726 lines
24 KiB
Python
726 lines
24 KiB
Python
#-------------------------------------------------------------------------------
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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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# Standard library imports.
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import warnings
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# SciPy imports.
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from scipy import linalg, special
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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, exp, pi,
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sqrt, ravel, power, atleast_1d, squeeze, sum, transpose,
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ones, cov)
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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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from ._stats import gaussian_kernel_estimate, gaussian_kernel_estimate_log
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__all__ = ['gaussian_kde']
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class gaussian_kde:
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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`. The square
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of `kde.factor` multiplies the covariance matrix of the data in the kde
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estimation.
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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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`gaussian_kde` does not currently support data that lies in a
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lower-dimensional subspace of the space in which it is expressed. For such
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data, consider performing principle component analysis / dimensionality
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reduction and using `gaussian_kde` with the transformed data.
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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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>>> import numpy as np
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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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# This can be converted to a warning once gh-10205 is resolved
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if self.d > self.n:
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msg = ("Number of dimensions is greater than number of samples. "
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"This results in a singular data covariance matrix, which "
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"cannot be treated using the algorithms implemented in "
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"`gaussian_kde`. Note that `gaussian_kde` interprets each "
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"*column* of `dataset` to be a point; consider transposing "
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"the input to `dataset`.")
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raise ValueError(msg)
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try:
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self.set_bandwidth(bw_method=bw_method)
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except linalg.LinAlgError as e:
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msg = ("The data appears to lie in a lower-dimensional subspace "
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"of the space in which it is expressed. This has resulted "
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"in a singular data covariance matrix, which cannot be "
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"treated using the algorithms implemented in "
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"`gaussian_kde`. Consider performing principle component "
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"analysis / dimensionality reduction and using "
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"`gaussian_kde` with the transformed data.")
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raise linalg.LinAlgError(msg) from e
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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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output_dtype, spec = _get_output_dtype(self.covariance, points)
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result = gaussian_kernel_estimate[spec](
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self.dataset.T, self.weights[:, None],
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points.T, self.cho_cov, output_dtype)
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return result[:, 0]
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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))
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warnings.warn(msg)
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return value
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def integrate_kde(self, other):
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"""
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Computes the integral of the product of this kernel density estimate
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with another.
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Parameters
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----------
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other : gaussian_kde instance
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The other kde.
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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 KDEs have different dimensionality.
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"""
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if other.d != self.d:
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raise ValueError("KDEs are not the same dimensionality")
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# we want to iterate over the smallest number of points
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if other.n < self.n:
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small = other
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large = self
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else:
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small = self
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large = other
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sum_cov = small.covariance + large.covariance
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sum_cov_chol = linalg.cho_factor(sum_cov)
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result = 0.0
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for i in range(small.n):
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mean = small.dataset[:, i, newaxis]
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diff = large.dataset - mean
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tdiff = linalg.cho_solve(sum_cov_chol, diff)
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energies = sum(diff * tdiff, axis=0) / 2.0
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result += sum(exp(-energies)*large.weights, axis=0)*small.weights[i]
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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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result /= norm_const
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return result
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def resample(self, size=None, seed=None):
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"""Randomly sample a dataset from the estimated pdf.
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Parameters
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----------
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size : int, optional
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The number of samples to draw. If not provided, then the size is
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the same as the effective number of samples in the underlying
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dataset.
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seed : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional
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If `seed` is None (or `np.random`), the `numpy.random.RandomState`
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singleton is used.
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If `seed` is an int, a new ``RandomState`` instance is used,
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seeded with `seed`.
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If `seed` is already a ``Generator`` or ``RandomState`` instance then
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that instance is used.
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Returns
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-------
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resample : (self.d, `size`) ndarray
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The sampled dataset.
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"""
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if size is None:
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size = int(self.neff)
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random_state = check_random_state(seed)
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norm = transpose(random_state.multivariate_normal(
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zeros((self.d,), float), self.covariance, size=size
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))
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indices = random_state.choice(self.n, size=size, p=self.weights)
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means = self.dataset[:, indices]
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return means + norm
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def scotts_factor(self):
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"""Compute Scott's factor.
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Returns
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-------
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s : float
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Scott's factor.
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"""
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return power(self.neff, -1./(self.d+4))
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def silverman_factor(self):
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"""Compute the Silverman factor.
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Returns
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-------
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s : float
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The silverman factor.
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"""
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return power(self.neff*(self.d+2.0)/4.0, -1./(self.d+4))
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# Default method to calculate bandwidth, can be overwritten by subclass
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covariance_factor = scotts_factor
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covariance_factor.__doc__ = """Computes the coefficient (`kde.factor`) that
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multiplies the data covariance matrix to obtain the kernel covariance
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matrix. The default is `scotts_factor`. A subclass can overwrite this
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method to provide a different method, or set it through a call to
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`kde.set_bandwidth`."""
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def set_bandwidth(self, bw_method=None):
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"""Compute the estimator bandwidth with given method.
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The new bandwidth calculated after a call to `set_bandwidth` is used
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for subsequent evaluations of the estimated density.
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Parameters
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----------
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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
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scalar, this will be used directly as `kde.factor`. If a callable,
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it should take a `gaussian_kde` instance as only parameter and
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return a scalar. If None (default), nothing happens; the current
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`kde.covariance_factor` method is kept.
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Notes
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-----
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.. versionadded:: 0.11
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Examples
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--------
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>>> import numpy as np
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>>> import scipy.stats as stats
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>>> x1 = np.array([-7, -5, 1, 4, 5.])
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>>> kde = stats.gaussian_kde(x1)
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>>> 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, str):
|
|
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 Cholesky decomp of covariance
|
|
if not hasattr(self, '_data_cho_cov'):
|
|
self._data_covariance = atleast_2d(cov(self.dataset, rowvar=1,
|
|
bias=False,
|
|
aweights=self.weights))
|
|
self._data_cho_cov = linalg.cholesky(self._data_covariance,
|
|
lower=True)
|
|
|
|
self.covariance = self._data_covariance * self.factor**2
|
|
self.cho_cov = (self._data_cho_cov * self.factor).astype(np.float64)
|
|
self.log_det = 2*np.log(np.diag(self.cho_cov
|
|
* np.sqrt(2*pi))).sum()
|
|
|
|
@property
|
|
def inv_cov(self):
|
|
# Re-compute from scratch each time because I'm not sure how this is
|
|
# used in the wild. (Perhaps users change the `dataset`, since it's
|
|
# not a private attribute?) `_compute_covariance` used to recalculate
|
|
# all these, so we'll recalculate everything now that this is a
|
|
# a property.
|
|
self.factor = self.covariance_factor()
|
|
self._data_covariance = atleast_2d(cov(self.dataset, rowvar=1,
|
|
bias=False, aweights=self.weights))
|
|
return linalg.inv(self._data_covariance) / self.factor**2
|
|
|
|
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 = (f"points have dimension {d}, "
|
|
f"dataset has dimension {self.d}")
|
|
raise ValueError(msg)
|
|
|
|
output_dtype, spec = _get_output_dtype(self.covariance, points)
|
|
result = gaussian_kernel_estimate_log[spec](
|
|
self.dataset.T, self.weights[:, None],
|
|
points.T, self.cho_cov, output_dtype)
|
|
|
|
return result[:, 0]
|
|
|
|
def marginal(self, dimensions):
|
|
"""Return a marginal KDE distribution
|
|
|
|
Parameters
|
|
----------
|
|
dimensions : int or 1-d array_like
|
|
The dimensions of the multivariate distribution corresponding
|
|
with the marginal variables, that is, the indices of the dimensions
|
|
that are being retained. The other dimensions are marginalized out.
|
|
|
|
Returns
|
|
-------
|
|
marginal_kde : gaussian_kde
|
|
An object representing the marginal distribution.
|
|
|
|
Notes
|
|
-----
|
|
.. versionadded:: 1.10.0
|
|
|
|
"""
|
|
|
|
dims = np.atleast_1d(dimensions)
|
|
|
|
if not np.issubdtype(dims.dtype, np.integer):
|
|
msg = ("Elements of `dimensions` must be integers - the indices "
|
|
"of the marginal variables being retained.")
|
|
raise ValueError(msg)
|
|
|
|
n = len(self.dataset) # number of dimensions
|
|
original_dims = dims.copy()
|
|
|
|
dims[dims < 0] = n + dims[dims < 0]
|
|
|
|
if len(np.unique(dims)) != len(dims):
|
|
msg = ("All elements of `dimensions` must be unique.")
|
|
raise ValueError(msg)
|
|
|
|
i_invalid = (dims < 0) | (dims >= n)
|
|
if np.any(i_invalid):
|
|
msg = (f"Dimensions {original_dims[i_invalid]} are invalid "
|
|
f"for a distribution in {n} dimensions.")
|
|
raise ValueError(msg)
|
|
|
|
dataset = self.dataset[dims]
|
|
weights = self.weights
|
|
|
|
return gaussian_kde(dataset, bw_method=self.covariance_factor(),
|
|
weights=weights)
|
|
|
|
@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
|
|
|
|
|
|
def _get_output_dtype(covariance, points):
|
|
"""
|
|
Calculates the output dtype and the "spec" (=C type name).
|
|
|
|
This was necessary in order to deal with the fused types in the Cython
|
|
routine `gaussian_kernel_estimate`. See gh-10824 for details.
|
|
"""
|
|
output_dtype = np.common_type(covariance, points)
|
|
itemsize = np.dtype(output_dtype).itemsize
|
|
if itemsize == 4:
|
|
spec = 'float'
|
|
elif itemsize == 8:
|
|
spec = 'double'
|
|
elif itemsize in (12, 16):
|
|
spec = 'long double'
|
|
else:
|
|
raise ValueError(
|
|
f"{output_dtype} has unexpected item size: {itemsize}"
|
|
)
|
|
|
|
return output_dtype, spec
|