knnsearch
Find k-nearest neighbors using searcher object
Description
searches for the nearest neighbor (i.e., the closest point, row, or observation) in
Idx
= knnsearch(Mdl
,Y
)Mdl.X
to each point (i.e., row or observation) in the query
data Y
using an exhaustive search or a
Kd-tree. knnsearch
returns
Idx
, which is a column vector of the indices in
Mdl.X
representing the nearest neighbors.
returns the indices of the closest points in Idx
= knnsearch(Mdl
,Y
,Name,Value
)Mdl.X
to
Y
with additional options specified by one or more
Name,Value
pair arguments. For example, specify the number of
nearest neighbors to search for, distance metric different from the one stored in
Mdl.Distance
. You can also specify which action to take if
the closest distances are tied.
[
additionally returns the matrix Idx
,D
]
= knnsearch(___)D
using any of the input
arguments in the previous syntaxes. D
contains the distances
between each observation in Y
that correspond to the closest
observations in Mdl.X
. By default, the function arranges the
columns of D
in ascending order by closeness, with respect to
the distance metric.
Examples
Search for Nearest Neighbors Using Kd-tree and Exhaustive Search
knnsearch
accepts ExhaustiveSearcher
or KDTreeSearcher
model objects to search the training data for the nearest neighbors to the query data. An ExhaustiveSearcher
model invokes the exhaustive searcher algorithm, and a KDTreeSearcher
model defines a Kd-tree, which knnsearch
uses to search for nearest neighbors.
Load Fisher's iris data set. Randomly reserve five observations from the data for query data.
load fisheriris rng(1); % For reproducibility n = size(meas,1); idx = randsample(n,5); X = meas(~ismember(1:n,idx),:); % Training data Y = meas(idx,:); % Query data
The variable meas
contains 4 predictors.
Grow a default four-dimensional Kd-tree.
MdlKDT = KDTreeSearcher(X)
MdlKDT = KDTreeSearcher with properties: BucketSize: 50 Distance: 'euclidean' DistParameter: [] X: [145x4 double]
MdlKDT
is a KDTreeSearcher
model object. You can alter its writable properties using dot notation.
Prepare an exhaustive nearest neighbor searcher.
MdlES = ExhaustiveSearcher(X)
MdlES = ExhaustiveSearcher with properties: Distance: 'euclidean' DistParameter: [] X: [145x4 double]
MdlKDT
is an ExhaustiveSearcher
model object. It contains the options, such as the distance metric, to use to find nearest neighbors.
Alternatively, you can grow a Kd-tree or prepare an exhaustive nearest neighbor searcher using createns
.
Search the training data for the nearest neighbors indices that correspond to each query observation. Conduct both types of searches using the default settings. By default, the number of neighbors to search for per query observation is 1
.
IdxKDT = knnsearch(MdlKDT,Y); IdxES = knnsearch(MdlES,Y); [IdxKDT IdxES]
ans = 5×2
17 17
6 6
1 1
89 89
124 124
In this case, the results of the search are the same.
Search for Nearest Neighbors of Query Data Using Minkowski Distance
Grow a Kd-tree nearest neighbor searcher object by using the createns
function. Pass the object and query data to the knnsearch
function to find k-nearest neighbors.
Load Fisher's iris data set.
load fisheriris
Remove five irises randomly from the predictor data to use as a query set.
rng(1) % For reproducibility n = size(meas,1); % Sample size qIdx = randsample(n,5); % Indices of query data tIdx = ~ismember(1:n,qIdx); % Indices of training data Q = meas(qIdx,:); X = meas(tIdx,:);
Grow a four-dimensional Kd-tree using the training data. Specify the Minkowski distance for finding nearest neighbors.
Mdl = createns(X,'Distance','minkowski')
Mdl = KDTreeSearcher with properties: BucketSize: 50 Distance: 'minkowski' DistParameter: 2 X: [145x4 double]
Because X
has four columns and the distance metric is Minkowski, createns
creates a KDTreeSearcher
model object by default. The Minkowski distance exponent is 2
by default.
Find the indices of the training data (Mdl.X
) that are the two nearest neighbors of each point in the query data (Q
).
IdxNN = knnsearch(Mdl,Q,'K',2)
IdxNN = 5×2
17 4
6 2
1 12
89 66
124 100
Each row of IdxNN
corresponds to a query data observation, and the column order corresponds to the order of the nearest neighbors, with respect to ascending distance. For example, based on the Minkowski distance, the second nearest neighbor of Q(3,:)
is X(12,:)
.
Include Ties in Nearest Neighbor Search
Load Fisher's iris data set.
load fisheriris
Remove five irises randomly from the predictor data to use as a query set.
rng(4); % For reproducibility n = size(meas,1); % Sample size qIdx = randsample(n,5); % Indices of query data X = meas(~ismember(1:n,qIdx),:); Y = meas(qIdx,:);
Grow a four-dimensional Kd-tree using the training data. Specify the Minkowski distance for finding nearest neighbors.
Mdl = KDTreeSearcher(X);
Mdl
is a KDTreeSearcher
model object. By default, the distance metric for finding nearest neighbors is the Euclidean metric.
Find the indices of the training data (X
) that are the seven nearest neighbors of each point in the query data (Y
).
[Idx,D] = knnsearch(Mdl,Y,'K',7,'IncludeTies',true);
Idx
and D
are five-element cell arrays of vectors, with each vector having at least seven elements.
Display the lengths of the vectors in Idx
.
cellfun('length',Idx)
ans = 5×1
8
7
7
7
7
Because cell 1
contains a vector with length greater than k = 7, query observation 1 (Y(1,:)
) is equally close to at least two observations in X
.
Display the indices of the nearest neighbors to Y(1,:)
and their distances.
nn5 = Idx{1}
nn5 = 1×8
91 98 67 69 71 93 88 95
nn5d = D{1}
nn5d = 1×8
0.1414 0.2646 0.2828 0.3000 0.3464 0.3742 0.3873 0.3873
Training observations 88
and 95
are 0.3873 cm away from query observation 1
.
Compare k-Nearest Neighbors Using Different Distance Metrics
Train two KDTreeSearcher
models using different distance metrics, and compare k-nearest neighbors of query data for the two models.
Load Fisher's iris data set. Consider the petal measurements as predictors.
load fisheriris X = meas(:,3:4); % Predictors Y = species; % Response
Train a KDTreeSearcher
model object by using the predictors. Specify the Minkowski distance with exponent 5.
KDTreeMdl = KDTreeSearcher(X,'Distance','minkowski','P',5)
KDTreeMdl = KDTreeSearcher with properties: BucketSize: 50 Distance: 'minkowski' DistParameter: 5 X: [150x2 double]
Find the 10 nearest neighbors from X
to a query point (newpoint
), first using Minkowski then Chebychev distance metrics. The query point must have the same column dimension as the data used to train the model.
newpoint = [5 1.45]; [IdxMk,DMk] = knnsearch(KDTreeMdl,newpoint,'k',10); [IdxCb,DCb] = knnsearch(KDTreeMdl,newpoint,'k',10,'Distance','chebychev');
IdxMk
and IdxCb
are 1-by-10 matrices containing the row indices of X
corresponding to the nearest neighbors to newpoint
using Minkowski and Chebychev distances, respectively. Element (1,1) is the nearest, element (1,2) is the next nearest, and so on.
Plot the training data, query point, and nearest neighbors.
figure; gscatter(X(:,1),X(:,2),Y); title('Fisher''s Iris Data -- Nearest Neighbors'); xlabel('Petal length (cm)'); ylabel('Petal width (cm)'); hold on plot(newpoint(1),newpoint(2),'kx','MarkerSize',10,'LineWidth',2); % Query point plot(X(IdxMk,1),X(IdxMk,2),'o','Color',[.5 .5 .5],'MarkerSize',10); % Minkowski nearest neighbors plot(X(IdxCb,1),X(IdxCb,2),'p','Color',[.5 .5 .5],'MarkerSize',10); % Chebychev nearest neighbors legend('setosa','versicolor','virginica','query point',... 'minkowski','chebychev','Location','Best');
Zoom in on the points of interest.
h = gca; % Get current axis handle. h.XLim = [4.5 5.5]; h.YLim = [1 2]; axis square;
Several observations are equal, which is why only eight nearest neighbors are identified in the plot.
Input Arguments
Mdl
— Nearest neighbor searcher
ExhaustiveSearcher
model object | KDTreeSearcher
model object
Nearest neighbor searcher, specified as an ExhaustiveSearcher
or KDTreeSearcher
model object, respectively.
If Mdl
is an ExhaustiveSearcher
model, then
knnsearch
searches for nearest neighbors using an exhaustive
search. Otherwise, knnsearch
uses the grown
Kd-tree to search for nearest neighbors.
Y
— Query data
numeric matrix
Query data, specified as a numeric matrix.
Y
is an m-by-K matrix.
Rows of Y
correspond to observations (i.e., examples),
and columns correspond to predictors (i.e., variables or features). Y
must
have the same number of columns as the training data stored in Mdl.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: 'K',2,'Distance','minkowski'
specifies to find the two
nearest neighbors of Mdl.X
to each point in Y
and to use the Minkowski distance metric.
Distance
— Distance metric
Mdl.Distance
(default) | 'cityblock'
| 'euclidean'
| 'mahalanobis'
| 'minkowski'
| 'seuclidean'
| function handle | ...
Distance metric used to find neighbors of the training data to the query observations, specified as one of the values in this table or function handle.
For both types of nearest neighbor searchers, knnsearch
supports these
distance metrics.
Value | Description |
---|---|
'chebychev' | Chebychev distance (maximum coordinate difference) |
'cityblock' | City block distance |
'euclidean' | Euclidean distance |
'minkowski' | Minkowski distance. The default exponent is 2. To specify a different exponent, use the
'P' name-value
argument. |
If Mdl
is an ExhaustiveSearcher
model object, then
knnsearch
also supports these distance metrics.
Value | Description |
---|---|
'correlation' | One minus the sample linear correlation between observations (treated as sequences of values) |
'cosine' | One minus the cosine of the included angle between observations (treated as row vectors) |
'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. |
'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 |
'mahalanobis' | Mahalanobis distance, computed using a positive definite covariance matrix. To change the
value of the covariance matrix, use the 'Cov' name-value
argument. |
'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. |
'spearman' | One minus the sample Spearman's rank correlation between observations (treated as sequences of values) |
If Mdl
is an ExhaustiveSearcher
model object, then you
can also specify a function handle for a custom distance metric
by using @
(for example,
@distfun
). The custom distance
function must:
Have the form
function D2 = distfun(ZI,ZJ)
.Take as arguments:
A 1-by-K vector
ZI
containing a single row fromMdl.X
orY
, where K is the number of columns ofMdl.X
.An m-by-K matrix
ZJ
containing multiple rows ofMdl.X
orY
, where m is a positive integer.
Return an m-by-1 vector of distances
D2
, whereD2(
is the distance between the observationsj
)ZI
andZJ(
.j
,:)
For more details, see Distance Metrics.
Example: 'Distance','minkowski'
Data Types: char
| string
| function_handle
IncludeTies
— Flag to include all nearest neighbors
false
(0
) (default) | true
(1
)
Flag to include nearest neighbors that have the same distance from
query observations, specified as the comma-separated pair consisting of
'IncludeTies'
and false
(0
) or true
(1
).
If IncludeTies
is true
, then:
knnsearch
includes all nearest neighbors whose distances are equal to the kth smallest distance in the output arguments, where k is the number of searched nearest neighbors specified by the'K'
name-value pair argument.Idx
andD
are m-by-1
cell arrays such that each cell contains a vector of at least k indices and distances, respectively. Each vector inD
contains arranged distances in ascending order. Each row inIdx
contains the indices of the nearest neighbors corresponding to the distances inD
.
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.
Example: 'IncludeTies',true
K
— Number of nearest neighbors
1
(default) | positive integer
Number of nearest neighbors to search for in the training data per
query observation, specified as the comma-separated pair consisting of
'K'
and a positive integer.
Example: 'K',2
Data Types: single
| double
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
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 inX
.Mdl
is aKDTreeSearcher
model object.IncludeTies
isfalse
.
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 Mdl
is an
ExhaustiveSearcher
model object or
IncludeTies
is true
, the
function always sorts the indices.
Example: 'SortIndices',false
Data Types: logical
Cov
— Covariance matrix for Mahalanobis distance metric
cov(Mdl.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.
Cov
is a K-by-K matrix,
where K is the number of columns of Mdl.X
. If you
specify Cov
and do not specify
'
Distance
','mahalanobis'
,
then knnsearch
returns an error message.
Example: 'Cov',eye(3)
Data Types: single
| double
Scale
— Scale parameter value for standardized Euclidean distance metric
std(Mdl.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 K, where
K is the number of columns of Mdl.X
.
The software scales each difference between the training and query data using the
corresponding element of Scale
. If you specify
Scale
and do not specify
'
Distance
','seuclidean'
,
then knnsearch
returns an error message.
Example: 'Scale',quantile(Mdl.X,0.75) -
quantile(Mdl.X,0.25)
Data Types: single
| double
Note
If you specify
'
Distance
'
,
'
Cov
'
,
'
P
'
, or
'
Scale
'
, then
Mdl.Distance
and Mdl.DistParameter
do
not change value.
Output Arguments
Idx
— Training data indices of nearest neighbors
numeric matrix | cell array of numeric vectors
Training data indices of nearest neighbors, returned as a numeric matrix or cell array of numeric vectors.
If you do not specify
IncludeTies
(false
by default), thenIdx
is an m-by-k numeric matrix, where m is the number of rows inY
and k is the number of searched nearest neighbors specified by the'K'
name-value pair argument.Idx(j,i)
indicates thatMdl.X(Idx(j,i),:)
is one of the k closest observations inMdl.X
to the query observationY(j,:)
.If you specify
'IncludeTies',true
, thenIdx
is an m-by-1
cell array such that cellj
(Idx{j}
) contains a vector of at least k indices of the closest observations inMdl.X
to the query observationY(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 data, returned as a numeric matrix or cell array of numeric vectors.
If you do not specify
IncludeTies
(false
by default), thenD
is an m-by-k numeric matrix, where m is the number of rows inY
and k is the number of searched nearest neighbors specified by the'K'
name-value pair argument.D(j,i)
is the distance betweenMdl.X(Idx(j,i),:)
and the query observationY(j,:)
with respect to the distance metric.If you specify
'IncludeTies',true
, thenD
is an m-by-1
cell array such that cellj
(D{j}
) contains a vector of at least k distances of the closest observations inMdl.X
to the query observationY(j,:)
.
If SortIndices
is true
, then
knnsearch
arranges the distances in ascending
order.
Tips
knnsearch
finds the k (positive integer)
points in Mdl.X
that are k-nearest for each
Y
point. In contrast, rangesearch
finds all the points in Mdl.X
that are
within distance r
(positive scalar) of each Y
point.
Algorithms
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:
The matrix 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].
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
knnsearch
is an object function that requires anExhaustiveSearcher
or aKDTreeSearcher
model object and query data. Under equivalent conditions, theknnsearch
object function returns the same results as theknnsearch
function when you specify the name-value pair argument'NSMethod','exhaustive'
or'NSMethod','kdtree'
, respectively.For k-nearest neighbors classification, see
fitcknn
andClassificationKNN
.
References
[1] Friedman, J. H., Bentley, J., and Finkel, R. A. (1977). “An Algorithm for Finding Best Matches in Logarithmic Expected Time.” ACM Transactions on Mathematical Software Vol. 3, Issue 3, Sept. 1977, pp. 209–226.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
This table contains notes about the arguments of
knnsearch
. Arguments not included in this table are fully supported.Argument Notes and Limitations Mdl
There are two ways to use
Mdl
in code generation. For an example, see Code Generation for Nearest Neighbor Searcher.Use
saveLearnerForCoder
,loadLearnerForCoder
, andcodegen
(MATLAB Coder) to generate code for theknnsearch
function. Save a trained model by usingsaveLearnerForCoder
. Define an entry-point function that loads the saved model by usingloadLearnerForCoder
and calls theknnsearch
function. Then usecodegen
to generate code for the entry-point function.Include
coder.Constant(Mdl)
in the-args
value ofcodegen
(MATLAB Coder).
If
Mdl
is aKDTreeSearcher
object, and the code generation build type is a MEX function, thencodegen
(MATLAB Coder) generates a MEX function using Intel® Threading Building Blocks (TBB) for parallel computation. Otherwise,codegen
generates code usingparfor
(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 theExtrinsicCalls
property of the MEX configuration object tofalse
. For details, seecoder.MexCodeConfig
(MATLAB Coder).MEX function for the exhaustive search algorithm and standalone C/C++ code for both algorithms — The generated code of
knnsearch
usesparfor
(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 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).
'Distance'
Cannot be a custom distance function.
Must be a compile-time constant; its value cannot change in the generated code.
knnsearch
does not support code generation for fast Euclidean distance computations, meaning those distance metrics whose names begin withfast
(for example,'fasteuclidean'
).
'IncludeTies'
Must be a compile-time constant; its value cannot change in the generated code.
'SortIndices'
Not supported. The output arguments are always sorted. Name-value pair arguments 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 ofcodegen
(MATLAB Coder).Idx
When you specify
'IncludeTies'
astrue
, the sorted order of tied distances in the generated code can be different from the order in MATLAB due to numerical precision.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, see Introduction to Code Generation and Code Generation for Nearest Neighbor Searcher.
Version History
Introduced in R2010aR2023a: 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).
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