Mahalanobis distances

Euclidean distance, (Euclid), city block distance, d(city), or Mahalanobis distance, (Mahalanobis). 

Constructing the library. First, one must choose the PRM to be used (e.g. correlation, wavelength distance, Mahalanobis distance, residual variance, etc.), the choice being dictated by the specific purpose of the library. Then, one must choose constraction parameters such as the math pretreatment (standard... 

In the first step of HCA, a distance matrix is calculated that contains the complete set of interspectral distances. The distance matrix is symmetric along its diagonal and has the dimension nxn, with n as the number of patterns. Spectral distance can be obtained in different ways depending on how the similarity of two patterns is calculated. Popular distance measures are Euclidean distances, including the city-block distance (Manhattan block distance), Mahalanobis distance, and so-called differentiation indices (D-values, see also Appendix B) . 

So far we have been considering leverage with respect to a point s Euclidean distance from an origin. But this is not the only measure of distance, nor is it necessarily the optimum measure of distance in this context. Consider the data set shown in Figure E4. Points C and D are located at approximately equal Euclidean distances from the centroid of the data set. However, while point C is clearly a typical member of the data set, point D may well be an outlier. It would be useful to have a measure of distance which relates more closely to the similarity/difference of a data point to/from a set of data points than simple Euclidean distance.The various Mahalanobis distances are one such family of such measures of distance. Thus, while the Euclidean distances of points C and D from the centroid of the data set are equal, the various Mahalanobis distances from the centroid of the data set are larger for point D than for point C. 

Mark, H. "Use of Mahalanobis Distances to Evaluate Sample Preparation Methods for Near-Infrared Reflectance Analysis", Anal. Chem. 1987 (59) 790-795. 

Fig. 30.4. Mahalanobis distance (a) object B is closer to centroid C of cluster G1 then object A (b) the distance between clusters G1 and G2 is smaller than between G3 and G4.

Euclidean distances (ordinary or standardized) are used very often for clustering purposes. This is not the case for Mahalanobis distance. An application of Mahalanobis distances can be found in Ref. [16]. 

Scaling is a very important operation in multivariate data analysis and we will treat the issues of scaling and normalisation in much more detail in Chapter 31. It should be noted that scaling has no impact (except when the log transform is used) on the correlation coefficient and that the Mahalanobis distance is also scale-invariant because the C matrix contains covariance (related to correlation) and variances (related to standard deviation). 

There is still another approach to explain LDA, namely by considering the Mahalanobis distance (see Chapter 30) to a class. All these approaches lead to the same result. The Mahalanobis distance is the distance to the centre of a class taking correlation into account and is the same for all points on the same probability ellipse. For equally probable classes, i.e. classes with the same number of training objects, a smaller Mahalanobis distance to class K than to class L, means that the probability that the object belongs to class K is larger than that it belongs to L. 

The Mahalanobis distance representation will help us to have a more general look at discriminant analysis. The multivariate normal distribution for w variables and class K can be described by... 

UNEQ can be applied when only a few variables must be considered. It is based on the Mahalanobis distance from the centroid of the class. When this distance exceeds a critical distance, the object is an outlier and therefore not part of the class. Since for each class one uses its own covariance matrix, it is somewhat related to QDA (Section 33.2.3). The situation described here is very similar to that discussed for multivariate quality control in Chapter 20. In eq. (20.10) the original variables are used. This equation can therefore also be used for UNEQ. For convenience it is repeated here. 

The Mahalanobis distance measures the degree to which data fit the calibration model. It is defined as... 

If a is a spectral vector (dimension / by 1) and A is the matrix of calibration spectra (of dimension n by /), then the Mahalanobis Distance is defined as ... 

If a weighted regression is used, the expression for the Mahalanobis Distance becomes equation 74-5b ... 

In MLR, if m is the vector (dimension by 1) of the selected absorbance values obtained from a spectral vector a, and M is the matrix of selected absorbance values for the calibration samples, then the Mahalanobis Distance is defined as equation 74-6a ... 

In PCR and PLS, the Mahalanobis distance for a sample with spectrum a is obtained by substituting the decomposition for PCR, or for PLS, into equation 74-5a. The statistic is expressed as equation 74-7a. 

The Mahalanobis Distance statistic provides a useful indication of the first type of extrapolation. For the calibration set, one sample will have a maximum Mahalanobis Distance, Z) ax. This is the most extreme sample in the calibration set, in that, it is the farthest from the center of the space defined by the spectral variables. If the Mahalanobis Distance for an unknown sample is greater than ZTax, then the estimate for the sample clearly represents an extrapolation of the model. Provided that outliers have been eliminated during the calibration, the distribution of Mahalanobis Distances should be representative of the calibration model, and ZEax can be used as an indication of extrapolation. 

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