Due to the extremely volatile nature of financial markets, it is commonly accepted that stock price prediction is a task filled with challenges. However, in order to make profits or understand the essence of equity market, numerous market participants or researchers try to forecast stock prices using various statistical, econometric or even neural network models. In this work, we survey and compare the predictive power of five neural network models, namely, back propagation (BP) neural network, radial basis function neural network, general regression neural network, support vector machine regression (SVMR) and least squares support vector machine regression. We apply the five models to make price predictions for three individual stocks, namely, Bank of China, Vanke A and Guizhou Maotai. Adopting mean square error and average absolute percentage error as criteria, we find that BP neural network consistently and robustly outperforms the other four models. Then some theoretical and practical implications have been discussed.
Keywords
- back propagation
- stock price prediction
- radial basis function
- support vector machine regression
- least squares support vector machine regression
MSC 2010
- 92B20
Price prediction in equity markets is of great practical and theoretical interest. On the one hand, relatively accurate prediction brings maximum profit to investors. Many market participants, especially institutional ones, spend a lot of time and money to collect and analyse relevant information before making investment decisions. On the other hand, researchers often use the fact of whether or not the price can be forecast as a tool to check market efficiency. Also, they invent, apply or adjust different models to improve the predictive power. Finding a good method to forecast stock price more accurately will be a topic forever in both the academic field and the financial industry. Equity price prediction is regarded as a challenging task in the financial time series prediction process since the stock market is essentially dynamic, nonlinear, complicated, nonparametric and chaotic in nature [1]. Besides, many macro-economic environments, such as political events, company policies, general economic conditions, commodity price index, interest rates, investor expectations, institutional investors choices and psychological factors of investors, are also the influencing factors [2]. In this paper, we apply five artificial intelligence (AI) models in the predicting research. Among the AI models, the back propagation neural networks (BPNN), radial basis neural networks (RBFNN), general regression neural network (GRNN), support vector machine regression (SVMR) and least squares support vector machine regression (LS-SVMR) are the most widely used and mature methods. The BPNN is successfully used in many fields, such as engineering [3], power forecasting [4], time series forecasting [5], stock index forecasting [6] and stock price variation prediction [7]. BPNN is also useful in the economic field. Lu and Bai [8] proposed a hybrid forecasting model [Wavelet Denoising-based Back Propagation (BP)], which firstly decomposed the original data into multiple layers by wavelet transform, and then established BPNN model using the low-frequency signal of each layer for predicting the Shanghai Composite Index (SCI) closing price. The radial basis function neural network (RBFNN) is a feed forward neural network with a simple structure, which has a single hidden layer. Mller et al. [9] applied RBFNN as a tool for nonlinear pattern recognition to correct the estimation error in the prediction of linear models in predicting two stock series in Shanghai and Shenzhen stock exchanges. In Osuna's study [10], the author demonstrated RBFNN's effectiveness in financial time series forecasting. RBFNN is proposed to overcome the main drawback of BPNN – namely, that of easily falling into local minima in the training process. RBFNN have also been used in various forecasting areas and achieve good forecasting performance, with demonstrated advantages over BPNN in some applications [11]. The GRNN, which is put forward by Specht [12], shows its effectiveness in pattern recognition [13], stock price prediction [14] and groundwater level prediction [15]. Tan et al. [16] showed the forecasting ability of GRNN in the prediction of closing stock price. However, their research is characterised by a lack of comparison with other data mining models, which is also the limitation of other references cited in this paper. Support Vector Machine (SVM), first developed by Vapnik [17], is based on statistical learning theory. Owing to its successful performance in classification tasks [18] and regression tasks, especially in time series prediction and finance-related applications, SVM has drawn significant attention and thus earned intensive study. By using the structural risk minimisation principle to turn the solving process into a convex quadratic programming problem, the SVM obtains better generation performance; and moreover, the solution is unique and globally optimal. The LS-SVMR, based on structural risk minimisation principle, is able to approximate any nonlinear system. As a reformulation of the SVM algorithm, LS-SVMR overcomes the drawbacks of local optima and overfitting in the traditional machine learning algorithm. To the best of our knowledge, there is a dearth in the literature of studies that are focused on comparing the effectiveness of the above-mentioned five algorithms reviewed in this paper. In this study, we present this comparative view by using data to compare the performance of these five neural networks, namely BPNN, RBFNN, GRNN, SVM and LS-SVMR, in predicting stock price. This paper is organised as follows. Section 2 XXX. Section 3 XXX.
A neutral network generally contains one input layer, one or many hidden layers and one output layer. Supposing that the total number of layers is
We use
To rewrite this expression in a matrix form, we define a weight matrix
With these notations in mind, (2.1) can be rewritten in the elegant and compact vectorised form
Let
The cost function is defined by the following quadratic form
We recall that the Hadamard products ⊙
With learning rate, the weights are learned by
Owing to the additivity of cost over sample, we can adopt the idea of stochastic gradient descent to speed up the learning. To make these ideas more precise, stochastic gradient descent works by randomly picking out a small number
Supposing that
Here the sums are over all the training examples
RBF networks typically have three layers: an input layer, a hidden layer with a nonlinear RBF activation function and a linear output layer. Let us suppose that the hidden layer has
The parameters
GRNN essentially belongs to radial basis neural networks. GRNN was suggested by D.F. Specht in 1991. Recalling the framework of RBF network, we recollect that the number of neurons in the hidden layer is the same as the sample size
If the number of neurons in the hidden layer stayed at the sample size
A version of SVM for regression (SVR) was proposed in 1996 by Vladimir N. The Vapnik et al. model aims to find a linear function of
Let
A modified problem is to solve
Here
The linear predictor (2.5) cannot reveal a possible nonlinear relation between
Different functions correspond to different kernel functions in application. The subsequent subsection provides information on more kernel functions.
For some function
Here
The solution of LS-SVM regression will be obtained after we construct the Lagrangian function:
The kernel function
Linear kernel:
Polynomial kernel of degree
RBF kernel:
MLP kernel:
In this work, we study the weekly adjusted closing price of three individual stocks: Bank of China (601988), Vanke A (000002) and Guizhou Maotai (600519). Each price data has a sample size of 427, ranging from 3 January 2006 to 11 March 2018. As usual, we split the whole data set into a training set (80%) and a test set (20%).
We intentionally select the three stocks based on such an observation: they are totally different in price scale. As shown in Table 1, the price of Bank of China is about within 2–5 RMB, Vanke A (000002) is approximately in the range 5–40 RMB and Guizhou Maotai has a wide range of 80–800 RMB. Actually, Guizhou Maotai ranks first in terms of price per share among all stocks listed in the only two stock exchanges of Mainland of China: Shanghai and Shenzhen.
Price range
2.00 | 5.65 | 81.13 | |
5.01 | 40.04 | 788.42 |
Let us use {
We adopt a neural network with three layers, which contains only one hidden layer. The input layer has three neurons, and the output layer has only one neuron which represents the predicted value. To determine the number of neurons in the hidden layer, by rule of thumb, we apply the following formula
For the implementation of RBF, SVMR and LS-SVMR, we use standard R packages. When applying GRNN, we choose
Table 2 shows the performance of the five neural network models. From these results, we can see all the five models have some predictive power. Even the worst one, GRNN, has MAPE not exceeding 5%, which is very satisfactory considering we are forecasting stock price rather than volatility.
Results of the five methods
MSE | 0.009 | 0.014 | 0.02 | 0.012 | 0.018 | |
MAPE | 0.019 | 0.025 | 0.024 | 0.023 | 0.028 | |
MSE | 2.976 | 4.686 | 6.036 | 3.422 | 5.472 | |
MAPE | 0.049 | 0.065 | 0.067 | 0.059 | 0.072 | |
MSE | 395.1 | 740.1 | 1103.6 | 407.4 | 405.5 | |
MAPE | 0.026 | 0.036 | 0.048 | 0.029 | 0.027 |
BP, back propagation; GRNN, general regression neural network;
LS-SVMR, least squares support vector machine regression;
RBF, radial basis function; SVMR, support vector machine regression.
Across all the three stocks, in terms of both MSE and MAPE, BP neural network outperforms the other four models. One may refer to Figure 2 in the next subsection for a more intuitive view of the accuracy of prediction for Bank of China using the BP method. SVMR ranks second consistently across the three stocks. However, in terms of both MSE and MAPE, results from SVMR are greater than that of BP by at least 10%. Moreover, on the prediction of Bank of China and Vanke A, BP surpasses SVMR by at least 20% under both criteria.
We cannot tell which one among RBF and LS-SVMR is better. As shown in Table 2, on the prediction of Bank of China and Vanke A, RBF is more accurate than LS-SVMR, while on the prediction of Guizhou Maotai, LS-SVMR has a better performance. Overall, they share a similar accuracy level of prediction. Finally, GRNN behaves the worst consistently across the three stocks.
One reason we could guess for the superior performance of BP over the other methods is that the latter four models all involve the mostly used kernel function: exp (−|
Comparison of four kernels in SVMR
MSE | 0.01 | 0.01 | 0.011 | 0.012 | |
MAPE | 0.019 | 0.02 | 0.021 | 0.023 | |
MSE | 2.993 | 3.292 | 3.515 | 3.422 | |
MAPE | 0.05 | 0.053 | 0.055 | 0.059 | |
MSE | 395.6 | 403.5 | 405.7 | 407.4 | |
MAPE | 0.027 | 0.028 | 0.028 | 0.029 |
RBF, radial basis function; SVMR, support vector machine regression.
We have two remarks on Table 3. On the one hand, linear kernel is the best in this prediction task, and consistently outperforms the other three kernels. Although RBF kernel is the default kernel in many packages due its flexibility to various data resources, it is not good enough here. Thus, we should try other kernels to make comparison when doing similar predicting projects. On the other hand, BP stills surpasses SVMR with linear kernel, even if the advantage is not obvious now. They share a similar prediction error, possibly resultant to the fact they both involve weighted average, which captures some linear relation in the network.
When implementing BP algorithm, it needs to initialise the weights randomly, which causes instability of the result. To show that the result of BP is stable, we train the neural network for 100 times, and compute its mean and standard deviation. Table 4 helps us eradiate this concern since the standard deviations are extremely small compared to the scale of their corresponding mean values. In other words, the result of every experiment is reliable.
100 times experiment
MSE | 0.009 | 4.8×10^{−5} | |
MAPE | 0.019 | 0.0001 | |
MSE | 2.976 | 0.0067 | |
MAPE | 0.049 | 0.0001 | |
MSE | 395.1 | 1.3728 | |
MAPE | 0.026 | 7.7×10^{−5} |
Figure 2 plots the observed and predicted prices of Bank of China. It can be seen clearly that the predicted values fit the observed ones well. Also, the turning points are forecasted quite timely. When there is a trend in the actual price, the predicted value follows accordingly and closely.
At a first glance, the network needs at least one period to react or assimilate new information. Actually, it is a false appearance that the predicted values lag one period of the observed values. More precisely, supposing that
To prove that the market is actually inefficient, we take difference
In this work, we have successfully demonstrated that the five neural network models are all able to effectively extract meaningful information from past price. With evidence from the forecast accuracy of three unrelated stocks, we find that BP surpasses the other four models consistently and robustly. Also, by implementing the algorithm many times and checking the standard deviation, the stability of BP is observed and confirmed. Based on our trial on different kernels, we advise readers of the current paper not take the default kernel for granted and ‘descrisized’ other kernels. For our own interest, we test the error series and destroy the market efficiency hypothesis. In our future research, we will investigate other more involved neural networks to complete the current tentative work.
Price range
2.00 | 5.65 | 81.13 | |
5.01 | 40.04 | 788.42 |
100 times experiment
MSE | 0.009 | 4.8×10^{−5} | |
MAPE | 0.019 | 0.0001 | |
MSE | 2.976 | 0.0067 | |
MAPE | 0.049 | 0.0001 | |
MSE | 395.1 | 1.3728 | |
MAPE | 0.026 | 7.7×10^{−5} |
Results of the five methods
MSE | 0.009 | 0.014 | 0.02 | 0.012 | 0.018 | |
MAPE | 0.019 | 0.025 | 0.024 | 0.023 | 0.028 | |
MSE | 2.976 | 4.686 | 6.036 | 3.422 | 5.472 | |
MAPE | 0.049 | 0.065 | 0.067 | 0.059 | 0.072 | |
MSE | 395.1 | 740.1 | 1103.6 | 407.4 | 405.5 | |
MAPE | 0.026 | 0.036 | 0.048 | 0.029 | 0.027 |
Comparison of four kernels in SVMR
MSE | 0.01 | 0.01 | 0.011 | 0.012 | |
MAPE | 0.019 | 0.02 | 0.021 | 0.023 | |
MSE | 2.993 | 3.292 | 3.515 | 3.422 | |
MAPE | 0.05 | 0.053 | 0.055 | 0.059 | |
MSE | 395.6 | 403.5 | 405.7 | 407.4 | |
MAPE | 0.027 | 0.028 | 0.028 | 0.029 |
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 The Spatial Form of Digital Nonlinear Landscape Architecture Design Based on Computer Big Data 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 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 Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China 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 The evaluation of college students’ innovation and entrepreneurship ability based on nonlinear model 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 First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe_{80}Si_{10}Nb_{6}B_{2}Cu_{2} 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