Source code for astropy.stats.histogram

"""
Methods for selecting the bin width of histograms

Ported from the astroML project: https://www.astroml.org/
"""

import numpy as np
from . import bayesian_blocks

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

def calculate_bin_edges(a, bins=10, range=None, weights=None):
"""
Calculate histogram bin edges like numpy.histogram_bin_edges.

Parameters
----------

a : array-like
Input data. The bin edges are calculated over the flattened array.

bins : int, list, or str, optional
If bins is an int, it is the number of bins. If it is a list
it is taken to be the bin edges. If it is a string, it must be one
of  'blocks', 'knuth', 'scott' or 'freedman'. See
~astropy.stats.histogram for a description of each method.

range : tuple or None, optional
The minimum and maximum range for the histogram.  If not specified,
it will be (a.min(), a.max()). However, if bins is a list it is
returned unmodified regardless of the range argument.

weights : array-like, optional
An array the same shape as a. If given, the histogram accumulates
the value of the weight corresponding to a instead of returning the
count of values. This argument does not affect determination of bin
edges, though they may be used in the future as new methods are added.
"""
# if range is specified, we need to truncate the data for
# the bin-finding routines
if range is not None:
a = a[(a >= range) & (a <= range)]

# if bins is a string, first compute bin edges with the desired heuristic
if isinstance(bins, str):
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 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(f"unrecognized bin code: '{bins}'")

if range:
# Check that the upper and lower edges are what was requested.
# The current implementation of the bin width estimators does not
# guarantee this, it only ensures that data outside the range is
# excluded from calculation of the bin widths.
if bins != range:
bins = range
if bins[-1] != range:
bins[-1] = range

elif np.ndim(bins) == 0:
# Number of bins was given
bins = np.histogram_bin_edges(a, bins, range=range, weights=weights)

return bins

[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, 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 An array the same shape as a. If given, the histogram accumulates the value of the weight corresponding to a instead of returning the count of values. This argument does not affect determination of bin edges. other keyword arguments are described in numpy.histogram(). Returns ------- hist : array The values of the histogram. See density 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 """ bins = calculate_bin_edges(a, bins=bins, range=range, weights=weights) # 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 _. References ---------- ..  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 _. References ---------- ..  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)) try: bins = dmin + dx * np.arange(Nbins + 1) except ValueError as e: if 'Maximum allowed size exceeded' in str(e): raise ValueError( 'The inter-quartile range of the data is too small: ' 'failed to construct histogram with {} bins. ' 'Please use another bin method, such as ' 'bins="scott"'.format(Nbins + 1)) else: # Something else # pragma: no cover raise 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 _. References ---------- ..  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) bins = knuthF.bins(M) dx = bins - bins if return_bins: return dx, bins else: return dx
class _KnuthF: 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 M number of bins""" return np.linspace(self.data, self.data[-1], int(M) + 1) def __call__(self, M): return self.eval(M) def eval(self, M): """Evaluate the Knuth function Parameters ---------- M : int Number of bins Returns ------- F : float evaluation of the negative Knuth loglikelihood 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)))