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 control1. It is defined in terms of the covariance matrix, which allows it to represent the dependence between variables effectively4.
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 correlated1.
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 attributes2.
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 MD4.
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 detection6.
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 PCs7.
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 distances8.
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 scenarios9.
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.
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Most relevant research papers on this topic
The Mahalanobis distance
The Mahalanobis distance is a key tool in multivariate chemometrics, aiding in multivariate calibration, pattern recognition, and process control.
Highly CitedHesitant Mahalanobis distance with applications to estimating the optimal number of clusters
The hesitant Mahalanobis distance measure effectively measures deviation between hesitant fuzzy sets, allowing adaptive weighting and eliminating correlation between attributes in hesitant fuzzy environments.
The Mahalanobis distance and elliptic distributions
The Mahalanobis distance is an appropriate measure of distance between elliptic distributions with different locations but a common shape, and can also be used for multivariate normal distributions differing only in location.
Highly CitedOn a comparison between Mahalanobis distance and Choquet integral: The Choquet-Mahalanobis operator
The Choquet-Mahalanobis integral is a generalization of the Choquet integral and the Mahalanobis distance, useful for representing dependence between variables in probability-density functions.
Three Mahalanobis distances and their role in assessing unidimensionality.
A residual-based Mahalanobis distance measure is the most effective in identifying and downweighting aberrant item response patterns, improving scale reliability and unidimensionality.
A note on Mahalanobis and related distance measures in WinISI and The Unscrambler
The relationship between squared Mahalanobis distance, T2, and leverage statistics in WinISI and The Unscrambler simplifies the translation of these distance measures for spectral outlier detection in near infrared calibration.
Re‐evaluating the role of the Mahalanobis distance measure
The reduced Mahalanobis distance, with fewer retained principal components, outperforms the full rank model for discriminating via soft models.
On the distribution of cross-validated Mahalanobis distances
Cross-validated Mahalanobis distances can effectively assess representational similarity in neural activity patterns measured by fMRI, enabling efficient comparisons with computational models.
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