Statistics Toolbox™

pdist - Pairwise distance

Syntax

y = pdist(X)
y = pdist(X,metric)
y = pdist(X,distfun)
y = pdist(X,'minkowski',p)

Description

y = pdist(X) computes the Euclidean distance between pairs of objects in n-by-p data matrix X. Rows of X correspond to observations; columns correspond to variables. y is a row vector of length n(n–1)/2, corresponding to pairs of observations in X. The distances are arranged in the order (2,1), (3,1), ..., (n,1), (3,2), ..., (n,2), ..., (n,n–1)). y is commonly used as a dissimilarity matrix in clustering or multidimensional scaling.

To save space and computation time, y is formatted as a vector. However, you can convert this vector into a square matrix using the squareform function so that element i, j in the matrix, where i < j, corresponds to the distance between objects i and j in the original data set.

y = pdist(X,metric) computes the distance between objects in the data matrix, X, using the method specified by metric, which can be any of the following character strings.

y = pdist(X,distfun) accepts a function handle distfun to a metric of the form

d = distfun(u,V)

which takes as arguments a 1-by-p vector u, corresponding to a single row of X, and an m-by-p matrix V, corresponding to multiple rows of X. distfun must accept a matrix V with an arbitrary number of rows. distfun must return an m-by-1 vector of distances d, whose kth element is the distance between u and V(k,:).

y = pdist(X,'minkowski',p) computes the distance between objects in the data matrix, X, using the Minkowski metric. p is the exponent used in the Minkowski computation which, by default, is 2.

Metrics

Given an m-by-n data matrix X, which is treated as m (1-by-n) row vectors x₁, x₂, ..., x_m, the various distances between the vector x_r and x_s are defined as follows:

• Euclidean distance

  

• Standardized Euclidean distance

  

  where D is the diagonal matrix with diagonal elements given by , which denotes the variance of the variable X_j over the m objects.

• Mahalanobis distance

  

  where V is the sample covariance matrix.

• City block metric

  

• Minkowski metric

  

  Notice that for the special case of p = 1, the Minkowski metric gives the City Block metric, and for the special case of p = 2, the Minkowski metric gives the Euclidean distance.

• Cosine distance

  

• Correlation distance

  

where

 and 

• Hamming distance

  

• Jaccard distance

  

Examples

X = [1 2; 1 3; 2 2; 3 1]
X =
   1   2
   1   3
   2   2
   3   1

Y = pdist(X,'mahal')
Y =
  2.3452  2.0000  2.3452  1.2247  2.4495  1.2247

Y = pdist(X)
Y =
  1.0000  1.0000  2.2361  1.4142  2.8284  1.4142

squareform(Y)
ans =
       0  1.0000  1.0000  2.2361
  1.0000       0  1.4142  2.8284
  1.0000  1.4142       0  1.4142
  2.2361  2.8284  1.4142       0

See Also

cluster, clusterdata, cmdscale, cophenet, dendrogram, inconsistent, linkage, silhouette, squareform

pdf

pearsrnd