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Keywords:
chi-square statistic
chi-square statistic
Summary:
This paper highlights advantageous properties of the Bhattacharyya metric over the chi-squared statistic for comparing frequency distributed data. The original interpretation of the Bhattacharyya metric as a geometric similarity measure is reviewed and it is pointed out that this derivation is independent of the use of the Bhattacharyya measure as an upper bound on the probability of misclassification in a two-class problem. The affinity between the Bhattacharyya and Matusita measures is described and we suggest use of the Bhattacharyya measure for comparing histogram data. We explain how the chi- squared statistic compensates for the implicit assumption of a Euclidean distance measure being the shortest path between two points in high dimensional space. By using the square-root transformation the Bhattacharyya metric requires no such standardization and by its multiplicative nature has no singularity problems (unlike those caused by the denominator of the chi- squared statistic) with zero count-data.
References:
[1] Aherne F. J., Thacker N. A., Rockett P. I.: Optimal pairwise geometric histograms. In: Proc. 8th British Machine Vision Conf., Colchester 1997, pp. 480–490
[2] Bhattacharyya A.: On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc. 35 (1943), 99–110 MR 0010358 | Zbl 0063.00364
[3] Christensen R.: Linear Models for Multivariate Time Series and Spatial Data. Springer–Verlag, New York 1991 MR 1081535 | Zbl 0717.62079
[4] Fukanaga K.: Introduction to Statistical Pattern Recognition. Second edition. Academic Press, New York 1990 MR 1075415
[5] Matusita K.: Decision rules based on distance for problems of fit, two samples and estimation. Ann. Math. Statist. 26 (1955), 631–641 DOI 10.1214/aoms/1177728422 | MR 0073899
[6] Thacker N. A., Abraham, I., Courtney P. G.: Supervised learning extensions to the CLAM network. Neural Network J. 10 (1997), 2, 315–326 DOI 10.1016/S0893-6080(96)00074-3
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