# pdist

Pairwise distance between pairs of observations

## Description

example

D = pdist(X) returns the Euclidean distance between pairs of observations in X.

example

D = pdist(X,Distance) returns the distance by using the method specified by Distance.

example

D = pdist(X,Distance,DistParameter) returns the distance by using the method specified by Distance and DistParameter. You can specify DistParameter only when Distance is 'seuclidean', 'minkowski', or 'mahalanobis'.

## Examples

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Compute the Euclidean distance between pairs of observations, and convert the distance vector to a matrix using squareform.

Create a matrix with three observations and two variables.

rng('default') % For reproducibility
X = rand(3,2);

Compute the Euclidean distance.

D = pdist(X)
D = 1×3

0.2954    1.0670    0.9448

The pairwise distances are arranged in the order (2,1), (3,1), (3,2). You can easily locate the distance between observations i and j by using squareform.

Z = squareform(D)
Z = 3×3

0    0.2954    1.0670
0.2954         0    0.9448
1.0670    0.9448         0

squareform returns a symmetric matrix where Z(i,j) corresponds to the pairwise distance between observations i and j. For example, you can find the distance between observations 2 and 3.

Z(2,3)
ans = 0.9448

Pass Z to the squareform function to reproduce the output of the pdist function.

y = squareform(Z)
y = 1×3

0.2954    1.0670    0.9448

The outputs y from squareform and D from pdist are the same.

Create a matrix with three observations and two variables.

rng('default') % For reproducibility
X = rand(3,2);

Compute the Minkowski distance with the default exponent 2.

D1 = pdist(X,'minkowski')
D1 = 1×3

0.2954    1.0670    0.9448

Compute the Minkowski distance with an exponent of 1, which is equal to the city block distance.

D2 = pdist(X,'minkowski',1)
D2 = 1×3

0.3721    1.5036    1.3136

D3 = pdist(X,'cityblock')
D3 = 1×3

0.3721    1.5036    1.3136

Define a custom distance function that ignores coordinates with NaN values, and compute pairwise distance by using the custom distance function.

Create a matrix with three observations and two variables.

rng('default') % For reproducibility
X = rand(3,2);

Assume that the first element of the first observation is missing.

X(1,1) = NaN;

Compute the Euclidean distance.

D1 = pdist(X)
D1 =

NaN       NaN    0.9448

If observation i or j contains NaN values, the function pdist returns NaN for the pairwise distance between i and j. Therefore, D1(1) and D1(2), the pairwise distances (2,1) and (3,1), are NaN values.

Define a custom distance function naneucdist that ignores coordinates with NaN values and returns the Euclidean distance.

function D2 = naneucdist(XI,XJ)
%NANEUCDIST Euclidean distance ignoring coordinates with NaNs
n = size(XI,2);
sqdx = (XI-XJ).^2;
nstar = sum(~isnan(sqdx),2); % Number of pairs that do not contain NaNs
nstar(nstar == 0) = NaN; % To return NaN if all pairs include NaNs
D2squared = nansum(sqdx,2).*n./nstar; % Correction for missing coordinates
D2 = sqrt(D2squared);

Compute the distance with naneucdist by passing the function handle as an input argument of pdist.

D2 = pdist(X,@naneucdist)
D2 =

0.3974    1.1538    0.9448

## Input Arguments

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Input data, specified as a numeric matrix of size m-by-n. Rows correspond to individual observations, and columns correspond to individual variables.

Data Types: single | double

Distance metric, specified as a character vector, string scalar, or function handle, as described in the following table.

ValueDescription
'euclidean'

Euclidean distance (default).

'squaredeuclidean'

Squared Euclidean distance. (This option is provided for efficiency only. It does not satisfy the triangle inequality.)

'seuclidean'

Standardized Euclidean distance. Each coordinate difference between observations is scaled by dividing by the corresponding element of the standard deviation, S = nanstd(X). Use DistParameter to specify another value for S.

'mahalanobis'

Mahalanobis distance using the sample covariance of X, C = nancov(X). Use DistParameter to specify another value for C, where the matrix C is symmetric and positive definite.

'cityblock'

City block distance.

'minkowski'

Minkowski distance. The default exponent is 2. Use DistParameter to specify a different exponent P, where P is a positive scalar value of the exponent.

'chebychev'

Chebychev distance (maximum coordinate difference).

'cosine'

One minus the cosine of the included angle between points (treated as vectors).

'correlation'

One minus the sample correlation between points (treated as sequences of values).

'hamming'

Hamming distance, which is the percentage of coordinates that differ.

'jaccard'

One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ.

'spearman'

One minus the sample Spearman's rank correlation between observations (treated as sequences of values).

@distfun

Custom distance function handle. A distance function has the form

function D2 = distfun(ZI,ZJ)
% calculation of distance
...
where

• ZI is a 1-by-n vector containing a single observation.

• ZJ is an m2-by-n matrix containing multiple observations. distfun must accept a matrix ZJ with an arbitrary number of observations.

• D2 is an m2-by-1 vector of distances, and D2(k) is the distance between observations ZI and ZJ(k,:).

If your data is not sparse, you can generally compute distance more quickly by using a built-in distance instead of a function handle.

For definitions, see Distance Metrics.

When you use 'seuclidean', 'minkowski', or 'mahalanobis', you can specify an additional input argument DistParameter to control these metrics. You can also use these metrics in the same way as the other metrics with a default value of DistParameter.

Example: 'minkowski'

Distance metric parameter values, specified as a positive scalar, numeric vector, or numeric matrix. This argument is valid only when you specify Distance as 'seuclidean', 'minkowski', or 'mahalanobis'.

• If Distance is 'seuclidean', DistParameter is a vector of scaling factors for each dimension, specified as a positive vector. The default value is nanstd(X).

• If Distance is 'minkowski', DistParameter is the exponent of Minkowski distance, specified as a positive scalar. The default value is 2.

• If Distance is 'mahalanobis', DistParameter is a covariance matrix, specified as a numeric matrix. The default value is nancov(X). DistParameter must be symmetric and positive definite.

Example: 'minkowski',3

Data Types: single | double

## Output Arguments

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Pairwise distances, returned as a numeric row vector of length m(m–1)/2, corresponding to pairs of observations, where m is the number of observations in X.

The distances are arranged in the order (2,1), (3,1), ..., (m,1), (3,2), ..., (m,2), ..., (m,m–1), i.e., the lower-left triangle of the m-by-m distance matrix in column order. The pairwise distance between observations i and j is in D((i-1)*(m-i/2)+j-i) for ij.

You can convert D into a symmetric matrix by using the squareform function. Z = squareform(D) returns an m-by-m matrix where Z(i,j) corresponds to the pairwise distance between observations i and j.

If observation i or j contains NaNs, then the corresponding value in D is NaN for the built-in distance functions.

D is commonly used as a dissimilarity matrix in clustering or multidimensional scaling. For details, see Hierarchical Clustering and the function reference pages for cmdscale, cophenet, linkage, mdscale, and optimalleaforder. These functions take D as an input argument.

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### Distance Metrics

A distance metric is a function that defines a distance between two observations. pdist supports various distance metrics: Euclidean distance, standardized Euclidean distance, Mahalanobis distance, city block distance, Minkowski distance, Chebychev distance, cosine distance, correlation distance, Hamming distance, Jaccard distance, and Spearman distance.

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

• Euclidean distance

${d}_{st}^{2}=\left({x}_{s}-{x}_{t}\right)\left({x}_{s}-{x}_{t}{\right)}^{\prime }.$

The Euclidean distance is a special case of the Minkowski distance, where p = 2.

• Standardized Euclidean distance

${d}_{st}^{2}=\left({x}_{s}-{x}_{t}\right){V}^{-1}\left({x}_{s}-{x}_{t}{\right)}^{\prime },$

where V is the n-by-n diagonal matrix whose jth diagonal element is (S(j))2, where S is a vector of scaling factors for each dimension.

• Mahalanobis distance

${d}_{st}^{2}=\left({x}_{s}-{x}_{t}\right){C}^{-1}\left({x}_{s}-{x}_{t}{\right)}^{\prime },$

where C is the covariance matrix.

• City block distance

${d}_{st}=\sum _{j=1}^{n}|{x}_{sj}-{x}_{tj}|.$

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

• Minkowski distance

${d}_{st}=\sqrt[p]{\sum _{j=1}^{n}{|{x}_{sj}-{x}_{tj}|}^{p}}.$

For the special case of p = 1, the Minkowski distance gives the city block distance. For the special case of p = 2, the Minkowski distance gives the Euclidean distance. For the special case of p = ∞, the Minkowski distance gives the Chebychev distance.

• Chebychev distance

${d}_{st}={\mathrm{max}}_{j}\left\{|{x}_{sj}-{x}_{tj}|\right\}.$

The Chebychev distance is a special case of the Minkowski distance, where p = ∞.

• Cosine distance

${d}_{st}=1-\frac{{x}_{s}{{x}^{\prime }}_{t}}{\sqrt{\left({x}_{s}{{x}^{\prime }}_{s}\right)\left({x}_{t}{{x}^{\prime }}_{t}\right)}}.$

• Correlation distance

${d}_{st}=1-\frac{\left({x}_{s}-{\overline{x}}_{s}\right){\left({x}_{t}-{\overline{x}}_{t}\right)}^{\prime }}{\sqrt{\left({x}_{s}-{\overline{x}}_{s}\right){\left({x}_{s}-{\overline{x}}_{s}\right)}^{\prime }}\sqrt{\left({x}_{t}-{\overline{x}}_{t}\right){\left({x}_{t}-{\overline{x}}_{t}\right)}^{\prime }}},$

where

${\overline{x}}_{s}=\frac{1}{n}\sum _{j}{x}_{sj}$ and ${\overline{x}}_{t}=\frac{1}{n}\sum _{j}{x}_{tj}$.

• Hamming distance

${d}_{st}=\left(#\left({x}_{sj}\ne {x}_{tj}\right)/n\right).$

• Jaccard distance

${d}_{st}=\frac{#\left[\left({x}_{sj}\ne {x}_{tj}\right)\cap \left(\left({x}_{sj}\ne 0\right)\cup \left({x}_{tj}\ne 0\right)\right)\right]}{#\left[\left({x}_{sj}\ne 0\right)\cup \left({x}_{tj}\ne 0\right)\right]}.$

• Spearman distance

${d}_{st}=1-\frac{\left({r}_{s}-{\overline{r}}_{s}\right){\left({r}_{t}-{\overline{r}}_{t}\right)}^{\prime }}{\sqrt{\left({r}_{s}-{\overline{r}}_{s}\right){\left({r}_{s}-{\overline{r}}_{s}\right)}^{\prime }}\sqrt{\left({r}_{t}-{\overline{r}}_{t}\right){\left({r}_{t}-{\overline{r}}_{t}\right)}^{\prime }}},$

where

• rsj is the rank of xsj taken over x1j, x2j, ...xmj, as computed by tiedrank.

• rs and rt are the coordinate-wise rank vectors of xs and xt, i.e., rs = (rs1, rs2, ... rsn).

• ${\overline{r}}_{s}=\frac{1}{n}\sum _{j}{r}_{sj}=\frac{\left(n+1\right)}{2}$.

• ${\overline{r}}_{t}=\frac{1}{n}\sum _{j}{r}_{tj}=\frac{\left(n+1\right)}{2}$.