# Classification Performance of Thresholding Methods in the Mahalanobis-Taguchi System
## 2 The Concept of Mahalanobis Distance (MD)
### Overview on Common Thresholding Methods in the Mahalanobis Taguchi System  
#### Quadratic Loss Function  
Quadratic Loss Function (QLF) was introduced by Dr. [PERSON] which aims to quantify the quality lost to society [30]. [PERSON] defines \"loss to society\" not only in  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & \\ \hline Run & 1 & 2 & 3 & 4 & 5 & 6 & 7 & & & & & \\ \hline
3 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & MD\({}_{1}\) & MD\({}_{2}\) & MD\({}_{3}\) & MD\({}_{4}\) & SNR\({}_{3}\) \\ \hline
4 & 1 & 2 & 2 & 2 & 2 & 1 & 1 & MD\({}_{1}\) & MD\({}_{2}\) & MD\({}_{3}\) & MD\({}_{4}\) & SNR\({}_{4}\) \\ \hline
5 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & MD\({}_{1}\) & MD\({}_{2}\) & MD\({}_{3}\) & MD\({}_{4}\) & SNR\({}_{3}\) \\ \hline
6 & 2 & 1 & 2 & 2 & 1 & 2 & 1 & MD\({}_{1}\) & MD\({}_{2}\) & MD\({}_{3}\) & MD\({}_{4}\) & SNR\({}_{3}\) \\ \hline
7 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & MD\({}_{1}\) & MD\({}_{2}\) & MD\({}_{3}\) & MD\({}_{4}\) & SNR\({}_{3}\) \\ \hline
8 & 2 & 2 & 1 & 2 & 1 & 1 & 2 & MD\({}_{1}\) & MD\({}_{2}\) & MD\({}_{3}\) & MD\({}_{4}\) & SNR\({}_{6}\) \\ \hline \multirow{2}{*}{Averaging} & SNR\({}_{1}\) & SNR\({}_{3}\) & SNR\({}_{1}\) & SNR\({}_{1}\) & SNR\({}_{1}\) & SNR\({}_{1}\) & SNR\({}_{1}\) & SNR\({}_{1}\) & & & \\ \cline{2-11}  & SNR\({}_{12}\) & SNR\({}_{2}\) & SNR\({}_{2}\) & SNR\({}_{2}\) & SNR\({}_{2}\) & SNR\({}_{2}\) & SNR\({}_{2}\) & & & & \\ \hline \multirow{2}{*}{Substraction} & Gain(+/-) & Gain(+/-) & Gain(+/-) & Gain(+/-) & Gain(+/-) & Gain(+/-) & Gain(+/-) & & & & \\ \hline \end{tabular}
\end{table}
Table 3: An example of useful feature selection using OA (\(L_{8}\) 2\({}^{7}\)) and signal-to-noise ratio (SNR).  
Figure 2: Fundamental four stages in Mahalanobis–Taguchi System (MTS) methodology.  
terms of operational problems such as rejections, scraps, or rework but also in terms of among others, pollution that is added to the environment, products that are worn out too quickly while in use, or other negative effects that could occur over the operational life of the products. In the context of Robust Engineering Design, QLF is used to determine the specification limits for a product. Ref. [30] provides a clear discussion of QLF. QLF promotes that any deviation on either side of the quality target incurs a monetary loss. This concept helps management understand the importance of robustness of a design, because the variation is expressed in monetary terms.  
The idea of QLF is applied in the MTS to determine the threshold values for the classification problem [8]. Take a medical diagnosis problem, for example, if the MD value of a patient's blood sample exceeds the threshold value, the patient is classified as unhealthy, and thus leading to a decision where the patient should be given a further complete medical examination. In Quadratic Loss Function, the optimal threshold (\(MD_{T}\)) is given by:  
\[MD_{T}=\sqrt{\frac{A}{A_{0}}}\,D \tag{13}\]  
where:  
* \(MD_{T}\) = the threshold (in MD term)
* \(A\) = the cost of the complete examination of patients who diagnose as unhealthy (including loss of time),
* \(A_{0}\) = the monetary loss caused by not taking the complete examination and having the disease show up before the next examination or the loss increase after having subjective symptoms followed by taking a complete examination,
* \(D\) = the mid-value of the MD of a patient group having the subjective symptoms  
The key element in the QLF concept is to balance between the cost of treating a patient and the cost of not treating a patient (as in the medical application). However, in real practical applications, even outside the medical diagnosis problems, obtaining the associated monetary information was seen to be impractical and difficult to obtain [10; 13; 31], hence several alternative approaches to determine the optimal threshold value have been reported in the literature of which several of them are discussed as follows.