""" Kernel Density Estimation ------------------------- """ # Author: Jake Vanderplas import numpy as np from scipy.special import gammainc from ..base import BaseEstimator from ..utils import check_array, check_random_state from ..utils.extmath import row_norms from .ball_tree import BallTree, DTYPE from .kd_tree import KDTree VALID_KERNELS = ['gaussian', 'tophat', 'epanechnikov', 'exponential', 'linear', 'cosine'] TREE_DICT = {'ball_tree': BallTree, 'kd_tree': KDTree} # TODO: implement a brute force version for testing purposes # TODO: bandwidth estimation # TODO: create a density estimation base class? class KernelDensity(BaseEstimator): """Kernel Density Estimation Read more in the :ref:`User Guide `. Parameters ---------- bandwidth : float The bandwidth of the kernel. algorithm : string The tree algorithm to use. Valid options are ['kd_tree'|'ball_tree'|'auto']. Default is 'auto'. kernel : string The kernel to use. Valid kernels are ['gaussian'|'tophat'|'epanechnikov'|'exponential'|'linear'|'cosine'] Default is 'gaussian'. metric : string The distance metric to use. Note that not all metrics are valid with all algorithms. Refer to the documentation of :class:`BallTree` and :class:`KDTree` for a description of available algorithms. Note that the normalization of the density output is correct only for the Euclidean distance metric. Default is 'euclidean'. atol : float The desired absolute tolerance of the result. A larger tolerance will generally lead to faster execution. Default is 0. rtol : float The desired relative tolerance of the result. A larger tolerance will generally lead to faster execution. Default is 1E-8. breadth_first : boolean If true (default), use a breadth-first approach to the problem. Otherwise use a depth-first approach. leaf_size : int Specify the leaf size of the underlying tree. See :class:`BallTree` or :class:`KDTree` for details. Default is 40. metric_params : dict Additional parameters to be passed to the tree for use with the metric. For more information, see the documentation of :class:`BallTree` or :class:`KDTree`. """ def __init__(self, bandwidth=1.0, algorithm='auto', kernel='gaussian', metric="euclidean", atol=0, rtol=0, breadth_first=True, leaf_size=40, metric_params=None): self.algorithm = algorithm self.bandwidth = bandwidth self.kernel = kernel self.metric = metric self.atol = atol self.rtol = rtol self.breadth_first = breadth_first self.leaf_size = leaf_size self.metric_params = metric_params # run the choose algorithm code so that exceptions will happen here # we're using clone() in the GenerativeBayes classifier, # so we can't do this kind of logic in __init__ self._choose_algorithm(self.algorithm, self.metric) if bandwidth <= 0: raise ValueError("bandwidth must be positive") if kernel not in VALID_KERNELS: raise ValueError("invalid kernel: '{0}'".format(kernel)) def _choose_algorithm(self, algorithm, metric): # given the algorithm string + metric string, choose the optimal # algorithm to compute the result. if algorithm == 'auto': # use KD Tree if possible if metric in KDTree.valid_metrics: return 'kd_tree' elif metric in BallTree.valid_metrics: return 'ball_tree' else: raise ValueError("invalid metric: '{0}'".format(metric)) elif algorithm in TREE_DICT: if metric not in TREE_DICT[algorithm].valid_metrics: raise ValueError("invalid metric for {0}: " "'{1}'".format(TREE_DICT[algorithm], metric)) return algorithm else: raise ValueError("invalid algorithm: '{0}'".format(algorithm)) def fit(self, X, y=None): """Fit the Kernel Density model on the data. Parameters ---------- X : array_like, shape (n_samples, n_features) List of n_features-dimensional data points. Each row corresponds to a single data point. """ algorithm = self._choose_algorithm(self.algorithm, self.metric) X = check_array(X, order='C', dtype=DTYPE) kwargs = self.metric_params if kwargs is None: kwargs = {} self.tree_ = TREE_DICT[algorithm](X, metric=self.metric, leaf_size=self.leaf_size, **kwargs) return self def score_samples(self, X): """Evaluate the density model on the data. Parameters ---------- X : array_like, shape (n_samples, n_features) An array of points to query. Last dimension should match dimension of training data (n_features). Returns ------- density : ndarray, shape (n_samples,) The array of log(density) evaluations. """ # The returned density is normalized to the number of points. # For it to be a probability, we must scale it. For this reason # we'll also scale atol. X = check_array(X, order='C', dtype=DTYPE) N = self.tree_.data.shape[0] atol_N = self.atol * N log_density = self.tree_.kernel_density( X, h=self.bandwidth, kernel=self.kernel, atol=atol_N, rtol=self.rtol, breadth_first=self.breadth_first, return_log=True) log_density -= np.log(N) return log_density def score(self, X, y=None): """Compute the total log probability under the model. Parameters ---------- X : array_like, shape (n_samples, n_features) List of n_features-dimensional data points. Each row corresponds to a single data point. Returns ------- logprob : float Total log-likelihood of the data in X. """ return np.sum(self.score_samples(X)) def sample(self, n_samples=1, random_state=None): """Generate random samples from the model. Currently, this is implemented only for gaussian and tophat kernels. Parameters ---------- n_samples : int, optional Number of samples to generate. Defaults to 1. random_state : int, RandomState instance or None. default to None If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by `np.random`. Returns ------- X : array_like, shape (n_samples, n_features) List of samples. """ # TODO: implement sampling for other valid kernel shapes if self.kernel not in ['gaussian', 'tophat']: raise NotImplementedError() data = np.asarray(self.tree_.data) rng = check_random_state(random_state) i = rng.randint(data.shape[0], size=n_samples) if self.kernel == 'gaussian': return np.atleast_2d(rng.normal(data[i], self.bandwidth)) elif self.kernel == 'tophat': # we first draw points from a d-dimensional normal distribution, # then use an incomplete gamma function to map them to a uniform # d-dimensional tophat distribution. dim = data.shape[1] X = rng.normal(size=(n_samples, dim)) s_sq = row_norms(X, squared=True) correction = (gammainc(0.5 * dim, 0.5 * s_sq) ** (1. / dim) * self.bandwidth / np.sqrt(s_sq)) return data[i] + X * correction[:, np.newaxis]