### Function Reference: `rangesearch`

statistics: idx = rangesearch (X, Y, r)
statistics: [idx, D] = rangesearch (X, Y, r)
statistics: […] = rangesearch (…, name, value)

Find all neighbors within specified distance from input data.

`idx = rangesearch (X, Y, r)` returns all the points in X that are within distance r from the points in Y. X must be an N×P numeric matrix of input data, where rows correspond to observations and columns correspond to features or variables. Y is an M×P numeric matrix with query points, which must have the same numbers of column as X. r must be a nonnegative scalar value. idx is an M×1 cell array, where M is the number of observations in Y. The vector `Idx{j}` contains the indices of observations (rows) in X whose distances to `Y(j,:)` are not greater than r.

`[idx, D] = rangesearch (X, Y, r)` also returns the distances, D, which correspond to the points in X that are within distance r from the points in Y. D is an M×1 cell array, where M is the number of observations in Y. The vector `D{j}` contains the indices of observations (rows) in X whose distances to `Y(j,:)` are not greater than r.

Additional parameters can be specified by `Name-Value` pair arguments.

Name |  | Value
--- | --- | ---
`"P"` |  | is the Minkowski distance exponent and it must be a positive scalar. This argument is only valid when the selected distance metric is `"minkowski"`. By default it is 2.
`"Scale"` |  | is the scale parameter for the standardized Euclidean distance and it must be a nonnegative numeric vector of equal length to the number of columns in X. This argument is only valid when the selected distance metric is `"seuclidean"`, in which case each coordinate of X is scaled by the corresponding element of `"scale"`, as is each query point in Y. By default, the scale parameter is the standard deviation of each coordinate in X.
`"Cov"` |  | is the covariance matrix for computing the mahalanobis distance and it must be a positive definite matrix matching the the number of columns in X. This argument is only valid when the selected distance metric is `"mahalanobis"`.
`"BucketSize"` |  | is the maximum number of data points in the leaf node of the Kd-tree and it must be a positive integer. This argument is only valid when the selected search method is `"kdtree"`.
`"SortIndices"` |  | is a boolean flag to sort the returned indices in ascending order by distance and it is `true` by default. When the selected search method is `"exhaustive"` or the `"IncludeTies"` flag is true, `rangesearch` always sorts the returned indices.
`"Distance"` |  | is the distance metric used by `rangesearch` as specified below:

| `"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.
| `"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"` | Cosine distance.
| `"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.
| @distfun | Custom distance function handle. A distance function of the form `function D2 = distfun (XI, YI)`, where XI is a 1×P vector containing a single observation in P-dimensional space, YI is an N×P matrix containing an arbitrary number of observations in the same P-dimensional space, and D2 is an N×P vector of distances, where `(D2k)` is the distance between observations XI and `(YIk,:)`.

`"NSMethod"` |  | is the nearest neighbor search method used by `rangesearch` as specified below.

| `"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"`, `"manhattan"`, `"chebychev"`, or `"minkowski"`. Otherwise, the default value is `"exhaustive"`. This argument is only valid when the distance metric is one of the four aforementioned metrics.
| `"exhaustive"` | Uses the exhaustive search algorithm by computing the distance values from all the points in X to each point in Y.

See also: knnsearch, pdist2

Source Code: rangesearch

### Example: 1

     ## Generate 1000 random 2D points from each of five distinct multivariate
     ## normal distributions that form five separate classes
     N = 1000;
     d = 10;
     randn ("seed", 5);
     X1 = mvnrnd (d * [0, 0], eye (2), 1000);
     randn ("seed", 6);
     X2 = mvnrnd (d * [1, 1], eye (2), 1000);
     randn ("seed", 7);
     X3 = mvnrnd (d * [-1, -1], eye (2), 1000);
     randn ("seed", 8);
     X4 = mvnrnd (d * [1, -1], eye (2), 1000);
     randn ("seed", 8);
     X5 = mvnrnd (d * [-1, 1], eye (2), 1000);
     X = [X1; X2; X3; X4; X5];

     ## For each point in X, find the points in X that are within a radius d
     ## away from the points in X.
     Idx = rangesearch (X, X, d, "NSMethod", "exhaustive");

     ## Select the first point in X (corresponding to the first class) and find
     ## its nearest neighbors within the radius d.  Display these points in
     ## one color and the remaining points in a different color.
     x = X(1,:);
     nearestPoints = X (Idx{1},:);
     nonNearestIdx = true (size (X, 1), 1);
     nonNearestIdx(Idx{1}) = false;

     scatter (X(nonNearestIdx,1), X(nonNearestIdx,2))
     hold on
     scatter (nearestPoints(:,1),nearestPoints(:,2))
     scatter (x(1), x(2), "black", "filled")
     hold off

     ## Select the last point in X (corresponding to the fifth class) and find
     ## its nearest neighbors within the radius d.  Display these points in
     ## one color and the remaining points in a different color.
     x = X(end,:);
     nearestPoints = X (Idx{1},:);
     nonNearestIdx = true (size (X, 1), 1);
     nonNearestIdx(Idx{1}) = false;

     figure
     scatter (X(nonNearestIdx,1), X(nonNearestIdx,2))
     hold on
     scatter (nearestPoints(:,1),nearestPoints(:,2))
     scatter (x(1), x(2), "black", "filled")
     hold off