Electronic Theses and Dissertations

Date

2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematical Sciences

Committee Chair

E. Olusegun George

Committee Member

Deepak Venugopal

Committee Member

Majid Naroozi

Committee Member

Mohamed Yeasin

Abstract

Orthogonal polynomials have had vast historical applications in applied mathematics, particularly in differential equations and numerical analysis. In recent years, they have become instrumental as a natural basis for constructing kernel systems in machine learning (ML). In this thesis, we orthogonalize a set of polynomials, the Logistic Polynomials, first developed by George and Rousseau, using the Gram-Schmidt process. We show that this new Logistic Orthogonal Polynomial System (LOPS) satisfies the hypergeometric differential equation and we relate the system to the Legendre polynomials. The connection to the Legendre polynomials has motivated this study to explore applications of the LOPS in ML, particularly in the context of Support Vector Machines (SVMs) and Neural Networks (NN). Our goal for this study is to evaluate the effectiveness of the Polynomial Kernel of the LOPS as kernels for SVM and NN. We will observe continuous real-valued data and use the sigmoid function to transform this data to the interval [0,1] to assess this kernel. The measures of effectiveness for this study are operating time and output accuracy. The implications of this study will have a large impact on artificial intelligence by observing its mathematical properties.

Comments

Data is provided by the student.”

Library Comment

Dissertation or thesis originally submitted to ProQuest.

Notes

Open Access

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