knnsearch

Find k-nearest neighbors using input data

Syntax

Idx = knnsearch(X,Y)

Idx = knnsearch(X,Y,Name,Value)

[Idx,D] = knnsearch(___)

Description

Idx = knnsearch(X,Y) finds the nearest neighbor in X for each query point in Y and returns the indices of the nearest neighbors in Idx, a column vector. Idx has the same number of rows as Y.

Idx = knnsearch(X,Y,Name,Value) returns Idx with additional options specified using one or more name-value pair arguments. For example, you can specify the number of nearest neighbors to search for and the distance metric used in the search.

[Idx,D] = knnsearch(___) additionally returns the matrix D, using any of the input arguments in the previous syntaxes. D contains the distances between each observation in Y and the corresponding closest observations in X.

Examples

Find Nearest Neighbors

Find the patients in the hospital data set that most closely resemble the patients in Y, according to age and weight.

Load the hospital data set.

load hospital;
X = [hospital.Age hospital.Weight];
Y = [20 162; 30 169; 40 168; 50 170; 60 171];   % New patients

Perform a knnsearch between X and Y to find indices of nearest neighbors.

Idx = knnsearch(X,Y);

Find the patients in X closest in age and weight to those in Y.

X(Idx,:)

ans = 5×2

    25   171
    25   171
    39   164
    49   170
    50   172

Find k-Nearest Neighbors Using Different Distance Metrics

Find the 10 nearest neighbors in X to each point in Y, first using the Minkowski distance metric and then using the Chebychev distance metric.

Load Fisher's iris data set.

load fisheriris
X = meas(:,3:4);    % Measurements of original flowers
Y = [5 1.45;6 2;2.75 .75];  % New flower data

Perform a knnsearch between X and the query points Y using Minkowski and Chebychev distance metrics.

[mIdx,mD] = knnsearch(X,Y,'K',10,'Distance','minkowski','P',5);
[cIdx,cD] = knnsearch(X,Y,'K',10,'Distance','chebychev');

Visualize the results of the two nearest neighbor searches. Plot the training data. Plot the query points with the marker X. Use circles to denote the Minkowski nearest neighbors. Use pentagrams to denote the Chebychev nearest neighbors.

gscatter(X(:,1),X(:,2),species)
line(Y(:,1),Y(:,2),'Marker','x','Color','k',...
   'Markersize',10,'Linewidth',2,'Linestyle','none')
line(X(mIdx,1),X(mIdx,2),'Color',[.5 .5 .5],'Marker','o',...
   'Linestyle','none','Markersize',10)
line(X(cIdx,1),X(cIdx,2),'Color',[.5 .5 .5],'Marker','p',...
   'Linestyle','none','Markersize',10)
legend('setosa','versicolor','virginica','query point',...
'minkowski','chebychev','Location','best')

Accelerate Distance Computation Using fasteuclidean Distance

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

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

standard = 25.2888

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

Idx2 = knnsearch(X,Y,Distance="fasteuclidean",CacheSize=100); % Warm up function
tic
Idx2 = knnsearch(X,Y,Distance="fasteuclidean",CacheSize=100);
accelerated = toc

accelerated = 2.2442

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

standard/accelerated

ans = 11.2685

The accelerated version is more than three times faster for this example.

Input Arguments

X — Input data

numeric matrix

Input data, specified as a numeric matrix. Rows of X correspond to observations, and columns correspond to variables.

Data Types: single | double

Y — Query points

numeric matrix

Query points, specified as a numeric matrix. Rows of Y correspond to observations, and columns correspond to variables. Y must have the same number of columns as X.

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: knnsearch(X,Y,'K',10,'IncludeTies',true,'Distance','cityblock') searches for 10 nearest neighbors, including ties and using the city block distance.

K — Number of nearest neighbors

1 (default) | positive integer

Number of nearest neighbors to find in X for each point in Y, specified as the comma-separated pair consisting of 'K' and a positive integer.

Example: 'K',10

Data Types: single | double

IncludeTies — Flag to include all nearest neighbors

false (0) (default) | true (1)

Flag to include all nearest neighbors that have the same distance from query points, specified as the comma-separated pair consisting of 'IncludeTies' and false (0) or true (1).

If 'IncludeTies' is false, then knnsearch chooses the observation with the smallest index among the observations that have the same distance from a query point.

If 'IncludeTies' is true, then:

• knnsearch includes all nearest neighbors whose distances are equal to the kth smallest distance in the output arguments. To specify k, use the 'K' name-value pair argument.

• Idx and D are m-by-1 cell arrays such that each cell contains a vector of at least k indices and distances, respectively. Each vector in D contains distances arranged in ascending order. Each row in Idx contains the indices of the nearest neighbors corresponding to the distances in D.

Example: 'IncludeTies',true

NSMethod — Nearest neighbor search method

'kdtree' | 'exhaustive'

Nearest neighbor search method, specified as the comma-separated pair consisting of 'NSMethod' and one of these values.

• 'kdtree' — Creates and uses a Kd-tree to find nearest neighbors. 'kdtree' is the default value when the number of columns in X is less than or equal to 10, X is not sparse, and the distance metric is 'euclidean', 'cityblock', 'chebychev', or 'minkowski'. Otherwise, the default value is 'exhaustive'.

  The value 'kdtree' is valid only when the distance metric is one of the four metrics noted above.

• 'exhaustive' — Uses the exhaustive search algorithm by computing the distance values from all the points in X to each point in Y.

Example: 'NSMethod','exhaustive'

Distance — Distance metric

'euclidean' (default) | 'seuclidean' | 'fasteuclidean' | 'fastseuclidean' | 'cityblock' | 'chebychev' | 'minkowski' | 'mahalanobis' | 'cosine' | 'correlation' | 'spearman' | 'hamming' | 'jaccard' | function handle | ...

Distance metric knnsearch uses, specified as one of the values in this table or a function handle.

Value	Description
'euclidean'	Euclidean distance
'seuclidean'	Standardized Euclidean distance. Each coordinate difference between the rows in X and the query matrix Y is scaled by dividing by the corresponding element of the standard deviation computed from X. To specify a different scaling, use the 'Scale' name-value argument.
'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. This distance metric is available only when NSMethod is 'exhaustive'. 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. This distance metric is available only when NSMethod is 'exhaustive'. Algorithms starting with 'fast' do not support sparse data. For details, see Algorithms.
'cityblock'	City block distance
'chebychev'	Chebychev distance (maximum coordinate difference)
'minkowski'	Minkowski distance. The default exponent is 2. To specify a different exponent, use the 'P' name-value argument.
'mahalanobis'	Mahalanobis distance, computed using a positive definite covariance matrix. To change the value of the covariance matrix, use the 'Cov' name-value argument.
'cosine'	One minus the cosine of the included angle between observations (treated as vectors)
'correlation'	One minus the sample linear correlation between observations (treated as sequences of values)
'spearman'	One minus the sample Spearman's rank correlation between observations (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

You can also specify a function handle for a custom distance metric by using @ (for example, @distfun). A custom distance function must:

• Have the form function D2 = distfun(ZI,ZJ).

• Take as arguments:

  • A 1-by-n vector ZI containing a single row from X or from the query points Y.

  • An m₂-by-n matrix ZJ containing multiple rows of X or Y.

• Return an m₂-by-1 vector of distances D2, whose jth element is the distance between the observations ZI and ZJ(j,:).

For more information, see Distance Metrics.

Example: 'Distance','chebychev'

Data Types: char | string | function_handle

CacheSize — Size of Gram matrix in megabytes

1e3 (default) | positive scalar | "maximal"

Size of the Gram matrix in megabytes, specified as a positive scalar or "maximal". The knnsearch function can use CacheSize only when the Distance name-value argument begins with fast and the NSMethod name-value argument is set to 'exhaustive'.

If you set CacheSize to "maximal", knnsearch tries 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, knnsearch uses the standard algorithm for computing Euclidean distance.

If the value of the Distance argument begins with fast, the value of NSMethod is 'exhaustive', and the value of CacheSize is too large or "maximal", knnsearch might try 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

P — Exponent for Minkowski distance metric

2 (default) | positive scalar

Exponent for the Minkowski distance metric, specified as the comma-separated pair consisting of 'P' and a positive scalar.

This argument is valid only if 'Distance' is 'minkowski'.

Example: 'P',3

Data Types: single | double

Cov — Covariance matrix for Mahalanobis distance metric

cov(X,'omitrows') (default) | positive definite matrix

Covariance matrix for the Mahalanobis distance metric, specified as the comma-separated pair consisting of 'Cov' and a positive definite matrix.

This argument is valid only if 'Distance' is 'mahalanobis'.

Example: 'Cov',eye(4)

Data Types: single | double

Scale — Scale parameter value for standardized Euclidean distance metric

std(X,'omitnan') (default) | nonnegative numeric vector

Scale parameter value for the standardized Euclidean distance metric, specified as the comma-separated pair consisting of 'Scale' and a nonnegative numeric vector. 'Scale' has length equal to the number of columns in X. When knnsearch computes the standardized Euclidean distance, each coordinate of X is scaled by the corresponding element of 'Scale', as is each query point. This argument is valid only when 'Distance' is 'seuclidean'.

Example: 'Scale',quantile(X,0.75) - quantile(X,0.25)

Data Types: single | double

BucketSize — Maximum number of data points in leaf node of Kd-tree

50 (default) | positive integer

Maximum number of data points in the leaf node of the Kd-tree, specified as the comma-separated pair consisting of 'BucketSize' and a positive integer. This argument is valid only when NSMethod is 'kdtree'.

Example: 'BucketSize',20

Data Types: single | double

SortIndices — Flag to sort returned indices according to distance

true (1) (default) | false (0)

Flag to sort returned indices according to distance, specified as the comma-separated pair consisting of 'SortIndices' and either true (1) or false (0).

For faster performance, you can set SortIndices to false when the following are true:

• Y contains many observations that have many nearest neighbors in X.

• NSMethod is 'kdtree'.

• IncludeTies is false.

In this case, knnsearch returns the indices of the nearest neighbors in no particular order. When SortIndices is true, the function arranges the nearest neighbor indices in ascending order by distance.

SortIndices is true by default. When NSMethod is 'exhaustive' or IncludeTies is true, the function always sorts the indices.

Example: 'SortIndices',false

Data Types: logical

Output Arguments

Idx — Input data indices of nearest neighbors

numeric matrix | cell array of numeric vectors

Input data indices of the nearest neighbors, returned as a numeric matrix or cell array of numeric vectors.

• If you do not specify IncludeTies (false by default), then Idx is an m-by-k numeric matrix, where m is the number of rows in Y and k is the number of searched nearest neighbors. Idx(j,i) indicates that X(Idx(j,i),:) is one of the k closest observations in X to the query point Y(j,:).

• If you specify 'IncludeTies',true, then Idx is an m-by-1 cell array such that cell j (Idx{j}) contains a vector of at least k indices of the closest observations in X to the query point Y(j,:).

If SortIndices is true, then knnsearch arranges the indices in ascending order by distance.

D — Distances of nearest neighbors

numeric matrix | cell array of numeric vectors

Distances of the nearest neighbors to the query points, returned as a numeric matrix or cell array of numeric vectors.

• If you do not specify IncludeTies (false by default), then D is an m-by-k numeric matrix, where m is the number of rows in Y and k is the number of searched nearest neighbors. D(j,i) is the distance between X(Idx(j,i),:) and Y(j,:) with respect to the distance metric.

• If you specify 'IncludeTies',true, then D is an m-by-1 cell array such that cell j (D{j}) contains a vector of at least k distances of the closest observations in X to the query point Y(j,:).

If SortIndices is true, then knnsearch arranges the distances in ascending order.

Tips

• For a fixed positive integer k, knnsearch finds the k points in X that are the nearest to each point in Y. To find all points in X within a fixed distance of each point in Y, use rangesearch.

• knnsearch does not save a search object. To create a search object, use createns.

Algorithms

For information on a specific search algorithm, see k-Nearest Neighbor Search and Radius Search.

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 x_i and x_j, where each x_i has n variables, the algorithm computes distance using the final line in the following equations:

$$\begin{array}{c}{D}_{i,j}^2=\Vert x_i-x_j{\Vert }^2\\ ={(}^{x_i}(x_i-x_j)\\ =\Vert x_i{\Vert }^2-2x_i^T{x}_j+\Vert x_j{\Vert }^2.\end{array}$$

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", knnsearch 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.

Alternative Functionality

If you set the knnsearch function's 'NSMethod' name-value pair argument to the appropriate value ('exhaustive' for an exhaustive search algorithm or 'kdtree' for a Kd-tree algorithm), then the search results are equivalent to the results obtained by conducting a distance search using the knnsearch object function. Unlike the knnsearch function, the knnsearch object function requires an ExhaustiveSearcher or a KDTreeSearcher model object.

References

[1] Friedman, J. H., J. Bentley, and R. A. Finkel. “An Algorithm for Finding Best Matches in Logarithmic Expected Time.” ACM Transactions on Mathematical Software 3, no. 3 (1977): 209–226.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

Usage notes and limitations:

• If X is a tall array, then Y cannot be a tall array. Similarly, if Y is a tall array, then X 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:

• For code generation, the default value of the 'NSMethod' name-value pair argument is 'exhaustive' when the number of columns in X is greater than 7.

• The value of the 'Distance' name-value pair argument must be a compile-time constant and cannot be a custom distance function.

• The value of the 'IncludeTies' name-value pair argument must be a compile-time constant.

• The 'SortIndices' name-value pair argument is not supported. The output arguments are always sorted.

• knnsearch does not support code generation for fast Euclidean distance computations, meaning those distance metrics whose names begin with fast (for example, 'fasteuclidean').

• Names in name-value arguments must be compile-time constants. For example, to allow a user-defined exponent for the Minkowski distance in the generated code, include {coder.Constant('Distance'),coder.Constant('Minkowski'),coder.Constant('P'),0} in the -args value of codegen (MATLAB Coder).

• When you specify 'IncludeTies' as true, the sorted order of tied distances in the generated code can be different from the order in MATLAB due to numerical precision.

• When knnsearch uses the kd-tree search algorithm, and the code generation build type is a MEX function, codegen (MATLAB Coder) generates a MEX function using Intel^® Threading Building Blocks (TBB) for parallel computation. Otherwise, codegen generates code using parfor (MATLAB Coder).

  • MEX function for the kd-tree search algorithm — codegen generates an optimized MEX function using Intel TBB for parallel computation on multicore platforms. You can use the MEX function to accelerate MATLAB algorithms. For details on Intel TBB, see https://www.intel.com/content/www/us/en/developer/tools/oneapi/onetbb.html.

    If you generate the MEX function to test the generated code of the parfor version, you can disable the usage of Intel TBB. Set the ExtrinsicCalls property of the MEX configuration object to false. For details, see coder.MexCodeConfig (MATLAB Coder).

  • MEX function for the exhaustive search algorithm and standalone C/C++ code for both algorithms — The generated code of knnsearch uses parfor (MATLAB Coder) 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 (MATLAB Coder).

• knnsearch returns integer-type (int32) indices in generated standalone C/C++ code. Therefore, the function allows for strict single-precision support when you use single-precision inputs. For MEX code generation, the function still returns double-precision indices to match the MATLAB behavior.

  Before R2020a: knnsearch returns double-precision indices in generated standalone C/C++ code.

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

GPU Arrays

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

Usage notes and limitations:

• The 'IncludeTies', 'NSMethod', and 'SortIndices' name-value pair arguments are not supported.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2010a

R2023a: Fast Euclidean distance using a cache

The 'fasteuclidean' and 'fastseuclidean' distance metrics accelerate the computation of Euclidean distances by using a cache and a different algorithm (see Algorithms). Set the size of the cache using the CacheSize name-value argument.