Chapter 2 - Mathematical and statistical concepts underlying big data analytics
El-Kafrawy , Passent
El-Kafrawy , Passent
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2026-01-09
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Abstract
This chapter provides a comprehensive introduction to the mathematical and statistical foundations that underpin big data analytics. It begins with an exploration of combinatorics, multivariable calculus, and integration techniques, which serve as the groundwork for understanding data mining and preprocessing. Key regression models, including linear regression, multiple linear regression, and logistic regression, are emphasized as essential tools for analyzing relationships between variables and making accurate predictions. Time-series analysis is presented as a robust methodology for examining sequential data, with a focus on identifying trends, seasonality, and cyclical patterns. Prominent models such as ARIMA and exponential smoothing are highlighted for their ability to uncover temporal dynamics and support forecasting. The chapter also introduces clustering algorithms, such as k-means and hierarchical clustering, which are fundamental for organizing data into coherent structures in unsupervised learning. Additionally, the chapter covers pivotal topics including probability theory, statistical inference, linear algebra, and optimization techniques. These areas are critical for analyzing and processing large datasets effectively. Bayesian statistics is explored as a dynamic framework for incorporating prior knowledge and updating probabilities, with applications ranging from Bayesian inference to Markov Chain Monte Carlo (MCMC) methods. Addressing the challenges inherent in big data analytics, such as high dimensionality and model overfitting, the chapter introduces techniques like dimensionality reduction (e.g., Principal Component Analysis and t-SNE), regularization methods (e.g., lasso, ridge, and elastic net regression), and robust validation strategies (e.g., cross-validation and bootstrapping). These approaches are essential for ensuring model reliability and interpretability in the context of large-scale and complex datasets. By bridging theoretical concepts with practical applications, this chapter equips readers with the necessary tools to navigate and address the intricacies of big data analytics. Special emphasis is placed on enhancing computational efficiency, enabling practitioners to extract meaningful insights from high-dimensional and voluminous data. Through this integration of foundational principles and applied techniques the chapter serves as a vital resource for effectively tackling the complexities of big data.
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Mathematical Modeling for Big Data Analytics