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Revision 11 as of 2011-09-02 08:51:50

location: FAQ / euclid

# What is Euclidean distance and how do I compute it ?

Euclidean distance measures the distance between two vectors of length t denoting t traits of various observations and is a specific example of Mahalanobis distance with an identity covariance matrix (ie uncorrelated traits).

ED = for vectors, observations with vectors $$x_text{i} = (x_text{1i}, ..., x_text{ti})text{T}$$ and $$x_text{j} = (x_text{1j}, ..., x_text{tj})text{T}$$ equals $$\sqrt{(x_text{i} - x_text{j})^text{T}(x_text{i} - x_text{j})}$$

This can be written in long hand as $$\sqrt{(x_text{1i}-x_text{1j})text{2} + .. + (x_text{ti}-x_text{tj})text{2}}$$

The Euclidean distance is the distance on a graph between two points. This is easily seen in two dimensions since by Pythagoras's theorem the linear distance (hypotenuse) between two points (x11, x21) and (x12, x22) equals the square root of the squared difference in x and y co-ordinates = square root of (x11-x12)(x11-x12) + (x21-x22)(x22-x21). See [attachment:euclide.bmp here.]

The Euclidean ditance is a special case of Mahalanobis distance which is used for measuring multivariate group distances or [:FAQ/mahal distance of an observation from its group means] ie with 2 or more predictors. In particular it is the square root of the Mahalanobis distance, D2, with the covariance matrix replaced by the identity matrix. D2 is defined in, for example, Campbell, Donner and Webster (1991).

Reference

Campbell MK, Donner, A and Webster, KM (1991) Are ordinal models useful for classification? Statistics in Medicine 10 383-394.