 0.0144 | 65.80±0.0091 | 67.28± 0.0270 | 66.33±0.0210 | |

IJK | 96.80± 0.0045 | 98.11±0.0091 | 98.14± 0.0087 | 98.30±0.0054 | |

W-T-L | 10-0-1 | 7-3-1 | 9-0-2 | 7-2-2 | — |

#### Results on the Checkerboard dataset.

Kernel | Poly | Gauss | ERBF | UKF | AtanK |
---|---|---|---|---|---|

Parameter | σ=0.5 | σ=0.7 | σ=1.1 | σ=0.7 | |

Training accuracy (%) | 67.14 | 98.18 | 97.18 | 94.15 | |

Testing accuracy (%) | 71.87 | 92.71 | 89.58 | 92.71 | |

Number of SV | 381 | 129 | 199 | 183 |

#### The values corresponding to kernel parameters in experiments.

Kernel | Parameters | Values |
---|---|---|

Poly | 1,2,3,4,5,6,7,8 | |

Gauss | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

ERBF | 2^{–4},2^{–3},2 ^{2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

AtanK | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

UKF | 2^{–4},2^{–3},2^{–2},2^{–1},2^{0},2^{1},2^{2},2^{3},2^{4} | |

1 |

#### The specification of data sets for experiments.

Data set | Number of features | Number of samples | Number of classes |
---|---|---|---|

Heart | 13 | 270 | 2 |

Ringnorm | 20 | 1000 | 2 |

German | 24 | 1000 | 2 |

Wdbc | 30 | 569 | 2 |

Ionosphere | 34 | 351 | 2 |

Sonar | 60 | 208 | 2 |

Splice | 60 | 1000 | 2 |

Australian | 14 | 690 | 2 |

Twonorm | 20 | 1000 | 2 |

Yeast | 8 | 1136 | 2 |

IJK | 16 | 2241 | 2 |

#### Results on the Cosexp dataset.

Kernel | Poly | Gauss | ERBF | UKF | AtanK |
---|---|---|---|---|---|

Parameter | σ=1.1 | σ=0.7 | σ=0.5 | σ=1.3 | |

Trainging accuracy (%) | 75.20 | 88.20 | 98.00 | 78.20 | |

Testing accuracy (%) | 71.00 | 71.00 | 70.00 | 72.00 | |

Number of SV | 281 | 193 | 134 | 128 |

Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Industrial transfer and regional economy coordination based on multiple regression model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Performance evaluation of college laboratories based on fusion of decision tree and BP neural network Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Analysis of the relationship between industrial agglomeration and regional economic growth based on the multi-objective optimisation model Constraint effect of enterprise productivity based on constrained form variational computing The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 TOPSIS missile target selection method supported by the posterior probability of target recognition Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model The incentive contract of subject librarians in university library under the non-linear task importance Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot Visual error correction of continuous aerobics action images based on graph difference function Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Radioactive source search problem and optimisation model based on meta-heuristic algorithm Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, product-market competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of e-portfolios suggest Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity Data mining of Chain convenience stores location The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics A generative design method of building layout generated by path Parameter Id of Metal Hi-pressure State Equation Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control P-Matrix Reasoning and Information Intelligent Mining Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Application of regression function model based on panel data in bank resource allocation financial risk management Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Badminton players’ trajectory under numerical calculation method Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Difference-in-differences test for micro effect of technological finance cooperation pilot in China Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanK-A New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Research on the influencing factors of agricultural product purchase willingness in social e-commerce situation Study of a linear-physical-programming-based approach for web service selection under uncertain service quality A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Fractional Differential Equations in the Exploration of Geological and Mineral Construction Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China The Law of Large Numbers in Children's Education Data structure simulation for the reform of the teaching process of university computer courses Calculating university education model based on finite element fractional differential equations and macro-control analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Good congruences on weakly U-abundant semigroups Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Mathematical simulation analysis of optimal detection of shot-putters’ best path Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation Fed-UserPro: A user profile construction method based on federated learning Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}Computer Vision Communication Technology in Mathematical Modeling The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption MCM of Student’s Physical Health Based on Mathematical Cone Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Informatisation of educational reform based on fractional differential equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression System dynamics model of output of ball mill Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Design of Morlet wavelet neural network to solve the non-linear influenza disease system Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the Jupiter-Europa system with oblateness Fractional Differential Equations in Electronic Information Models Badminton players’ trajectory under numerical calculation method BIM Engineering Management Oriented to Curve Equation Model Optimal preview repetitive control for impulse-free continuous-time descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decision-making Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Nonlinear strategic human resource management based on organisational mathematical model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model