# 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/
"""

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).

--------
numpy.histogram
"""
# 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 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

--------
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

--------
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

--------
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:
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.

--------
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)))