The Shape of Data: Topology Meets Analytics. A Practical Introduction to Topological Analytics and the Stability Index (TSI) in Business

Modern business and economic datasets often exhibit nonlinear, multi-scale structures that traditional linear tools under-represent. Topological Data Analysis (TDA) offers a geometric lens for uncovering robust patterns, such as connected components, loops and voids, across scales. This paper provides an intuitive, figure-driven introduction to persistent homology and a practical, reproducible TDA pipeline for applied analysts. Through comparative case studies in consumer behavior, equity markets (SAX/eSAX vs.\ TDA) and foreign exchange dynamics, we demonstrate how topological features can reveal segmentation patterns and structural relationships beyond classical statistical methods. We discuss methodological choices regarding distance metrics, complex construction and interpretation, and we introduce the \textit{Topological Stability Index} (TSI), a simple yet interpretable indicator of structural variability derived from persistence lifetimes. We conclude with practical guidelines for TDA implementation, visualization and communication in business and economic analytics.