# scipy/scipy

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 #------------------------------------------------------------------------------- # # Define classes for (uni/multi)-variate kernel density estimation. # # Currently, only Gaussian kernels are implemented. # # Written by: Robert Kern # # Date: 2004-08-09 # # Modified: 2005-02-10 by Robert Kern. # Contributed to Scipy # 2005-10-07 by Robert Kern. # Some fixes to match the new scipy_core # # Copyright 2004-2005 by Enthought, Inc. # #------------------------------------------------------------------------------- from __future__ import division, print_function, absolute_import # Standard library imports. import warnings # Scipy imports. from scipy._lib.six import callable, string_types from scipy import linalg, special from scipy.special import logsumexp from numpy import atleast_2d, reshape, zeros, newaxis, dot, exp, pi, sqrt, \ ravel, power, atleast_1d, squeeze, sum, transpose import numpy as np from numpy.random import randint, multivariate_normal # Local imports. from . import mvn __all__ = ['gaussian_kde'] class gaussian_kde(object): """Representation of a kernel-density estimate using Gaussian kernels. Kernel density estimation is a way to estimate the probability density function (PDF) of a random variable in a non-parametric way. `gaussian_kde` works for both uni-variate and multi-variate data. It includes automatic bandwidth determination. The estimation works best for a unimodal distribution; bimodal or multi-modal distributions tend to be oversmoothed. Parameters ---------- dataset : array_like Datapoints to estimate from. In case of univariate data this is a 1-D array, otherwise a 2-D array with shape (# of dims, # of data). bw_method : str, scalar or callable, optional The method used to calculate the estimator bandwidth. This can be 'scott', 'silverman', a scalar constant or a callable. If a scalar, this will be used directly as `kde.factor`. If a callable, it should take a `gaussian_kde` instance as only parameter and return a scalar. If None (default), 'scott' is used. See Notes for more details. Attributes ---------- dataset : ndarray The dataset with which `gaussian_kde` was initialized. d : int Number of dimensions. n : int Number of datapoints. factor : float The bandwidth factor, obtained from `kde.covariance_factor`, with which the covariance matrix is multiplied. covariance : ndarray The covariance matrix of `dataset`, scaled by the calculated bandwidth (`kde.factor`). inv_cov : ndarray The inverse of `covariance`. Methods ------- evaluate __call__ integrate_gaussian integrate_box_1d integrate_box integrate_kde pdf logpdf resample set_bandwidth covariance_factor Notes ----- Bandwidth selection strongly influences the estimate obtained from the KDE (much more so than the actual shape of the kernel). Bandwidth selection can be done by a "rule of thumb", by cross-validation, by "plug-in methods" or by other means; see [3]_, [4]_ for reviews. `gaussian_kde` uses a rule of thumb, the default is Scott's Rule. Scott's Rule [1]_, implemented as `scotts_factor`, is:: n**(-1./(d+4)), with ``n`` the number of data points and ``d`` the number of dimensions. Silverman's Rule [2]_, implemented as `silverman_factor`, is:: (n * (d + 2) / 4.)**(-1. / (d + 4)). Good general descriptions of kernel density estimation can be found in [1]_ and [2]_, the mathematics for this multi-dimensional implementation can be found in [1]_. References ---------- .. [1] D.W. Scott, "Multivariate Density Estimation: Theory, Practice, and Visualization", John Wiley & Sons, New York, Chicester, 1992. .. [2] B.W. Silverman, "Density Estimation for Statistics and Data Analysis", Vol. 26, Monographs on Statistics and Applied Probability, Chapman and Hall, London, 1986. .. [3] B.A. Turlach, "Bandwidth Selection in Kernel Density Estimation: A Review", CORE and Institut de Statistique, Vol. 19, pp. 1-33, 1993. .. [4] D.M. Bashtannyk and R.J. Hyndman, "Bandwidth selection for kernel conditional density estimation", Computational Statistics & Data Analysis, Vol. 36, pp. 279-298, 2001. Examples -------- Generate some random two-dimensional data: >>> from scipy import stats >>> def measure(n): ... "Measurement model, return two coupled measurements." ... m1 = np.random.normal(size=n) ... m2 = np.random.normal(scale=0.5, size=n) ... return m1+m2, m1-m2 >>> m1, m2 = measure(2000) >>> xmin = m1.min() >>> xmax = m1.max() >>> ymin = m2.min() >>> ymax = m2.max() Perform a kernel density estimate on the data: >>> X, Y = np.mgrid[xmin:xmax:100j, ymin:ymax:100j] >>> positions = np.vstack([X.ravel(), Y.ravel()]) >>> values = np.vstack([m1, m2]) >>> kernel = stats.gaussian_kde(values) >>> Z = np.reshape(kernel(positions).T, X.shape) Plot the results: >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> ax.imshow(np.rot90(Z), cmap=plt.cm.gist_earth_r, ... extent=[xmin, xmax, ymin, ymax]) >>> ax.plot(m1, m2, 'k.', markersize=2) >>> ax.set_xlim([xmin, xmax]) >>> ax.set_ylim([ymin, ymax]) >>> plt.show() """ def __init__(self, dataset, bw_method=None): self.dataset = atleast_2d(dataset) if not self.dataset.size > 1: raise ValueError("`dataset` input should have multiple elements.") self.d, self.n = self.dataset.shape self.set_bandwidth(bw_method=bw_method) def evaluate(self, points): """Evaluate the estimated pdf on a set of points. Parameters ---------- points : (# of dimensions, # of points)-array Alternatively, a (# of dimensions,) vector can be passed in and treated as a single point. Returns ------- values : (# of points,)-array The values at each point. Raises ------ ValueError : if the dimensionality of the input points is different than the dimensionality of the KDE. """ points = atleast_2d(points) d, m = points.shape if d != self.d: if d == 1 and m == self.d: # points was passed in as a row vector points = reshape(points, (self.d, 1)) m = 1 else: msg = "points have dimension %s, dataset has dimension %s" % (d, self.d) raise ValueError(msg) result = zeros((m,), dtype=float) if m >= self.n: # there are more points than data, so loop over data for i in range(self.n): diff = self.dataset[:, i, newaxis] - points tdiff = dot(self.inv_cov, diff) energy = sum(diff*tdiff,axis=0) / 2.0 result = result + exp(-energy) else: # loop over points for i in range(m): diff = self.dataset - points[:, i, newaxis] tdiff = dot(self.inv_cov, diff) energy = sum(diff * tdiff, axis=0) / 2.0 result[i] = sum(exp(-energy), axis=0) result = result / self._norm_factor return result __call__ = evaluate def integrate_gaussian(self, mean, cov): """ Multiply estimated density by a multivariate Gaussian and integrate over the whole space. Parameters ---------- mean : aray_like A 1-D array, specifying the mean of the Gaussian. cov : array_like A 2-D array, specifying the covariance matrix of the Gaussian. Returns ------- result : scalar The value of the integral. Raises ------ ValueError If the mean or covariance of the input Gaussian differs from the KDE's dimensionality. """ mean = atleast_1d(squeeze(mean)) cov = atleast_2d(cov) if mean.shape != (self.d,): raise ValueError("mean does not have dimension %s" % self.d) if cov.shape != (self.d, self.d): raise ValueError("covariance does not have dimension %s" % self.d) # make mean a column vector mean = mean[:, newaxis] sum_cov = self.covariance + cov # This will raise LinAlgError if the new cov matrix is not s.p.d # cho_factor returns (ndarray, bool) where bool is a flag for whether # or not ndarray is upper or lower triangular sum_cov_chol = linalg.cho_factor(sum_cov) diff = self.dataset - mean tdiff = linalg.cho_solve(sum_cov_chol, diff) sqrt_det = np.prod(np.diagonal(sum_cov_chol[0])) norm_const = power(2 * pi, sum_cov.shape[0] / 2.0) * sqrt_det energies = sum(diff * tdiff, axis=0) / 2.0 result = sum(exp(-energies), axis=0) / norm_const / self.n return result def integrate_box_1d(self, low, high): """ Computes the integral of a 1D pdf between two bounds. Parameters ---------- low : scalar Lower bound of integration. high : scalar Upper bound of integration. Returns ------- value : scalar The result of the integral. Raises ------ ValueError If the KDE is over more than one dimension. """ if self.d != 1: raise ValueError("integrate_box_1d() only handles 1D pdfs") stdev = ravel(sqrt(self.covariance))[0] normalized_low = ravel((low - self.dataset) / stdev) normalized_high = ravel((high - self.dataset) / stdev) value = np.mean(special.ndtr(normalized_high) - special.ndtr(normalized_low)) return value def integrate_box(self, low_bounds, high_bounds, maxpts=None): """Computes the integral of a pdf over a rectangular interval. Parameters ---------- low_bounds : array_like A 1-D array containing the lower bounds of integration. high_bounds : array_like A 1-D array containing the upper bounds of integration. maxpts : int, optional The maximum number of points to use for integration. Returns ------- value : scalar The result of the integral. """ if maxpts is not None: extra_kwds = {'maxpts': maxpts} else: extra_kwds = {} value, inform = mvn.mvnun(low_bounds, high_bounds, self.dataset, self.covariance, **extra_kwds) if inform: msg = ('An integral in mvn.mvnun requires more points than %s' % (self.d * 1000)) warnings.warn(msg) return value def integrate_kde(self, other): """ Computes the integral of the product of this kernel density estimate with another. Parameters ---------- other : gaussian_kde instance The other kde. Returns ------- value : scalar The result of the integral. Raises ------ ValueError If the KDEs have different dimensionality. """ if other.d != self.d: raise ValueError("KDEs are not the same dimensionality") # we want to iterate over the smallest number of points if other.n < self.n: small = other large = self else: small = self large = other sum_cov = small.covariance + large.covariance sum_cov_chol = linalg.cho_factor(sum_cov) result = 0.0 for i in range(small.n): mean = small.dataset[:, i, newaxis] diff = large.dataset - mean tdiff = linalg.cho_solve(sum_cov_chol, diff) energies = sum(diff * tdiff, axis=0) / 2.0 result += sum(exp(-energies), axis=0) sqrt_det = np.prod(np.diagonal(sum_cov_chol[0])) norm_const = power(2 * pi, sum_cov.shape[0] / 2.0) * sqrt_det result /= norm_const * large.n * small.n return result def resample(self, size=None): """ Randomly sample a dataset from the estimated pdf. Parameters ---------- size : int, optional The number of samples to draw. If not provided, then the size is the same as the underlying dataset. Returns ------- resample : (self.d, `size`) ndarray The sampled dataset. """ if size is None: size = self.n norm = transpose(multivariate_normal(zeros((self.d,), float), self.covariance, size=size)) indices = randint(0, self.n, size=size) means = self.dataset[:, indices] return means + norm def scotts_factor(self): return power(self.n, -1./(self.d+4)) def silverman_factor(self): return power(self.n*(self.d+2.0)/4.0, -1./(self.d+4)) # Default method to calculate bandwidth, can be overwritten by subclass covariance_factor = scotts_factor covariance_factor.__doc__ = """Computes the coefficient (`kde.factor`) that multiplies the data covariance matrix to obtain the kernel covariance matrix. The default is `scotts_factor`. A subclass can overwrite this method to provide a different method, or set it through a call to `kde.set_bandwidth`.""" def set_bandwidth(self, bw_method=None): """Compute the estimator bandwidth with given method. The new bandwidth calculated after a call to `set_bandwidth` is used for subsequent evaluations of the estimated density. Parameters ---------- bw_method : str, scalar or callable, optional The method used to calculate the estimator bandwidth. This can be 'scott', 'silverman', a scalar constant or a callable. If a scalar, this will be used directly as `kde.factor`. If a callable, it should take a `gaussian_kde` instance as only parameter and return a scalar. If None (default), nothing happens; the current `kde.covariance_factor` method is kept. Notes ----- .. versionadded:: 0.11 Examples -------- >>> import scipy.stats as stats >>> x1 = np.array([-7, -5, 1, 4, 5.]) >>> kde = stats.gaussian_kde(x1) >>> xs = np.linspace(-10, 10, num=50) >>> y1 = kde(xs) >>> kde.set_bandwidth(bw_method='silverman') >>> y2 = kde(xs) >>> kde.set_bandwidth(bw_method=kde.factor / 3.) >>> y3 = kde(xs) >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots() >>> ax.plot(x1, np.ones(x1.shape) / (4. * x1.size), 'bo', ... label='Data points (rescaled)') >>> ax.plot(xs, y1, label='Scott (default)') >>> ax.plot(xs, y2, label='Silverman') >>> ax.plot(xs, y3, label='Const (1/3 * Silverman)') >>> ax.legend() >>> plt.show() """ if bw_method is None: pass elif bw_method == 'scott': self.covariance_factor = self.scotts_factor elif bw_method == 'silverman': self.covariance_factor = self.silverman_factor elif np.isscalar(bw_method) and not isinstance(bw_method, string_types): self._bw_method = 'use constant' self.covariance_factor = lambda: bw_method elif callable(bw_method): self._bw_method = bw_method self.covariance_factor = lambda: self._bw_method(self) else: msg = "`bw_method` should be 'scott', 'silverman', a scalar " \ "or a callable." raise ValueError(msg) self._compute_covariance() def _compute_covariance(self): """Computes the covariance matrix for each Gaussian kernel using covariance_factor(). """ self.factor = self.covariance_factor() # Cache covariance and inverse covariance of the data if not hasattr(self, '_data_inv_cov'): self._data_covariance = atleast_2d(np.cov(self.dataset, rowvar=1, bias=False)) self._data_inv_cov = linalg.inv(self._data_covariance) self.covariance = self._data_covariance * self.factor**2 self.inv_cov = self._data_inv_cov / self.factor**2 self._norm_factor = sqrt(linalg.det(2*pi*self.covariance)) * self.n def pdf(self, x): """ Evaluate the estimated pdf on a provided set of points. Notes ----- This is an alias for `gaussian_kde.evaluate`. See the ``evaluate`` docstring for more details. """ return self.evaluate(x) def logpdf(self, x): """ Evaluate the log of the estimated pdf on a provided set of points. """ points = atleast_2d(x) d, m = points.shape if d != self.d: if d == 1 and m == self.d: # points was passed in as a row vector points = reshape(points, (self.d, 1)) m = 1 else: msg = "points have dimension %s, dataset has dimension %s" % (d, self.d) raise ValueError(msg) result = zeros((m,), dtype=float) if m >= self.n: # there are more points than data, so loop over data energy = zeros((self.n, m), dtype=float) for i in range(self.n): diff = self.dataset[:, i, newaxis] - points tdiff = dot(self.inv_cov, diff) energy[i] = sum(diff*tdiff,axis=0) / 2.0 result = logsumexp(-energy, b=1/self._norm_factor, axis=0) else: # loop over points for i in range(m): diff = self.dataset - points[:, i, newaxis] tdiff = dot(self.inv_cov, diff) energy = sum(diff * tdiff, axis=0) / 2.0 result[i] = logsumexp(-energy, b=1/self._norm_factor) return result