A Generalized Coherence Framework for Quantifying Input Contributions in Multi-input Systems with Correlated or Uncorrelated Inputs

In multi-input systems, it is often necessary to quantify the contribution of each input to an output. Such contribution analysis is frequently performed using coherence. However, when correlation is present between inputs, existing coherence measures do not accurately quantify the contribution of individual inputs, except in special cases. Here we propose an expanded coherence framework that enables contribution analysis in any multi-input system, regardless of input correlation. We bridged the gap by defining three new coherence measures: component, excluded, and isolated coherence. Component coherence is an intermediate measure that decomposes measured output power into components attributable to inputs directly vs to interference between inputs. Strategically summing component coherence terms yields contributions from individual inputs, defined as either excluded coherence (the portion of the output that would be removed if a given input were excluded) or isolated coherence (the portion of the output that would remain if a given input were isolated). To demonstrate, we simulated a three-input system and compared both existing and novel coherence measures to the known contributions at varying levels of input correlation. We also demonstrated a real-world application of these measures in a case study on Essential Tremor. Only excluded and isolated coherence accurately estimated the true contributions at all levels of input correlation. Even when existing coherence measures accurately estimated true contributions, novel measures did the same, but with less random error. These new coherence measures represent a generalization of the existing framework that enables accurate contribution analysis in multi-input systems regardless of input correlation.