The National Basketball Association (NBA) is a professional basketball league in the North American continent. Its enormous influence attracts many fans from all over the world. Because of the significant amount of money and the vast number of fans involved in the NBA league, there have been many studies attempting to predict the outcome of NBA games by simulating the winning team and analysing their players’ abilities so as to assist the coach in team organisation. NBA games have accumulated much historical game data and statistical analysis data. Even with this situation being what it is, analysing and predicting these games is still very complicated .
Statistics have always been a must for basketball player evaluation, from simple field-goal average to overall efficiency indicators such as the attack score introduced by Oliver (2004) . Generally, a professional sport network is staffed with a professional analysis team, which is responsible for collecting and interpreting data from each game, and establishing statistical indicators based on the player performance in the actual play to measure the realisation level of players and teams. In predicting the performance of individual athletes, only a few statistical indicators can be used. In order to simplify the game analysis and accurately predict game results by using data, related technologies such as machine learning have been applied to predict outcomes of NBA games.
In this paper, machine learning was used to predict the performance of players before they played in the regulation game, by using FPTS collected from professional sports sites and their NBA basketball player statistical indicators, to predict outcomes of NBA games. Such prediction will only define the winning team, without considering its final scores in the game. Moreover, the effects of feature variables of basketball matches on game prediction were analysed, and feature selection was performed. The machine learning models adopted in this paper included the linear regression, extreme gradient boosting (
The paper uses a machine learning algorithm to predict the player performance based on past player data, and to predict team performance and the outcome of basketball games. Specifically, the data adopted, the feature variables constructed and the predicted evaluation objectives were analysed. First, the statistics of player games and player salary and position information during three regular seasons of NBA from 2015 to 2018 were used. For missing values in player data, the mean value or
Second, indicators were defined to quantify a player's ability, including total points (
There were other variables affecting player performance, such as home-court advantage, rest days, team sheets, player positions and salary, which were collected from the DraftKings’ algorithm or coaches’ decision-making system . The variable (value) of a player was constructed as the ratio between salary divided by 1,000 and FPTS, which was treated as a heuristic algorithm. When the value is higher than 5, it indicates that the player is in a good state, with higher ability evaluation . Moreover, the dimensionality resulting from more feature variables is solved by reducing the correlation between variables and selected important features. In this paper, considering the predictive effectiveness of variables for players, the following models were established to represent the quantitative relationship between variables and players’ ability values. The ability value
Finally, this paper aims to predict players’ ability value
Considering the prediction goal of this paper, it is obvious that the prediction is closely related to the regression. Therefore, the regression algorithm was adopted. The model comparison in this paper applies relevant machine learning regression algorithms, including linear regression,
For the linear regression, if a linear relationship exists between independent variables and dependent variables, it will meet the following equation:
The least square method is used to estimate the parameters. This method can deduce the optimised parameters after the model training, so as to predict outcomes by regression.
Boosting is a method to transform a weak classifier into a strong classifier, whose function model is superposed. To be specific:
The objective function is a feature of
Optimising the solution:
When the objective function is determined, it moves to the training process. For each iteration process, the training of the objective function of a tree can be written as follows:
The input is the predicted value after the (
After a series of calculations, the smallest objective function
After putting it into the original equation, the minimum value obtained is:
So, the finally obtained
The neural network algorithms are widely used in all subdomains of artificial intelligence. They are briefly introduced in the literature, including the explained version.
Data were pre-processed, including filling the missing value, making the variable name uniform and carrying out variable standardisation. The mean value was used to fill in the missing data. The data from different sources were uniformly processed and the player data were standardised for
Except for the variables directly used to calculate FPTS, other variables were selected for models to quantify the player ability by using more advanced statistical methods. The details are discussed in part 2.
Using data from recent games allows a more objective and accurate prediction of the results of players’ abilities. Therefore, the weighted mean of the past 10 games was used to obtain each variable. The relevant theory shows that the mean weight of games increases linearly with the number of games. In this paper, the square root and linear and square modes were used for quantitative evaluation. It is necessary to normalise the weights such that the sum of the weights is 1. As shown in Figure 1, the weighted square mode is a better weighting method. Accordingly, it was considered as the best option in this paper.
For the consistency of feature variables, the standard deviation of the FPTS variable over the last 10 games was defined as FPTS_std, while salary information from DraftKings was also defined as the Salary variable. Before a game, the participation of a player in the game cannot be determined from the model. Thus, it is necessary to calculate the value of the model's feature variable according to the published player sheet  before the game.
Because of the different predictive abilities of features on game outcomes and correlations between features, the features should be filtered and ranked. Taking the field goal (FG) as an example, it is highly correlated with effective field-goal percentage (eFG%) because the latter considered far fewer free throw points. Furthermore, some variables have multiple collinearity, such as three pointers (3P), three pointers (3PA) and three-point percentage (3P%). In this paper, the Pearson correlation coefficient was used for screening features. With the setting correlation threshold of 0.90, the following six features were screened out: three-point shot (3PA), defensive rebound (DRB), field goals attempted (FAG), field-goal percentage (FG%) and offensive rating (ORtg). In addition, variables without predictive ability are directly ignored in models. The gradient descent method was used in models to evaluate and quantify the feature significance of 34 variables. Using feature ranking, such features as dummy variables of position (SG, F, C), three pointers (3P) and three doubles (TD) were excepted. Finally, the remaining 29 variables were used as selected features for regression, gradient enhancement and deep learning.
The results of linear regression for all variables, using three different datasets, are shown in Table 1. A 5-fold cross validation was implemented when the linear regression model was trained. According to the linear model regression prediction, the minimum value of
Prediction effect of linear model
MAE, Mean absolute error;
RMSE, Root mean square error.
Max_depth: 5, n_estimators: 354, n_child_weight: 0, gamma: 0.8, learning_rate: 0.0152. The use of these parametric adjustment models will result in better performance (
In terms of the neural network model, Figure 4 shows the learning process of the model. A total of 20% of the training data was retained as validation data. It can be seen that the model soon starts to overfit, and verification losses are different from training losses. To prevent the model from overfitting, a loss layer was added, which randomly ignored 40% of the data points before feeding them forward to the last layer. In this paper, the EarlyStopping method in the Keras.callbacks model was applied. If verification is lost for no more than five periods, learning is terminated. This solution improved the model performance, with the RMSE reduced from 9.0678 in the original model to 9.0387.
Finally, the performance of these three models was compared, as shown in Table 2, where the models in the first and third rows calculated the mean value by using the defined linear combination of coefficients. In the fourth and fifth lines, the
Comparison of three machine learning models
MAE, mean absolute error;
RMSE, root mean square error;
XGBoost, extreme gradient boosting.
After the prediction calculation by models, and 5-fold cross validation, the consistent RMSEs (8.9910, 9.0522, 9.0148, 8.9351, and 9.0831) were obtained. In order to further improve the robustness of the model, the effect of small changes in input data on the performance of the model was also studied. First, the Gaussian noise was created by using mean 0 and variance 1, which was then scaled to the range of [0, 0.2] and added to the continuous variable in the original input. When the 5-fold cross validation was performed on the independent variable evaluated by using the noise, the losses seem to have barely budged, with
For the model,
The statistical significance was 0.1%, with four degrees of freedom and a critical value of 15.54. Therefore, the weighted mean, feature engineering and
In this paper, the steps and processes of solving application problems by using machine learning methods are discussed, i.e., analysing problems (predicting NBA games), searching research and processing data, feature selection, model training, evaluation and optimisation, etc. The indicators from DraftKings were used to predict how NBA players perform in regular games. The model prediction is run to minimise the RMSE between the predicted value and the actual FPTS statistics. It started from the basis linear model, where averages of the seasons’ past statistical data were used, together with weighting methods, to extract important features from selected features. The key feature variables for the model to predict a player's ability included salary information, team, player sheet provided by DraftKings, and other important statistical factors such as total rebounds and individual points scored. After feature selection and data normalisation, the
Moreover, the comparison between the predictions by models and FPTS actual statistics was verified in relevant games selected from the training data. The algorithm was used in five games broadcast on 10 March 2018, which produced the following eight player lineups, with an expected FPTS of 242.2643 and a total salary of US$ 49,900. The blue, orange and green bars stand for the actual FPTS, predictions by final model and predictions by basis liner mode, respectively. Below the names of the players their positions are mentioned. The final model predicted that the actual FPTS were much better than that of the basis liner modes of certain athletes such as Dillon Brooks (SF) and Dwight Powell (C). However, it tends to overestimate the FPTS of players such as Tomas Satoransky (G) and Kobi Simmons (F). Overall, the predictions from the match were very accurate with losses of 6.2836 (MAE) and 7.6538 (RMSE).
Throughout the data mining exercise, the feature importance and feature correlation matrix were essential to understanding how statistical indicators affect the predicated outcome of competition. Most importantly, data processing and feature extraction in this project have a great impact on the predicted results, which must be focused on. In particular, unifying names, handling missing values in data processing, and ranking and combination of important features in the feature selection process will greatly affect the final model training.
While the models used in this paper generally followed the performance comparisons of the algorithms themselves, the final improvement in RMSE was less than 10%. If, the opponents’ defensive data, such as a team's defensive rating and the positions of opposing players, are considered, the accuracy of prediction of the model might be improved. Furthermore, there is another important factor – namely, a coach determines when a player enters the field. When simulated players are used, it becomes possible to observe how the number of minutes varies under different game managements and, especially, how the formation tactics change during games. These factors can be modelled for quantitative evaluation and included within models. DraftKings also provides the views of news outlets and professional reviewers, which can be combined through natural language processing to be useful for performance prediction and formation optimisation. In conclusion, it is highly necessary to use the machine learning algorithm in the basketball game prediction and player ability quantitative evaluation system, and thus this usage is worthy of further research.
Prediction effect of linear model
Comparison of three machine learning models
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and implementation of visual question and answer system based on deep learning Títul Research on Wireless Network Coverage for Transformation and Upgrading of Exhibition Management with Artificial Intelligence Technology The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Adaptive fusion of multi-exposure images based on perceptron model Study on the application of multi-angle imaging related technology in the construction process Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Exploring the Path of Integrating Excellent Traditional Culture into Civic Education in Universities in the Internet+ Era Economic statistical strategies for the transportation industry based on time-varying parametric state space models Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Design and research of environmentally friendly field sewage purification mechanical device Public Management Model Based on Mathematical Equation of Statistics of Probability Theory 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System An analysis of time commitment for college students to online English writing and peer assessment learning Optimization path of corporate ESG performance and compensation performance management under carbon neutral target Research on network security management monitoring platform based on Internet of things technology New anergy tide control strategy based on Eviews econometric model Study of frictional wear properties of materials for mechanical seals Optimization of vocal singing training method based on intelligent big data technology Fractional Differential Equations in Electronic Information Models Exploring the construction of an all-round cost management system for investment control in university engineering construction The construction of blockchain platform for engineering museums based on vectorized image processing technology BIM Engineering Management Oriented to Curve Equation Model Relationship Between Physical Activity Sedentary Behavior and Body Mass Index in Children and Adolescents Based on FIT Modelo Green supply chain innovation management strategy based on the combination of low carbon economy and e-commerce with big data technology Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Grid data asset relationship and intelligent classification integrating knowledge graph and Internet of Things Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Analysis of digital humanistic knowledge production based on natural language processing Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering The Construction of Judgment Model for Cultivating Intelligent Accounting Professionals in Universities Based on Logarithmic Periodic Power Law Model Action of Aut( G) on the set of maximal subgroups of p-groups Digital Regeneration and Database Construction of Hunan Embroidery Needlework under the Perspective of Artificial Intelligence Research on the practice of innovation and entrepreneurship education in universities under the environment of big data Application of digital technology in environmental art design Research on loyalty prediction of e-commerce customer based on data mining Research on the evaluation system of tourism management classroom teaching based on multimedia big data analysis The application of ARCS motivation model in college oral English teaching Research on Innovation of Translation Teaching and Translation Strategies for College Students in Multimedia Background A multi-channel convolutional neural network based on attention mechanism fusion for facial expression recognition Exploring the development of cultural and creative tourism under the perspective of ecological theory of tourism destination management Modeling Analysis of User Consumption Data of Internet Culture Industry in the Context of Emerging Technologies Reference: An algorithm for recognizing the main melody of orchestral music based on artificial intelligence of music melody contour Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A study of local smoothness-informed convolutional neural network models for image inpainting The construction of a practical platform for party building and ideological and political education based on information fusion A study on the influence of the spread of Yangming Studies in Japan on the psychology of the Japanese people based on big data analysis Research on personalized recommendation algorithm for micromechanical sensors based on cloud model Research on the value of modern vernacular literature in contemporary rural construction based on ADDIE model Robust speech recognition based on deep learning for sports game review IoT-based interaction design of smart home products for elderly families Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Model Construction of Influencing Factors of Physical Education Core Quality Cultivation Based on Multiple Linear Regression A multidimensional interpretation analysis of Marxism globalization theory based on distributed Internet Application of Nonlinear Differential Equation in Electric Automation Control System Research on the operation and management mechanism of cross-border e-commerce shared practice training base Strategy for financing mode optimization in international trade supply chain based on deep learning model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Strategies for improving Chinese language proficiency based on artificial intelligence technology Early warning and detection of geological disasters based on intelligent genetic algorithm The success rate model of college students’ new entrepreneurship based on nonlinear differential equations Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Research on the application of marketing strategy of national fitness exercise and dance events in the construction of sports culture based on big data technology Exploring the Application of Modern Fine Arts Artistic Modeling and Color in Modern Art Design Based on Information Fusion Technology Decisions of competing supply chain with altruistic retailer under risk aversion Strategies for constructing mathematical models of nonlinear systems based on multiple linear regression models Research on China’s tourism marketing model in the data perspective: based on ecological perspective Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations Research on human resource management performance evaluation based on BP algorithm The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network 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 Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Badminton players’ trajectory under numerical calculation method Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis The Mental Health Education Management of Higher Vocational Students Based on Fractional Differential Equations Application of regression function model based on panel data in financial risk management of bank resource allocation University Ideological and Political Learning Model Based on Statistical Memory Curve Mathematical Equation Research on Resonance Properties of Semantic Wave Fractal Fractals Based on Quantitative Analysis of English Corpus Research on urban landscape big data information processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis 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 Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on urban energy Internet planning based on fuzzy comprehensive method Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the intelligent path of college students’ network ideological and political education based on big data mining technology Research on the application of PLC technology in electrical automation engineering Application of matrix multiplication in signal sensor image perception Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence VR-based computer maintenance practical training platform development design and application research Design and implementation of an integrated management system for discipline construction and postgraduate education in universities Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Research on educational resource recommendation system based on MRLG Rec Visual analysis of the trend of mental health changes among Chinese college students in the context of the normalisation of the new crown epidemic A study on the phenomenon of anaphoric correction in college students’ English conversation Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model Research on the innovation of collaborative education model of scientific research and teaching in colleges and universities based on the theory of process reengineering AtanK-A New SVM Kernel for Classification The innovative path of diversified vocal music education reform in colleges and universities under the advanced educational concepts of China and the West Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Rhino application in ceramic packaging design teaching Study of a linear-physical-programming-based approach for web service selection under uncertain service quality Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory Virtual Museum Scene Design Based on VRAR Realistic Interaction under PMC Artificial Intelligence Model A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Research on innovative cultivation mode of English and American literature translation talents based on internet environment Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Analysis of quantitative management of online intelligent monitoring of tailing ponds based on the perspective of safety prevention and control Optimization of college physical education curriculum based on G-DINA Optimization of local ethnic music teaching transmission path based on logistic regression model RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning Design and evaluation of intelligent teaching system on basic movements in PE Research on identifying psychological health problems of college students by logistic regression model based on data mining Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Retraction: Wei, Wei and Lu, Liu. “The influence of X fuzzy mathematics method in basketball tactics scoring” Applied Mathematics and Nonlinear Sciences, vol.7, no.1, 2022, pp.677-684 Retraction Retraction: Retraction of: Mundewadi, R. A. and S, Kumbinarasaiah. “Numerical Solution of Abel's Integral Equations using Hermite Wavelet” Applied Mathematics and Nonlinear Sciences, vol.4, no.2, 2019, pp.395-406