FAQ/euclid - CBU statistics Wiki

Upload page content

You can upload content for the page named below. If you change the page name, you can also upload content for another page. If the page name is empty, we derive the page name from the file name.

File to load page content from
Page name
Comment
Type the missing letters from: He's no ded, he's jus resing hs eys

Revision 10 as of 2011-09-02 08:51:30

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

Campbell,