# -*- coding: utf-8 -*- # Authors: Alexandre Gramfort # Mathieu Blondel # Robert Layton # Andreas Mueller # Philippe Gervais # Lars Buitinck # Joel Nothman # License: BSD 3 clause import itertools import numpy as np from scipy.spatial import distance from scipy.sparse import csr_matrix from scipy.sparse import issparse from ..utils import check_array from ..utils import gen_even_slices from ..utils import gen_batches from ..utils.fixes import partial from ..utils.extmath import row_norms, safe_sparse_dot from ..preprocessing import normalize from ..externals.joblib import Parallel from ..externals.joblib import delayed from ..externals.joblib.parallel import cpu_count from .pairwise_fast import _chi2_kernel_fast, _sparse_manhattan # 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 = np.float return X, Y, dtype def check_pairwise_arrays(X, Y, precomputed=False, dtype=None): """ 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}, shape (n_samples_a, n_features) Y : {array-like, sparse matrix}, shape (n_samples_b, n_features) precomputed : bool True if X is to be treated as precomputed distances to the samples in Y. dtype : string, type, list of types or None (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 Returns ------- safe_X : {array-like, sparse matrix}, shape (n_samples_a, n_features) An array equal to X, guaranteed to be a numpy array. safe_Y : {array-like, sparse matrix}, shape (n_samples_b, 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) warn_on_dtype = dtype is not None 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='csr', dtype=dtype, warn_on_dtype=warn_on_dtype, estimator=estimator) else: X = check_array(X, accept_sparse='csr', dtype=dtype, warn_on_dtype=warn_on_dtype, estimator=estimator) Y = check_array(Y, accept_sparse='csr', dtype=dtype, warn_on_dtype=warn_on_dtype, 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}, shape (n_samples_a, n_features) Y : {array-like, sparse matrix}, shape (n_samples_b, n_features) Returns ------- safe_X : {array-like, sparse matrix}, shape (n_samples_a, n_features) An array equal to X, guaranteed to be a numpy array. safe_Y : {array-like, sparse matrix}, shape (n_samples_b, 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): """ Considering the rows of X (and Y=X) as vectors, compute the distance matrix between each pair of vectors. 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, and 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}, shape (n_samples_1, n_features) Y : {array-like, sparse matrix}, shape (n_samples_2, n_features) Y_norm_squared : array-like, shape (n_samples_2, ), optional Pre-computed dot-products of vectors in Y (e.g., ``(Y**2).sum(axis=1)``) squared : boolean, optional Return squared Euclidean distances. X_norm_squared : array-like, shape = [n_samples_1], optional Pre-computed dot-products of vectors in X (e.g., ``(X**2).sum(axis=1)``) Returns ------- distances : {array, sparse matrix}, shape (n_samples_1, n_samples_2) 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]]) See also -------- paired_distances : distances betweens pairs of elements of X and Y. """ X, Y = check_pairwise_arrays(X, Y) if X_norm_squared is not None: XX = check_array(X_norm_squared) if XX.shape == (1, X.shape[0]): XX = XX.T elif XX.shape != (X.shape[0], 1): raise ValueError( "Incompatible dimensions for X and X_norm_squared") else: XX = row_norms(X, squared=True)[:, np.newaxis] if X is Y: # shortcut in the common case euclidean_distances(X, X) YY = XX.T elif Y_norm_squared is not None: YY = np.atleast_2d(Y_norm_squared) if YY.shape != (1, Y.shape[0]): raise ValueError( "Incompatible dimensions for Y and Y_norm_squared") else: YY = row_norms(Y, squared=True)[np.newaxis, :] distances = safe_sparse_dot(X, Y.T, dense_output=True) distances *= -2 distances += XX distances += YY np.maximum(distances, 0, 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. distances.flat[::distances.shape[0] + 1] = 0.0 return distances if squared else np.sqrt(distances, out=distances) def pairwise_distances_argmin_min(X, Y, axis=1, metric="euclidean", batch_size=500, 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, Y : {array-like, sparse matrix} Arrays containing points. Respective shapes (n_samples1, n_features) and (n_samples2, n_features) batch_size : integer To reduce memory consumption over the naive solution, data are processed in batches, comprising batch_size rows of X and batch_size rows of Y. The default value is quite conservative, but can be changed for fine-tuning. The larger the number, the larger the memory usage. metric : string 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', 'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'] See the documentation for scipy.spatial.distance for details on these metrics. metric_kwargs : dict, optional Keyword arguments to pass to specified metric function. axis : int, optional, default 1 Axis along which the argmin and distances are to be computed. Returns ------- argmin : numpy.ndarray Y[argmin[i], :] is the row in Y that is closest to X[i, :]. distances : numpy.ndarray distances[i] is the distance between the i-th row in X and the argmin[i]-th row in Y. See also -------- sklearn.metrics.pairwise_distances sklearn.metrics.pairwise_distances_argmin """ dist_func = None if metric in PAIRWISE_DISTANCE_FUNCTIONS: dist_func = PAIRWISE_DISTANCE_FUNCTIONS[metric] elif not callable(metric) and not isinstance(metric, str): raise ValueError("'metric' must be a string or a callable") X, Y = check_pairwise_arrays(X, Y) if metric_kwargs is None: metric_kwargs = {} if axis == 0: X, Y = Y, X # Allocate output arrays indices = np.empty(X.shape[0], dtype=np.intp) values = np.empty(X.shape[0]) values.fill(np.infty) for chunk_x in gen_batches(X.shape[0], batch_size): X_chunk = X[chunk_x, :] for chunk_y in gen_batches(Y.shape[0], batch_size): Y_chunk = Y[chunk_y, :] if dist_func is not None: if metric == 'euclidean': # special case, for speed d_chunk = safe_sparse_dot(X_chunk, Y_chunk.T, dense_output=True) d_chunk *= -2 d_chunk += row_norms(X_chunk, squared=True)[:, np.newaxis] d_chunk += row_norms(Y_chunk, squared=True)[np.newaxis, :] np.maximum(d_chunk, 0, d_chunk) else: d_chunk = dist_func(X_chunk, Y_chunk, **metric_kwargs) else: d_chunk = pairwise_distances(X_chunk, Y_chunk, metric=metric, **metric_kwargs) # Update indices and minimum values using chunk min_indices = d_chunk.argmin(axis=1) min_values = d_chunk[np.arange(chunk_x.stop - chunk_x.start), min_indices] flags = values[chunk_x] > min_values indices[chunk_x][flags] = min_indices[flags] + chunk_y.start values[chunk_x][flags] = min_values[flags] if metric == "euclidean" and not metric_kwargs.get("squared", False): np.sqrt(values, values) return indices, values def pairwise_distances_argmin(X, Y, axis=1, metric="euclidean", batch_size=500, 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 Arrays containing points. Respective shapes (n_samples1, n_features) and (n_samples2, n_features) Y : array-like Arrays containing points. Respective shapes (n_samples1, n_features) and (n_samples2, n_features) batch_size : integer To reduce memory consumption over the naive solution, data are processed in batches, comprising batch_size rows of X and batch_size rows of Y. The default value is quite conservative, but can be changed for fine-tuning. The larger the number, the larger the memory usage. metric : string or callable 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', 'matching', 'minkowski', 'rogerstanimoto', 'russellrao', 'seuclidean', 'sokalmichener', 'sokalsneath', 'sqeuclidean', 'yule'] See the documentation for scipy.spatial.distance for details on these metrics. metric_kwargs : dict keyword arguments to pass to specified metric function. axis : int, optional, default 1 Axis along which the argmin and distances are to be computed. Returns ------- argmin : numpy.ndarray Y[argmin[i], :] is the row in Y that is closest to X[i, :]. See also -------- sklearn.metrics.pairwise_distances sklearn.metrics.pairwise_distances_argmin_min """ if metric_kwargs is None: metric_kwargs = {} return pairwise_distances_argmin_min(X, Y, axis, metric, batch_size, metric_kwargs)[0] def manhattan_distances(X, Y=None, sum_over_features=True, size_threshold=5e8): """ 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 An array with shape (n_samples_X, n_features). Y : array_like, optional An array with shape (n_samples_Y, n_features). 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. size_threshold : int, default=5e8 Unused parameter. Returns ------- D : array 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. Examples -------- >>> from sklearn.metrics.pairwise import manhattan_distances >>> manhattan_distances([[3]], [[3]])#doctest:+ELLIPSIS array([[ 0.]]) >>> manhattan_distances([[3]], [[2]])#doctest:+ELLIPSIS array([[ 1.]]) >>> manhattan_distances([[2]], [[3]])#doctest:+ELLIPSIS array([[ 1.]]) >>> manhattan_distances([[1, 2], [3, 4]],\ [[1, 2], [0, 3]])#doctest:+ELLIPSIS array([[ 0., 2.], [ 4., 4.]]) >>> import numpy as np >>> X = np.ones((1, 2)) >>> y = 2 * np.ones((2, 2)) >>> manhattan_distances(X, y, sum_over_features=False)#doctest:+ELLIPSIS array([[ 1., 1.], [ 1., 1.]]...) """ 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) D = np.zeros((X.shape[0], Y.shape[0])) _sparse_manhattan(X.data, X.indices, X.indptr, Y.data, Y.indices, Y.indptr, X.shape[1], 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 with shape (n_samples_X, n_features). Y : array_like, sparse matrix (optional) with shape (n_samples_Y, n_features). Returns ------- distance matrix : array An array with shape (n_samples_X, n_samples_Y). See also -------- sklearn.metrics.pairwise.cosine_similarity scipy.spatial.distance.cosine (dense matrices only) """ # 1.0 - cosine_similarity(X, Y) without copy S = cosine_similarity(X, Y) S *= -1 S += 1 return S # Paired distances def paired_euclidean_distances(X, Y): """ Computes the paired euclidean distances between X and Y Read more in the :ref:`User Guide `. Parameters ---------- X : array-like, shape (n_samples, n_features) Y : array-like, shape (n_samples, n_features) Returns ------- distances : ndarray (n_samples, ) """ X, Y = check_paired_arrays(X, Y) return row_norms(X - Y) def paired_manhattan_distances(X, Y): """Compute the L1 distances between the vectors in X and Y. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like, shape (n_samples, n_features) Y : array-like, shape (n_samples, n_features) Returns ------- distances : ndarray (n_samples, ) """ 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): """ Computes the paired cosine distances between X and Y Read more in the :ref:`User Guide `. Parameters ---------- X : array-like, shape (n_samples, n_features) Y : array-like, shape (n_samples, n_features) Returns ------- distances : ndarray, shape (n_samples, ) 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 .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): """ Computes the paired distances between X and Y. Computes the distances between (X[0], Y[0]), (X[1], Y[1]), etc... Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray (n_samples, n_features) Array 1 for distance computation. Y : ndarray (n_samples, n_features) Array 2 for distance computation. metric : string or callable 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. Returns ------- distances : ndarray (n_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.]) See also -------- pairwise_distances : pairwise distances. """ 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): """ Compute the linear kernel between X and Y. Read more in the :ref:`User Guide `. Parameters ---------- X : array of shape (n_samples_1, n_features) Y : array of shape (n_samples_2, n_features) Returns ------- Gram matrix : array of shape (n_samples_1, n_samples_2) """ X, Y = check_pairwise_arrays(X, Y) return safe_sparse_dot(X, Y.T, dense_output=True) def polynomial_kernel(X, Y=None, degree=3, gamma=None, coef0=1): """ Compute the polynomial kernel between X and Y:: K(X, Y) = (gamma + coef0)^degree Read more in the :ref:`User Guide `. Parameters ---------- X : ndarray of shape (n_samples_1, n_features) Y : ndarray of shape (n_samples_2, n_features) degree : int, default 3 gamma : float, default None if None, defaults to 1.0 / n_samples_1 coef0 : int, default 1 Returns ------- Gram matrix : array of shape (n_samples_1, n_samples_2) """ 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_1, n_features) Y : ndarray of shape (n_samples_2, n_features) gamma : float, default None If None, defaults to 1.0 / n_samples_1 coef0 : int, default 1 Returns ------- Gram matrix: array of shape (n_samples_1, n_samples_2) """ 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 : array of shape (n_samples_X, n_features) Y : array of shape (n_samples_Y, n_features) gamma : float, default None If None, defaults to 1.0 / n_samples_X Returns ------- kernel_matrix : array of shape (n_samples_X, n_samples_Y) """ 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 : array of shape (n_samples_X, n_features) Y : array of shape (n_samples_Y, n_features) gamma : float, default None If None, defaults to 1.0 / n_samples_X Returns ------- kernel_matrix : array of shape (n_samples_X, n_samples_Y) """ 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 or sparse array, shape: (n_samples_X, n_features) Input data. Y : ndarray or sparse array, shape: (n_samples_Y, n_features) Input data. If ``None``, the output will be the pairwise similarities between all samples in ``X``. dense_output : boolean (optional), 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 : array An array with shape (n_samples_X, n_samples_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): """Computes 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 `. Notes ----- As the negative of a distance, this kernel is only conditionally positive definite. Parameters ---------- X : array-like of shape (n_samples_X, n_features) Y : array of shape (n_samples_Y, n_features) Returns ------- kernel_matrix : array of shape (n_samples_X, n_samples_Y) 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 http://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf See also -------- chi2_kernel : The exponentiated version of the kernel, which is usually preferable. sklearn.kernel_approximation.AdditiveChi2Sampler : A Fourier approximation to this kernel. """ 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.): """Computes the exponential chi-squared kernel 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) Y : array of shape (n_samples_Y, n_features) gamma : float, default=1. Scaling parameter of the chi2 kernel. Returns ------- kernel_matrix : array of shape (n_samples_X, n_samples_Y) 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 http://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf 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. """ 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, 'l2': euclidean_distances, 'l1': manhattan_distances, 'manhattan': manhattan_distances, 'precomputed': None, # HACK: precomputed is always allowed, never called } 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 'l1' metrics.pairwise.manhattan_distances 'l2' metrics.pairwise.euclidean_distances 'manhattan' metrics.pairwise.manhattan_distances ============ ==================================== Read more in the :ref:`User Guide `. """ return PAIRWISE_DISTANCE_FUNCTIONS def _parallel_pairwise(X, Y, func, n_jobs, **kwds): """Break the pairwise matrix in n_jobs even slices and compute them in parallel""" if n_jobs < 0: n_jobs = max(cpu_count() + 1 + n_jobs, 1) if Y is None: Y = X if n_jobs == 1: # Special case to avoid picklability checks in delayed return func(X, Y, **kwds) # TODO: in some cases, backend='threading' may be appropriate fd = delayed(func) ret = Parallel(n_jobs=n_jobs, verbose=0)( fd(X, Y[s], **kwds) for s in gen_even_slices(Y.shape[0], n_jobs)) return np.hstack(ret) def _pairwise_callable(X, Y, metric, **kwds): """Handle the callable case for pairwise_{distances,kernels} """ X, Y = check_pairwise_arrays(X, Y) 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"] def pairwise_distances(X, Y=None, metric="euclidean", n_jobs=1, **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. - From scipy.spatial.distance: ['braycurtis', 'canberra', 'chebyshev', 'correlation', 'dice', 'hamming', 'jaccard', 'kulsinski', 'mahalanobis', 'matching', '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 the __doc__ of the sklearn.pairwise.distance_metrics function. Read more in the :ref:`User Guide `. Parameters ---------- X : array [n_samples_a, n_samples_a] if metric == "precomputed", or, \ [n_samples_a, n_features] otherwise Array of pairwise distances between samples, or a feature array. Y : array [n_samples_b, n_features], optional An optional second feature array. Only allowed if metric != "precomputed". metric : string, or callable 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 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. If -1 all CPUs are used. If 1 is given, no parallel computing code is used at all, which is useful for debugging. For n_jobs below -1, (n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one are 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. Returns ------- D : array [n_samples_a, n_samples_a] or [n_samples_a, n_samples_b] 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. """ 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) return X elif metric in PAIRWISE_DISTANCE_FUNCTIONS: func = PAIRWISE_DISTANCE_FUNCTIONS[metric] elif callable(metric): func = partial(_pairwise_callable, metric=metric, **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 X, Y = check_pairwise_arrays(X, Y, dtype=dtype) if 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 recquire 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 `. """ return PAIRWISE_KERNEL_FUNCTIONS KERNEL_PARAMS = { "additive_chi2": (), "chi2": (), "cosine": (), "exp_chi2": frozenset(["gamma"]), "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=1, **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:: ['rbf', 'sigmoid', 'polynomial', 'poly', 'linear', 'cosine'] Read more in the :ref:`User Guide `. Parameters ---------- X : array [n_samples_a, n_samples_a] if metric == "precomputed", or, \ [n_samples_a, n_features] otherwise Array of pairwise kernels between samples, or a feature array. Y : array [n_samples_b, n_features] A second feature array only if X has shape [n_samples_a, n_features]. metric : string, or callable 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 arrays from X as input and return a value indicating the distance between them. n_jobs : int 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. If -1 all CPUs are used. If 1 is given, no parallel computing code is used at all, which is useful for debugging. For n_jobs below -1, (n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one are used. filter_params: boolean Whether to filter invalid parameters or not. `**kwds` : optional keyword parameters Any further parameters are passed directly to the kernel function. Returns ------- K : array [n_samples_a, n_samples_a] or [n_samples_a, n_samples_b] 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 = dict((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)