# Classification Performance of Thresholding Methods in the Mahalanobis-Taguchi System
## 2 The Concept of Mahalanobis Distance (MD)
### Mahalanobis-Taguchi System (MTS) Procedures  
[PERSON] extended the MD methodology with his robust engineering concepts to become an efficient and effective strategy for prediction and forecasting in multidimensional systems. In the MTS methodology, the formulation of MD is \"scaled\" where the existing MD formulation stated in Equation (1) is divided by a term \"\(k\)\" that denotes the number of variables or features of a recognition system. Therefore, the equation for calculating the scaled MD in the MTS methodology becomes:  
\[MD_{j}=\ D_{j}^{2}=\frac{1}{k}Z_{ij}^{T}C^{-1}Z_{ij} \tag{2}\]  
From this point onwards, the MD computation will be based on Equation (2). The MD offers a statistical measure to diagnose unknown sample conditions with known samples and provides information to make future predictions. The fundamental steps in the MTS methodology are explained in the next section. Figure 1: Example illustration of Mahalanobis Distance with two features. To construct a measurement scale, a homogeneous dataset from normal observations needs to be collected to build a reference group called the normal group [22]. It is used as a base or reference point in the scale. The collected normal datasets need to be standardized to obtain a dimensionless unit vector followed by the MD computation. Practically, the MD for unknown data is interpreted as the nearness to the mean of the normal group. As a countercheck, the average value of the MDs for the normal group must always be close to unity; therefore they are called the normal space or Mahalanobis Space (MS) [23]. The steps for the construction of the MS are outlined below:  
* Calculate the mean characteristic in the normal dataset as: \[\overline{x}_{i}=\frac{\sum_{j=1}^{n}X_{ij}}{n}\] (3)
* Then, calculate the standard deviation for each characteristic: \[s_{i}=\sqrt{\frac{\sum_{j=1}^{n}\left(X_{ij}-\overline{x}_{i}\right)^{2}}{n- 1}}\] (4)
* Next, standardise each characteristic to form the normalized data matrix (\(Z_{ij}\)) and its transpose (\(Z_{ij}^{T}\)): \[Z_{ij}=\frac{\left(X_{ij}-\overline{x}_{i}\right)}{s_{i}}\] (5)
* Then, verify that the mean of the normalized data is zero: \[\overline{z}_{i}=\frac{\sum_{j=1}^{n}Z_{ij}}{n}=0\] (6)
* Verify that the standard deviation of the normalized data is one: \[s_{z}=\sqrt{\frac{\sum_{j=1}^{n}\left(Z_{ij}-\overline{z}_{i}\right)^{2}}{n- 1}}=1\] (7)
* Form the correlation coefficient matrix (\(C\)) of the normalized data. The element matrix (\(c_{ij}\)) is calculated as follows: \[c_{ij}=\frac{\sum_{m=1}^{n}\left(Z_{m}Z_{jm}\right)}{n-1}\] (8)
* Compute inverse correlation coefficient matrix (\(C^{-1}\)) where: \[C=\frac{Cov(X,Y)}{V(X)V(Y)}\] (9) where: \[Cov\left(X,Y\right)=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right) \left(Y_{i}-\overline{Y}\right)\] (10)  
\(n\) is the number of samples, \(X\) and \(Y\) are two different features being correlated, \(X\)_bar_ and \(Y\)_bar_ are the averages among the data in each variable, and \(V(X)\) and \(V(Y)\) are the variances of \(X\) and \(Y\). Finally, calculate the MD using Equation (2). STAGE2: Assessment of the measurement scale  
To evaluate the measurement scale, observations outside the MS or abnormal datasets are used. The same mathematical calculation is repeated to calculate the same goal (MD value) using the abnormal sample data. However, the abnormal data are normalized based on the mean, standard deviation and correlation matrix of the normal group. The normal MDs and abnormal MDs are then compared. An acceptable measurement scale should demonstrate significant discrimination between the normal and abnormal MD values. STAGE 3: Identify significant features  
In the third stage, the system is optimized by means of selecting only the features that are known to be significant or \"useful\" for the system. This is where the Orthogonal Array (OA) and signal-to-noise ratio (SNR) are utilized. The features are assigned to an orthogonal array experimental run of two-level, in which \"used\" is signified as level 1 and \"not used\" as level 2.