Illustration of the difference between the Euclidean distance and the Hellinger distance for one-dimensional Gaussian distributions. Two Gaussian distributions are shown as black lines for different choices of μ and σ. The grey area indicates the overlap between both distributions. |μ1−μ2| is the Euclidean distance between the centers of the Gaussians, DH is the Hellinger distance (equation 1). Both values are indicated in the title of panels A-D. A: For μ1 = μ2 = 0, σ1 = σ2 = 1, the Euclidean distance and the Hellinger distance are both zero. B: For μ1 = μ2 = 0, σ1 =1, σ2 = 5 the Euclidean distance is zero, whereas the Hellinger distance is larger than zero because the distributions do not overlap perfectly (the second Gaussian is wider than the first). C: For μ1 =0, μ2 = 5, σ1 = σ2 = 1, the Euclidean distance is five, whereas the Hellinger distance almost attains its maximum because the distributions only overlap little. D: For μ1 =0, μ2 = 5, σ1 =1, σ2 =5, the Euclidean distance is still five as in C because the means did not change. However, the Hellinger distance is larger than in C because the second Gaussian is wider, which leads to a larger overlap between the distributions.
Paramasivam et al. BMC Genomics 2012 13:510 doi:10.1186/1471-2164-13-510