Mathematical Data Science with Applications in Business, Industry, and Medicine

Mathematical data science is a field that combines mathematical techniques with data science methods to extract insights and knowledge from data. It includes working with data at all stages of the data life cycle, from collection and storage to cleansing and processing, analysis and visualization of data, and communication of the results and findings. Data scientists use a variety of tools and techniques to analyze data, including mathematical concepts and models, artificial intelligence techniques, machine learning algorithms, statistical analysis, and data visualization. Data science can be used to make predictions, identify patterns, and draw conclusions from data, and it is applied in a variety of areas, including business, industry, and medicine. It is a rapidly evolving field, and data scientists are expected to stay up to date with new tools, techniques, and technologies.

We are pleased to present this Special Issue of $Mathematics$, titled “Mathematical Data Science with Applications in Business, Industry, and Medicine”. This special issue brings together ten insightful papers that highlight innovative mathematical data science techniques in solving complex problems across various domains, including business, industry, and medicine. These publications are listed below, ordered by date of publication:

• Haj Ahmad, H.; Almetwally, E.M.; Ramadan, D.A. Investigating the Relationship between Processor and Memory Reliability in Data Science: A Bivariate Model Approach. Mathematics 2023, 11, 2142. https://doi.org/10.3390/math11092142.
• Hesamian, G.; Johannssen, A.; Chukhrova, N. A Three-Stage Nonparametric Kernel-Based Time Series Model Based on Fuzzy Data. Mathematics 2023, 11, 2800. https://doi.org/10.3390/math11132800.
• Alevizakos, V. Process Capability and Performance Indices for Discrete Data. Mathematics 2023, 11, 3457. https://doi.org/10.3390/math11163457.
• Sabahno, H.; Niaki, S.T.A. New Machine-Learning Control Charts for Simultaneous Monitoring of Multivariate Normal Process Parameters with Detection and Identification. Mathematics 2023, 11, 3566. https://doi.org/10.3390/math11163566.
• Netshiozwi, U.; Yeganeh, A.; Shongwe, S.C.; Hakimi, A. Data-Driven Surveillance of Internet Usage Using a Polynomial Profile Monitoring Scheme. Mathematics 2023, 11, 3650. https://doi.org/10.3390/math11173650.
• Mamzeridou, E.; Rakitzis, A.C. A Combined Runs Rules Scheme for Monitoring General Inflated Poisson Processes. Mathematics 2023, 11, 4671. https://doi.org/10.3390/math11224671.
• Triantafyllou, I.S. Archimedean Copulas-Based Estimation under One-Parameter Distributions in Coherent Systems. Mathematics 2024, 12, 334. https://doi.org/10.3390/math12020334.
• Marambakuyana, W.A.; Shongwe, S.C. Composite and Mixture Distributions for Heavy-Tailed Data—An Application to Insurance Claims. Mathematics 2024, 12, 335. https://doi.org/10.3390/math12020335.
• Enck, D.; Beruvides, M.; Tercero-Gómez, V.G.; Cordero-Franco, A.E. Addressing Concerns about Single Path Analysis in Business Cycle Turning Points: The Case of Learning Vector Quantization. Mathematics 2024, 12, 678. https://doi.org/10.3390/math12050678.
• Pakzad, A.; Yeganeh, A.; Noorossana, R.; Shongwe, S.C. Process Capability Index for Simple Linear Profile in the Presence of Within- and Between-Profile Autocorrelation. Mathematics 2024, 12, 2549. https://doi.org/10.3390/math12162549.

The remainder of this editorial contains a summary of the contributions to this special issue.

Haj Ahmad et al. (1.) introduce a bivariate model using a copula function to model the failure times of processors and memories in computers. Properties of the model are analyzed, and inferential statistics for the parameters are performed under the assumption of a Type-II censored sampling scheme. The parameters are estimated by both maximum likelihood and Bayesian estimation methods, and the estimations are compared. The model’s efficiency was validated with real data, demonstrating its excellent fit compared to other bivariate models.

Hesamian et al. (2.) present a nonlinear time series model tailored for data reported as $LR$ fuzzy numbers, using a three-stage nonparametric kernel-based estimation procedure. The Nadaraya–Watson estimator is employed in each stage to estimate the center and spreads of the fuzzy smooth function, with a hybrid algorithm determining the optimal bandwidths and autoregressive order. The model’s effectiveness is validated through a simulation study and real-life applications, demonstrating its improved performance over other fuzzy time series models.

The paper from Alevizakos (3.) focuses on adapting classical process capability indices (PCIs) and performance indices for discrete data following Poisson, binomial, or negative binomial distributions using various transformation techniques. The methodology simplifies the computation of these indices, allowing for a straightforward assessment of process capability and performance. The paper includes a simulation study comparing these indices with existing ones for discrete data and provides three examples to demonstrate the practical application of the transformation techniques.

Sabahno and Niaki (4.) propose three control charts based on artificial neural networks, support vector machines, and random forests for monitoring process parameters in multivariate normal processes. These tools are designed to classify processes as in-control or out-of-control variables and are evaluated under different input scenarios, training methods, and process control objectives, including detection and identification of out-of-control variables. The developed control charts demonstrate very good performance compared to traditional memory-less statistical control charts, with an illustrative example provided for application in a healthcare process.

Netshiozwi et al. (5.) study a data-driven monitoring framework based on profile monitoring to assess internet usage patterns in a telecom company. By defining a polynomial model between the hours of each day and internet usage, the framework aims to detect unnatural patterns, assess the impact of policies like discounts, and investigate social behavior variations in usage. The study compares charting statistics, showing that the MEWMA scheme outperforms Hoteling $T^2$ chart in detecting small shifts, offering quicker detection capabilities.

Mamzeridou and Rakitzis (6.) propose a Shewhart-type control chart with multiple runs rules to monitor inflated processes, particularly when the process mean level is very low, making it difficult to detect decreases in the mean. The chart is supplemented with two runs rules: one for detecting decreases in the process mean and another for enhancing sensitivity to small and moderate increases in the mean. Using the Markov chain method, the study evaluates the performance of these schemes, showing their effectiveness in detecting shifts in process parameters, with practical applications demonstrated through two examples based on healthcare data.

Triantafyllou (7.) deals with a signature-based framework for estimating the mean lifetime and variance of a continuous distribution in a coherent system with exchangeable components. The dependency among components is modeled using Archimedean multivariate copulas, specifically the Frank and Joe copulas. Numerical experiments are conducted to illustrate the methodology across all possible coherent systems with three components.

Marambakuyana and Shongwe (8.) analyze two-component non–Gaussian composite and mixture models for insurance claims data, emphasizing their flexibility in curve fitting. A total of 256 composite and 256 mixture models, derived from 16 popular parametric distributions, are evaluated. The study applies these models to two real insurance datasets, using model selection criteria and risk metrics to identify the top 20 models in each category. Results indicate that composite models offer superior risk estimates compared to mixture models for both datasets.

Enck et al. (9.) highlight the limitations of single path analysis (SPA) in economic analysis, particularly in identifying business cycle (BC) turning points using machine learning. SPA fails to account for temporal dependence in BCs, which can lead to inadequate evaluation and calibration of algorithms. Using learning vector quantization (LVQ) as a case study, the study employs a multivariate Monte Carlo simulation to incorporate change points, autocorrelations, and cross-correlations, offering a more robust understanding of LVQ’s uncertainties. The results reveal the shortcomings of SPA and underscore the importance of considering temporal dependence to improve the robustness of data-driven approaches in economic and financial analysis.

The paper from Pakzad et al. (10.) addresses the issue of autocorrelation in simple linear profiles (SLPs) when assessing process capability, which is often overlooked in traditional PCIs. The study introduces novel methods to evaluate SLP capability under within-profile, between-profile, and simultaneous autocorrelation, using an AR(1) model. A new functional index is proposed and modified to account for these autocorrelation effects, with simulation results showing improved performance in terms of bias and mean square error. Bootstrap confidence intervals are provided, and an illustrative example from the chemical industry demonstrates the method’s practical applicability.

In conclusion, the papers presented in this Special Issue highlight the crucial role of mathematical data science in tackling complex problems across various fields such as business, industry, and medicine. As the landscape of data science continues to evolve, the integration of advanced mathematical techniques with emerging technologies will be pivotal in driving innovation. Future research should focus on enhancing the scalability and adaptability of these methods to address increasingly diverse and large-scale datasets. We anticipate that the insights and methodologies discussed in this Special Issue will inspire further exploration and contribute to the ongoing advancement of mathematical data science, ultimately leading to more robust solutions and transformative applications across different domains.