Directional Mahalanobis Distance and Parameter Sensitivities 2016-01-0289
Mahalanobis Distance (MD) is gaining momentum in many fields where classification, statistical pattern recognition, and forecasting are primary focus. It is a multivariate method and considers correlation relationships among parameters for computing generalized distance measure to separate groups or populations. MD is a useful statistic in multivariate analysis to test that an observed random sample is from a multivariate normal distribution. This capability alone enables engineers to determine if an observed sample is an outlier (defect) that falls outside the constructed (good) multivariate normal distribution. In Mahalanobis-Taguchi System (MTS), MD is suitably scaled and used as a measure of severity in abnormality assessment.
It is obvious that computed MD depends on values of parameters observed on a random sample. All parameters may not equally impact MD. MD could be highly sensitive with respect to some parameters and less sensitive to some other parameters. Knowledge of parameter sensitivities help develop variation control plan in manufacturing so all produced parts belong to the good normal distribution and the scrap (waste) is eliminated.
In this paper, the author has developed a formulation to calculate parameter sensitivities in terms of Eigenvalues and Eigenvectors of the characteristic (A-1) matrix where A is the correlation matrix of parameters. The formulation is further extended to develop Directional Mahalanobis Distance (DMD) where MD is measured in a desired direction to assess goodness of a random sample. This feature of the DMD method enhances discrimination power and has a huge potential for continuous monitoring of patient health or online product quality. Usefulness of this formulation is illustrated with an example.