Mahalonobis distance

NIR spectroscopy was utilized by Aldridge and coworkers86 to determine, in a rapid manner, the polymorphic quality of a solid drug substance. Two computational methods, Mahalonobis distance and soft independent modeling of class analogy (SIMCA) residual variance, were used to distinguish between acceptable and unacceptable samples. The authors not only determined that the Mahalonobis distance classification yielded the best results, they addressed one of the key implementation issues regarding NIR as a PAT tool. 

Gemperline, P. J. and Boyer, N. T. (1995). Classification of near-infrared spectra using wavelength distances comparison to the Mahalonobis distance and residual variance methods. Ana/. Chem., 67, 160-6. [130]... 

Originally limited to ellipsoids, the use of Mahalonobis distances allows the use of more variables as the confidence ellipsoid can be transformed to a confidence or tolerance hypersphere. These ideas were examined using the microecosystem test method developed by Kersting for the examination of multispecies systems. These three-compartment microecosystems are comprised of an autotrophic, herbivore, and decomposer subsystems that are connected by tubing and pumps. Although relatively simple and small, these systems are operable over a number of years. 

Si is the covariation matrix, and rq is the sample size of class i. Thus, based on these equations we can calculate the coefficients of the LDF and the threshold constant c using the values of characteristics of site and non-site sequences from the training sets and then test the accuracy of the LDF on the test set data. The significance of a given characteristic or set of characteristics can be estimated by the generalized distance between two classes (called the Mahalonobis distance or D2) ... 

All ORF regions that were flanked by GT (on the left) and finished with a stop codon were considered as potential last exons. The structure of such exons is presented in Figure 3.9. The characteristics of their discrimimant function and their Mahalonobis distances are presented in Table 3.12. The accuracy of the discriminant function was tested on the recognition of the last 322 exon and 247 644 pseudo-exon sequences. The gene sequences were scanned and the 3 -exon with the maximal weight was selected for each of them. The function can identify exactly 60% of the annotated last exons. 

It is recommended to associate to d(M) a probability P which will delineate the contour of the group as a function of x2(a ri). In fact, the Mahalonobis distance gives a default classification since it does not take into account the random errors of measurement and the non Gaussian contribution in the distribution of the data. 

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