# pdist2

Pairwise distance between two sets of observations

## Description

example

D = pdist2(X,Y,Distance) returns the distance between each pair of observations in X and Y using the metric specified by Distance.

example

D = pdist2(X,Y,Distance,DistParameter) returns the distance using the metric specified by Distance and DistParameter. You can specify DistParameter only when Distance is 'seuclidean', 'minkowski', or 'mahalanobis'.

example

D = pdist2(___,Name,Value), for any previous arguments, modifies the computation using name-value parameters. For example,

• D = pdist2(X,Y,Distance,'Smallest',K) computes the distance using the metric specified by Distance and returns the K smallest pairwise distances to observations in X for each observation in Y in ascending order.

• D = pdist2(X,Y,Distance,DistParameter,'Largest',K) computes the distance using the metric specified by Distance and DistParameter and returns the K largest pairwise distances in descending order.

example

[D,I] = pdist2(___) also returns the matrix I. The matrix I contains the indices of the observations in X corresponding to the distances in D.

## Examples

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Create two matrices with three observations and two variables.

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

Compute the Euclidean distance. The default value of the input argument Distance is 'euclidean'. When computing the Euclidean distance without using a name-value pair argument, you do not need to specify Distance.

D = pdist2(X,Y)
D = 3×3

0.5387    0.8018    0.1538
0.7100    0.5951    0.3422
0.8805    0.4242    1.2050

D(i,j) corresponds to the pairwise distance between observation i in X and observation j in Y.

Create two matrices with three observations and two variables.

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

Compute the Minkowski distance with the default exponent 2.

D1 = pdist2(X,Y,'minkowski')
D1 = 3×3

0.5387    0.8018    0.1538
0.7100    0.5951    0.3422
0.8805    0.4242    1.2050

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

D2 = pdist2(X,Y,'minkowski',1)
D2 = 3×3

0.5877    1.0236    0.2000
0.9598    0.8337    0.3899
1.0189    0.4800    1.7036

D3 = pdist2(X,Y,'cityblock')
D3 = 3×3

0.5877    1.0236    0.2000
0.9598    0.8337    0.3899
1.0189    0.4800    1.7036

Create two matrices with three observations and two variables.

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

Find the two smallest pairwise Euclidean distances to observations in X for each observation in Y.

[D,I] = pdist2(X,Y,'euclidean','Smallest',2)
D = 2×3

0.5387    0.4242    0.1538
0.7100    0.5951    0.3422

I = 2×3

1     3     1
2     2     2

For each observation in Y, pdist2 finds the two smallest distances by computing and comparing the distance values to all the observations in X. The function then sorts the distances in each column of D in ascending order. I contains the indices of the observations in X corresponding to the distances in D.

Create two large matrices of points, and then measure the time used by pdist2 with the default "euclidean" distance metric.

rng default % For reproducibility
N = 10000;
X = randn(N,1000);
Y = randn(N,1000);
D = pdist2(X,Y); % Warm up function for more reliable timing information
tic
D = pdist2(X,Y);
standard = toc
standard = 8.6606

Next, measure the time used by pdist2 with the "fasteuclidean" distance metric. Specify a cache size of 100.

D = pdist2(X,Y,"fasteuclidean",CacheSize=100); % Warm up function
tic
D2 = pdist2(X,Y,"fasteuclidean",CacheSize=100);
accelerated = toc
accelerated = 1.4911

Evaluate how many times faster the accelerated computation is compared to the standard.

standard/accelerated
ans = 5.8082

The accelerated version is more than twice as fast for this example.

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

Create two matrices with three observations and three variables.

rng('default') % For reproducibility
X = rand(3,3)
Y = [X(:,1:2) rand(3,1)]
X =

0.8147    0.9134    0.2785
0.9058    0.6324    0.5469
0.1270    0.0975    0.9575

Y =

0.8147    0.9134    0.9649
0.9058    0.6324    0.1576
0.1270    0.0975    0.9706

The first two columns of X and Y are identical. Assume that X(1,1) is missing.

X(1,1) = NaN
X =

NaN    0.9134    0.2785
0.9058    0.6324    0.5469
0.1270    0.0975    0.9575

Compute the Hamming distance.

D1 = pdist2(X,Y,'hamming')
D1 =

NaN       NaN       NaN
1.0000    0.3333    1.0000
1.0000    1.0000    0.3333

If observation i in X or observation j in Y contains NaN values, the function pdist2 returns NaN for the pairwise distance between i and j. Therefore, D1(1,1), D1(1,2), and D1(1,3) are NaN values.

Define a custom distance function nanhamdist that ignores coordinates with NaN values and computes the Hamming distance. When working with a large number of observations, you can compute the distance more quickly by looping over coordinates of the data.

function D2 = nanhamdist(XI,XJ)
%NANHAMDIST Hamming distance ignoring coordinates with NaNs
[m,p] = size(XJ);
nesum = zeros(m,1);
pstar = zeros(m,1);
for q = 1:p
notnan = ~(isnan(XI(q)) | isnan(XJ(:,q)));
nesum = nesum + ((XI(q) ~= XJ(:,q)) & notnan);
pstar = pstar + notnan;
end
D2 = nesum./pstar;

Compute the distance with nanhamdist by passing the function handle as an input argument of pdist2.

D2 = pdist2(X,Y,@nanhamdist)
D2 =

0.5000    1.0000    1.0000
1.0000    0.3333    1.0000
1.0000    1.0000    0.3333

kmeans performs k-means clustering to partition data into k clusters. When you have a new data set to cluster, you can create new clusters that include the existing data and the new data by using kmeans. The kmeans function supports C/C++ code generation, so you can generate code that accepts training data and returns clustering results, and then deploy the code to a device. In this workflow, you must pass training data, which can be of considerable size. To save memory on the device, you can separate training and prediction by using kmeans and pdist2, respectively.

Use kmeans to create clusters in MATLAB® and use pdist2 in the generated code to assign new data to existing clusters. For code generation, define an entry-point function that accepts the cluster centroid positions and the new data set, and returns the index of the nearest cluster. Then, generate code for the entry-point function.

Generating C/C++ code requires MATLAB® Coder™.

Perform k-Means Clustering

Generate a training data set using three distributions.

rng('default') % For reproducibility
X = [randn(100,2)*0.75+ones(100,2);
randn(100,2)*0.5-ones(100,2);
randn(100,2)*0.75];

Partition the training data into three clusters by using kmeans.

[idx,C] = kmeans(X,3);

Plot the clusters and the cluster centroids.

figure
gscatter(X(:,1),X(:,2),idx,'bgm')
hold on
plot(C(:,1),C(:,2),'kx')
legend('Cluster 1','Cluster 2','Cluster 3','Cluster Centroid')

Assign New Data to Existing Clusters

Generate a test data set.

Xtest = [randn(10,2)*0.75+ones(10,2);
randn(10,2)*0.5-ones(10,2);
randn(10,2)*0.75];

Classify the test data set using the existing clusters. Find the nearest centroid from each test data point by using pdist2.

[~,idx_test] = pdist2(C,Xtest,'euclidean','Smallest',1);

Plot the test data and label the test data using idx_test by using gscatter.

gscatter(Xtest(:,1),Xtest(:,2),idx_test,'bgm','ooo')
legend('Cluster 1','Cluster 2','Cluster 3','Cluster Centroid', ...
'Data classified to Cluster 1','Data classified to Cluster 2', ...
'Data classified to Cluster 3')

Generate Code

Generate C code that assigns new data to the existing clusters. Note that generating C/C++ code requires MATLAB® Coder™.

Define an entry-point function named findNearestCentroid that accepts centroid positions and new data, and then find the nearest cluster by using pdist2.

Add the %#codegen compiler directive (or pragma) to the entry-point function after the function signature to indicate that you intend to generate code for the MATLAB algorithm. Adding this directive instructs the MATLAB Code Analyzer to help you diagnose and fix violations that would cause errors during code generation.

type findNearestCentroid % Display contents of findNearestCentroid.m
function idx = findNearestCentroid(C,X) %#codegen
[~,idx] = pdist2(C,X,'euclidean','Smallest',1); % Find the nearest centroid

Note: If you click the button located in the upper-right section of this page and open this example in MATLAB®, then MATLAB® opens the example folder. This folder includes the entry-point function file.

Generate code by using codegen (MATLAB Coder). Because C and C++ are statically typed languages, you must determine the properties of all variables in the entry-point function at compile time. To specify the data type and array size of the inputs of findNearestCentroid, pass a MATLAB expression that represents the set of values with a certain data type and array size by using the -args option. For details, see Specify Variable-Size Arguments for Code Generation.

codegen findNearestCentroid -args {C,Xtest}
Code generation successful.

codegen generates the MEX function findNearestCentroid_mex with a platform-dependent extension.

Verify the generated code.

myIndx = findNearestCentroid(C,Xtest);
myIndex_mex = findNearestCentroid_mex(C,Xtest);
verifyMEX = isequal(idx_test,myIndx,myIndex_mex)
verifyMEX = logical
1

isequal returns logical 1 (true), which means all the inputs are equal. The comparison confirms that the pdist2 function, the findNearestCentroid function, and the MEX function return the same index.

You can also generate optimized CUDA® code using GPU Coder™.

cfg = coder.gpuConfig('mex');
codegen -config cfg findNearestCentroid -args {C,Xtest}

For more information on code generation, see General Code Generation Workflow. For more information on GPU coder, see Get Started with GPU Coder (GPU Coder) and Supported Functions (GPU Coder).

## Input Arguments

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Input data, specified as a numeric matrix. X is an mx-by-n matrix and Y is an my-by-n matrix. 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 = std(X,'omitnan'). Use DistParameter to specify a different value for S.

'fasteuclidean'Euclidean distance computed by using an alternative algorithm that saves time when the number of predictors is at least 10. In some cases, this faster algorithm can reduce accuracy. Algorithms starting with 'fast' do not support sparse data. For details, see Algorithms.
'fastsquaredeuclidean'Squared Euclidean distance computed by using an alternative algorithm that saves time when the number of predictors is at least 10. In some cases, this faster algorithm can reduce accuracy. Algorithms starting with 'fast' do not support sparse data. For details, see Algorithms.
'fastseuclidean'Standardized Euclidean distance computed by using an alternative algorithm that saves time when the number of predictors is at least 10. In some cases, this faster algorithm can reduce accuracy. Algorithms starting with 'fast' do not support sparse data. For details, see Algorithms.
'mahalanobis'

Mahalanobis distance, computed using the sample covariance of X, C = cov(X,'omitrows'). Use DistParameter to specify a different 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 distances more quickly by using a built-in distance metric 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 the default value of DistParameter.

Example: 'minkowski'

Data Types: char | string | function_handle

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 std(X,'omitnan').

• 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 cov(X,'omitrows'). DistParameter must be symmetric and positive definite.

Example: 'minkowski',3

Data Types: single | double

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: Either 'Smallest',K or 'Largest',K. You cannot use both 'Smallest' and 'Largest'.

Size of the Gram matrix in megabytes, specified as a positive scalar or 'maximal'. The pdist2 function can use CacheSize only when the Distance argument begins with fast.

If 'maximal', pdist2 attempts to allocate enough memory for an entire intermediate matrix whose size is MX-by-MY, where MX is the number of rows of the input data X, and MY is the number of rows of the input data Y. The cache size does not have to be large enough for an entire intermediate matrix, but must be at least large enough to hold an MX-by-1 vector. Otherwise, pdist2 uses the regular algorithm for computing Euclidean distance.

If the distance argument begins with fast and CacheSize is too large or is 'maximal', pdist2 can attempt to allocate a Gram matrix that exceeds the available memory. In this case, MATLAB® issues an error.

Example: CacheSize='maximal'

Data Types: double | char | string

Number of smallest distances to find, specified as the comma-separated pair consisting of 'Smallest' and a positive integer. If you specify 'Smallest', then pdist2 sorts the distances in each column of D in ascending order. You can use only one of the arguments Smallest and Largest.

Example: 'Smallest',3

Data Types: single | double

Number of largest distances to find, specified as the comma-separated pair consisting of 'Largest' and a positive integer. If you specify 'Largest', then pdist2 sorts the distances in each column of D in descending order. You can use only one of the arguments Smallest and Largest.

Example: 'Largest',3

Data Types: single | double

## Output Arguments

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Pairwise distances, returned as a numeric matrix.

If you do not specify either 'Smallest' or 'Largest', then D is an mx-by-my matrix, where mx and my are the number of observations in X and Y, respectively. D(i,j) is the distance between observation i in X and observation j in Y. If observation i in X or observation j in Y contains NaN, then D(i,j) is NaN for the built-in distance functions.

If you specify either 'Smallest' or 'Largest' as K, then D is a K-by-my matrix. D contains either the K smallest or K largest pairwise distances to observations in X for each observation in Y. For each observation in Y, pdist2 finds the K smallest or largest distances by computing and comparing the distance values to all the observations in X. If K is greater than mx, pdist2 returns an mx-by-my matrix.

Sort index, returned as a positive integer matrix. I is the same size as D. I contains the indices of the observations in X corresponding to the distances in D.

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### Distance Metrics

A distance metric is a function that defines a distance between two observations. pdist2 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 mx-by-n data matrix X, which is treated as mx (1-by-n) row vectors x1, x2, ..., xmx, and an my-by-n data matrix Y, which is treated as my (1-by-n) row vectors y1, y2, ...,ymy, the various distances between the vector xs and yt are defined as follows:

• Euclidean distance

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

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

• Standardized Euclidean distance

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

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

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

where C is the covariance matrix.

• City block distance

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

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

• Minkowski distance

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

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

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

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

• Cosine distance

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

• Correlation distance

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

where

${\overline{x}}_{s}=\frac{1}{n}\sum _{j}{x}_{sj}$

and

${\overline{y}}_{t}=\frac{1}{n}\sum _{j}{y}_{tj}.$

• Hamming distance

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

• Jaccard distance

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

• 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, ...xmx,j, as computed by tiedrank.

• rtj is the rank of ytj taken over y1j, y2j, ...ymy,j, as computed by tiedrank.

• rs and rt are the coordinate-wise rank vectors of xs and yt, that is, rs = (rs1, rs2, ... rsn) and rt = (rt1, rt2, ... rtn).

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

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

## Algorithms

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### Fast Euclidean Distance Algorithm

The values of the Distance argument that begin fast (such as 'fasteuclidean' and 'fastseuclidean') calculate Euclidean distances using an algorithm that uses extra memory to save computational time. This algorithm is named "Euclidean Distance Matrix Trick" in Albanie [1] and elsewhere. Internal testing shows that this algorithm saves time when the number of predictors is at least 10. Algorithms starting with 'fast' do not support sparse data.

To find the matrix D of distances between all the points xi and xj, where each xi has n variables, the algorithm computes distance using the final line in the following equations:

$\begin{array}{c}{D}_{i,j}^{2}=‖{x}_{i}-{x}_{j}{‖}^{2}\\ ={\left(}^{{x}_{i}}\left({x}_{i}-{x}_{j}\right)\\ =‖{x}_{i}{‖}^{2}-2{x}_{i}^{T}{x}_{j}+‖{x}_{j}{‖}^{2}.\end{array}$

The matrix ${x}_{i}^{T}{x}_{j}$ in the last line of the equations is called the Gram matrix. Computing the set of squared distances is faster, but slightly less numerically stable, when you compute and use the Gram matrix instead of computing the squared distances by squaring and summing. For a discussion, see Albanie [1].

To store the Gram matrix, the software uses a cache with the default size of 1e3 megabytes. You can set the cache size using the CacheSize name-value argument. If the value of CacheSize is too large or "maximal", pdist2 might try to allocate a Gram matrix that exceeds the available memory. In this case, MATLAB issues an error.

## References

[1] Albanie, Samuel. Euclidean Distance Matrix Trick. June, 2019. Available at https://www.robots.ox.ac.uk/%7Ealbanie/notes/Euclidean_distance_trick.pdf.

## Version History

Introduced in R2010a

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