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from __future__ import division, print_function, absolute_import
import numpy as np
from scipy._lib.six import callable, xrange
from scipy._lib._numpy_compat import suppress_warnings
from collections import namedtuple
__all__ = ['binned_statistic',
'binned_statistic_2d',
'binned_statistic_dd']
BinnedStatisticResult = namedtuple('BinnedStatisticResult',
('statistic', 'bin_edges', 'binnumber'))
def binned_statistic(x, values, statistic='mean',
bins=10, range=None):
"""
Compute a binned statistic for one or more sets of data.
This is a generalization of a histogram function. A histogram divides
the space into bins, and returns the count of the number of points in
each bin. This function allows the computation of the sum, mean, median,
or other statistic of the values (or set of values) within each bin.
Parameters
----------
x : (N,) array_like
A sequence of values to be binned.
values : (N,) array_like or list of (N,) array_like
The data on which the statistic will be computed. This must be
the same shape as `x`, or a set of sequences - each the same shape as
`x`. If `values` is a set of sequences, the statistic will be computed
on each independently.
statistic : string or callable, optional
The statistic to compute (default is 'mean').
The following statistics are available:
* 'mean' : compute the mean of values for points within each bin.
Empty bins will be represented by NaN.
* 'median' : compute the median of values for points within each
bin. Empty bins will be represented by NaN.
* 'count' : compute the count of points within each bin. This is
identical to an unweighted histogram. `values` array is not
referenced.
* 'sum' : compute the sum of values for points within each bin.
This is identical to a weighted histogram.
* 'min' : compute the minimum of values for points within each bin.
Empty bins will be represented by NaN.
* 'max' : compute the maximum of values for point within each bin.
Empty bins will be represented by NaN.
* function : a user-defined function which takes a 1D array of
values, and outputs a single numerical statistic. This function
will be called on the values in each bin. Empty bins will be
represented by function([]), or NaN if this returns an error.
bins : int or sequence of scalars, optional
If `bins` is an int, it defines the number of equal-width bins in the
given range (10 by default). If `bins` is a sequence, it defines the
bin edges, including the rightmost edge, allowing for non-uniform bin
widths. Values in `x` that are smaller than lowest bin edge are
assigned to bin number 0, values beyond the highest bin are assigned to
``bins[-1]``. If the bin edges are specified, the number of bins will
be, (nx = len(bins)-1).
range : (float, float) or [(float, float)], optional
The lower and upper range of the bins. If not provided, range
is simply ``(x.min(), x.max())``. Values outside the range are
ignored.
Returns
-------
statistic : array
The values of the selected statistic in each bin.
bin_edges : array of dtype float
Return the bin edges ``(length(statistic)+1)``.
binnumber: 1-D ndarray of ints
Indices of the bins (corresponding to `bin_edges`) in which each value
of `x` belongs. Same length as `values`. A binnumber of `i` means the
corresponding value is between (bin_edges[i-1], bin_edges[i]).
See Also
--------
numpy.digitize, numpy.histogram, binned_statistic_2d, binned_statistic_dd
Notes
-----
All but the last (righthand-most) bin is half-open. In other words, if
`bins` is ``[1, 2, 3, 4]``, then the first bin is ``[1, 2)`` (including 1,
but excluding 2) and the second ``[2, 3)``. The last bin, however, is
``[3, 4]``, which *includes* 4.
.. versionadded:: 0.11.0
Examples
--------
>>> from scipy import stats
>>> import matplotlib.pyplot as plt
First some basic examples:
Create two evenly spaced bins in the range of the given sample, and sum the
corresponding values in each of those bins:
>>> values = [1.0, 1.0, 2.0, 1.5, 3.0]
>>> stats.binned_statistic([1, 1, 2, 5, 7], values, 'sum', bins=2)
(array([ 4. , 4.5]), array([ 1., 4., 7.]), array([1, 1, 1, 2, 2]))
Multiple arrays of values can also be passed. The statistic is calculated
on each set independently:
>>> values = [[1.0, 1.0, 2.0, 1.5, 3.0], [2.0, 2.0, 4.0, 3.0, 6.0]]
>>> stats.binned_statistic([1, 1, 2, 5, 7], values, 'sum', bins=2)
(array([[ 4. , 4.5], [ 8. , 9. ]]), array([ 1., 4., 7.]),
array([1, 1, 1, 2, 2]))
>>> stats.binned_statistic([1, 2, 1, 2, 4], np.arange(5), statistic='mean',
... bins=3)
(array([ 1., 2., 4.]), array([ 1., 2., 3., 4.]),
array([1, 2, 1, 2, 3]))
As a second example, we now generate some random data of sailing boat speed
as a function of wind speed, and then determine how fast our boat is for
certain wind speeds:
>>> windspeed = 8 * np.random.rand(500)
>>> boatspeed = .3 * windspeed**.5 + .2 * np.random.rand(500)
>>> bin_means, bin_edges, binnumber = stats.binned_statistic(windspeed,
... boatspeed, statistic='median', bins=[1,2,3,4,5,6,7])
>>> plt.figure()
>>> plt.plot(windspeed, boatspeed, 'b.', label='raw data')
>>> plt.hlines(bin_means, bin_edges[:-1], bin_edges[1:], colors='g', lw=5,
... label='binned statistic of data')
>>> plt.legend()
Now we can use ``binnumber`` to select all datapoints with a windspeed
below 1:
>>> low_boatspeed = boatspeed[binnumber == 0]
As a final example, we will use ``bin_edges`` and ``binnumber`` to make a
plot of a distribution that shows the mean and distribution around that
mean per bin, on top of a regular histogram and the probability
distribution function:
>>> x = np.linspace(0, 5, num=500)
>>> x_pdf = stats.maxwell.pdf(x)
>>> samples = stats.maxwell.rvs(size=10000)
>>> bin_means, bin_edges, binnumber = stats.binned_statistic(x, x_pdf,
... statistic='mean', bins=25)
>>> bin_width = (bin_edges[1] - bin_edges[0])
>>> bin_centers = bin_edges[1:] - bin_width/2
>>> plt.figure()
>>> plt.hist(samples, bins=50, normed=True, histtype='stepfilled',
... alpha=0.2, label='histogram of data')
>>> plt.plot(x, x_pdf, 'r-', label='analytical pdf')
>>> plt.hlines(bin_means, bin_edges[:-1], bin_edges[1:], colors='g', lw=2,
... label='binned statistic of data')
>>> plt.plot((binnumber - 0.5) * bin_width, x_pdf, 'g.', alpha=0.5)
>>> plt.legend(fontsize=10)
>>> plt.show()
"""
try:
N = len(bins)
except TypeError:
N = 1
if N != 1:
bins = [np.asarray(bins, float)]
if range is not None:
if len(range) == 2:
range = [range]
medians, edges, binnumbers = binned_statistic_dd(
[x], values, statistic, bins, range)
return BinnedStatisticResult(medians, edges[0], binnumbers)
BinnedStatistic2dResult = namedtuple('BinnedStatistic2dResult',
('statistic', 'x_edge', 'y_edge',
'binnumber'))
def binned_statistic_2d(x, y, values, statistic='mean',
bins=10, range=None, expand_binnumbers=False):
"""
Compute a bidimensional binned statistic for one or more sets of data.
This is a generalization of a histogram2d function. A histogram divides
the space into bins, and returns the count of the number of points in
each bin. This function allows the computation of the sum, mean, median,
or other statistic of the values (or set of values) within each bin.
Parameters
----------
x : (N,) array_like
A sequence of values to be binned along the first dimension.
y : (N,) array_like
A sequence of values to be binned along the second dimension.
values : (N,) array_like or list of (N,) array_like
The data on which the statistic will be computed. This must be
the same shape as `x`, or a list of sequences - each with the same
shape as `x`. If `values` is such a list, the statistic will be
computed on each independently.
statistic : string or callable, optional
The statistic to compute (default is 'mean').
The following statistics are available:
* 'mean' : compute the mean of values for points within each bin.
Empty bins will be represented by NaN.
* 'median' : compute the median of values for points within each
bin. Empty bins will be represented by NaN.
* 'count' : compute the count of points within each bin. This is
identical to an unweighted histogram. `values` array is not
referenced.
* 'sum' : compute the sum of values for points within each bin.
This is identical to a weighted histogram.
* 'min' : compute the minimum of values for points within each bin.
Empty bins will be represented by NaN.
* 'max' : compute the maximum of values for point within each bin.
Empty bins will be represented by NaN.
* function : a user-defined function which takes a 1D array of
values, and outputs a single numerical statistic. This function
will be called on the values in each bin. Empty bins will be
represented by function([]), or NaN if this returns an error.
bins : int or [int, int] or array_like or [array, array], optional
The bin specification:
* the number of bins for the two dimensions (nx = ny = bins),
* the number of bins in each dimension (nx, ny = bins),
* the bin edges for the two dimensions (x_edge = y_edge = bins),
* the bin edges in each dimension (x_edge, y_edge = bins).
If the bin edges are specified, the number of bins will be,
(nx = len(x_edge)-1, ny = len(y_edge)-1).
range : (2,2) array_like, optional
The leftmost and rightmost edges of the bins along each dimension
(if not specified explicitly in the `bins` parameters):
[[xmin, xmax], [ymin, ymax]]. All values outside of this range will be
considered outliers and not tallied in the histogram.
expand_binnumbers : bool, optional
'False' (default): the returned `binnumber` is a shape (N,) array of
linearized bin indices.
'True': the returned `binnumber` is 'unraveled' into a shape (2,N)
ndarray, where each row gives the bin numbers in the corresponding
dimension.
See the `binnumber` returned value, and the `Examples` section.
.. versionadded:: 0.17.0
Returns
-------
statistic : (nx, ny) ndarray
The values of the selected statistic in each two-dimensional bin.
x_edge : (nx + 1) ndarray
The bin edges along the first dimension.
y_edge : (ny + 1) ndarray
The bin edges along the second dimension.
binnumber : (N,) array of ints or (2,N) ndarray of ints
This assigns to each element of `sample` an integer that represents the
bin in which this observation falls. The representation depends on the
`expand_binnumbers` argument. See `Notes` for details.
See Also
--------
numpy.digitize, numpy.histogram2d, binned_statistic, binned_statistic_dd
Notes
-----
Binedges:
All but the last (righthand-most) bin is half-open. In other words, if
`bins` is ``[1, 2, 3, 4]``, then the first bin is ``[1, 2)`` (including 1,
but excluding 2) and the second ``[2, 3)``. The last bin, however, is
``[3, 4]``, which *includes* 4.
`binnumber`:
This returned argument assigns to each element of `sample` an integer that
represents the bin in which it belongs. The representation depends on the
`expand_binnumbers` argument. If 'False' (default): The returned
`binnumber` is a shape (N,) array of linearized indices mapping each
element of `sample` to its corresponding bin (using row-major ordering).
If 'True': The returned `binnumber` is a shape (2,N) ndarray where
each row indicates bin placements for each dimension respectively. In each
dimension, a binnumber of `i` means the corresponding value is between
(D_edge[i-1], D_edge[i]), where 'D' is either 'x' or 'y'.
.. versionadded:: 0.11.0
Examples
--------
>>> from scipy import stats
Calculate the counts with explicit bin-edges:
>>> x = [0.1, 0.1, 0.1, 0.6]
>>> y = [2.1, 2.6, 2.1, 2.1]
>>> binx = [0.0, 0.5, 1.0]
>>> biny = [2.0, 2.5, 3.0]
>>> ret = stats.binned_statistic_2d(x, y, None, 'count', bins=[binx,biny])
>>> ret.statistic
array([[ 2., 1.],
[ 1., 0.]])
The bin in which each sample is placed is given by the `binnumber`
returned parameter. By default, these are the linearized bin indices:
>>> ret.binnumber
array([5, 6, 5, 9])
The bin indices can also be expanded into separate entries for each
dimension using the `expand_binnumbers` parameter:
>>> ret = stats.binned_statistic_2d(x, y, None, 'count', bins=[binx,biny],
... expand_binnumbers=True)
>>> ret.binnumber
array([[1, 1, 1, 2],
[1, 2, 1, 1]])
Which shows that the first three elements belong in the xbin 1, and the
fourth into xbin 2; and so on for y.
"""
# This code is based on np.histogram2d
try:
N = len(bins)
except TypeError:
N = 1
if N != 1 and N != 2:
xedges = yedges = np.asarray(bins, float)
bins = [xedges, yedges]
medians, edges, binnumbers = binned_statistic_dd(
[x, y], values, statistic, bins, range,
expand_binnumbers=expand_binnumbers)
return BinnedStatistic2dResult(medians, edges[0], edges[1], binnumbers)
BinnedStatisticddResult = namedtuple('BinnedStatisticddResult',
('statistic', 'bin_edges',
'binnumber'))
def binned_statistic_dd(sample, values, statistic='mean',
bins=10, range=None, expand_binnumbers=False):
"""
Compute a multidimensional binned statistic for a set of data.
This is a generalization of a histogramdd function. A histogram divides
the space into bins, and returns the count of the number of points in
each bin. This function allows the computation of the sum, mean, median,
or other statistic of the values within each bin.
Parameters
----------
sample : array_like
Data to histogram passed as a sequence of D arrays of length N, or
as an (N,D) array.
values : (N,) array_like or list of (N,) array_like
The data on which the statistic will be computed. This must be
the same shape as `x`, or a list of sequences - each with the same
shape as `x`. If `values` is such a list, the statistic will be
computed on each independently.
statistic : string or callable, optional
The statistic to compute (default is 'mean').
The following statistics are available:
* 'mean' : compute the mean of values for points within each bin.
Empty bins will be represented by NaN.
* 'median' : compute the median of values for points within each
bin. Empty bins will be represented by NaN.
* 'count' : compute the count of points within each bin. This is
identical to an unweighted histogram. `values` array is not
referenced.
* 'sum' : compute the sum of values for points within each bin.
This is identical to a weighted histogram.
* 'min' : compute the minimum of values for points within each bin.
Empty bins will be represented by NaN.
* 'max' : compute the maximum of values for point within each bin.
Empty bins will be represented by NaN.
* function : a user-defined function which takes a 1D array of
values, and outputs a single numerical statistic. This function
will be called on the values in each bin. Empty bins will be
represented by function([]), or NaN if this returns an error.
bins : sequence or int, optional
The bin specification must be in one of the following forms:
* A sequence of arrays describing the bin edges along each dimension.
* The number of bins for each dimension (nx, ny, ... = bins).
* The number of bins for all dimensions (nx = ny = ... = bins).
range : sequence, optional
A sequence of lower and upper bin edges to be used if the edges are
not given explicitely in `bins`. Defaults to the minimum and maximum
values along each dimension.
expand_binnumbers : bool, optional
'False' (default): the returned `binnumber` is a shape (N,) array of
linearized bin indices.
'True': the returned `binnumber` is 'unraveled' into a shape (D,N)
ndarray, where each row gives the bin numbers in the corresponding
dimension.
See the `binnumber` returned value, and the `Examples` section of
`binned_statistic_2d`.
.. versionadded:: 0.17.0
Returns
-------
statistic : ndarray, shape(nx1, nx2, nx3,...)
The values of the selected statistic in each two-dimensional bin.
bin_edges : list of ndarrays
A list of D arrays describing the (nxi + 1) bin edges for each
dimension.
binnumber : (N,) array of ints or (D,N) ndarray of ints
This assigns to each element of `sample` an integer that represents the
bin in which this observation falls. The representation depends on the
`expand_binnumbers` argument. See `Notes` for details.
See Also
--------
numpy.digitize, numpy.histogramdd, binned_statistic, binned_statistic_2d
Notes
-----
Binedges:
All but the last (righthand-most) bin is half-open in each dimension. In
other words, if `bins` is ``[1, 2, 3, 4]``, then the first bin is
``[1, 2)`` (including 1, but excluding 2) and the second ``[2, 3)``. The
last bin, however, is ``[3, 4]``, which *includes* 4.
`binnumber`:
This returned argument assigns to each element of `sample` an integer that
represents the bin in which it belongs. The representation depends on the
`expand_binnumbers` argument. If 'False' (default): The returned
`binnumber` is a shape (N,) array of linearized indices mapping each
element of `sample` to its corresponding bin (using row-major ordering).
If 'True': The returned `binnumber` is a shape (D,N) ndarray where
each row indicates bin placements for each dimension respectively. In each
dimension, a binnumber of `i` means the corresponding value is between
(bin_edges[D][i-1], bin_edges[D][i]), for each dimension 'D'.
.. versionadded:: 0.11.0
"""
known_stats = ['mean', 'median', 'count', 'sum', 'std','min','max']
if not callable(statistic) and statistic not in known_stats:
raise ValueError('invalid statistic %r' % (statistic,))
# `Ndim` is the number of dimensions (e.g. `2` for `binned_statistic_2d`)
# `Dlen` is the length of elements along each dimension.
# This code is based on np.histogramdd
try:
# `sample` is an ND-array.
Dlen, Ndim = sample.shape
except (AttributeError, ValueError):
# `sample` is a sequence of 1D arrays.
sample = np.atleast_2d(sample).T
Dlen, Ndim = sample.shape
# Store initial shape of `values` to preserve it in the output
values = np.asarray(values)
input_shape = list(values.shape)
# Make sure that `values` is 2D to iterate over rows
values = np.atleast_2d(values)
Vdim, Vlen = values.shape
# Make sure `values` match `sample`
if(statistic != 'count' and Vlen != Dlen):
raise AttributeError('The number of `values` elements must match the '
'length of each `sample` dimension.')
nbin = np.empty(Ndim, int) # Number of bins in each dimension
edges = Ndim * [None] # Bin edges for each dim (will be 2D array)
dedges = Ndim * [None] # Spacing between edges (will be 2D array)
try:
M = len(bins)
if M != Ndim:
raise AttributeError('The dimension of bins must be equal '
'to the dimension of the sample x.')
except TypeError:
bins = Ndim * [bins]
# Select range for each dimension
# Used only if number of bins is given.
if range is None:
smin = np.atleast_1d(np.array(sample.min(axis=0), float))
smax = np.atleast_1d(np.array(sample.max(axis=0), float))
else:
smin = np.zeros(Ndim)
smax = np.zeros(Ndim)
for i in xrange(Ndim):
smin[i], smax[i] = range[i]
# Make sure the bins have a finite width.
for i in xrange(len(smin)):
if smin[i] == smax[i]:
smin[i] = smin[i] - .5
smax[i] = smax[i] + .5
# Create edge arrays
for i in xrange(Ndim):
if np.isscalar(bins[i]):
nbin[i] = bins[i] + 2 # +2 for outlier bins
edges[i] = np.linspace(smin[i], smax[i], nbin[i] - 1)
else:
edges[i] = np.asarray(bins[i], float)
nbin[i] = len(edges[i]) + 1 # +1 for outlier bins
dedges[i] = np.diff(edges[i])
nbin = np.asarray(nbin)
# Compute the bin number each sample falls into, in each dimension
sampBin = [
np.digitize(sample[:, i], edges[i])
for i in xrange(Ndim)
]
# Using `digitize`, values that fall on an edge are put in the right bin.
# For the rightmost bin, we want values equal to the right
# edge to be counted in the last bin, and not as an outlier.
for i in xrange(Ndim):
# Find the rounding precision
decimal = int(-np.log10(dedges[i].min())) + 6
# Find which points are on the rightmost edge.
on_edge = np.where(np.around(sample[:, i], decimal) ==
np.around(edges[i][-1], decimal))[0]
# Shift these points one bin to the left.
sampBin[i][on_edge] -= 1
# Compute the sample indices in the flattened statistic matrix.
binnumbers = np.ravel_multi_index(sampBin, nbin)
result = np.empty([Vdim, nbin.prod()], float)
if statistic == 'mean':
result.fill(np.nan)
flatcount = np.bincount(binnumbers, None)
a = flatcount.nonzero()
for vv in xrange(Vdim):
flatsum = np.bincount(binnumbers, values[vv])
result[vv, a] = flatsum[a] / flatcount[a]
elif statistic == 'std':
result.fill(0)
flatcount = np.bincount(binnumbers, None)
a = flatcount.nonzero()
for vv in xrange(Vdim):
flatsum = np.bincount(binnumbers, values[vv])
flatsum2 = np.bincount(binnumbers, values[vv] ** 2)
result[vv, a] = np.sqrt(flatsum2[a] / flatcount[a] -
(flatsum[a] / flatcount[a]) ** 2)
elif statistic == 'count':
result.fill(0)
flatcount = np.bincount(binnumbers, None)
a = np.arange(len(flatcount))
result[:, a] = flatcount[np.newaxis, :]
elif statistic == 'sum':
result.fill(0)
for vv in xrange(Vdim):
flatsum = np.bincount(binnumbers, values[vv])
a = np.arange(len(flatsum))
result[vv, a] = flatsum
elif statistic == 'median':
result.fill(np.nan)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = np.median(values[vv, binnumbers == i])
elif statistic == 'min':
result.fill(np.nan)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = np.min(values[vv, binnumbers == i])
elif statistic == 'max':
result.fill(np.nan)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = np.max(values[vv, binnumbers == i])
elif callable(statistic):
with np.errstate(invalid='ignore'), suppress_warnings() as sup:
sup.filter(RuntimeWarning)
try:
null = statistic([])
except:
null = np.nan
result.fill(null)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = statistic(values[vv, binnumbers == i])
# Shape into a proper matrix
result = result.reshape(np.append(Vdim, nbin))
# Remove outliers (indices 0 and -1 for each bin-dimension).
core = [slice(None)] + Ndim * [slice(1, -1)]
result = result[core]
# Unravel binnumbers into an ndarray, each row the bins for each dimension
if(expand_binnumbers and Ndim > 1):
binnumbers = np.asarray(np.unravel_index(binnumbers, nbin))
if np.any(result.shape[1:] != nbin - 2):
raise RuntimeError('Internal Shape Error')
# Reshape to have output (`reulst`) match input (`values`) shape
result = result.reshape(input_shape[:-1] + list(nbin-2))
return BinnedStatisticddResult(result, edges, binnumbers)