# Multivariate Distance and Similarity

Multivariate Distance and Similarity Robert F. Murphy Cytometry Development Workshop 2000 General Multivariate Dataset We are given values of p variables for n independent observations Construct an n x p matrix M consisting of vectors X1 through Xn

each of length p Multivariate Sample Mean Define mean vector I of length p n n M(i, j) I( j) = i=1

or n matrix notation Xi i=1 I= n vector notation

Multivariate Variance Define variance vector of length p 2 n 2 ( j) = (M(i, j) I( j))

i=1 n1 matrix notation Multivariate Variance or 2 n 2 =

(Xi I) i=1 n1 vector notation Covariance Matrix Define a p x p matrix cov (called the covariance matrix) analogous to 2 n

(M(i, j) I(j ))(M(i,k) I(k)) cov( j,k) = i=1 n1 Covariance Matrix Note that the covariance of a variable with itself is simply the variance of that variable 2

cov( j, j) = ( j) Univariate Distance The simple distance between the values of a single variable j for two observations i and l is M(i, j) M(l, j) Univariate z-score Distance To measure distance in units of standard deviation between the values

of a single variable j for two observations i and l we define the z-score distance M(i, j) M(l, j) ( j) Bivariate Euclidean Distance The most commonly used measure of distance between two observations i and l on two variables j and k is the Euclidean distance 2

2 (M(i, j) M(l, j)) +(M(i,k) M(l,k)) Multivariate Euclidean Distance This can be extended to more than two variables p 2 (M(i, j) M(l, j))

j=1 Effects of variance and covariance on Euclidean distance B A The ellipse shows the 50% contour of a hypothetical population.

Points A and B have similar Euclidean distances from the mean, but point B is clearly more different from the population than point A. Mahalanobis Distance To account for differences in variance between the variables, and to account for correlations between variables, we use the Mahalanobis distance 2 -1

T D =(Xi Xl )cov (Xi Xl )

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