649 lines
21 KiB
Python
649 lines
21 KiB
Python
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#-------------------------------------------------------------------------------
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#
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# Define classes for (uni/multi)-variate kernel density estimation.
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#
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# Currently, only Gaussian kernels are implemented.
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#
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# Written by: Robert Kern
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#
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# Date: 2004-08-09
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#
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# Modified: 2005-02-10 by Robert Kern.
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# Contributed to SciPy
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# 2005-10-07 by Robert Kern.
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# Some fixes to match the new scipy_core
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#
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# Copyright 2004-2005 by Enthought, Inc.
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#
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#-------------------------------------------------------------------------------
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from __future__ import division, print_function, absolute_import
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# Standard library imports.
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import warnings
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# SciPy imports.
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from scipy._lib.six import callable, string_types
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from scipy import linalg, special
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from scipy.special import logsumexp
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from scipy._lib._numpy_compat import cov
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from scipy._lib._util import check_random_state
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from numpy import (asarray, atleast_2d, reshape, zeros, newaxis, dot, exp, pi,
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sqrt, ravel, power, atleast_1d, squeeze, sum, transpose,
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ones)
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import numpy as np
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# Local imports.
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from . import mvn
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__all__ = ['gaussian_kde']
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class gaussian_kde(object):
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"""Representation of a kernel-density estimate using Gaussian kernels.
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Kernel density estimation is a way to estimate the probability density
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function (PDF) of a random variable in a non-parametric way.
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`gaussian_kde` works for both uni-variate and multi-variate data. It
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includes automatic bandwidth determination. The estimation works best for
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a unimodal distribution; bimodal or multi-modal distributions tend to be
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oversmoothed.
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Parameters
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----------
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dataset : array_like
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Datapoints to estimate from. In case of univariate data this is a 1-D
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array, otherwise a 2-D array with shape (# of dims, # of data).
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bw_method : str, scalar or callable, optional
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The method used to calculate the estimator bandwidth. This can be
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'scott', 'silverman', a scalar constant or a callable. If a scalar,
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this will be used directly as `kde.factor`. If a callable, it should
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take a `gaussian_kde` instance as only parameter and return a scalar.
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If None (default), 'scott' is used. See Notes for more details.
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weights : array_like, optional
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weights of datapoints. This must be the same shape as dataset.
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If None (default), the samples are assumed to be equally weighted
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Attributes
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----------
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dataset : ndarray
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The dataset with which `gaussian_kde` was initialized.
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d : int
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Number of dimensions.
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n : int
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Number of datapoints.
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neff : int
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Effective number of datapoints.
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.. versionadded:: 1.2.0
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factor : float
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The bandwidth factor, obtained from `kde.covariance_factor`, with which
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the covariance matrix is multiplied.
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covariance : ndarray
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The covariance matrix of `dataset`, scaled by the calculated bandwidth
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(`kde.factor`).
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inv_cov : ndarray
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The inverse of `covariance`.
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Methods
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-------
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evaluate
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__call__
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integrate_gaussian
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integrate_box_1d
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integrate_box
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integrate_kde
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pdf
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logpdf
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resample
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set_bandwidth
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covariance_factor
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Notes
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-----
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Bandwidth selection strongly influences the estimate obtained from the KDE
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(much more so than the actual shape of the kernel). Bandwidth selection
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can be done by a "rule of thumb", by cross-validation, by "plug-in
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methods" or by other means; see [3]_, [4]_ for reviews. `gaussian_kde`
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uses a rule of thumb, the default is Scott's Rule.
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Scott's Rule [1]_, implemented as `scotts_factor`, is::
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n**(-1./(d+4)),
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with ``n`` the number of data points and ``d`` the number of dimensions.
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In the case of unequally weighted points, `scotts_factor` becomes::
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neff**(-1./(d+4)),
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with ``neff`` the effective number of datapoints.
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Silverman's Rule [2]_, implemented as `silverman_factor`, is::
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(n * (d + 2) / 4.)**(-1. / (d + 4)).
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or in the case of unequally weighted points::
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(neff * (d + 2) / 4.)**(-1. / (d + 4)).
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Good general descriptions of kernel density estimation can be found in [1]_
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and [2]_, the mathematics for this multi-dimensional implementation can be
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found in [1]_.
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With a set of weighted samples, the effective number of datapoints ``neff``
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is defined by::
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neff = sum(weights)^2 / sum(weights^2)
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as detailed in [5]_.
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References
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----------
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.. [1] D.W. Scott, "Multivariate Density Estimation: Theory, Practice, and
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Visualization", John Wiley & Sons, New York, Chicester, 1992.
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.. [2] B.W. Silverman, "Density Estimation for Statistics and Data
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Analysis", Vol. 26, Monographs on Statistics and Applied Probability,
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Chapman and Hall, London, 1986.
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.. [3] B.A. Turlach, "Bandwidth Selection in Kernel Density Estimation: A
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Review", CORE and Institut de Statistique, Vol. 19, pp. 1-33, 1993.
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.. [4] D.M. Bashtannyk and R.J. Hyndman, "Bandwidth selection for kernel
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conditional density estimation", Computational Statistics & Data
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Analysis, Vol. 36, pp. 279-298, 2001.
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.. [5] Gray P. G., 1969, Journal of the Royal Statistical Society.
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Series A (General), 132, 272
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Examples
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--------
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Generate some random two-dimensional data:
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>>> from scipy import stats
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>>> def measure(n):
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... "Measurement model, return two coupled measurements."
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... m1 = np.random.normal(size=n)
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... m2 = np.random.normal(scale=0.5, size=n)
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... return m1+m2, m1-m2
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>>> m1, m2 = measure(2000)
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>>> xmin = m1.min()
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>>> xmax = m1.max()
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>>> ymin = m2.min()
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>>> ymax = m2.max()
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Perform a kernel density estimate on the data:
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>>> X, Y = np.mgrid[xmin:xmax:100j, ymin:ymax:100j]
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>>> positions = np.vstack([X.ravel(), Y.ravel()])
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>>> values = np.vstack([m1, m2])
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>>> kernel = stats.gaussian_kde(values)
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>>> Z = np.reshape(kernel(positions).T, X.shape)
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Plot the results:
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>>> import matplotlib.pyplot as plt
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>>> fig, ax = plt.subplots()
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>>> ax.imshow(np.rot90(Z), cmap=plt.cm.gist_earth_r,
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... extent=[xmin, xmax, ymin, ymax])
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>>> ax.plot(m1, m2, 'k.', markersize=2)
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>>> ax.set_xlim([xmin, xmax])
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>>> ax.set_ylim([ymin, ymax])
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>>> plt.show()
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"""
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def __init__(self, dataset, bw_method=None, weights=None):
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self.dataset = atleast_2d(asarray(dataset))
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if not self.dataset.size > 1:
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raise ValueError("`dataset` input should have multiple elements.")
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self.d, self.n = self.dataset.shape
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if weights is not None:
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self._weights = atleast_1d(weights).astype(float)
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self._weights /= sum(self._weights)
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if self.weights.ndim != 1:
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raise ValueError("`weights` input should be one-dimensional.")
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if len(self._weights) != self.n:
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raise ValueError("`weights` input should be of length n")
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self._neff = 1/sum(self._weights**2)
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self.set_bandwidth(bw_method=bw_method)
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def evaluate(self, points):
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"""Evaluate the estimated pdf on a set of points.
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Parameters
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----------
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points : (# of dimensions, # of points)-array
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Alternatively, a (# of dimensions,) vector can be passed in and
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treated as a single point.
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Returns
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-------
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values : (# of points,)-array
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The values at each point.
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Raises
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------
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ValueError : if the dimensionality of the input points is different than
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the dimensionality of the KDE.
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"""
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points = atleast_2d(asarray(points))
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d, m = points.shape
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if d != self.d:
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if d == 1 and m == self.d:
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# points was passed in as a row vector
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points = reshape(points, (self.d, 1))
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m = 1
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else:
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msg = "points have dimension %s, dataset has dimension %s" % (d,
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self.d)
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raise ValueError(msg)
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result = zeros((m,), dtype=float)
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whitening = linalg.cholesky(self.inv_cov)
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scaled_dataset = dot(whitening, self.dataset)
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scaled_points = dot(whitening, points)
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if m >= self.n:
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# there are more points than data, so loop over data
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for i in range(self.n):
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diff = scaled_dataset[:, i, newaxis] - scaled_points
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energy = sum(diff * diff, axis=0) / 2.0
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result += self.weights[i]*exp(-energy)
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else:
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# loop over points
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for i in range(m):
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diff = scaled_dataset - scaled_points[:, i, newaxis]
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energy = sum(diff * diff, axis=0) / 2.0
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result[i] = sum(exp(-energy)*self.weights, axis=0)
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result = result / self._norm_factor
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return result
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__call__ = evaluate
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def integrate_gaussian(self, mean, cov):
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"""
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Multiply estimated density by a multivariate Gaussian and integrate
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over the whole space.
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Parameters
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----------
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mean : aray_like
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A 1-D array, specifying the mean of the Gaussian.
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cov : array_like
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A 2-D array, specifying the covariance matrix of the Gaussian.
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Returns
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-------
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result : scalar
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The value of the integral.
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Raises
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------
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ValueError
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If the mean or covariance of the input Gaussian differs from
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the KDE's dimensionality.
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"""
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mean = atleast_1d(squeeze(mean))
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cov = atleast_2d(cov)
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if mean.shape != (self.d,):
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raise ValueError("mean does not have dimension %s" % self.d)
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if cov.shape != (self.d, self.d):
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raise ValueError("covariance does not have dimension %s" % self.d)
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# make mean a column vector
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mean = mean[:, newaxis]
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sum_cov = self.covariance + cov
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# This will raise LinAlgError if the new cov matrix is not s.p.d
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# cho_factor returns (ndarray, bool) where bool is a flag for whether
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# or not ndarray is upper or lower triangular
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sum_cov_chol = linalg.cho_factor(sum_cov)
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diff = self.dataset - mean
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tdiff = linalg.cho_solve(sum_cov_chol, diff)
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sqrt_det = np.prod(np.diagonal(sum_cov_chol[0]))
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norm_const = power(2 * pi, sum_cov.shape[0] / 2.0) * sqrt_det
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energies = sum(diff * tdiff, axis=0) / 2.0
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result = sum(exp(-energies)*self.weights, axis=0) / norm_const
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return result
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def integrate_box_1d(self, low, high):
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"""
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Computes the integral of a 1D pdf between two bounds.
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Parameters
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----------
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low : scalar
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Lower bound of integration.
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high : scalar
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Upper bound of integration.
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Returns
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-------
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value : scalar
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The result of the integral.
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Raises
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------
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ValueError
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If the KDE is over more than one dimension.
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"""
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if self.d != 1:
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raise ValueError("integrate_box_1d() only handles 1D pdfs")
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stdev = ravel(sqrt(self.covariance))[0]
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normalized_low = ravel((low - self.dataset) / stdev)
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normalized_high = ravel((high - self.dataset) / stdev)
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value = np.sum(self.weights*(
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special.ndtr(normalized_high) -
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special.ndtr(normalized_low)))
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return value
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def integrate_box(self, low_bounds, high_bounds, maxpts=None):
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"""Computes the integral of a pdf over a rectangular interval.
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Parameters
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----------
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low_bounds : array_like
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A 1-D array containing the lower bounds of integration.
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high_bounds : array_like
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A 1-D array containing the upper bounds of integration.
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maxpts : int, optional
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The maximum number of points to use for integration.
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Returns
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-------
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value : scalar
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The result of the integral.
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"""
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if maxpts is not None:
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extra_kwds = {'maxpts': maxpts}
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else:
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extra_kwds = {}
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value, inform = mvn.mvnun_weighted(low_bounds, high_bounds,
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self.dataset, self.weights,
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self.covariance, **extra_kwds)
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if inform:
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msg = ('An integral in mvn.mvnun requires more points than %s' %
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(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)*large.weights, axis=0)*small.weights[i]
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
return result
|
||
|
|
|
||
|
|
def resample(self, size=None, seed=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 effective number of samples in the underlying
|
||
|
|
dataset.
|
||
|
|
seed : None or int or `np.random.RandomState`, optional
|
||
|
|
If `seed` is None, random variates are drawn by the RandomState
|
||
|
|
singleton used by np.random.
|
||
|
|
If `seed` is an int, a new `np.random.RandomState` instance is used,
|
||
|
|
seeded with seed.
|
||
|
|
If `seed` is already a `np.random.RandomState instance`, then that
|
||
|
|
`np.random.RandomState` instance is used.
|
||
|
|
Specify `seed` for reproducible drawing of random variates.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
resample : (self.d, `size`) ndarray
|
||
|
|
The sampled dataset.
|
||
|
|
|
||
|
|
"""
|
||
|
|
if size is None:
|
||
|
|
size = int(self.neff)
|
||
|
|
|
||
|
|
random_state = check_random_state(seed)
|
||
|
|
norm = transpose(random_state.multivariate_normal(
|
||
|
|
zeros((self.d,), float), self.covariance, size=size
|
||
|
|
))
|
||
|
|
indices = random_state.choice(self.n, size=size, p=self.weights)
|
||
|
|
means = self.dataset[:, indices]
|
||
|
|
|
||
|
|
return means + norm
|
||
|
|
|
||
|
|
def scotts_factor(self):
|
||
|
|
"""Compute Scott's factor.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
s : float
|
||
|
|
Scott's factor.
|
||
|
|
"""
|
||
|
|
return power(self.neff, -1./(self.d+4))
|
||
|
|
|
||
|
|
def silverman_factor(self):
|
||
|
|
"""Compute the Silverman factor.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
s : float
|
||
|
|
The silverman factor.
|
||
|
|
"""
|
||
|
|
return power(self.neff*(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.full(x1.shape, 1 / (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(cov(self.dataset, rowvar=1,
|
||
|
|
bias=False,
|
||
|
|
aweights=self.weights))
|
||
|
|
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))
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
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.T,
|
||
|
|
b=self.weights / self._norm_factor, axis=1)
|
||
|
|
else:
|
||
|
|
# loop over points
|
||
|
|
result = zeros((m,), dtype=float)
|
||
|
|
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=self.weights /
|
||
|
|
self._norm_factor)
|
||
|
|
|
||
|
|
return result
|
||
|
|
|
||
|
|
@property
|
||
|
|
def weights(self):
|
||
|
|
try:
|
||
|
|
return self._weights
|
||
|
|
except AttributeError:
|
||
|
|
self._weights = ones(self.n)/self.n
|
||
|
|
return self._weights
|
||
|
|
|
||
|
|
@property
|
||
|
|
def neff(self):
|
||
|
|
try:
|
||
|
|
return self._neff
|
||
|
|
except AttributeError:
|
||
|
|
self._neff = 1/sum(self.weights**2)
|
||
|
|
return self._neff
|