## 2 Related Study
### Mahalanobis-Taguchi System (MTS)  
#### 2.2.1 Four Steps in MTS  
The data were analysed using the Mahalanobis-Taguchi System (MTS) [40], a combinations of MD and Taguchi strategies. MD is a generalised distance that is useful for identifying similarities between healthy and unhealthy sample sets and a scalar value is used to represent a multivariate system [39]. Taguchi strategies are statistical approaches used to enhance the engineered quality and to make the system more powerful [24; 41; 42]. There are four steps in a MTS [40]:  
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 the standard data set; their MDs constitute a reference space that is called MS. Their MDs are approximately equal to one. In this study, a feature data set 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 deviation 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 normalised by the mean (\(\overline{H_{j}}\)) and the standard deviation (\(S_{j}\)). Hence, the normalised 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 \(z=\left[z_{i1},z_{i2}\ldots,z_{in}\right]\), \(z_{i}^{T}\) 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 confronted by 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 an inaccurate MD [43]. MDA may be utilised to address this issue; MDA is MD corresponding to the adjoint matrix of the covariance coefficient matrix,  
\[MDA_{I}=\frac{1}{n}z_{i}C_{ad}z_{i}^{T} \tag{6}\]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  
Trials of ACLR subjects are selected [19]. The abnormal data set is symbolised as P; \(p_{ij}\) is the \(i\)_th_ observation of the \(j\)th feature, where \(i=1,2,\ldots,\;m\), and \(j=1,2,\;\ldots,\;n\). It is normalised by applying the mean and standard deviation of the healthy data set while the MDs are estimated using the feature information and the data's coefficient matrix. MDs corresponding to ACLRs will be outside of the MS if the MS is appropriately constructed. In other words, the MDs associated with abnormal conditions will have higher values [20; 21]. Step III: Identification of useful features  
By utilising the OAs and S/N ratios, useful features are selected. In MTS, OAs are used to recognise significant features by decreasing the number feature combinations established initially. The number of features determines the true number of columns in the OA. Two levels of factors are used: Level-1 means that the feature is included, while Level-2 means that the feature is not included. To measure the accuracy of the MS predictions, S/N ratios are used; these are calculated using only the ACLR conditions.