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 (
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
The Lagrange dual operation converts the optimization problem above into the dual formulation, which can be rewritten as:
To solve real-world classification problems, the dot product <
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
Fig. 1
Trends near the zero for AtanK kernel with different σ values.

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.
If 0 <
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:
Fig. 2
The trends near and far from the zero for Gauss, ERBF, UKF and AtanK kernels with different σ values.

Where
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 =
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.
Fig. 3
Two-dimensional Checkerboard dataset and classification boundaries of 5 sets of kernels on Checkerboard dataset.

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 {2–4,2–3,2–2,2–1,20,21, 22,23,24}. And the description of the hyperparameter values of kernels are summarized in Table 4.
Fig. 4
Boundary between domains of positive and negative samples corresponding to Cosexp dataset and the classification boundaries of 5 sets of kernels on Cosexp dataset.

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 | 2–4,2–3,2–2,2–1,20,21,22,23,24 | |
ERBF | 2–4,2–3,2 2,2–1,20,21,22,23,24 | |
AtanK | 2–4,2–3,2–2,2–1,20,21,22,23,24 | |
UKF | 2–4,2–3,2–2,2–1,20,21,22,23,24 | |
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 | — |
Fig. 5
Result of Friedman’s test of the testing accuracy.

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.
Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

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 | 2–4,2–3,2–2,2–1,20,21,22,23,24 | |
ERBF | 2–4,2–3,2 2,2–1,20,21,22,23,24 | |
AtanK | 2–4,2–3,2–2,2–1,20,21,22,23,24 | |
UKF | 2–4,2–3,2–2,2–1,20,21,22,23,24 | |
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 |
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and field-based on nonlinear mathematical equations Visual error correction of continuous aerobics action images based on graph difference function Ecological balance model of effective utilization of agricultural water resources based on fractional differential equations Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Precision algorithms in second-order fractional differential equations Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Fractional differential equations in National Sports Training in Colleges and Universities Radioactive source search problem and optimisation model based on meta-heuristic algorithm Visualized calculation of regional power grid power data based on multiple linear regression equation Application of mathematical probabilistic statistical model of base – FFCA financial data processing Least-squares method and deep learning in the identification and analysis of name-plates of power equipment Research on a method of completeness index based on complex model Distribution network monitoring and management system based on intelligent recognition and judgement 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 Topological optimisation technology of gravity dam section structure based on ANSYS partial differential equation operation 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 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 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 Analysing the action techniques of basketball players’ shooting training using calculus method Engineering project management based on multiple regression equation and building information modelling technology 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 Mathematical model of back propagation for stock price forecasting Application of regression function model based on panel data in bank resource allocation financial risk management Application of Logical Regression Function Model in Credit Business of Commercial Banks 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 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 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 Application of Lane-Emden differential equation numerical method in fair value analysis of financial accounting Regression function model in risk management of bank resource allocation Application of numerical method of functional differential equations in fair value of financial accounting Optimal solution of fractional differential equations in solving the relief of college students’ mental obstacles Risk contagion in financial markets based on copula model Calculating university education model based on finite element fractional differential equations and macro-control analysis Automatic parameter selection ZVD shaping algorithm for crane vibration suppression based on particle swarm optimisation 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 Calculation of tourism development income index based on finite element ordinary differential mathematical equation Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Radar system simulation and non-Gaussian mathematical model under virtual reality technology Comparison of compression estimations under the penalty functions of different violent crimes on campus through deep learning and linear spatial autoregressive models Research and application of constructing football training linear programming based on multiple linear regression equation Research on management evaluation of enterprise sales cash flow percentage method based on the application of quadratic linear regression equations Mathematical simulation analysis of optimal detection of shot-putters’ best path Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Mathematical modelling of enterprise financial risk assessment based on risk conduction model Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Mathematical simulation experiment based on optimisation of heat treatment process of aluminium alloy materials Mathematical model of transforming image elements to structured data based on BP neural network Educational reform informatisation based on fractional differential equation 3D Mathematical Modelling Technology in Visual Rehearsal System of Sports Dance MCM of Student’s Physical Health Based on Mathematical Cone 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 Informationisation of teaching model for track and field education based on finite element higher-order fractional differential equation Research on identifying psychological health problems of college students by logistic regression model based on data mining Information technology of preschool education reform of fine arts based on fractional differential equation Information Teaching Model of Preschool Art Education in Colleges and Universities Based on Finite Element Higher-Order Fractional Differential Equation A mathematical model of PCNN for image fusion with non-sampled contourlet transform Application of artificial intelligence algorithm in mathematical modelling and solving College Students’ Mental Health Climbing Consumption Model Based on Nonlinear Differential Equations Communication architecture of power monitoring system based on incidence matrix model Differential equation to verify the validity of the model of the whole-person mental health education activity in Universities Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Analysis and Prediction of College Students’ Mental Health Based on K-means Clustering Algorithm Mathematical function data model analysis and synthesis system based on short-term human movement Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement Energy-saving technology of BIM green buildings using fractional differential equation Study on the training model of football movement trajectory drop point based on fractional differential equation Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation User online consumption behaviour based on fractional differential equation Differential equation model of financial market stability based on Internet big data Multi-attribute Decision Method Based on Normal Random Variable in Economic Management Risk Control Children’s cognitive function and mental health based on finite element nonlinear mathematical model 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 Stock price analysis based on the research of multiple linear regression macroeconomic variables Fractional Linear Regression Equation in Agricultural Disaster Assessment Model Based on Geographic Information System Analysis Technology