According to some paper I am reading, Jeffries and Matusita distance is commonly used. But I couldn't find much information on it except for the formula below
JMD(x,y)=$\sqrt[2]{\sum(\sqrt[2]{x_i}-\sqrt[2]{y_i})^2}$
It is similar to Euclidean distance except for the square root
E(x,y)=$\sqrt[2]{\sum(x_i-y_i)^2}$
JM distance is claimed to be more reliable than Euclidean distance in term of classification. Can anyone explain why this difference make JM distance better?
Best Answer
Some key differences, preceding a longer explanation below, are that:
The Jeffries-Matusita distance, which seems to be particularly popular in the Remote Sensing literature, is a transformation of the Bhattacharrya distance (a popular measure of the dissimilarity between two distributions, denoted here as $b_{p,q}$) from the range $[0, \inf)$ to the fixed range $[0, \sqrt{2}]$:
$$ JM_{p,q}=\sqrt{2(1-\exp(-b(p,q))} $$
A practical advantage of the J-M distance, according to this paper is that this measure "tends to suppress high separability values, whilst overemphasising low separability values".
The Bhattacharrya distance measures the dissimilarity of two distributions $p$ and $q$ in the following abstract continuous sense: $$ b(p,q)=-\ln\int{\sqrt{p(x)q(x)}}dx $$ If the distributions $p$ and $q$ are captured by histograms, represented by unit length vectors (where the $i$th element is the normalised count for $i$th of $N$ bins) this becomes: $$ b(p,q)=-\ln\sum_{i=1}^{N}\sqrt{p_i\cdot q_i} $$ And consequently the J-M distance for the two histograms is: $$ JM_{p,q}=\sqrt{2\left(1-\sum_{i=1}^{N}{\sqrt{p_i\cdot q_i}}\right)} $$ Which, noting that for normalised histograms $\sum_{i}{p_i}=1$, is the same as the formula you gave above: $$ JM_{p,q}=\sqrt{\sum_{i=1}^{N}{\left(\sqrt{p_i} - \sqrt{q_i}\right)^2}}=\sqrt{\sum_{i=1}^{N}{\left(p_i -2 \sqrt{p_i}\sqrt{q_i} + q_i \right)}}=\sqrt{2\left(1-\sum_{i=1}^{N}{\sqrt{p_i\cdot q_i}}\right)} $$