# knnsearch

Find k-nearest neighbors using input data

## Syntax

``Idx = knnsearch(X,Y)``
``Idx = knnsearch(X,Y,Name,Value)``
``[Idx,D] = knnsearch(___)``

## Description

example

````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`.```

example

````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.```

example

````[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

collapse all

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 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 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')``` 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 = 20.8135 ```

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 = 2.0801 ```

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

`standard/accelerated`
```ans = 10.0062 ```

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

## Input Arguments

collapse all

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

Data Types: `single` | `double`

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.

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`

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`

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

ValueDescription
`'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 `j`th element is the distance between the observations `ZI` and `ZJ(j,:)`.

Example: `'Distance','chebychev'`

Data Types: `char` | `string` | `function_handle`

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`

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`

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 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`

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`

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:

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

collapse all

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.

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.

## Algorithms

collapse all

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  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:

`$\begin{array}{c}{D}_{i,j}^{2}=‖{x}_{i}-{x}_{j}{‖}^{2}\\ ={\left(}^{{x}_{i}}\left({x}_{i}-{x}_{j}\right)\\ =‖{x}_{i}{‖}^{2}-2{x}_{i}^{T}{x}_{j}+‖{x}_{j}{‖}^{2}.\end{array}$`

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 .

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.

 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.

To integrate a `k`-nearest neighbor search into Simulink®, you can use the KNN Search block in the Statistics and Machine Learning Toolbox™ library or a MATLAB Function block with the `knnsearch` function. For an example, see Predict Class Labels Using MATLAB Function Block.

When deciding which approach to use, consider the following:

• If you use the Statistics and Machine Learning Toolbox library block, you can use the Fixed-Point Tool (Fixed-Point Designer) to convert a floating-point model to fixed point.

• Support for variable-size arrays must be enabled for a MATLAB Function block with the `knnsearch` function.

## References

 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.

## Version History

Introduced in R2010a

expand all