The examination of real-world systems through the lens of complexity science has highlighted the need for a transition from qualitative analysis to quantitative evaluation. The well-established "Complexity Loss Theory" serves as a theoretical framework that links changes in structural complexity within dynamic systems with the manifestation of stress at macro levels. This thesis aims to expand upon the concept of complexity loss to compute and model stress levels in two prototypical real-world systems: biological systems in humans and economic systems in finance.

Recent studies in complexity science have underscored the vital role of nonlinear features in discerning between system states. Entropy-based algorithms have garnered significant attention in this context, however, despite substantial progress, practical applications for real-world signal analysis are still underexplored. Given the constraints of traditional entropy measures, this thesis introduces innovative entropy metrics: "ClassA Entropy," "Variational Embedding Multiscale Sample Entropy," and "Multivariate Multiscale Cosines Similarity Entropy". Analysis across several case studies demonstrates that the new measures effectively detect an increase in structural complexity, signifying the occurrence of stress-related stimuli in systems.

The assessment of economic stress through complexity science is a burgeoning field. Various complexity assessment techniques are employed to examine and estimate the non-representational stress in financial indices. This thesis introduces an innovative approach termed "Young's Modulus for Finance" to quantify the robustness of individual stocks/equities. This novel measure can identify the pre-crisis stage, enabling the detection of economic crises and providing a valuable indicator for performance evaluation.

The innovative algorithms presented in this thesis address various limitations of existing entropy measures. The presence of 'open systems' in real-world data sets poses a challenge to many parametric analytic models. Non-parametric and model-free entropy estimation plays unique and crucial roles in the analysis of real-world systems, forming the core focus of this thesis.