# Beyond Correlation: Mutual Information to Detect Damage in Nonlinear Systems
###### Abstract  
Analyzing and measuring the similarity between two signals is a common task in many vibration-based structural health monitoring applications. Coherence between input and response signals serves as a convenient indicator of damage, based on the premise that nonlinearity due to damage in a linear system manifests as a loss of coherence in specific frequency bands. Because input excitations in civil structures are difficult to measure, damage indicators based on the coherence between two response signals have been developed. These indicators have shown promise in detecting nonlinear behavior in structures that were initially linear. This paper proposes a new damage indicator based on Mutual Information, a nonlinear extension  
second assumption limits its applicability to a small class of damages, where the structure continues to behave in a linear fashion after damage, albeit with altered properties. The coherence function, originally introduced by [PERSON] in the late 1920s [5], measures the similarity between two signals in the frequency domain. As a complex function, coherence function represents both amplitude and phase relationships between two signals. Coherence between input and response signals is a convenient indicator of damage, based on the premise that nonlinearity due to damage in a linear system manifests as a loss of coherence in specific frequency bands. This approach has been successful in detecting damages in vibrating machinery in mechanical engineering [6; 7]. Some researchers attempted to detect and quantify nonlinear structural behavior by defining higher-order frequency response functions. Although this route has shown some promise for weakly nonlinear systems [8; 9], its applicability to complex, in-service civil engineering structures remains to be established. Damage detection in complex civil-engineering structures presents challenges that pose little to no concern in vibrating machinery. These include unknown boundary conditions, uncertainties in material and geometric properties, complex or unknown damage scenarios, environmental loads (e.g., traffic, temperature changes, and wind), the lack of data from both baseline and damaged states, inherent nonlinear behavior, and non-stationarity in the excitation and response processes. Because the concept of output-only transmissibility assumes a linear system under stationary excitation, its applicability to civil-engineering structures has not been widely demonstrated. Nonetheless, recently introduced extensions and variants show promise in addressing these challenges. For example, [PERSON] and [PERSON] [10] use a finite-element model-updating method based on a transmissibility sensitivity equation and apply it to numerical data from a two-bay, two-story frame where damage is simulated as reduced stiffness. Their method accurately identifies the number, location, and severity of damaged elements. Entropy-related methods offer significant potential for structural damage detection by measuring the complexity of vibration signals. [PERSON] and [PERSON] [11] developed an output-only structural health monitoring (SHM) method using information entropy and the inverse transmissibility function (ITF) to facilitate damage detection. Their study suggested that sample entropy, as measured from ITF signals, is highly sensitive to the level of damage as well as the type of structural damage. Their model tracks variations in complexity point by point. Results show that ITF-based entropy is significantly more sensitive to damage, especially nonlinear damage, than directly calculated entropy from raw acceleration. To address non-stationary vibrations, [PERSON] et al. [12] defined a transmissibility-coherence metric using the wavelet transform. Unlike the Fourier based power spectral density (PSD), which distributes power over frequency, a wavelet-based power spectrum distributes power across time and frequency jointly. The resulting time-frequency coherence function was used to detect damage in a building.