# Beyond Correlation: Mutual Information to Detect Damage in Nonlinear Systems
## 4 Proposed Approach  
In this section, we use the concept of mutual information to infer changes in the vibration behavior of structures. Such changes are often characterized as structural damage. Figure 1 shows a flowchart for detecting damage using the proposed MI-based metric. The proposed method can be considered an extension of the Transmissibility Modal Assurance Criterion (TMAC), where the similarity between two processes is measured in terms of the information shared between them. Mutual information is an elegant statistic that captures all types of dependence, as opposed to the correlation coefficient which only measures linear dependence. The main ingredient of the proposed approach is a set of vibration signals measured at different points on the structure. These signals can be acquired using either contact-based sensors, such as accelerometers and strain gauges, or non-contact methods, such as vision-based techniques that extract displacement data from video recordings. Treating the response quantities as realizations of random variables, we estimate the information shared between response quantities and use its deviation from a reference (undamaged) state as a damage indicator. Given a pair of response signals \((u_{i},u_{j})\) measured at locations \(i\) and \(j\), respectively, in the reference (undamaged) state of the structure, and a pair of response signals \((d_{i},d_{j})\) measured at the same locations in a new (potentially damaged) state, the mutual information score \(MI(i,j)\) between the two states is calculated as follows: First, we estimate the auto- and cross-power spectra of \(u_{i}\) and \(u_{j}\), and normalize the cross-spectra using Equation (3). Call the resulting vector \(u_{ij}\). Repeat the same using \(d_{i}\) and \(d_{j}\) from the new state. Call the resulting vector \(d_{ij}\). Next, we estimate the joint and marginal probability mass functions of \(u_{ij}\) and \(d_{ij}\) empirically, using a two-dimensional histogram. Finally, we determine the mutual information between \(u_{ij}\) and \(d_{ij}\) using Equation (7). Changes in the \(MI\) scores between a reference (undamaged) state and a new state can be used to infer changes in the dynamic properties of structures. The rationale is as follows: mutual information between two random variables is zero if and only if the two random  
Figure 3: Comparison of the coefficient of determination (\(R^{2}\)) and mutual information (MI) for four bivariate cases: (**a**) perfect linear, (**b**) linear with noise, (**c**) linear with outliers, and (**d**) nonlinear. variables are independent and thus do not share any information, and it is maximized when one of the random variables is a deterministic function of the other. When there is a change in the structure that affects the underlying relationship between response processes at different locations, that change will be reflected in the mutual information scores. The proposed approach is similar to the correlation-based methods where a decrease in correlation is interpreted as damage but goes beyond assessing the strength of linear associations. As the maximum value of mutual information depends on the entropies of the random variables, interpretation of mutual information requires either comparison to a reference state or normalization using an upper bound. For example, mutual information can be normalized to the range [0,1] by dividing the raw MI score by the summation of the marginal entropies of the random variables. The normalized MI score has the following interpretation: 0 indicates independence, and 1 indicates existence of a deterministic, not necessarily linear, relationship. Although normalization of MI makes it easier to interpret, we use the raw MI scores for the following reasons: First, normalized MI might not be directly comparable across different damage states as the total entropy is not necessarily constant in different states.