Determining the approximate value of MD-Mahalanobis-Taguchi-Gram-Schmidt process by (K-MTGS) process

Introduction

Mahalanobis Distance MD used to find the "nearness" of unknown points from the mean point of a group. Mahalanobis Space (MS) is a database containing the means, standard deviations, and correlation structure of the variables in the reference group which has average unit distance. Taguchi Methods is used the system not only for diagnosis but also for forecasting/prediction of occurrence of particular events systems also used to develop and optimize a system of multivariable diagnosis, pattern recognition. Another alternative way of MTS to evaluating MD with the help of the Gram-Schmidt orthogonalization process (GSP) is MTGS. Method (MTS and MTGS) have the same result, but there are a different steps to evaluate the MD value. And MS has the zero point at its mean with an average MD equal to unity [2,4,6] . The new modification method denoted by K-MTGS is suggested to calculate (MD). In this paper evaluating the value of Mahalanobis Distance MD to compare the result of two methods, K-MTGS with (MTGS and MTS) in order to know the efficiency of this method, therefor evaluating MD by two different methods (MTGS and K-MTGS) on different random sample size with different number of variables were selected randomly and one can say that the MS, reference group is obtained using the standardized variables of the healthy or normal group data.

There are two main methods to determine the values of MD. 1. The Mahalanobis-Taguchi System (MTS). 2. Mahalanobois-Taguchi-Gram-Schmidt (MTGS).

The main objective of his application was to make statistical judgments to distinguish one group from another. The Mahalanobis Distance is a squared distance (D 2 ) calculated for the i th observation in as ample of size (n) with (m) variables as the following formula. [1,3,6,12] .

Where: j=1 to n The Mahalanobis-Taguchi System (MTS) [7,2] . In the MTS, the MD obtained from equation ( 1) is scaled by dividing in to the number of variables (m). Therefore the equation.

Taguchi introduced the Mahalanobis Distance into the process of defining a reference group and measuring individual subsets MTS was developed by Taguchi as a diagnostic and for casting using multivariate data. Eq. (2) becomes as:

Where Zi is a Vectors of (𝑍 1 , 𝑍 2 , 𝑍 3 , . . . . 𝑍 𝑚 )

And 𝐶 -1 is inverse matrix of correlation matrix Gram-Schmidt Orthgonalization Process [5,12,13] . MD can also be computed by using Gram-Schmidte Process. Which is discretion as follows: Given linearly Independent Vectors(𝑍 1 , 𝑍 2 , 𝑍 3 , . . . . 𝑍 𝑚 ) there exist mutually perpendicular Vectors (𝑈 1 , 𝑈 2 , 𝑈 3 , . . . . 𝑈 𝑚 ) with the same linear span.

Then the Gram -Schmidt vectors are constructed sequentially by setting:

When MD is calculated using Gram-Schmidt Process (GSP), standardized value of variables are used. Therefore in the above set of equation 𝑍

Where Z 1 , Z 2 ,….Zm are the set of standardized vectors And the GSP orthogonal vector as:

Where j= 1, …,n , the values of MD obtained from (2) and ( 7) are exactly the same .

To prove that the equation ( 2) and ( 7) are the same it is known that the Gram -Schmidt vectors(𝑈 1 , 𝑈 2 , 𝑈 3 , . . . . 𝑈 𝑚 ) are mutually perpendicular vectors, and (𝑆 1 , 𝑆 2 , 𝑆 3 , . . . . 𝑆 𝑚 ) are be the standard deviation of (𝑈 1 , 𝑈 2 , 𝑈 3 , . . . . 𝑈 𝑚 ) these vectors are obtained from standardized vectors (𝑍 1 , 𝑍 2 , 𝑍 3 , . . . . 𝑍 𝑚 ) and the mean of these vectors are equal to zero , To compare two methods, (MTGS and MTS) with K-MTGS in order to know the efficiency of this method, samples with different sizes will be used and different number of variables were selected randomly. Read phonetically It is clear that the new value of MD when calculated by this method is very close to the values of (MD) By (MTS & MTGS), as well as the proposed method is simpler than (MTS & MTGS) methods [2,3,6,12] .

Dictionary-VIEW DETAILED DICTIONARY

Step Where Z1, Z2,….Zm are the set of standardized vectors and standardized matrix as:

Step 2 Calculate the correlation coefficient matrix of Xij observations and can be written as matrix:

Then K Vectors are constructed sequentially by setting

The K1, K2 ,…Km are vectors:

Then the DM value by using the K-MTGS method:

Where, (m) number of the variables or factors, j denote the number of repeating samples. And 𝑠denote the standard deviation𝑘 𝑚𝑗 [7, 8, 9, 11, 12] .

Numerical illustration Application

Standardizing the normal data.

Step 2 Calculating the correlation coefficient of Standardizing data.

Step 3

Calculating the U1, U2,…Um and K1, K2,…Km vector as equation ( 7)

Calculate the value of MD by using K-MTGS methods using the equation (11).

The average of MD (MTGS and K-MTGS

) is close to one. 2 is equal to one 3. The difference of between two methods MD (MTGS and K-MTGS) is very small.

Comparison between (MTGS and K-MTGS)

In this paper, we have six numerical applications to evaluate the (MTGS and K-MTGS ) method then compare the result with it, each case of application contain a different data was simulated randomly with different sample size and variables. The average of (𝜇 1𝐽 , 𝜇 2𝐽 , . . . . 𝜇 𝐾𝐽 ) are equal to zero, and the mean of MD-values is Closed to one and the average of

B: Testing the example from Mahalanobis-Taguchi Strategy book

To more compare between (MTGS and K-MTGS) methods used the data from Mahalanobis-Taguchi Strategy book page 51: [11] .

Calculating the MD value of the data by two methods (MTGS and k-MTGS) as shows in table (7) and fig ( 6), it seen that the result of two methods are simile equal. Determining the MD-MTS methods is very difficult, the mathematical calculation is not easy, exactly finding the inverse of correlation matrix, so that the value of the inverted leads to incorrect results. 2. The MD-MTGS method have complex steps to determine the MD with the knowledge that the two methods have the same complications and they have the same results. We conclude that the two methods have the same results but applying the procedures are not easy. 3. But the K-MTGS method is an alternative method to use instead of MTS and MTGS methods. 4. From all Cases it clears that: i) The difference between MD value by (MTGS and K-MTGS) is very small. ii) The average of MD value by using K-MTGS is equal to one or close to one the same as MTS and MTGS.

iii) The average of (𝑢 𝑖𝑗 , 𝑢 2𝑗 , . . . 𝑢 𝑘𝑗 ) are equal to zero and With these results obtained by using the modified method (K-MTGS) needs further investigation to study in the future.

Conclusion

This study evaluated the efficiency of the Mahalanobis Distance (MD) using two methods: Mahalanobis-Taguchi-Gram-Schmidt (MTGS) and the proposed K-MTGS method. Both methods yielded similar results, demonstrating that K-MTGS is a viable alternative to MTGS and MTS for calculating MD. The analysis showed that the average MD values from both methods were close to one, indicating consistency in results. Additionally, the K-MTGS method simplifies the calculation process while maintaining accuracy. Future research should further explore the K-MTGS method's applications and its potential advantages in various statistical analyses, particularly in multivariate diagnostics and pattern recognition.

References

1. Alwan LC. Statistical process 1 st Ed,. 2000. ISBN: 0-256-11939-2.
2. Cudney EA, Hong J, Taguchi G. An evaluation of Mahalanobis-Taguchi system and neural network for multivariate pattern recognition. Journal of Industrial and Systems Engineering. 2007;1(2):139-150.
3. Chen Y, Phillips J. ECU software abnormal behavior detection based on Mahalanobis-Taguchi technique. World Congress Detroit Michigan, 2008 April.
4. Montgomery DC. Introduction to statistical quality control. 6 th Ed, 2000.
5. Elizabeth A, Paryani K, Ragsdell KM. Applying the Mahalanobis-Taguchi system to vehicle handling. Concurrent Engineering: Research and Applications. 2006;12:1-12.
6. Kawa MJ, Rashid. Optimize the Taguchi method, the signal-to-noise ratio, and the sensitivity. International Journal of Statistics and Applied Mathematics. 2023;8(6):64-70.
7. Roy R. A primer on the Taguchi method. New York, USA, 1990.
8. Quarton R, Bloomfield H. Basic design of experiments: The Taguchi approach. Presented by Nutek Inc, 2001. Available from: http://nutek-us.com.
9. Taguchi G, Chowdhury S, Wu Y. The Mahalanobis-Taguchi system. New York, USA: McGraw-Hill, 2001.
10. Taguchi G, Jugulum R. The Mahalanobis-Taguchi strategy. New York, USA: John Wiley & Sons, 2002.
11. Taguchi G, Chowdhury S, Wu Y. Taguchi's quality engineering handbook. USA: Wiley, 2005.
12. Taguchi G, Rajesh J. New trends in multivariate diagnosis. The Indian Journal of Statistics. 2000;62(2):233-248.