pdist - Pairwise distance between pairs of objects

Syntax

Description

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

To save space and computation time, D 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.

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

d2 = distfun(XI,XJ)

The output D is arranged in the order of ((2,1),(3,1),..., (m,1),(3,2),...(m,2),.....(m,m–1)), i.e. the lower left triangle of the full m-by-m distance matrix in column order. To get the distance between the ith and jth observations (i < j), either use the formula D((i–1)*(m–i/2)+j–i), or use the helper function Z = squareform(D), which returns an m-by-m square symmetric matrix, with the (i,j) entry equal to distance between observation i and observation j.

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_s and x_t are defined as follows:

• Euclidean distance

  

  Notice that the Euclidean distance is a special case of the Minkowski metric, where p = 2.

• Standardized Euclidean distance

  

  where V is the n-by-n diagonal matrix whose jth diagonal element is S(j)², where S is the vector of standard deviations.

• Mahalanobis distance

  

  where C is the covariance matrix.

• City block metric

  

  Notice that the city block distance is a special case of the Minkowski metric, where p=1.

• Minkowski metric

  

  Notice that for the special case of p = 1, the Minkowski metric gives the city block metric, for the special case of p = 2, the Minkowski metric gives the Euclidean distance, and for the special case of p = ∞, the Minkowski metric gives the Chebychev distance.

• Chebychev distance

  

  Notice that the Chebychev distance is a special case of the Minkowski metric, where p = ∞.

• Cosine distance

  

• Correlation distance

  

  where

  and

• Hamming distance

  

• Jaccard distance

  

• Spearman distance

  

  where

  • r_sj is the rank of x_sj taken over x_1j, x_2j, ...x_mj, as computed by tiedrank

  • r_s and r_t are the coordinate-wise rank vectors of x_s and x_t, i.e., r_s = (r_s1, r_s2, ... r_sn)

  • 

  • 

Examples

Generate random data and find the unweighted Euclidean distance and then find the weighted distance using two different methods:

% Compute the ordinary Euclidean distance.
X = randn(100, 5);
D = pdist(X,'euclidean');  % euclidean distance
 
% Compute the Euclidean distance with each coordinate
% difference scaled by the standard deviation.
Dstd = pdist(X,'seuclidean');
 
% Use a function handle to compute a distance that weights
% each coordinate contribution differently.
Wgts = [.1 .3 .3 .2 .1];     % coordinate weights
weuc = @(XI,XJ,W)(sqrt(bsxfun(@minus,XI,XJ).^2 * W'));
Dwgt = pdist(X, @(Xi,Xj) weuc(Xi,Xj,Wgts));

See Also

cluster | clusterdata | cmdscale | cophenet | dendrogram | inconsistent | linkage | pdist2 | silhouette | squareform