The Kernel Function of Reproducing Kernel Hilbert Space and Its Application on Support Vector Machine
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Keywords:
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Linear Kernel Function, Polynomial Kernel Function, Gaussian Kernel Function, Reproducing Kernel Hilbert Space
Abstract
Reproducing Kernel Hilbert Space (RKHS) is a Hilbert space consisting of functions that can be represented or reproduced by a kernel function. The development of data science has made RKHS a method that refers to an approach or technique using the concept of reproducing kernels in certain applications, especially machine learning. Support Vector Machine (SVM) is one of the machine learning methods included in the supervised learning category for classification and regression tasks. This research aims to determine the form of linear kernel functions, polynomial kernel functions, and Gaussian kernel functions in Support Vector Machine analysis and analyze their performance in Support Vector Machine classification and regression. Application of the RKHS method in SVM classification analysis using World Disaster Risk Dataset data published by Institute for International Law of Peace and Armed Conflict (IFHV) from Ruhr-University Bochum in 2022 obtained results that are based on the results by comparing the predictions of training data and testing data using linear kernel functions, polynomial kernels and Gaussian kernels, it is recommended that classification using linear kernels provides the best prediction performance.
References
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Anjani, S. D., W. Widiarti, B. H. S. Utami, M. Usman, and V. A. Handayani (2024). Georaphically Weighted Ridge Regression Modelling on 2023 Poverty Indicators Data in the Provinces of West Kalimantan and Central Kalimantan. Integra: Journal of Integrated Mathematics and Computer Science, 1(3); 73–80
Arqub, O. A., M. Al-Smadi, and N. Shawagfeh (2013). Solving Fredholm IntegroDifferential Equations Using Reproducing Kernel Hilbert Space Method. Applied Mathematics and Computation, 219(17); 8938–8948
Awan, D. A., L. G. C. Renato, M. Yukawa, and S. Stanczak (2020). Adaptive Learning for Symbol Detection: A Reproducing Kernel Hilbert Space Approach. In Machine Learning for Future Wireless Communications. pages 197–211
Azarnavid, B., F. Parvaneh, and S. Abbasbandy (2015). Picard-Reproducing Kernel Hilbert Space Method for Solving Generalized Singular Nonlinear Lane-Emden Type Equations. Mathematical Modelling and Analysis, 20(6); 754–767
Baltagi, B. H., P. H. Egger, and M. Kesina (2012). Small Sample Properties and Pretest Estimation of a Spatial Hausman-Taylor Model. In Advances in Econometrics, volume 29. pages 215–236
Berlinet, A. and C. Thomas-Agnan (2004). Reproducing Kernel Hilbert Spaces in Probability and Statistics. Springer, New York, USA
Bowman, A. W. and A. Azzalini (1997). Applied Smoothing Techniques for Data Analysis: The Kernel Approach. Clarendon Press, Oxford, USA
Cervantes, J., G.-L. Farid, L. Rodríguez-Mazahua, and A. Lopez (2020). A Comprehensive Survey on Support Vector Machine Classification: Applications, Challenges, and Trends. Neurocomputing, 408(1); 189–215
Chakraborty, S. (2012). Bayesian Multiple Response Kernel Regression Model for High Dimensional Data and Its Practical Applications in Near Infrared Spectroscopy. Computational Statistics and Data Analysis, 56(9); 2742–2755
Chen, S. B., S. Soradi-Zeid, H. Dutta, M. Mesrizadeh, H. Jahanshahi, and Y. M. Chu (2021). Reproducing Kernel Hilbert Space Method for Nonlinear Second Order Singularly Perturbed Boundary Value Problems with Time-Delay. Chaos, Solitons and Fractals, 144(1); 1–10
Ghojogh, B., A. Ghodsi, F. Karray, and M. Crowley (2021). Reproducing Kernel Hilbert Space, Mercer’s Theorem, Eigenfunctions, Nystrom Method, and Use of Kernels in Machine Learning: Tutorial and Survey. CoRR, 2106; 1–31
Ghosh, S., A. Dasgupta, and A. Swetapadma (2019). A Study on Support Vector Machine Based Linear and Non-Linear Pattern Classification. In Proceedings of the International Conference on Intelligent Sustainable Systems, volume 1. pages 24–28
Lestari, W., W. Widiarti, B. H. S. Utami, M. Usman, and V. A. Handayani (2024). Robust Panel Data Regression Analysis Using the Least Trimmed Squares (LTS) Estimator on Poverty Line Data in Lampung Province. Integra: Journal of Integrated Mathematics and Computer Science, 1(2); 35–42
Mohan, L., J. Pant, P. Suyal, and A. Kumar (2020). Support Vector Machine Accuracy Improvement with Classification. In Proceedings 12th International Conference on Computational Intelligence and Communication Networks, volume 12. pages 477–481
Muthukrishnan, S., H. Krishnaswamy, S. Thanikodi, D. Sundaresan, and V. Venkatraman (2020). Support Vector Machine for Modelling and Simulation of Heat Exchangers. Thermal Science, 24(1B); 499–503
Paiva, A. R. C., I. Park, and J. C. Principe (2019). A Reproducing Kernel Hilbert Space Framework for Spike Train Signal Processing. Neural Computation, 21(2); 424–449
Paulsen, V. I. and M. Raghupathi (2009). An Introducing to The Theory of Reproducing Kernel Hilbert Spaces. University of Houston, Texas, USA
Pisner, D. A. and D. M. Schnyer (2020). Support Vector Machine. In Machine Learning: Methods and Applications to Brain Disorders. Elsevier Inc., New York, USA
Rampisela, T. V. and Z. Rustam (2018). Classification of Schizophrenia Data Using Support Vector Machine (SVM). Journal of Physics: Conference Series, 1108; 1–7
Rincón, M. and M. D. Ruiz-Medina (2012). Wavelet-RKHS-Based Functional Statistical Classification. Adv Data Anal Classif, 6; 201–217
Rosipal, R. and L. J. Trejo (2001). Kernel Partial Least Square Regression in Reproducing Kernel Hilbert Space. Journal of Machine Learning Research, 2; 97–123
Shawe-Taylor, J. and N. Cristianini (2004). Kernel Methods for Pattern Analysis. Cambridge University Press, New York, USA
Shukla, S., N. Badal, and B. K. Thakur (2023). Sustainable Computing: Transforming Industry 4.0 to Society 5.0. Springer, New York, USA
Small, C. G. and D. L. McLeish (1994). Hilbert Space Methods in Probability and Statistical Inference. John Wiley and Sons, New Jersey, USA
Tuo, R., S. He, A. Pourhabib, Y. Ding, and J. Z. Huang (2023). A Reproducing Kernel Hilbert Space Approach to Functional Calibration of Computer Models. Journal of the American Statistical Association, 118(542); 883–897
Utami, B. H. S., M. Usman, and F.Warsono (2023). The Form of σ-Algebra on Probability Hilbert Space. Mathematics and Statistics, 11(3); 446–453
Yang, H. (2016). Learning Methods in Reproducing Kernel Hilbert Space Based on High-Dimensional Features. Ph.D. thesis, University of California, California, USA
Zhu, W., Y. Song, and Y. Xiao (2020). Support Vector Machine Classifier with Huberized Pinball Loss. Engineering Applications of Artificial Intelligence, 91; 1–9
Authors
Utami, B. H. S. ., Warsono, Usman, M. ., & Fitriani. (2025). The Kernel Function of Reproducing Kernel Hilbert Space and Its Application on Support Vector Machine. Science and Technology Indonesia, 10(4), 1096–1108. https://doi.org/10.26554/sti.2025.10.4.1096-1108
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