Resting‐state functional MRI (rsfMRI) analysis relies on complex mathematical operations whose properties and pitfalls are often poorly understood, leading to interpretational errors and suboptimal processing choices. This work presents novel mathematical insights for rsfMRI analysis through three key contributions: (1) a unified geometric framework showing that all common preprocessing and analysis operations can be understood as rotations in time‐series vector space, (2) identification of rotationally invariant properties that remain stable across different processing choices, and (3) mathematical equivalences between seemingly different connectivity measures. We demonstrate how the hemodynamic response function acts as a rotation operator in frequency space, derive closed‐form expressions for the impact of preprocessing on effective degrees of freedom, and show that correlation and coherence measures can be unified through frequency‐weighted integration. Common mathematical errors in the literature are identified and corrected with worked examples. This framework provides practical guidance for choosing connectivity measures, ordering preprocessing steps, and understanding the mathematical constraints imposed by operations such as global signal regression. By connecting abstract mathematical concepts to concrete rsfMRI applications, this work serves as both a theoretical foundation and a practical guide for researchers using functional connectivity methods. Practical applications include detecting network disruption in neuropsychiatric disorders (e.g., schizophrenia) with dramatically improved sensitivity, harmonizing multisite data without complex corrections, optimizing scan protocols for specific effect sizes, and providing robust quality control metrics that outperform traditional approaches.
Chisondi S. Warioba (2026) studied this question.