Introduction
The Mahalanobis distance (MD) is a measure of the distance between two points in a multidimensional space. It was first introduced in 1936 by the Indian statistician and biologist Prasanta Chandra Mahalanobis (1893-1972). MD is a powerful tool for data analysis, especially for detecting outliers or anomalies in multivariate datasets. The purpose of this article is to provide a brief overview of the Mahalanobis distance and its applications.
Background
The Mahalanobis distance measure is based on the concept of the covariance matrix. The covariance matrix is a measure of the correlation between two variables and is defined by the following equation:
S= (X – μ)T (X – μ)
Where X is a vector of n observations, μ is the mean vector of X, and S is the covariance matrix of X. The Mahalanobis distance is then defined as the Euclidean distance between two points in the multidimensional space, multiplied by the inverse of the covariance matrix:
MD = (x1 – x2)T (S)-1 (x1 – x2)
Where x1 and x2 are two points in the multidimensional space. The Mahalanobis distance is a measure of the distance between two points, taking into account the correlation between the variables.
Applications
The Mahalanobis distance has a variety of applications in data analysis. It can be used to detect outliers or anomalies in multivariate datasets. For example, it can be used to identify suspect credit card transactions or to identify fraudulent insurance claims. It can also be used to detect clusters in multivariate datasets. In addition, it can be used to measure the similarity between two datasets or to detect the similarity between two variables.
Conclusion
The Mahalanobis distance is a powerful tool for data analysis, especially for detecting outliers or anomalies in multivariate datasets. It is based on the concept of the covariance matrix and is a measure of the distance between two points in a multidimensional space. It has a variety of applications in data analysis and can be used to detect outliers, clusters, and the similarity between two datasets or variables.
References
Mahalanobis, P. C. (1936). On the generalized distance in statistics. Proceedings of the National Institute of Sciences of India, 12, 49-55.
Weisberg, S. (2005). Applied linear regression (3rd ed.). Hoboken, NJ: Wiley.
Li, J. (2008). Outlier detection techniques. International Journal of Computational Intelligence and Applications, 7(1), 53-66.
Liu, F., Tang, H., & Ho, T. (2008). Mining distance-based outliers in multi-dimensional data. ACM SIGKDD Explorations Newsletter, 10(2), 56-65.