# Authors: Alexandre Gramfort # Mathieu Blondel # Robert Layton # Andreas Mueller # Philippe Gervais # Lars Buitinck # Joel Nothman # License: BSD 3 clause import itertools from functools import partial import warnings import numpy as np from scipy.spatial import distance from scipy.sparse import csr_matrix from scipy.sparse import issparse from joblib import effective_n_jobs from .. import config_context from ..utils.validation import _num_samples from ..utils.validation import check_non_negative from ..utils import check_array from ..utils import gen_even_slices from ..utils import gen_batches, get_chunk_n_rows from ..utils import is_scalar_nan from ..utils.extmath import row_norms, safe_sparse_dot from ..preprocessing import normalize from ..utils._mask import _get_mask from ..utils.parallel import delayed, Parallel from ..utils.fixes import sp_version, parse_version from ._pairwise_distances_reduction import ArgKmin from ._pairwise_fast import _chi2_kernel_fast, _sparse_manhattan from ..exceptions import DataConversionWarning # Utility Functions def _return_float_dtype(X, Y): """ 1. If dtype of X and Y is float32, then dtype float32 is returned. 2. Else dtype float is returned. """ if not issparse(X) and not isinstance(X, np.ndarray): X = np.asarray(X) if Y is None: Y_dtype = X.dtype elif not issparse(Y) and not isinstance(Y, np.ndarray): Y = np.asarray(Y) Y_dtype = Y.dtype else: Y_dtype = Y.dtype if X.dtype == Y_dtype == np.float32: dtype = np.float32 else: dtype = float return X, Y, dtype def check_pairwise_arrays( X, Y, *, precomputed=False, dtype=None, accept_sparse="csr", force_all_finite=True, copy=False, ): """Set X and Y appropriately and checks inputs. If Y is None, it is set as a pointer to X (i.e. not a copy). If Y is given, this does not happen. All distance metrics should use this function first to assert that the given parameters are correct and safe to use. Specifically, this function first ensures that both X and Y are arrays, then checks that they are at least two dimensional while ensuring that their elements are floats (or dtype if provided). Finally, the function checks that the size of the second dimension of the two arrays is equal, or the equivalent check for a precomputed distance matrix. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples_X, n_features) Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features) precomputed : bool, default=False True if X is to be treated as precomputed distances to the samples in Y. dtype : str, type, list of type, default=None Data type required for X and Y. If None, the dtype will be an appropriate float type selected by _return_float_dtype. .. versionadded:: 0.18 accept_sparse : str, bool or list/tuple of str, default='csr' String[s] representing allowed sparse matrix formats, such as 'csc', 'csr', etc. If the input is sparse but not in the allowed format, it will be converted to the first listed format. True allows the input to be any format. False means that a sparse matrix input will raise an error. force_all_finite : bool or 'allow-nan', default=True Whether to raise an error on np.inf, np.nan, pd.NA in array. The possibilities are: - True: Force all values of array to be finite. - False: accepts np.inf, np.nan, pd.NA in array. - 'allow-nan': accepts only np.nan and pd.NA values in array. Values cannot be infinite. .. versionadded:: 0.22 ``force_all_finite`` accepts the string ``'allow-nan'``. .. versionchanged:: 0.23 Accepts `pd.NA` and converts it into `np.nan`. copy : bool, default=False Whether a forced copy will be triggered. If copy=False, a copy might be triggered by a conversion. .. versionadded:: 0.22 Returns ------- safe_X : {array-like, sparse matrix} of shape (n_samples_X, n_features) An array equal to X, guaranteed to be a numpy array. safe_Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features) An array equal to Y if Y was not None, guaranteed to be a numpy array. If Y was None, safe_Y will be a pointer to X. """ X, Y, dtype_float = _return_float_dtype(X, Y) estimator = "check_pairwise_arrays" if dtype is None: dtype = dtype_float if Y is X or Y is None: X = Y = check_array( X, accept_sparse=accept_sparse, dtype=dtype, copy=copy, force_all_finite=force_all_finite, estimator=estimator, ) else: X = check_array( X, accept_sparse=accept_sparse, dtype=dtype, copy=copy, force_all_finite=force_all_finite, estimator=estimator, ) Y = check_array( Y, accept_sparse=accept_sparse, dtype=dtype, copy=copy, force_all_finite=force_all_finite, estimator=estimator, ) if precomputed: if X.shape[1] != Y.shape[0]: raise ValueError( "Precomputed metric requires shape " "(n_queries, n_indexed). Got (%d, %d) " "for %d indexed." % (X.shape[0], X.shape[1], Y.shape[0]) ) elif X.shape[1] != Y.shape[1]: raise ValueError( "Incompatible dimension for X and Y matrices: " "X.shape[1] == %d while Y.shape[1] == %d" % (X.shape[1], Y.shape[1]) ) return X, Y def check_paired_arrays(X, Y): """Set X and Y appropriately and checks inputs for paired distances. All paired distance metrics should use this function first to assert that the given parameters are correct and safe to use. Specifically, this function first ensures that both X and Y are arrays, then checks that they are at least two dimensional while ensuring that their elements are floats. Finally, the function checks that the size of the dimensions of the two arrays are equal. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples_X, n_features) Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features) Returns ------- safe_X : {array-like, sparse matrix} of shape (n_samples_X, n_features) An array equal to X, guaranteed to be a numpy array. safe_Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features) An array equal to Y if Y was not None, guaranteed to be a numpy array. If Y was None, safe_Y will be a pointer to X. """ X, Y = check_pairwise_arrays(X, Y) if X.shape != Y.shape: raise ValueError( "X and Y should be of same shape. They were respectively %r and %r long." % (X.shape, Y.shape) ) return X, Y # Pairwise distances def euclidean_distances( X, Y=None, *, Y_norm_squared=None, squared=False, X_norm_squared=None ): """ Compute the distance matrix between each pair from a vector array X and Y. For efficiency reasons, the euclidean distance between a pair of row vector x and y is computed as:: dist(x, y) = sqrt(dot(x, x) - 2 * dot(x, y) + dot(y, y)) This formulation has two advantages over other ways of computing distances. First, it is computationally efficient when dealing with sparse data. Second, if one argument varies but the other remains unchanged, then `dot(x, x)` and/or `dot(y, y)` can be pre-computed. However, this is not the most precise way of doing this computation, because this equation potentially suffers from "catastrophic cancellation". Also, the distance matrix returned by this function may not be exactly symmetric as required by, e.g., ``scipy.spatial.distance`` functions. Read more in the :ref:`User Guide `. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples_X, n_features) An array where each row is a sample and each column is a feature. Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features), \ default=None An array where each row is a sample and each column is a feature. If `None`, method uses `Y=X`. Y_norm_squared : array-like of shape (n_samples_Y,) or (n_samples_Y, 1) \ or (1, n_samples_Y), default=None Pre-computed dot-products of vectors in Y (e.g., ``(Y**2).sum(axis=1)``) May be ignored in some cases, see the note below. squared : bool, default=False Return squared Euclidean distances. X_norm_squared : array-like of shape (n_samples_X,) or (n_samples_X, 1) \ or (1, n_samples_X), default=None Pre-computed dot-products of vectors in X (e.g., ``(X**2).sum(axis=1)``) May be ignored in some cases, see the note below. Returns ------- distances : ndarray of shape (n_samples_X, n_samples_Y) Returns the distances between the row vectors of `X` and the row vectors of `Y`. See Also -------- paired_distances : Distances between pairs of elements of X and Y. Notes ----- To achieve a better accuracy, `X_norm_squared` and `Y_norm_squared` may be unused if they are passed as `np.float32`. Examples -------- >>> from sklearn.metrics.pairwise import euclidean_distances >>> X = [[0, 1], [1, 1]] >>> # distance between rows of X >>> euclidean_distances(X, X) array([[0., 1.], [1., 0.]]) >>> # get distance to origin >>> euclidean_distances(X, [[0, 0]]) array([[1. ], [1.41421356]]) """ X, Y = check_pairwise_arrays(X, Y) if X_norm_squared is not None: X_norm_squared = check_array(X_norm_squared, ensure_2d=False) original_shape = X_norm_squared.shape if X_norm_squared.shape == (X.shape[0],): X_norm_squared = X_norm_squared.reshape(-1, 1) if X_norm_squared.shape == (1, X.shape[0]): X_norm_squared = X_norm_squared.T if X_norm_squared.shape != (X.shape[0], 1): raise ValueError( f"Incompatible dimensions for X of shape {X.shape} and " f"X_norm_squared of shape {original_shape}." ) if Y_norm_squared is not None: Y_norm_squared = check_array(Y_norm_squared, ensure_2d=False) original_shape = Y_norm_squared.shape if Y_norm_squared.shape == (Y.shape[0],): Y_norm_squared = Y_norm_squared.reshape(1, -1) if Y_norm_squared.shape == (Y.shape[0], 1): Y_norm_squared = Y_norm_squared.T if Y_norm_squared.shape != (1, Y.shape[0]): raise ValueError( f"Incompatible dimensions for Y of shape {Y.shape} and " f"Y_norm_squared of shape {original_shape}." ) return _euclidean_distances(X, Y, X_norm_squared, Y_norm_squared, squared) def _euclidean_distances(X, Y, X_norm_squared=None, Y_norm_squared=None, squared=False): """Computational part of euclidean_distances Assumes inputs are already checked. If norms are passed as float32, they are unused. If arrays are passed as float32, norms needs to be recomputed on upcast chunks. TODO: use a float64 accumulator in row_norms to avoid the latter. """ if X_norm_squared is not None: if X_norm_squared.dtype == np.float32: XX = None else: XX = X_norm_squared.reshape(-1, 1) elif X.dtype == np.float32: XX = None else: XX = row_norms(X, squared=True)[:, np.newaxis] if Y is X: YY = None if XX is None else XX.T else: if Y_norm_squared is not None: if Y_norm_squared.dtype == np.float32: YY = None else: YY = Y_norm_squared.reshape(1, -1) elif Y.dtype == np.float32: YY = None else: YY = row_norms(Y, squared=True)[np.newaxis, :] if X.dtype == np.float32: # To minimize precision issues with float32, we compute the distance # matrix on chunks of X and Y upcast to float64 distances = _euclidean_distances_upcast(X, XX, Y, YY) else: # if dtype is already float64, no need to chunk and upcast distances = -2 * safe_sparse_dot(X, Y.T, dense_output=True) distances += XX distances += YY np.maximum(distances, 0, out=distances) # Ensure that distances between vectors and themselves are set to 0.0. # This may not be the case due to floating point rounding errors. if X is Y: np.fill_diagonal(distances, 0) return distances if squared else np.sqrt(distances, out=distances) def nan_euclidean_distances( X, Y=None, *, squared=False, missing_values=np.nan, copy=True ): """Calculate the euclidean distances in the presence of missing values. Compute the euclidean distance between each pair of samples in X and Y, where Y=X is assumed if Y=None. When calculating the distance between a pair of samples, this formulation ignores feature coordinates with a missing value in either sample and scales up the weight of the remaining coordinates: dist(x,y) = sqrt(weight * sq. distance from present coordinates) where, weight = Total # of coordinates / # of present coordinates For example, the distance between ``[3, na, na, 6]`` and ``[1, na, 4, 5]`` is: .. math:: \\sqrt{\\frac{4}{2}((3-1)^2 + (6-5)^2)} If all the coordinates are missing or if there are no common present coordinates then NaN is returned for that pair. Read more in the :ref:`User Guide `. .. versionadded:: 0.22 Parameters ---------- X : array-like of shape (n_samples_X, n_features) An array where each row is a sample and each column is a feature. Y : array-like of shape (n_samples_Y, n_features), default=None An array where each row is a sample and each column is a feature. If `None`, method uses `Y=X`. squared : bool, default=False Return squared Euclidean distances. missing_values : np.nan or int, default=np.nan Representation of missing value. copy : bool, default=True Make and use a deep copy of X and Y (if Y exists). Returns ------- distances : ndarray of shape (n_samples_X, n_samples_Y) Returns the distances between the row vectors of `X` and the row vectors of `Y`. See Also -------- paired_distances : Distances between pairs of elements of X and Y. References ---------- * John K. Dixon, "Pattern Recognition with Partly Missing Data", IEEE Transactions on Systems, Man, and Cybernetics, Volume: 9, Issue: 10, pp. 617 - 621, Oct. 1979. http://ieeexplore.ieee.org/abstract/document/4310090/ Examples -------- >>> from sklearn.metrics.pairwise import nan_euclidean_distances >>> nan = float("NaN") >>> X = [[0, 1], [1, nan]] >>> nan_euclidean_distances(X, X) # distance between rows of X array([[0. , 1.41421356], [1.41421356, 0. ]]) >>> # get distance to origin >>> nan_euclidean_distances(X, [[0, 0]]) array([[1. ], [1.41421356]]) """ force_all_finite = "allow-nan" if is_scalar_nan(missing_values) else True X, Y = check_pairwise_arrays( X, Y, accept_sparse=False, force_all_finite=force_all_finite, copy=copy ) # Get missing mask for X missing_X = _get_mask(X, missing_values) # Get missing mask for Y missing_Y = missing_X if Y is X else _get_mask(Y, missing_values) # set missing values to zero X[missing_X] = 0 Y[missing_Y] = 0 distances = euclidean_distances(X, Y, squared=True) # Adjust distances for missing values XX = X * X YY = Y * Y distances -= np.dot(XX, missing_Y.T) distances -= np.dot(missing_X, YY.T) np.clip(distances, 0, None, out=distances) if X is Y: # Ensure that distances between vectors and themselves are set to 0.0. # This may not be the case due to floating point rounding errors. np.fill_diagonal(distances, 0.0) present_X = 1 - missing_X present_Y = present_X if Y is X else ~missing_Y present_count = np.dot(present_X, present_Y.T) distances[present_count == 0] = np.nan # avoid divide by zero np.maximum(1, present_count, out=present_count) distances /= present_count distances *= X.shape[1] if not squared: np.sqrt(distances, out=distances) return distances def _euclidean_distances_upcast(X, XX=None, Y=None, YY=None, batch_size=None): """Euclidean distances between X and Y. Assumes X and Y have float32 dtype. Assumes XX and YY have float64 dtype or are None. X and Y are upcast to float64 by chunks, which size is chosen to limit memory increase by approximately 10% (at least 10MiB). """ n_samples_X = X.shape[0] n_samples_Y = Y.shape[0] n_features = X.shape[1] distances = np.empty((n_samples_X, n_samples_Y), dtype=np.float32) if batch_size is None: x_density = X.nnz / np.prod(X.shape) if issparse(X) else 1 y_density = Y.nnz / np.prod(Y.shape) if issparse(Y) else 1 # Allow 10% more memory than X, Y and the distance matrix take (at # least 10MiB) maxmem = max( ( (x_density * n_samples_X + y_density * n_samples_Y) * n_features + (x_density * n_samples_X * y_density * n_samples_Y) ) / 10, 10 * 2**17, ) # The increase amount of memory in 8-byte blocks is: # - x_density * batch_size * n_features (copy of chunk of X) # - y_density * batch_size * n_features (copy of chunk of Y) # - batch_size * batch_size (chunk of distance matrix) # Hence x² + (xd+yd)kx = M, where x=batch_size, k=n_features, M=maxmem # xd=x_density and yd=y_density tmp = (x_density + y_density) * n_features batch_size = (-tmp + np.sqrt(tmp**2 + 4 * maxmem)) / 2 batch_size = max(int(batch_size), 1) x_batches = gen_batches(n_samples_X, batch_size) for i, x_slice in enumerate(x_batches): X_chunk = X[x_slice].astype(np.float64) if XX is None: XX_chunk = row_norms(X_chunk, squared=True)[:, np.newaxis] else: XX_chunk = XX[x_slice] y_batches = gen_batches(n_samples_Y, batch_size) for j, y_slice in enumerate(y_batches): if X is Y and j < i: # when X is Y the distance matrix is symmetric so we only need # to compute half of it. d = distances[y_slice, x_slice].T else: Y_chunk = Y[y_slice].astype(np.float64) if YY is None: YY_chunk = row_norms(Y_chunk, squared=True)[np.newaxis, :] else: YY_chunk = YY[:, y_slice] d = -2 * safe_sparse_dot(X_chunk, Y_chunk.T, dense_output=True) d += XX_chunk d += YY_chunk distances[x_slice, y_slice] = d.astype(np.float32, copy=False) return distances def _argmin_min_reduce(dist, start): # `start` is specified in the signature but not used. This is because the higher # order `pairwise_distances_chunked` function needs reduction functions that are # passed as argument to have a two arguments signature. indices = dist.argmin(axis=1) values = dist[np.arange(dist.shape[0]), indices] return indices, values def _argmin_reduce(dist, start): # `start` is specified in the signature but not used. This is because the higher # order `pairwise_distances_chunked` function needs reduction functions that are # passed as argument to have a two arguments signature. return dist.argmin(axis=1) def pairwise_distances_argmin_min( X, Y, *, axis=1, metric="euclidean", metric_kwargs=None ): """Compute minimum distances between one point and a set of points. This function computes for each row in X, the index of the row of Y which is closest (according to the specified distance). The minimal distances are also returned. This is mostly equivalent to calling: (pairwise_distances(X, Y=Y, metric=metric).argmin(axis=axis), pairwise_distances(X, Y=Y, metric=metric).min(axis=axis)) but uses much less memory, and is faster for large arrays. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples_X, n_features) Array containing points. Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features) Array containing points. axis : int, default=1 Axis along which the argmin and distances are to be computed. metric : str or callable, default='euclidean' Metric to use for distance computation. Any metric from scikit-learn or scipy.spatial.distance can be used. If metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays as input and return one value indicating the distance between them. This works for Scipy's metrics, but is less efficient than passing the metric name as a string. Distance matrices are not supported. Valid values for metric are: - from scikit-learn: ['cityblock', 'cosine', 'euclidean', 'l1', 'l2', 'manhattan'] - from scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev', 'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'] See the documentation for scipy.spatial.distance for details on these metrics. metric_kwargs : dict, default=None Keyword arguments to pass to specified metric function. Returns ------- argmin : ndarray Y[argmin[i], :] is the row in Y that is closest to X[i, :]. distances : ndarray The array of minimum distances. `distances[i]` is the distance between the i-th row in X and the argmin[i]-th row in Y. See Also -------- pairwise_distances : Distances between every pair of samples of X and Y. pairwise_distances_argmin : Same as `pairwise_distances_argmin_min` but only returns the argmins. """ X, Y = check_pairwise_arrays(X, Y) if axis == 0: X, Y = Y, X if metric_kwargs is None: metric_kwargs = {} if ArgKmin.is_usable_for(X, Y, metric): # This is an adaptor for one "sqeuclidean" specification. # For this backend, we can directly use "sqeuclidean". if metric_kwargs.get("squared", False) and metric == "euclidean": metric = "sqeuclidean" metric_kwargs = {} values, indices = ArgKmin.compute( X=X, Y=Y, k=1, metric=metric, metric_kwargs=metric_kwargs, strategy="auto", return_distance=True, ) values = values.flatten() indices = indices.flatten() else: # Joblib-based backend, which is used when user-defined callable # are passed for metric. # This won't be used in the future once PairwiseDistancesReductions support: # - DistanceMetrics which work on supposedly binary data # - CSR-dense and dense-CSR case if 'euclidean' in metric. # Turn off check for finiteness because this is costly and because arrays # have already been validated. with config_context(assume_finite=True): indices, values = zip( *pairwise_distances_chunked( X, Y, reduce_func=_argmin_min_reduce, metric=metric, **metric_kwargs ) ) indices = np.concatenate(indices) values = np.concatenate(values) return indices, values def pairwise_distances_argmin(X, Y, *, axis=1, metric="euclidean", metric_kwargs=None): """Compute minimum distances between one point and a set of points. This function computes for each row in X, the index of the row of Y which is closest (according to the specified distance). This is mostly equivalent to calling: pairwise_distances(X, Y=Y, metric=metric).argmin(axis=axis) but uses much less memory, and is faster for large arrays. This function works with dense 2D arrays only. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples_X, n_features) Array containing points. Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features) Arrays containing points. axis : int, default=1 Axis along which the argmin and distances are to be computed. metric : str or callable, default="euclidean" Metric to use for distance computation. Any metric from scikit-learn or scipy.spatial.distance can be used. If metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays as input and return one value indicating the distance between them. This works for Scipy's metrics, but is less efficient than passing the metric name as a string. Distance matrices are not supported. Valid values for metric are: - from scikit-learn: ['cityblock', 'cosine', 'euclidean', 'l1', 'l2', 'manhattan'] - from scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev', 'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'] See the documentation for scipy.spatial.distance for details on these metrics. metric_kwargs : dict, default=None Keyword arguments to pass to specified metric function. Returns ------- argmin : numpy.ndarray Y[argmin[i], :] is the row in Y that is closest to X[i, :]. See Also -------- pairwise_distances : Distances between every pair of samples of X and Y. pairwise_distances_argmin_min : Same as `pairwise_distances_argmin` but also returns the distances. """ if metric_kwargs is None: metric_kwargs = {} X, Y = check_pairwise_arrays(X, Y) if axis == 0: X, Y = Y, X if metric_kwargs is None: metric_kwargs = {} if ArgKmin.is_usable_for(X, Y, metric): # This is an adaptor for one "sqeuclidean" specification. # For this backend, we can directly use "sqeuclidean". if metric_kwargs.get("squared", False) and metric == "euclidean": metric = "sqeuclidean" metric_kwargs = {} indices = ArgKmin.compute( X=X, Y=Y, k=1, metric=metric, metric_kwargs=metric_kwargs, strategy="auto", return_distance=False, ) indices = indices.flatten() else: # Joblib-based backend, which is used when user-defined callable # are passed for metric. # This won't be used in the future once PairwiseDistancesReductions support: # - DistanceMetrics which work on supposedly binary data # - CSR-dense and dense-CSR case if 'euclidean' in metric. # Turn off check for finiteness because this is costly and because arrays # have already been validated. with config_context(assume_finite=True): indices = np.concatenate( list( # This returns a np.ndarray generator whose arrays we need # to flatten into one. pairwise_distances_chunked( X, Y, reduce_func=_argmin_reduce, metric=metric, **metric_kwargs ) ) ) return indices def haversine_distances(X, Y=None): """Compute the Haversine distance between samples in X and Y. The Haversine (or great circle) distance is the angular distance between two points on the surface of a sphere. The first coordinate of each point is assumed to be the latitude, the second is the longitude, given in radians. The dimension of the data must be 2. .. math:: D(x, y) = 2\\arcsin[\\sqrt{\\sin^2((x1 - y1) / 2) + \\cos(x1)\\cos(y1)\\sin^2((x2 - y2) / 2)}] Parameters ---------- X : array-like of shape (n_samples_X, 2) A feature array. Y : array-like of shape (n_samples_Y, 2), default=None An optional second feature array. If `None`, uses `Y=X`. Returns ------- distance : ndarray of shape (n_samples_X, n_samples_Y) The distance matrix. Notes ----- As the Earth is nearly spherical, the haversine formula provides a good approximation of the distance between two points of the Earth surface, with a less than 1% error on average. Examples -------- We want to calculate the distance between the Ezeiza Airport (Buenos Aires, Argentina) and the Charles de Gaulle Airport (Paris, France). >>> from sklearn.metrics.pairwise import haversine_distances >>> from math import radians >>> bsas = [-34.83333, -58.5166646] >>> paris = [49.0083899664, 2.53844117956] >>> bsas_in_radians = [radians(_) for _ in bsas] >>> paris_in_radians = [radians(_) for _ in paris] >>> result = haversine_distances([bsas_in_radians, paris_in_radians]) >>> result * 6371000/1000 # multiply by Earth radius to get kilometers array([[ 0. , 11099.54035582], [11099.54035582, 0. ]]) """ from ..metrics import DistanceMetric return DistanceMetric.get_metric("haversine").pairwise(X, Y) def manhattan_distances(X, Y=None, *, sum_over_features="deprecated"): """Compute the L1 distances between the vectors in X and Y. With sum_over_features equal to False it returns the componentwise distances. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples_X, n_features) An array where each row is a sample and each column is a feature. Y : array-like of shape (n_samples_Y, n_features), default=None An array where each row is a sample and each column is a feature. If `None`, method uses `Y=X`. sum_over_features : bool, default=True If True the function returns the pairwise distance matrix else it returns the componentwise L1 pairwise-distances. Not supported for sparse matrix inputs. .. deprecated:: 1.2 ``sum_over_features`` was deprecated in version 1.2 and will be removed in 1.4. Returns ------- D : ndarray of shape (n_samples_X * n_samples_Y, n_features) or \ (n_samples_X, n_samples_Y) If sum_over_features is False shape is (n_samples_X * n_samples_Y, n_features) and D contains the componentwise L1 pairwise-distances (ie. absolute difference), else shape is (n_samples_X, n_samples_Y) and D contains the pairwise L1 distances. Notes ----- When X and/or Y are CSR sparse matrices and they are not already in canonical format, this function modifies them in-place to make them canonical. Examples -------- >>> from sklearn.metrics.pairwise import manhattan_distances >>> manhattan_distances([[3]], [[3]]) array([[0.]]) >>> manhattan_distances([[3]], [[2]]) array([[1.]]) >>> manhattan_distances([[2]], [[3]]) array([[1.]]) >>> manhattan_distances([[1, 2], [3, 4]],\ [[1, 2], [0, 3]]) array([[0., 2.], [4., 4.]]) """ # TODO(1.4): remove sum_over_features if sum_over_features != "deprecated": warnings.warn( "`sum_over_features` is deprecated in version 1.2 and will be" " removed in version 1.4.", FutureWarning, ) else: sum_over_features = True X, Y = check_pairwise_arrays(X, Y) if issparse(X) or issparse(Y): if not sum_over_features: raise TypeError( "sum_over_features=%r not supported for sparse matrices" % sum_over_features ) X = csr_matrix(X, copy=False) Y = csr_matrix(Y, copy=False) X.sum_duplicates() # this also sorts indices in-place Y.sum_duplicates() D = np.zeros((X.shape[0], Y.shape[0])) _sparse_manhattan(X.data, X.indices, X.indptr, Y.data, Y.indices, Y.indptr, D) return D if sum_over_features: return distance.cdist(X, Y, "cityblock") D = X[:, np.newaxis, :] - Y[np.newaxis, :, :] D = np.abs(D, D) return D.reshape((-1, X.shape[1])) def cosine_distances(X, Y=None): """Compute cosine distance between samples in X and Y. Cosine distance is defined as 1.0 minus the cosine similarity. Read more in the :ref:`User Guide `. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples_X, n_features) Matrix `X`. Y : {array-like, sparse matrix} of shape (n_samples_Y, n_features), \ default=None Matrix `Y`. Returns ------- distance matrix : ndarray of shape (n_samples_X, n_samples_Y) Returns the cosine distance between samples in X and Y. See Also -------- cosine_similarity : Compute cosine similarity between samples in X and Y. scipy.spatial.distance.cosine : Dense matrices only. """ # 1.0 - cosine_similarity(X, Y) without copy S = cosine_similarity(X, Y) S *= -1 S += 1 np.clip(S, 0, 2, out=S) if X is Y or Y is None: # Ensure that distances between vectors and themselves are set to 0.0. # This may not be the case due to floating point rounding errors. S[np.diag_indices_from(S)] = 0.0 return S # Paired distances def paired_euclidean_distances(X, Y): """Compute the paired euclidean distances between X and Y. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples, n_features) Input array/matrix X. Y : array-like of shape (n_samples, n_features) Input array/matrix Y. Returns ------- distances : ndarray of shape (n_samples,) Output array/matrix containing the calculated paired euclidean distances. """ X, Y = check_paired_arrays(X, Y) return row_norms(X - Y) def paired_manhattan_distances(X, Y): """Compute the paired L1 distances between X and Y. Distances are calculated between (X[0], Y[0]), (X[1], Y[1]), ..., (X[n_samples], Y[n_samples]). Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples, n_features) An array-like where each row is a sample and each column is a feature. Y : array-like of shape (n_samples, n_features) An array-like where each row is a sample and each column is a feature. Returns ------- distances : ndarray of shape (n_samples,) L1 paired distances between the row vectors of `X` and the row vectors of `Y`. Examples -------- >>> from sklearn.metrics.pairwise import paired_manhattan_distances >>> import numpy as np >>> X = np.array([[1, 1, 0], [0, 1, 0], [0, 0, 1]]) >>> Y = np.array([[0, 1, 0], [0, 0, 1], [0, 0, 0]]) >>> paired_manhattan_distances(X, Y) array([1., 2., 1.]) """ X, Y = check_paired_arrays(X, Y) diff = X - Y if issparse(diff): diff.data = np.abs(diff.data) return np.squeeze(np.array(diff.sum(axis=1))) else: return np.abs(diff).sum(axis=-1) def paired_cosine_distances(X, Y): """ Compute the paired cosine distances between X and Y. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples, n_features) An array where each row is a sample and each column is a feature. Y : array-like of shape (n_samples, n_features) An array where each row is a sample and each column is a feature. Returns ------- distances : ndarray of shape (n_samples,) Returns the distances between the row vectors of `X` and the row vectors of `Y`, where `distances[i]` is the distance between `X[i]` and `Y[i]`. Notes ----- The cosine distance is equivalent to the half the squared euclidean distance if each sample is normalized to unit norm. """ X, Y = check_paired_arrays(X, Y) return 0.5 * row_norms(normalize(X) - normalize(Y), squared=True) PAIRED_DISTANCES = { "cosine": paired_cosine_distances, "euclidean": paired_euclidean_distances, "l2": paired_euclidean_distances, "l1": paired_manhattan_distances, "manhattan": paired_manhattan_distances, "cityblock": paired_manhattan_distances, } def paired_distances(X, Y, *, metric="euclidean", **kwds): """ Compute the paired distances between X and Y. Compute the distances between (X[0], Y[0]), (X[1], Y[1]), etc... Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples, n_features) Array 1 for distance computation. Y : ndarray of shape (n_samples, n_features) Array 2 for distance computation. metric : str or callable, default="euclidean" The metric to use when calculating distance between instances in a feature array. If metric is a string, it must be one of the options specified in PAIRED_DISTANCES, including "euclidean", "manhattan", or "cosine". Alternatively, if metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays from `X` as input and return a value indicating the distance between them. **kwds : dict Unused parameters. Returns ------- distances : ndarray of shape (n_samples,) Returns the distances between the row vectors of `X` and the row vectors of `Y`. See Also -------- pairwise_distances : Computes the distance between every pair of samples. Examples -------- >>> from sklearn.metrics.pairwise import paired_distances >>> X = [[0, 1], [1, 1]] >>> Y = [[0, 1], [2, 1]] >>> paired_distances(X, Y) array([0., 1.]) """ if metric in PAIRED_DISTANCES: func = PAIRED_DISTANCES[metric] return func(X, Y) elif callable(metric): # Check the matrix first (it is usually done by the metric) X, Y = check_paired_arrays(X, Y) distances = np.zeros(len(X)) for i in range(len(X)): distances[i] = metric(X[i], Y[i]) return distances else: raise ValueError("Unknown distance %s" % metric) # Kernels def linear_kernel(X, Y=None, dense_output=True): """ Compute the linear kernel between X and Y. Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples_X, n_features) A feature array. Y : ndarray of shape (n_samples_Y, n_features), default=None An optional second feature array. If `None`, uses `Y=X`. dense_output : bool, default=True Whether to return dense output even when the input is sparse. If ``False``, the output is sparse if both input arrays are sparse. .. versionadded:: 0.20 Returns ------- Gram matrix : ndarray of shape (n_samples_X, n_samples_Y) The Gram matrix of the linear kernel, i.e. `X @ Y.T`. """ X, Y = check_pairwise_arrays(X, Y) return safe_sparse_dot(X, Y.T, dense_output=dense_output) def polynomial_kernel(X, Y=None, degree=3, gamma=None, coef0=1): """ Compute the polynomial kernel between X and Y. :math:`K(X, Y) = (gamma + coef0)^{degree}` Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples_X, n_features) A feature array. Y : ndarray of shape (n_samples_Y, n_features), default=None An optional second feature array. If `None`, uses `Y=X`. degree : int, default=3 Kernel degree. gamma : float, default=None Coefficient of the vector inner product. If None, defaults to 1.0 / n_features. coef0 : float, default=1 Constant offset added to scaled inner product. Returns ------- Gram matrix : ndarray of shape (n_samples_X, n_samples_Y) The polynomial kernel. """ X, Y = check_pairwise_arrays(X, Y) if gamma is None: gamma = 1.0 / X.shape[1] K = safe_sparse_dot(X, Y.T, dense_output=True) K *= gamma K += coef0 K **= degree return K def sigmoid_kernel(X, Y=None, gamma=None, coef0=1): """Compute the sigmoid kernel between X and Y. K(X, Y) = tanh(gamma + coef0) Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples_X, n_features) A feature array. Y : ndarray of shape (n_samples_Y, n_features), default=None An optional second feature array. If `None`, uses `Y=X`. gamma : float, default=None Coefficient of the vector inner product. If None, defaults to 1.0 / n_features. coef0 : float, default=1 Constant offset added to scaled inner product. Returns ------- Gram matrix : ndarray of shape (n_samples_X, n_samples_Y) Sigmoid kernel between two arrays. """ X, Y = check_pairwise_arrays(X, Y) if gamma is None: gamma = 1.0 / X.shape[1] K = safe_sparse_dot(X, Y.T, dense_output=True) K *= gamma K += coef0 np.tanh(K, K) # compute tanh in-place return K def rbf_kernel(X, Y=None, gamma=None): """Compute the rbf (gaussian) kernel between X and Y. K(x, y) = exp(-gamma ||x-y||^2) for each pair of rows x in X and y in Y. Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples_X, n_features) A feature array. Y : ndarray of shape (n_samples_Y, n_features), default=None An optional second feature array. If `None`, uses `Y=X`. gamma : float, default=None If None, defaults to 1.0 / n_features. Returns ------- kernel_matrix : ndarray of shape (n_samples_X, n_samples_Y) The RBF kernel. """ X, Y = check_pairwise_arrays(X, Y) if gamma is None: gamma = 1.0 / X.shape[1] K = euclidean_distances(X, Y, squared=True) K *= -gamma np.exp(K, K) # exponentiate K in-place return K def laplacian_kernel(X, Y=None, gamma=None): """Compute the laplacian kernel between X and Y. The laplacian kernel is defined as:: K(x, y) = exp(-gamma ||x-y||_1) for each pair of rows x in X and y in Y. Read more in the :ref:`User Guide `. .. versionadded:: 0.17 Parameters ---------- X : ndarray of shape (n_samples_X, n_features) A feature array. Y : ndarray of shape (n_samples_Y, n_features), default=None An optional second feature array. If `None`, uses `Y=X`. gamma : float, default=None If None, defaults to 1.0 / n_features. Returns ------- kernel_matrix : ndarray of shape (n_samples_X, n_samples_Y) The kernel matrix. """ X, Y = check_pairwise_arrays(X, Y) if gamma is None: gamma = 1.0 / X.shape[1] K = -gamma * manhattan_distances(X, Y) np.exp(K, K) # exponentiate K in-place return K def cosine_similarity(X, Y=None, dense_output=True): """Compute cosine similarity between samples in X and Y. Cosine similarity, or the cosine kernel, computes similarity as the normalized dot product of X and Y: K(X, Y) = / (||X||*||Y||) On L2-normalized data, this function is equivalent to linear_kernel. Read more in the :ref:`User Guide `. Parameters ---------- X : {ndarray, sparse matrix} of shape (n_samples_X, n_features) Input data. Y : {ndarray, sparse matrix} of shape (n_samples_Y, n_features), \ default=None Input data. If ``None``, the output will be the pairwise similarities between all samples in ``X``. dense_output : bool, default=True Whether to return dense output even when the input is sparse. If ``False``, the output is sparse if both input arrays are sparse. .. versionadded:: 0.17 parameter ``dense_output`` for dense output. Returns ------- kernel matrix : ndarray of shape (n_samples_X, n_samples_Y) Returns the cosine similarity between samples in X and Y. """ # to avoid recursive import X, Y = check_pairwise_arrays(X, Y) X_normalized = normalize(X, copy=True) if X is Y: Y_normalized = X_normalized else: Y_normalized = normalize(Y, copy=True) K = safe_sparse_dot(X_normalized, Y_normalized.T, dense_output=dense_output) return K def additive_chi2_kernel(X, Y=None): """Compute the additive chi-squared kernel between observations in X and Y. The chi-squared kernel is computed between each pair of rows in X and Y. X and Y have to be non-negative. This kernel is most commonly applied to histograms. The chi-squared kernel is given by:: k(x, y) = -Sum [(x - y)^2 / (x + y)] It can be interpreted as a weighted difference per entry. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples_X, n_features) A feature array. Y : ndarray of shape (n_samples_Y, n_features), default=None An optional second feature array. If `None`, uses `Y=X`. Returns ------- kernel_matrix : ndarray of shape (n_samples_X, n_samples_Y) The kernel matrix. See Also -------- chi2_kernel : The exponentiated version of the kernel, which is usually preferable. sklearn.kernel_approximation.AdditiveChi2Sampler : A Fourier approximation to this kernel. Notes ----- As the negative of a distance, this kernel is only conditionally positive definite. References ---------- * Zhang, J. and Marszalek, M. and Lazebnik, S. and Schmid, C. Local features and kernels for classification of texture and object categories: A comprehensive study International Journal of Computer Vision 2007 https://hal.archives-ouvertes.fr/hal-00171412/document """ if issparse(X) or issparse(Y): raise ValueError("additive_chi2 does not support sparse matrices.") X, Y = check_pairwise_arrays(X, Y) if (X < 0).any(): raise ValueError("X contains negative values.") if Y is not X and (Y < 0).any(): raise ValueError("Y contains negative values.") result = np.zeros((X.shape[0], Y.shape[0]), dtype=X.dtype) _chi2_kernel_fast(X, Y, result) return result def chi2_kernel(X, Y=None, gamma=1.0): """Compute the exponential chi-squared kernel between X and Y. The chi-squared kernel is computed between each pair of rows in X and Y. X and Y have to be non-negative. This kernel is most commonly applied to histograms. The chi-squared kernel is given by:: k(x, y) = exp(-gamma Sum [(x - y)^2 / (x + y)]) It can be interpreted as a weighted difference per entry. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like of shape (n_samples_X, n_features) A feature array. Y : ndarray of shape (n_samples_Y, n_features), default=None An optional second feature array. If `None`, uses `Y=X`. gamma : float, default=1 Scaling parameter of the chi2 kernel. Returns ------- kernel_matrix : ndarray of shape (n_samples_X, n_samples_Y) The kernel matrix. See Also -------- additive_chi2_kernel : The additive version of this kernel. sklearn.kernel_approximation.AdditiveChi2Sampler : A Fourier approximation to the additive version of this kernel. References ---------- * Zhang, J. and Marszalek, M. and Lazebnik, S. and Schmid, C. Local features and kernels for classification of texture and object categories: A comprehensive study International Journal of Computer Vision 2007 https://hal.archives-ouvertes.fr/hal-00171412/document """ K = additive_chi2_kernel(X, Y) K *= gamma return np.exp(K, K) # Helper functions - distance PAIRWISE_DISTANCE_FUNCTIONS = { # If updating this dictionary, update the doc in both distance_metrics() # and also in pairwise_distances()! "cityblock": manhattan_distances, "cosine": cosine_distances, "euclidean": euclidean_distances, "haversine": haversine_distances, "l2": euclidean_distances, "l1": manhattan_distances, "manhattan": manhattan_distances, "precomputed": None, # HACK: precomputed is always allowed, never called "nan_euclidean": nan_euclidean_distances, } def distance_metrics(): """Valid metrics for pairwise_distances. This function simply returns the valid pairwise distance metrics. It exists to allow for a description of the mapping for each of the valid strings. The valid distance metrics, and the function they map to, are: =============== ======================================== metric Function =============== ======================================== 'cityblock' metrics.pairwise.manhattan_distances 'cosine' metrics.pairwise.cosine_distances 'euclidean' metrics.pairwise.euclidean_distances 'haversine' metrics.pairwise.haversine_distances 'l1' metrics.pairwise.manhattan_distances 'l2' metrics.pairwise.euclidean_distances 'manhattan' metrics.pairwise.manhattan_distances 'nan_euclidean' metrics.pairwise.nan_euclidean_distances =============== ======================================== Read more in the :ref:`User Guide `. Returns ------- distance_metrics : dict Returns valid metrics for pairwise_distances. """ return PAIRWISE_DISTANCE_FUNCTIONS def _dist_wrapper(dist_func, dist_matrix, slice_, *args, **kwargs): """Write in-place to a slice of a distance matrix.""" dist_matrix[:, slice_] = dist_func(*args, **kwargs) def _parallel_pairwise(X, Y, func, n_jobs, **kwds): """Break the pairwise matrix in n_jobs even slices and compute them in parallel.""" if Y is None: Y = X X, Y, dtype = _return_float_dtype(X, Y) if effective_n_jobs(n_jobs) == 1: return func(X, Y, **kwds) # enforce a threading backend to prevent data communication overhead fd = delayed(_dist_wrapper) ret = np.empty((X.shape[0], Y.shape[0]), dtype=dtype, order="F") Parallel(backend="threading", n_jobs=n_jobs)( fd(func, ret, s, X, Y[s], **kwds) for s in gen_even_slices(_num_samples(Y), effective_n_jobs(n_jobs)) ) if (X is Y or Y is None) and func is euclidean_distances: # zeroing diagonal for euclidean norm. # TODO: do it also for other norms. np.fill_diagonal(ret, 0) return ret def _pairwise_callable(X, Y, metric, force_all_finite=True, **kwds): """Handle the callable case for pairwise_{distances,kernels}.""" X, Y = check_pairwise_arrays(X, Y, force_all_finite=force_all_finite) if X is Y: # Only calculate metric for upper triangle out = np.zeros((X.shape[0], Y.shape[0]), dtype="float") iterator = itertools.combinations(range(X.shape[0]), 2) for i, j in iterator: out[i, j] = metric(X[i], Y[j], **kwds) # Make symmetric # NB: out += out.T will produce incorrect results out = out + out.T # Calculate diagonal # NB: nonzero diagonals are allowed for both metrics and kernels for i in range(X.shape[0]): x = X[i] out[i, i] = metric(x, x, **kwds) else: # Calculate all cells out = np.empty((X.shape[0], Y.shape[0]), dtype="float") iterator = itertools.product(range(X.shape[0]), range(Y.shape[0])) for i, j in iterator: out[i, j] = metric(X[i], Y[j], **kwds) return out _VALID_METRICS = [ "euclidean", "l2", "l1", "manhattan", "cityblock", "braycurtis", "canberra", "chebyshev", "correlation", "cosine", "dice", "hamming", "jaccard", "kulsinski", "mahalanobis", "matching", "minkowski", "rogerstanimoto", "russellrao", "seuclidean", "sokalmichener", "sokalsneath", "sqeuclidean", "yule", "wminkowski", "nan_euclidean", "haversine", ] _NAN_METRICS = ["nan_euclidean"] def _check_chunk_size(reduced, chunk_size): """Checks chunk is a sequence of expected size or a tuple of same.""" if reduced is None: return is_tuple = isinstance(reduced, tuple) if not is_tuple: reduced = (reduced,) if any(isinstance(r, tuple) or not hasattr(r, "__iter__") for r in reduced): raise TypeError( "reduce_func returned %r. Expected sequence(s) of length %d." % (reduced if is_tuple else reduced[0], chunk_size) ) if any(_num_samples(r) != chunk_size for r in reduced): actual_size = tuple(_num_samples(r) for r in reduced) raise ValueError( "reduce_func returned object of length %s. " "Expected same length as input: %d." % (actual_size if is_tuple else actual_size[0], chunk_size) ) def _precompute_metric_params(X, Y, metric=None, **kwds): """Precompute data-derived metric parameters if not provided.""" if metric == "seuclidean" and "V" not in kwds: # There is a bug in scipy < 1.5 that will cause a crash if # X.dtype != np.double (float64). See PR #15730 dtype = np.float64 if sp_version < parse_version("1.5") else None if X is Y: V = np.var(X, axis=0, ddof=1, dtype=dtype) else: raise ValueError( "The 'V' parameter is required for the seuclidean metric " "when Y is passed." ) return {"V": V} if metric == "mahalanobis" and "VI" not in kwds: if X is Y: VI = np.linalg.inv(np.cov(X.T)).T else: raise ValueError( "The 'VI' parameter is required for the mahalanobis metric " "when Y is passed." ) return {"VI": VI} return {} def pairwise_distances_chunked( X, Y=None, *, reduce_func=None, metric="euclidean", n_jobs=None, working_memory=None, **kwds, ): """Generate a distance matrix chunk by chunk with optional reduction. In cases where not all of a pairwise distance matrix needs to be stored at once, this is used to calculate pairwise distances in ``working_memory``-sized chunks. If ``reduce_func`` is given, it is run on each chunk and its return values are concatenated into lists, arrays or sparse matrices. Parameters ---------- X : ndarray of shape (n_samples_X, n_samples_X) or \ (n_samples_X, n_features) Array of pairwise distances between samples, or a feature array. The shape the array should be (n_samples_X, n_samples_X) if metric='precomputed' and (n_samples_X, n_features) otherwise. Y : ndarray of shape (n_samples_Y, n_features), default=None An optional second feature array. Only allowed if metric != "precomputed". reduce_func : callable, default=None The function which is applied on each chunk of the distance matrix, reducing it to needed values. ``reduce_func(D_chunk, start)`` is called repeatedly, where ``D_chunk`` is a contiguous vertical slice of the pairwise distance matrix, starting at row ``start``. It should return one of: None; an array, a list, or a sparse matrix of length ``D_chunk.shape[0]``; or a tuple of such objects. Returning None is useful for in-place operations, rather than reductions. If None, pairwise_distances_chunked returns a generator of vertical chunks of the distance matrix. metric : str or callable, default='euclidean' The metric to use when calculating distance between instances in a feature array. If metric is a string, it must be one of the options allowed by scipy.spatial.distance.pdist for its metric parameter, or a metric listed in pairwise.PAIRWISE_DISTANCE_FUNCTIONS. If metric is "precomputed", X is assumed to be a distance matrix. Alternatively, if metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays from X as input and return a value indicating the distance between them. n_jobs : int, default=None The number of jobs to use for the computation. This works by breaking down the pairwise matrix into n_jobs even slices and computing them in parallel. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. working_memory : int, default=None The sought maximum memory for temporary distance matrix chunks. When None (default), the value of ``sklearn.get_config()['working_memory']`` is used. **kwds : optional keyword parameters Any further parameters are passed directly to the distance function. If using a scipy.spatial.distance metric, the parameters are still metric dependent. See the scipy docs for usage examples. Yields ------ D_chunk : {ndarray, sparse matrix} A contiguous slice of distance matrix, optionally processed by ``reduce_func``. Examples -------- Without reduce_func: >>> import numpy as np >>> from sklearn.metrics import pairwise_distances_chunked >>> X = np.random.RandomState(0).rand(5, 3) >>> D_chunk = next(pairwise_distances_chunked(X)) >>> D_chunk array([[0. ..., 0.29..., 0.41..., 0.19..., 0.57...], [0.29..., 0. ..., 0.57..., 0.41..., 0.76...], [0.41..., 0.57..., 0. ..., 0.44..., 0.90...], [0.19..., 0.41..., 0.44..., 0. ..., 0.51...], [0.57..., 0.76..., 0.90..., 0.51..., 0. ...]]) Retrieve all neighbors and average distance within radius r: >>> r = .2 >>> def reduce_func(D_chunk, start): ... neigh = [np.flatnonzero(d < r) for d in D_chunk] ... avg_dist = (D_chunk * (D_chunk < r)).mean(axis=1) ... return neigh, avg_dist >>> gen = pairwise_distances_chunked(X, reduce_func=reduce_func) >>> neigh, avg_dist = next(gen) >>> neigh [array([0, 3]), array([1]), array([2]), array([0, 3]), array([4])] >>> avg_dist array([0.039..., 0. , 0. , 0.039..., 0. ]) Where r is defined per sample, we need to make use of ``start``: >>> r = [.2, .4, .4, .3, .1] >>> def reduce_func(D_chunk, start): ... neigh = [np.flatnonzero(d < r[i]) ... for i, d in enumerate(D_chunk, start)] ... return neigh >>> neigh = next(pairwise_distances_chunked(X, reduce_func=reduce_func)) >>> neigh [array([0, 3]), array([0, 1]), array([2]), array([0, 3]), array([4])] Force row-by-row generation by reducing ``working_memory``: >>> gen = pairwise_distances_chunked(X, reduce_func=reduce_func, ... working_memory=0) >>> next(gen) [array([0, 3])] >>> next(gen) [array([0, 1])] """ n_samples_X = _num_samples(X) if metric == "precomputed": slices = (slice(0, n_samples_X),) else: if Y is None: Y = X # We get as many rows as possible within our working_memory budget to # store len(Y) distances in each row of output. # # Note: # - this will get at least 1 row, even if 1 row of distances will # exceed working_memory. # - this does not account for any temporary memory usage while # calculating distances (e.g. difference of vectors in manhattan # distance. chunk_n_rows = get_chunk_n_rows( row_bytes=8 * _num_samples(Y), max_n_rows=n_samples_X, working_memory=working_memory, ) slices = gen_batches(n_samples_X, chunk_n_rows) # precompute data-derived metric params params = _precompute_metric_params(X, Y, metric=metric, **kwds) kwds.update(**params) for sl in slices: if sl.start == 0 and sl.stop == n_samples_X: X_chunk = X # enable optimised paths for X is Y else: X_chunk = X[sl] D_chunk = pairwise_distances(X_chunk, Y, metric=metric, n_jobs=n_jobs, **kwds) if (X is Y or Y is None) and PAIRWISE_DISTANCE_FUNCTIONS.get( metric, None ) is euclidean_distances: # zeroing diagonal, taking care of aliases of "euclidean", # i.e. "l2" D_chunk.flat[sl.start :: _num_samples(X) + 1] = 0 if reduce_func is not None: chunk_size = D_chunk.shape[0] D_chunk = reduce_func(D_chunk, sl.start) _check_chunk_size(D_chunk, chunk_size) yield D_chunk def pairwise_distances( X, Y=None, metric="euclidean", *, n_jobs=None, force_all_finite=True, **kwds ): """Compute the distance matrix from a vector array X and optional Y. This method takes either a vector array or a distance matrix, and returns a distance matrix. If the input is a vector array, the distances are computed. If the input is a distances matrix, it is returned instead. This method provides a safe way to take a distance matrix as input, while preserving compatibility with many other algorithms that take a vector array. If Y is given (default is None), then the returned matrix is the pairwise distance between the arrays from both X and Y. Valid values for metric are: - From scikit-learn: ['cityblock', 'cosine', 'euclidean', 'l1', 'l2', 'manhattan']. These metrics support sparse matrix inputs. ['nan_euclidean'] but it does not yet support sparse matrices. - From scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev', 'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'] See the documentation for scipy.spatial.distance for details on these metrics. These metrics do not support sparse matrix inputs. Note that in the case of 'cityblock', 'cosine' and 'euclidean' (which are valid scipy.spatial.distance metrics), the scikit-learn implementation will be used, which is faster and has support for sparse matrices (except for 'cityblock'). For a verbose description of the metrics from scikit-learn, see :func:`sklearn.metrics.pairwise.distance_metrics` function. Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples_X, n_samples_X) or \ (n_samples_X, n_features) Array of pairwise distances between samples, or a feature array. The shape of the array should be (n_samples_X, n_samples_X) if metric == "precomputed" and (n_samples_X, n_features) otherwise. Y : ndarray of shape (n_samples_Y, n_features), default=None An optional second feature array. Only allowed if metric != "precomputed". metric : str or callable, default='euclidean' The metric to use when calculating distance between instances in a feature array. If metric is a string, it must be one of the options allowed by scipy.spatial.distance.pdist for its metric parameter, or a metric listed in ``pairwise.PAIRWISE_DISTANCE_FUNCTIONS``. If metric is "precomputed", X is assumed to be a distance matrix. Alternatively, if metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two arrays from X as input and return a value indicating the distance between them. n_jobs : int, default=None The number of jobs to use for the computation. This works by breaking down the pairwise matrix into n_jobs even slices and computing them in parallel. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. force_all_finite : bool or 'allow-nan', default=True Whether to raise an error on np.inf, np.nan, pd.NA in array. Ignored for a metric listed in ``pairwise.PAIRWISE_DISTANCE_FUNCTIONS``. The possibilities are: - True: Force all values of array to be finite. - False: accepts np.inf, np.nan, pd.NA in array. - 'allow-nan': accepts only np.nan and pd.NA values in array. Values cannot be infinite. .. versionadded:: 0.22 ``force_all_finite`` accepts the string ``'allow-nan'``. .. versionchanged:: 0.23 Accepts `pd.NA` and converts it into `np.nan`. **kwds : optional keyword parameters Any further parameters are passed directly to the distance function. If using a scipy.spatial.distance metric, the parameters are still metric dependent. See the scipy docs for usage examples. Returns ------- D : ndarray of shape (n_samples_X, n_samples_X) or \ (n_samples_X, n_samples_Y) A distance matrix D such that D_{i, j} is the distance between the ith and jth vectors of the given matrix X, if Y is None. If Y is not None, then D_{i, j} is the distance between the ith array from X and the jth array from Y. See Also -------- pairwise_distances_chunked : Performs the same calculation as this function, but returns a generator of chunks of the distance matrix, in order to limit memory usage. paired_distances : Computes the distances between corresponding elements of two arrays. """ if ( metric not in _VALID_METRICS and not callable(metric) and metric != "precomputed" ): raise ValueError( "Unknown metric %s. Valid metrics are %s, or 'precomputed', or a callable" % (metric, _VALID_METRICS) ) if metric == "precomputed": X, _ = check_pairwise_arrays( X, Y, precomputed=True, force_all_finite=force_all_finite ) whom = ( "`pairwise_distances`. Precomputed distance " " need to have non-negative values." ) check_non_negative(X, whom=whom) return X elif metric in PAIRWISE_DISTANCE_FUNCTIONS: func = PAIRWISE_DISTANCE_FUNCTIONS[metric] elif callable(metric): func = partial( _pairwise_callable, metric=metric, force_all_finite=force_all_finite, **kwds ) else: if issparse(X) or issparse(Y): raise TypeError("scipy distance metrics do not support sparse matrices.") dtype = bool if metric in PAIRWISE_BOOLEAN_FUNCTIONS else None if dtype == bool and (X.dtype != bool or (Y is not None and Y.dtype != bool)): msg = "Data was converted to boolean for metric %s" % metric warnings.warn(msg, DataConversionWarning) X, Y = check_pairwise_arrays( X, Y, dtype=dtype, force_all_finite=force_all_finite ) # precompute data-derived metric params params = _precompute_metric_params(X, Y, metric=metric, **kwds) kwds.update(**params) if effective_n_jobs(n_jobs) == 1 and X is Y: return distance.squareform(distance.pdist(X, metric=metric, **kwds)) func = partial(distance.cdist, metric=metric, **kwds) return _parallel_pairwise(X, Y, func, n_jobs, **kwds) # These distances require boolean arrays, when using scipy.spatial.distance PAIRWISE_BOOLEAN_FUNCTIONS = [ "dice", "jaccard", "kulsinski", "matching", "rogerstanimoto", "russellrao", "sokalmichener", "sokalsneath", "yule", ] # Helper functions - distance PAIRWISE_KERNEL_FUNCTIONS = { # If updating this dictionary, update the doc in both distance_metrics() # and also in pairwise_distances()! "additive_chi2": additive_chi2_kernel, "chi2": chi2_kernel, "linear": linear_kernel, "polynomial": polynomial_kernel, "poly": polynomial_kernel, "rbf": rbf_kernel, "laplacian": laplacian_kernel, "sigmoid": sigmoid_kernel, "cosine": cosine_similarity, } def kernel_metrics(): """Valid metrics for pairwise_kernels. This function simply returns the valid pairwise distance metrics. It exists, however, to allow for a verbose description of the mapping for each of the valid strings. The valid distance metrics, and the function they map to, are: =============== ======================================== metric Function =============== ======================================== 'additive_chi2' sklearn.pairwise.additive_chi2_kernel 'chi2' sklearn.pairwise.chi2_kernel 'linear' sklearn.pairwise.linear_kernel 'poly' sklearn.pairwise.polynomial_kernel 'polynomial' sklearn.pairwise.polynomial_kernel 'rbf' sklearn.pairwise.rbf_kernel 'laplacian' sklearn.pairwise.laplacian_kernel 'sigmoid' sklearn.pairwise.sigmoid_kernel 'cosine' sklearn.pairwise.cosine_similarity =============== ======================================== Read more in the :ref:`User Guide `. Returns ------- kernal_metrics : dict Returns valid metrics for pairwise_kernels. """ return PAIRWISE_KERNEL_FUNCTIONS KERNEL_PARAMS = { "additive_chi2": (), "chi2": frozenset(["gamma"]), "cosine": (), "linear": (), "poly": frozenset(["gamma", "degree", "coef0"]), "polynomial": frozenset(["gamma", "degree", "coef0"]), "rbf": frozenset(["gamma"]), "laplacian": frozenset(["gamma"]), "sigmoid": frozenset(["gamma", "coef0"]), } def pairwise_kernels( X, Y=None, metric="linear", *, filter_params=False, n_jobs=None, **kwds ): """Compute the kernel between arrays X and optional array Y. This method takes either a vector array or a kernel matrix, and returns a kernel matrix. If the input is a vector array, the kernels are computed. If the input is a kernel matrix, it is returned instead. This method provides a safe way to take a kernel matrix as input, while preserving compatibility with many other algorithms that take a vector array. If Y is given (default is None), then the returned matrix is the pairwise kernel between the arrays from both X and Y. Valid values for metric are: ['additive_chi2', 'chi2', 'linear', 'poly', 'polynomial', 'rbf', 'laplacian', 'sigmoid', 'cosine'] Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples_X, n_samples_X) or (n_samples_X, n_features) Array of pairwise kernels between samples, or a feature array. The shape of the array should be (n_samples_X, n_samples_X) if metric == "precomputed" and (n_samples_X, n_features) otherwise. Y : ndarray of shape (n_samples_Y, n_features), default=None A second feature array only if X has shape (n_samples_X, n_features). metric : str or callable, default="linear" The metric to use when calculating kernel between instances in a feature array. If metric is a string, it must be one of the metrics in pairwise.PAIRWISE_KERNEL_FUNCTIONS. If metric is "precomputed", X is assumed to be a kernel matrix. Alternatively, if metric is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two rows from X as input and return the corresponding kernel value as a single number. This means that callables from :mod:`sklearn.metrics.pairwise` are not allowed, as they operate on matrices, not single samples. Use the string identifying the kernel instead. filter_params : bool, default=False Whether to filter invalid parameters or not. n_jobs : int, default=None The number of jobs to use for the computation. This works by breaking down the pairwise matrix into n_jobs even slices and computing them in parallel. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. **kwds : optional keyword parameters Any further parameters are passed directly to the kernel function. Returns ------- K : ndarray of shape (n_samples_X, n_samples_X) or (n_samples_X, n_samples_Y) A kernel matrix K such that K_{i, j} is the kernel between the ith and jth vectors of the given matrix X, if Y is None. If Y is not None, then K_{i, j} is the kernel between the ith array from X and the jth array from Y. Notes ----- If metric is 'precomputed', Y is ignored and X is returned. """ # import GPKernel locally to prevent circular imports from ..gaussian_process.kernels import Kernel as GPKernel if metric == "precomputed": X, _ = check_pairwise_arrays(X, Y, precomputed=True) return X elif isinstance(metric, GPKernel): func = metric.__call__ elif metric in PAIRWISE_KERNEL_FUNCTIONS: if filter_params: kwds = {k: kwds[k] for k in kwds if k in KERNEL_PARAMS[metric]} func = PAIRWISE_KERNEL_FUNCTIONS[metric] elif callable(metric): func = partial(_pairwise_callable, metric=metric, **kwds) else: raise ValueError("Unknown kernel %r" % metric) return _parallel_pairwise(X, Y, func, n_jobs, **kwds)