Pairwise distance between pairs of observations
returns the distance by using the method specified by D
= pdist(X
,Distance
,DistParameter
)Distance
and DistParameter
. You can specify
DistParameter
only when Distance
is
'seuclidean'
, 'minkowski'
, or
'mahalanobis'
.
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 = 1×3
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 = (XIXJ).^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 = sum(sqdx,2,'omitnan').*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 = 1×3
0.3974 1.1538 0.9448
X
— Input dataInput data, specified as a numeric matrix of size mbyn. Rows correspond to individual observations, and columns correspond to individual variables.
Data Types: single
 double
Distance
— Distance metricDistance 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 valuesDistance 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
D
— Pairwise distancesPairwise 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 lowerleft triangle of the mbym distance matrix in column order. The pairwise distance between observations i and j is in D((i1)*(mi/2)+ji) for i≤j.
You can convert D
into a symmetric matrix by using
the squareform
function.
Z = squareform(D)
returns an
mbym matrix where
Z(i,j)
corresponds to the pairwise distance between
observations i and j.
If observation i or j contains
NaN
s, then the corresponding value in
D
is NaN
for the builtin
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.
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 mbyn data matrix
X
, which is treated as m
(1byn) row vectors
x_{1},
x_{2}, ...,
x_{m}, the various distances between
the vector x_{s} and
x_{t} are defined as follows:
Euclidean distance
$${d}_{st}^{2}=({x}_{s}{x}_{t})({x}_{s}{x}_{t}{)}^{\prime}.$$
The Euclidean distance is a special case of the Minkowski distance, where p = 2.
Standardized Euclidean distance
$${d}_{st}^{2}=({x}_{s}{x}_{t}){V}^{1}({x}_{s}{x}_{t}{)}^{\prime},$$
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
$${d}_{st}^{2}=({x}_{s}{x}_{t}){C}^{1}({x}_{s}{x}_{t}{)}^{\prime},$$
where C is the covariance matrix.
City block distance
$${d}_{st}={\displaystyle \sum _{j=1}^{n}\left{x}_{sj}{x}_{tj}\right}.$$
The city block distance is a special case of the Minkowski distance, where p = 1.
Minkowski distance
$${d}_{st}=\sqrt[p]{{\displaystyle \sum _{j=1}^{n}{\left{x}_{sj}{x}_{tj}\right}^{p}}}.$$
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
$${d}_{st}={\mathrm{max}}_{j}\left\{\left{x}_{sj}{x}_{tj}\right\right\}.$$
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
$${\overline{x}}_{s}=\frac{1}{n}{\displaystyle \sum _{j}{x}_{sj}}$$ and $${\overline{x}}_{t}=\frac{1}{n}{\displaystyle \sum _{j}{x}_{tj}}$$.
Hamming distance
$${d}_{st}=(\#({x}_{sj}\ne {x}_{tj})/n).$$
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
r_{sj} is the rank
of x_{sj} taken over
x_{1}_{j},
x_{2}_{j},
...x_{mj}, as
computed by tiedrank
.
r_{s} and r_{t} are the coordinatewise rank vectors of x_{s} and x_{t}, i.e., r_{s} = (r_{s}_{1}, r_{s}_{2}, ... r_{sn}).
$${\overline{r}}_{s}=\frac{1}{n}{\displaystyle \sum _{j}{r}_{sj}}=\frac{\left(n+1\right)}{2}$$.
$${\overline{r}}_{t}=\frac{1}{n}{\displaystyle \sum _{j}{r}_{tj}}=\frac{\left(n+1\right)}{2}$$.
Usage notes and limitations:
The distance input argument value (Distance
) must
be a compiletime constant. For example, to use the Minkowski distance,
include coder.Constant('Minkowski')
in the
args
value of codegen
.
The distance input argument value (Distance
)
cannot be a custom distance function.
The generated code of
pdist
uses parfor
(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 the parfor
loops as for
loops. To find supported compilers, see Supported Compilers.
To disable OpenMP library, set the EnableOpenMP
property of the
configuration object to false
. For
details, see coder.CodeConfig
(MATLAB Coder).
For more information on code generation, see Introduction to Code Generation and General Code Generation Workflow.
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.
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).
cluster
 clusterdata
 cmdscale
 cophenet
 dendrogram
 inconsistent
 linkage
 pdist2
 silhouette
 squareform
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