## 2 Method
### Mathematical, Index-Based Health Management System
#### 2.1.1 Mahalanobis-Taguchi System (MTS)  
The Mahalanobis-Taguchi System (MTS), which was developed by [PERSON], is a diagnosis and forecasting technique for multivariate data [22; 30]. MTS combines the Mahalanobis distance (MD) with [PERSON]'s method (TM), and is used to optimize multidimensional systems [19]. Recent developments in the MTS have enhanced the diagnostic techniques with an assessment scale for data classification. In general, MD provides two categories, namely, 'healthy' or 'normal' and 'abnormal' groups of elements. MD is employed to distinguish the level of abnormality between the groups of elements. The abnormal group is different from the normal group in a few aspects. Elements that are in the normal or desirable form are considered to be normal data, whereas those that are undesirable are considered to be abnormal data. Depending on the acuteness levels of their abnormalities, abnormal elements may occasionally be categorized into groups [11]. The Signal to Noise (S/N) ratio is utilized to assess the efficiency of the system in the MTS, and OAs are employed to maximize the potential of the features that directly optimize the S/N ratio. The data are analyzed using the Mahalanobis-Taguchi System (MTS) [18], i.e., a combination of MD and Taguchi strategies. MD is a generalized distance that is useful to identify similarities between healthy and unhealthy sample sets. A scalar value is used to represent a multivariate system [33]. Taguchi strategies are statistical approaches used to enhance the engineered quality to make the system more powerful [28; 34; 35]. There are four steps in MTS [18]:  
Step 0: Identification of assessment criteria and collection of patients' spatiotemporal data  
Step I: Mahalanobis space (MS) creation  
Data from the healthy subjects are collected to create a standard data set. Their MDs constitute a reference space, called MS. Their MDs are approximately equal to one. In this study, a feature dataset consisting of spatiotemporal data from healthy subjects was used to create the MS. The healthy data set is denoted as H, \(h_{ij}\) is the \(i\)th observation on the \(j\)th feature, where \(i=1,2,\ldots,m\), and \(j=1,2,\ \ldots,\ n\). Then, \(\overline{H}_{j}\) and \(S_{j}\) are the mean and standard deviations of the \(j\)th feature \(\left(H_{j}\right)\) respectively, where \(j=1,2,\ \ldots,\ n\). Each individual feature of each data vector (\(H_{j}\)) is normalized by the mean (\(\overline{H_{j}}\)) and the standard deviation (\(S_{j}\)). Hence, the normalized values are as follows:  
\[z_{ij}=\frac{h_{ij}-\overline{H_{j}}}{S_{j}}\ i=1,2,\ldots,m,j=1,2,\ldots,n \tag{1}\]  
where  
\[\overline{H_{j}}=\frac{1}{m}\sum_{i=1}^{m}h_{ij} \tag{2}\]  
and  
\[S_{j}=\sqrt{\frac{\sum_{i=1}^{m}\left(h_{ij}-\overline{H_{j}}\right)^{2}}{m-1}} \tag{3}\]  
The MDs of the healthy dataset are calculated with the following formula:  
\[MD_{i}=\frac{1}{n}z_{i}C^{-1}z_{i}^{T} \tag{4}\]  
where \(z=[z_{i1},z_{i2}\ldots,z_{in}]\), \(z_{i}^{T}\) is the transpose vector and \(z_{i}\)and \(C^{-1}\) is the inverse of the covariance coefficient matrix \(C\). We compute \(C\) as:  
\[C=\frac{1}{m-1}\sum_{i=1}^{m}z_{i}^{T}z_{i} \tag{5}\]  
Multicollinearity (strong correlations among features) is one issue that can be tackled with MD. Multicollinearities will have a problematic impact within a singular covariance coefficient matrix, resulting in an imprecise inverse of the covariance coefficient matrix, and subsequently producing an inaccurate MD [29]. MDA may be utilized to address this issue. MDA is MD corresponding to the adjoint matrix of the covariance coefficient matrix, as shown below:  
\[MDA_{I}=\frac{1}{n}z_{i}C_{adj}z_{i}^{T} \tag{6}\]where \(C_{adj}\) is the adjoint matrix of the covariance coefficient matrix \(C\). Since \(C^{-1}=C_{adj}/|C|\), the relationship between MD and MDA is given by:  
\[MD_{i}=\frac{1}{|C|}MDA_{i} \tag{7}\]  
Step II: Validation of MS  
In this step, the trials of ACLR subjects are selected [13].