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Pairwise distance between pairs of observations

`D = pdist(X)`

`D = pdist(X,Distance)`

`D = pdist(X,Distance,DistParameter)`

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 = 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

`X`

— Input datanumeric matrix

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`

— Distance metriccharacter 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 ... `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'`

`DistParameter`

— Distance metric parameter valuespositive 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 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`

`D`

— Pairwise distancesnumeric row vector

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 *i*≤*j*.

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
`NaN`

s, 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.

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
*x _{1}*,

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*n*-by-*n*diagonal matrix whose*j*th 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*is the rank of_{sj}*x*taken over_{sj}*x*_{1},_{j}*x*_{2}, ..._{j}*x*, as computed by_{mj}`tiedrank`

.*r*and_{s}*r*are the coordinate-wise rank vectors of_{t}*x*and_{s}*x*, i.e.,_{t}*r*= (_{s}*r*_{s}_{1},*r*_{s}_{2}, ...*r*)._{sn}$${\overline{r}}_{s}=\frac{1}{n}{\displaystyle \sum _{j}{r}_{sj}}=\frac{\left(n+1\right)}{2}$$.

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

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

The distance input argument value (

`Distance`

) must be a compile-time 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`

to create loops that run in parallel on supported shared-memory 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`

.You can generate optimized CUDA

^{®}code using GPU Coder™. The supported distance input argument values (`Distance`

) for optimized CUDA code are`'euclidean'`

,`'squaredeuclidean'`

,`'seuclidean'`

,`'cityblock'`

,`'minkowski'`

,`'chebychev'`

,`'cosine'`

,`'correlation'`

,`'hamming'`

, and`'jaccard'`

. For more information on GPU coder, see Getting Started with GPU Coder (GPU Coder) and Supported Functions (GPU Coder).

For more information on code generation, see Introduction to Code Generation and General Code Generation Workflow.

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

The

`Distance`

argument must be specified as a character vector.

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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