Soft set theory applications: Time series as soft sets and ratio based similarity measure on soft sets

Soft set theory, originally introduced by Molodtsov in 1999, offers a versatile and flexible mathematical framework for handling uncertainties, vagueness, and incomplete information in complex systems. Over the past two decades, it has found extensive applications in diverse domains such as decision making, data analysis, engineering, and medical diagnosis. However, despite this wide applicability, its potential in the field of time series analysis remains relatively unexplored and underutilized. Time series data, which capture the evolution of phenomena over time, often involve uncertainty, noise, and nonstationary pattern features that make it an ideal candidate for soft set based modeling. Building on this, we propose two distinct approaches, amplitude threshold soft set and time derivative threshold soft set for representing time series data within the soft set framework. The effectiveness of the proposed methodology is demonstrated through two comprehensive real world applications. First, 58 year temperature dataset from six Brazilian cities spanning from 1967 to 2019 is analyzed, demonstrating how the soft set framework can capture long-term climatological patterns and facilitate analysis across different geographical regions. Second, we consider heart sound classification using phonocardiogram (PCG) data, showing how soft set based time series representation can effectively distinguish between normal and abnormal heart sounds with promising classification performance with the help of a novel ratio based similarity measure specifically designed for soft sets. The methodology's effectiveness is demonstrated through medical signal processing applications, where it achieved competitive performance on heart sound classification (challenge score: 0.776). By integrating soft set theory with time series analysis, and applying the proposed similarity measure in this context, the study bridges an existing gap between the two fields. This approach offers a fresh perspective for pattern recognition, comparative analysis, and uncertainty modeling in temporal data, opening new avenues for future research.