What are the distance measures that can be compared to the Mahalanobis distance?

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Comparison of Distance Measures to Mahalanobis Distance

Introduction to Mahalanobis Distance

The Mahalanobis distance (MD) is a multivariate measure that accounts for the correlations between variables, making it particularly useful in fields such as chemometrics, pattern recognition, and process control. It is defined in terms of the covariance matrix, which allows it to represent the dependence between variables effectively.

Euclidean Distance

One of the most straightforward comparisons is with the Euclidean distance (ED). Unlike MD, which considers the covariance between variables, ED treats all variables as independent and measures the straight-line distance between points in space. This makes ED less effective in scenarios where variables are correlated.

Hesitant Fuzzy Sets Distance Measures

Recent advancements have introduced distance measures for hesitant fuzzy sets (HFSs), which include the hesitant Mahalanobis distance. This measure adapts the traditional MD to account for the variance, covariance, and correlation coefficients within HFSs, providing an adaptive weight to attributes and eliminating the influence of correlation between attributes.

Choquet Integral-Based Distance

The Choquet integral is another method used to measure distances in scenarios where variables are not independent. It shares similarities with MD in representing variable dependence. A Choquet integral-based distance has been introduced, and an operator known as the Choquet-Mahalanobis integral has been developed to generalize both the Choquet integral and MD.

Squared Mahalanobis Distance and Related Measures

In spectral analysis, several related measures are used, such as the squared Mahalanobis distance (D²), GH distance, T², and leverage (L) statistics. These measures are interrelated and can be translated between different software packages, which is crucial for setting thresholds for outlier detection.

Reduced Mahalanobis Distance

A variation known as the reduced Mahalanobis distance has been explored, where the number of principal components (PCs) retained is less than the full rank model. This approach has shown superior performance in discriminating via soft models by choosing the most discriminatory PCs.

Cross-Validated Mahalanobis Distance

The cross-validated Mahalanobis distance is particularly useful in representational similarity analysis (RSA) of neural activity patterns. Analytical expressions for the means and covariances of this measure allow for powerful inference on the measured statistics, enabling the statistical assessment of differences between distances.

Generalized Mahalanobis Distance

The Mahalanobis distance has also been generalized for distributions in the exponential family. This generalization provides a definition in terms of the data density function and a computable version, showing its performance in various data scenarios.

Conclusion

The Mahalanobis distance is a versatile and powerful measure that can be compared to several other distance measures, each with its unique applications and advantages. From the straightforward Euclidean distance to more complex measures like the Choquet integral-based distance and hesitant fuzzy sets distance measures, the Mahalanobis distance remains a cornerstone in multivariate analysis due to its ability to account for variable correlations effectively.