pdist2
Pairwise distance between two sets of observations
Syntax
Description
returns the distance using the metric specified by D
= pdist2(X,Y
,Distance
,DistParameter
)Distance
and DistParameter
. You can specify
DistParameter
only when Distance
is 'seuclidean'
, 'minkowski'
, or
'mahalanobis'
.
specifies an additional option using one of the namevalue pair arguments
D
= pdist2(___,Name,Value
)'Smallest'
or 'Largest'
in
addition to any of the arguments in the previous syntaxes.
For example,
D = pdist2(X,Y,Distance,'Smallest',K)
computes the distance using the metric specified byDistance
and returns theK
smallest pairwise distances to observations inX
for each observation inY
in ascending order.D = pdist2(X,Y,Distance,DistParameter,'Largest',K)
computes the distance using the metric specified byDistance
andDistParameter
and returns theK
largest pairwise distances in descending order.
[
also returns the matrix D
,I
] = pdist2(___,Name,Value
)I
. The matrix
I
contains the indices of the observations in
X
corresponding to the distances in
D
.
Examples
Compute Euclidean Distance
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 namevalue 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
.
Compute Minkowski Distance
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
Find the Two Smallest Pairwise Distances
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
.
Compute Pairwise Distance with Missing Elements Using a Custom Distance Function
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
Assign New Data to Existing Clusters and Generate C/C++ Code
kmeans
performs kmeans 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 entrypoint 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 entrypoint function.
Generating C/C++ code requires MATLAB® Coder™.
Perform kMeans 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)*0.5ones(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)*0.5ones(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 entrypoint 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 entrypoint 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 upperright section of this page and open this example in MATLAB®, then MATLAB® opens the example folder. This folder includes the entrypoint 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 entrypoint 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 VariableSize Arguments for Code Generation.
codegen findNearestCentroid args {C,Xtest}
Code generation successful.
codegen
generates the MEX function findNearestCentroid_mex
with a platformdependent 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
X,Y
— Input data
numeric matrix
Input data, specified as a numeric matrix. X
is an
mxbyn matrix and
Y
is an
mybyn matrix. Rows correspond to
individual observations, and columns correspond to individual
variables.
Data Types: single
 double
Distance
— Distance metric
character vector  string scalar  function handle
Distance metric, specified as a character vector, string scalar, or function handle, as described in the following table.
Value  Description 

'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,

'mahalanobis'  Mahalanobis distance using the sample covariance of

'cityblock'  City block distance. 
'minkowski'  Minkowski distance. The default exponent is 2. Use 
'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). 
@ 
Custom distance function handle. A distance function has the form function D2 = distfun(ZI,ZJ) % calculation of distance ...
If your data is not sparse, you can generally compute distance more quickly by using a builtin 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'
DistParameter
— Distance metric parameter values
positive scalar  numeric vector  numeric matrix
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 isstd(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 iscov(X,'omitrows')
.DistParameter
must be symmetric and positive definite.
Example:
'minkowski',3
Data Types: single
 double
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue 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'
.
Smallest
— Number of smallest distances to find
positive integer
Number of smallest distances to find, specified as the commaseparated
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.
Example: 'Smallest',3
Data Types: single
 double
Largest
— Number of largest distances to find
positive integer
Number of largest distances to find, specified as the commaseparated
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.
Example: 'Largest',3
Data Types: single
 double
Output Arguments
D
— Pairwise distances
numeric matrix
Pairwise distances, returned as a numeric matrix.
If you do not specify either 'Smallest'
or
'Largest'
, then D
is an
mxbymy 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 builtin distance functions.
If you specify either 'Smallest'
or
'Largest'
as K
, then
D
is a
K
bymy 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
mxbymy matrix.
I
— Sort index
positive integer 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
.
More About
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 mxbyn data matrix X, which is treated as mx (1byn) row vectors x_{1}, x_{2}, ..., x_{mx}, and an mybyn data matrix Y, which is treated as my (1byn) row vectors y_{1}, y_{2}, ...,y_{my}, the various distances between the vector x_{s} and y_{t} are defined as follows:
Euclidean distance
$${d}_{st}^{2}=({x}_{s}{y}_{t})({x}_{s}{y}_{t}{)}^{\prime}.$$
The Euclidean distance is a special case of the Minkowski distance, where p = 2.
Standardized Euclidean distance
$${d}_{st}^{2}=({x}_{s}{y}_{t}){V}^{1}({x}_{s}{y}_{t}{)}^{\prime},$$
where V is the nbyn diagonal matrix whose jth diagonal element is (S(j))^{2}, where S is a vector of scaling factors for each dimension.
Mahalanobis distance
$${d}_{st}^{2}=({x}_{s}{y}_{t}){C}^{1}({x}_{s}{y}_{t}{)}^{\prime},$$
where C is the covariance matrix.
City block distance
$${d}_{st}={\displaystyle \sum _{j=1}^{n}\left{x}_{sj}{y}_{tj}\right}.$$
The city block distance is a special case of the Minkowski distance, where p = 1.
Minkowski distance
$${d}_{st}=\sqrt[p]{{\displaystyle \sum _{j=1}^{n}{\left{x}_{sj}{y}_{tj}\right}^{p}}}.$$
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
$${d}_{st}={\mathrm{max}}_{j}\left\{\left{x}_{sj}{y}_{tj}\right\right\}.$$
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
$${\overline{x}}_{s}=\frac{1}{n}{\displaystyle \sum _{j}{x}_{sj}}$$
and
$${\overline{y}}_{t}=\frac{1}{n}{\displaystyle \sum _{j}{y}_{tj}}.$$
Hamming distance
$${d}_{st}=(\#({x}_{sj}\ne {y}_{tj})/n).$$
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
r_{sj} is the rank of x_{sj} taken over x_{1j}, x_{2j}, ...x_{mx,j}, as computed by
tiedrank
.r_{tj} is the rank of y_{tj} taken over y_{1j}, y_{2j}, ...y_{my,j}, as computed by
tiedrank
.r_{s} and r_{t} are the coordinatewise rank vectors of x_{s} and y_{t}, that is, r_{s} = (r_{s}_{1}, r_{s}_{2}, ... r_{sn}) and r_{t} = (r_{t1}, r_{t2}, ... r_{tn}).
$${\overline{r}}_{s}=\frac{1}{n}{\displaystyle \sum _{j}{r}_{sj}}=\frac{\left(n+1\right)}{2}$$.
$${\overline{r}}_{t}=\frac{1}{n}{\displaystyle \sum _{j}{r}_{tj}}=\frac{\left(n+1\right)}{2}$$.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
Usage notes and limitations:
The first input
X
must be a tall array. InputY
cannot be a tall array.
For more information, see Tall Arrays.
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
The distance input argument value (
Distance
) must be a compiletime constant. For example, to use the Minkowski distance, includecoder.Constant('Minkowski')
in theargs
value ofcodegen
.The distance input argument value (
Distance
) cannot be a custom distance function.Names in namevalue arguments must be compiletime constants. For example, to use the
'Smallest'
namevalue pair argument in the generated code, include{coder.Constant('Smallest'),0}
in theargs
value ofcodegen
(MATLAB Coder).The sorted order of tied distances in the generated code can be different from the order in MATLAB^{®} due to numerical precision.
The generated code of
pdist2
usesparfor
(MATLAB Coder) to create loops that run in parallel on supported sharedmemory multicore platforms in the generated code. If your compiler does not support the Open Multiprocessing (OpenMP) application interface or you disable OpenMP library, MATLAB Coder™ treats theparfor
loops asfor
loops. To find supported compilers, see Supported Compilers. To disable OpenMP library, set theEnableOpenMP
property of the configuration object tofalse
. For details, seecoder.CodeConfig
(MATLAB Coder).Starting in R2020a,
pdist2
returns integertype (int32
) indices, rather than doubleprecision indices, in generated standalone C/C++ code. Therefore, the function allows for strict singleprecision support when you use singleprecision inputs. For MEX code generation, the function still returns doubleprecision indices to match the MATLAB behavior.
For more information on code generation, see Introduction to Code Generation and General Code Generation Workflow.
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Usage notes and limitations:
The supported distance input argument values (
Distance
) for optimized CUDA code are'euclidean'
,'squaredeuclidean'
,'seuclidean'
,'cityblock'
,'minkowski'
,'chebychev'
,'cosine'
,'correlation'
,'hamming'
, and'jaccard'
.Distance
cannot be a custom distance function.Distance
must be a compiletime constant.Names in namevalue pair arguments must be compiletime constants.
The sorted order of tied distances in the generated code can be different from the order in MATLAB due to numerical precision.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
The distance input argument value (
Distance
) cannot be a custom distance function.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2010a
See Also
pdist
 createns
 knnsearch
 ExhaustiveSearcher
 KDTreeSearcher
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