In order to solve problems and provide practical solutions, researchers attempt to accurately describe societal difficulties and obstacles. An efficient method for handling complicated real-world data is rough set theory. The method finds confirmed and likely data from subsets using rough approximation operators. In order to increase accuracy while following Pawlak’s conventional approximation axioms, earlier research has created rough approximation models based on neighborhood systems. Based on cardinality rough neighborhoods and grills, we present new rough set notions in this study. These models are a suitable approach for a number of scenarios, including computational analysis, comparisons on medical datasets, real-world data analysis challenges, and classification accuracy metrics.We thoroughly examine the fundamental components of these ideas and clarify how they relate to one another as well as to earlier paradigms. Next, we describe boundary regions and assess the accuracy of the data using a topological technique. Additionally, we look at how well our models handle heart failure disease in certain individuals and come to the conclusion that the suggested rough set concepts improve upon the characteristics of the earlier approach spaces.Finally, we identify the shortcomings of the current concepts and show their advantages in terms of extending the verified information gleaned from subsets of data while preserving the key elements of Pawlak’s original paradigms that were destroyed by the models that went before them.

This work introduces a new correlation measure of quadri-partitioned neutrosophic refined sets (QPNRS) to address the problem of ambiguity and bias that may exist in the medical diagnostic. The model is a patient-symptom-disease interaction model that uses four membership functions—truth, relative truth, relative falsehood, and falsehood—to illustrate its interactions. Table III concludes the results by demonstrating that there is clear disease and estimator variance and that correlations between QPNRS measures Q and R are consistently strong and positive (0.743 -0.901) across four medical conditions. In particular, P2 is more extensively distributed, while P1 is constantly high in all environments, reaching its maximum value of 0.901 in the case of tuberculosis. These findings support the combined clinical use of Q and R by demonstrating a strong, primarily favorable reliance between them. We apply a multifaceted unsupervised learning pipeline to two distinct datasets—the conventional Iris dataset and patient disease-correlation profiles—in order to augment the correlation analysis. In both cases, the consistently strongly separated clusters support the process and imply that the detected groupings are not the result of labeling or modeling artifacts.

This research presents a new theoretical framework and shows how to use it e ectively to data analysis. First, we introduce a novel method to complex neutrosophic soft sets (CNSS) and ensure the generalize ability of their use by de ning basic operations such as union, intersection, and complement. We go on to construct a complete one-value neutrosophic soft topology, de ning the most important topological properties such as interior, closure, and other related theoretical aspects, to give a strong mathematical foundation. An example is then provided to illustrate this frameworks analytical capabilities. Using a comprehensive visualizing package that includes spline-smoothed functional curves, 3D surface plots, and 2D Heatmaps, we apply our method to address a signal-template alignment problem. The Cotangent Similarity Measure (Cot SM) is used to systematically examine three input signals (S1 S3) and three templates (T1 T3). The ndings are striking and unambiguous: a worldwide maximum indicates a high degree of condence, and the correlation between S2 T1 (CotSM = 06653) is widely recognized as the strongest and most conclusive connection. Subsequent analysis shows that while signal S2 is the most important overall, signals S1 and S3 are helpful for particular targets.

A variety of grill-based topologies are developed and contrasted with earlier topologies. The results demonstrate that the present ones exceed their predecessors. This study distinguishes itself by highlighting the advantages of certain topologies and identifying both the minimum and maximum values. These structural topologies are later utilized to conduct a more thorough investigation of extended rough sets. Compared to earlier models, the suggested approximate models reduce vagueness and uncertainty, which makes them especially important when applied to rough sets (rhss). Furthermore, the suggested models differ from their predecessors in that they exhibit all of Pawlak’s properties, including the capability of contrasting various approximations (Aprs), and have the quality of monotonicity across all relations. In addition, the importance of new discoveries was highlighted by demonstrating their use for human health. Besides examining its limitations, the benefits of the chosen technique were assessed. The paper ends with a summary of the main ideas of the proposed methodology and recommendations for future research paths.