Source code for astropy.stats.histogram

# Licensed under a 3-clause BSD style license - see LICENSE.rst

"""
Methods for selecting the bin width of histograms

Ported from the astroML project: http://astroML.org/
"""

from __future__ import (absolute_import, division, print_function,
                        unicode_literals)

from ..extern import six

import numpy as np
from . import bayesian_blocks

__all__ = ['histogram', 'scott_bin_width', 'freedman_bin_width',
           'knuth_bin_width']


[docs]def histogram(a, bins=10, range=None, weights=None, **kwargs): """Enhanced histogram function, providing adaptive binnings This is a histogram function that enables the use of more sophisticated algorithms for determining bins. Aside from the ``bins`` argument allowing a string specified how bins are computed, the parameters are the same as ``numpy.histogram()``. Parameters ---------- a : array_like array of data to be histogrammed bins : int or list or str (optional) If bins is a string, then it must be one of: - 'blocks' : use bayesian blocks for dynamic bin widths - 'knuth' : use Knuth's rule to determine bins - 'scott' : use Scott's rule to determine bins - 'freedman' : use the Freedman-Diaconis rule to determine bins range : tuple or None (optional) the minimum and maximum range for the histogram. If not specified, it will be (x.min(), x.max()) weights : array_like, optional Not Implemented other keyword arguments are described in numpy.histogram(). Returns ------- hist : array The values of the histogram. See ``normed`` and ``weights`` for a description of the possible semantics. bin_edges : array of dtype float Return the bin edges ``(length(hist)+1)``. See Also -------- numpy.histogram """ # if bins is a string, first compute bin edges with the desired heuristic if isinstance(bins, six.string_types): a = np.asarray(a).ravel() # TODO: if weights is specified, we need to modify things. # e.g. we could use point measures fitness for Bayesian blocks if weights is not None: raise NotImplementedError("weights are not yet supported " "for the enhanced histogram") # if range is specified, we need to truncate the data for # the bin-finding routines if range is not None: a = a[(a >= range[0]) & (a <= range[1])] if bins == 'blocks': bins = bayesian_blocks(a) elif bins == 'knuth': da, bins = knuth_bin_width(a, True) elif bins == 'scott': da, bins = scott_bin_width(a, True) elif bins == 'freedman': da, bins = freedman_bin_width(a, True) else: raise ValueError("unrecognized bin code: '{}'".format(bins)) # Now we call numpy's histogram with the resulting bin edges return np.histogram(a, bins=bins, range=range, weights=weights, **kwargs)
[docs]def scott_bin_width(data, return_bins=False): r"""Return the optimal histogram bin width using Scott's rule Scott's rule is a normal reference rule: it minimizes the integrated mean squared error in the bin approximation under the assumption that the data is approximately Gaussian. Parameters ---------- data : array-like, ndim=1 observed (one-dimensional) data return_bins : bool (optional) if True, then return the bin edges Returns ------- width : float optimal bin width using Scott's rule bins : ndarray bin edges: returned if ``return_bins`` is True Notes ----- The optimal bin width is .. math:: \Delta_b = \frac{3.5\sigma}{n^{1/3}} where :math:`\sigma` is the standard deviation of the data, and :math:`n` is the number of data points [1]_. References ---------- .. [1] Scott, David W. (1979). "On optimal and data-based histograms". Biometricka 66 (3): 605-610 See Also -------- knuth_bin_width freedman_bin_width bayesian_blocks histogram """ data = np.asarray(data) if data.ndim != 1: raise ValueError("data should be one-dimensional") n = data.size sigma = np.std(data) dx = 3.5 * sigma / (n ** (1 / 3)) if return_bins: Nbins = np.ceil((data.max() - data.min()) / dx) Nbins = max(1, Nbins) bins = data.min() + dx * np.arange(Nbins + 1) return dx, bins else: return dx
[docs]def freedman_bin_width(data, return_bins=False): r"""Return the optimal histogram bin width using the Freedman-Diaconis rule The Freedman-Diaconis rule is a normal reference rule like Scott's rule, but uses rank-based statistics for results which are more robust to deviations from a normal distribution. Parameters ---------- data : array-like, ndim=1 observed (one-dimensional) data return_bins : bool (optional) if True, then return the bin edges Returns ------- width : float optimal bin width using the Freedman-Diaconis rule bins : ndarray bin edges: returned if ``return_bins`` is True Notes ----- The optimal bin width is .. math:: \Delta_b = \frac{2(q_{75} - q_{25})}{n^{1/3}} where :math:`q_{N}` is the :math:`N` percent quartile of the data, and :math:`n` is the number of data points [1]_. References ---------- .. [1] D. Freedman & P. Diaconis (1981) "On the histogram as a density estimator: L2 theory". Probability Theory and Related Fields 57 (4): 453-476 See Also -------- knuth_bin_width scott_bin_width bayesian_blocks histogram """ data = np.asarray(data) if data.ndim != 1: raise ValueError("data should be one-dimensional") n = data.size if n < 4: raise ValueError("data should have more than three entries") v25, v75 = np.percentile(data, [25, 75]) dx = 2 * (v75 - v25) / (n ** (1 / 3)) if return_bins: dmin, dmax = data.min(), data.max() Nbins = max(1, np.ceil((dmax - dmin) / dx)) bins = dmin + dx * np.arange(Nbins + 1) return dx, bins else: return dx
[docs]def knuth_bin_width(data, return_bins=False, quiet=True): r"""Return the optimal histogram bin width using Knuth's rule. Knuth's rule is a fixed-width, Bayesian approach to determining the optimal bin width of a histogram. Parameters ---------- data : array-like, ndim=1 observed (one-dimensional) data return_bins : bool (optional) if True, then return the bin edges quiet : bool (optional) if True (default) then suppress stdout output from scipy.optimize Returns ------- dx : float optimal bin width. Bins are measured starting at the first data point. bins : ndarray bin edges: returned if ``return_bins`` is True Notes ----- The optimal number of bins is the value M which maximizes the function .. math:: F(M|x,I) = n\log(M) + \log\Gamma(\frac{M}{2}) - M\log\Gamma(\frac{1}{2}) - \log\Gamma(\frac{2n+M}{2}) + \sum_{k=1}^M \log\Gamma(n_k + \frac{1}{2}) where :math:`\Gamma` is the Gamma function, :math:`n` is the number of data points, :math:`n_k` is the number of measurements in bin :math:`k` [1]_. References ---------- .. [1] Knuth, K.H. "Optimal Data-Based Binning for Histograms". arXiv:0605197, 2006 See Also -------- freedman_bin_width scott_bin_width bayesian_blocks histogram """ # import here because of optional scipy dependency from scipy import optimize knuthF = _KnuthF(data) dx0, bins0 = freedman_bin_width(data, True) M = optimize.fmin(knuthF, len(bins0), disp=not quiet)[0] bins = knuthF.bins(M) dx = bins[1] - bins[0] if return_bins: return dx, bins else: return dx
class _KnuthF(object): r"""Class which implements the function minimized by knuth_bin_width Parameters ---------- data : array-like, one dimension data to be histogrammed Notes ----- the function F is given by .. math:: F(M|x,I) = n\log(M) + \log\Gamma(\frac{M}{2}) - M\log\Gamma(\frac{1}{2}) - \log\Gamma(\frac{2n+M}{2}) + \sum_{k=1}^M \log\Gamma(n_k + \frac{1}{2}) where :math:`\Gamma` is the Gamma function, :math:`n` is the number of data points, :math:`n_k` is the number of measurements in bin :math:`k`. See Also -------- knuth_bin_width """ def __init__(self, data): self.data = np.array(data, copy=True) if self.data.ndim != 1: raise ValueError("data should be 1-dimensional") self.data.sort() self.n = self.data.size # import here rather than globally: scipy is an optional dependency. # Note that scipy is imported in the function which calls this, # so there shouldn't be any issue importing here. from scipy import special # create a reference to gammaln to use in self.eval() self.gammaln = special.gammaln def bins(self, M): """Return the bin edges given a width dx""" return np.linspace(self.data[0], self.data[-1], int(M) + 1) def __call__(self, M): return self.eval(M) def eval(self, M): """Evaluate the Knuth function Parameters ---------- dx : float Width of bins Returns ------- F : float evaluation of the negative Knuth likelihood function: smaller values indicate a better fit. """ M = int(M) if M <= 0: return np.inf bins = self.bins(M) nk, bins = np.histogram(self.data, bins) return -(self.n * np.log(M) + self.gammaln(0.5 * M) - M * self.gammaln(0.5) - self.gammaln(self.n + 0.5 * M) + np.sum(self.gammaln(nk + 0.5)))