Measures of distance in the space

Once the classification space has been defined, it is then necessary to define the distance in the space that will be used to assess the similarity of prediction samples and calibration samples. The most straightforward distance that can be used for this purpose is the Euclidean distance between two vectors (Dab), which is defined as  

To address this issue, another type of distance measure, called the Mahalanobis distance, has been proposed. This distance is defined as  

Although Euclidean and Mahalanobis distances are the ones most commonly used in analytical chemistry applications, there are other distance measures that might be more appropriate for specific applications. For example, there are standardized Euclidean distances, where each of the dimensions is inversely weighted by the standard deviation of that dimension in the calibration data (standard deviation-standardized), or the range of that dimension in the calibration data (range-standardized). 

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