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Rice histogram bin estimator.
Another simple estimator with no normality assumption. It has better
performance for large data than Sturges, but tends to overestimate
the number of bins. The number of bins is proportional to the cube
root of data size (asymptotically optimal). The estimate depends
only on size of the data.
Parameters
----------
x : array_like
Input data that is to be histogrammed, trimmed to range. May not
be empty.
Returns
-------
h : An estimate of the optimal bin width for the given data.
@UUUUUU?)r r r s r _hist_bin_ricer L s$ ( 77cAFw // 00r c h ~dt j dz z | j z dz t j | z S )a
Scott histogram bin estimator.
The binwidth is proportional to the standard deviation of the data
and inversely proportional to the cube root of data size
(asymptotically optimal).
Parameters
----------
x : array_like
Input data that is to be histogrammed, trimmed to range. May not
be empty.
Returns
-------
h : An estimate of the optimal bin width for the given data.
g 8@ ?r )r pir stdr s r _hist_bin_scottr$ d s1 $ 25#: &) 4rvayy @@r c R j t dk sdk rdS fd}t dt t j }t
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Histogram bin estimator based on minimizing the estimated integrated squared error (ISE).
The number of bins is chosen by minimizing the estimated ISE against the unknown true distribution.
The ISE is estimated using cross-validation and can be regarded as a generalization of Scott's rule.
https://en.wikipedia.org/wiki/Histogram#Scott.27s_normal_reference_rule
This paper by Stone appears to be the origination of this rule.
http://digitalassets.lib.berkeley.edu/sdtr/ucb/text/34.pdf
Parameters
----------
x : array_like
Input data that is to be histogrammed, trimmed to range. May not
be empty.
range : (float, float)
The lower and upper range of the bins.
Returns
-------
h : An estimate of the optimal bin width for the given data.
r c | z }t j | d z }ddz | | z z
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}t j ||| t j |d| t j | }t | dt j | j z t j dt j | |z z z z S dS )a
Doane's histogram bin estimator.
Improved version of Sturges' formula which works better for
non-normal data. See
stats.stackexchange.com/questions/55134/doanes-formula-for-histogram-binning
Parameters
----------
x : array_like
Input data that is to be histogrammed, trimmed to range. May not
be empty.
Returns
-------
h : An estimate of the optimal bin width for the given data.
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The Freedman-Diaconis histogram bin estimator.
The Freedman-Diaconis rule uses interquartile range (IQR) to
estimate binwidth. It is considered a variation of the Scott rule
with more robustness as the IQR is less affected by outliers than
the standard deviation. However, the IQR depends on fewer points
than the standard deviation, so it is less accurate, especially for
long tailed distributions.
If the IQR is 0, this function returns 0 for the bin width.
Binwidth is inversely proportional to the cube root of data size
(asymptotically optimal).
Parameters
----------
x : array_like
Input data that is to be histogrammed, trimmed to range. May not
be empty.
Returns
-------
h : An estimate of the optimal bin width for the given data.
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Histogram bin estimator that uses the minimum width of the
Freedman-Diaconis and Sturges estimators if the FD bin width is non-zero.
If the bin width from the FD estimator is 0, the Sturges estimator is used.
The FD estimator is usually the most robust method, but its width
estimate tends to be too large for small `x` and bad for data with limited
variance. The Sturges estimator is quite good for small (<1000) datasets
and is the default in the R language. This method gives good off-the-shelf
behaviour.
.. versionchanged:: 1.15.0
If there is limited variance the IQR can be 0, which results in the
FD bin width being 0 too. This is not a valid bin width, so
``np.histogram_bin_edges`` chooses 1 bin instead, which may not be optimal.
If the IQR is 0, it's unlikely any variance-based estimators will be of
use, so we revert to the Sturges estimator, which only uses the size of the
dataset in its calculation.
Parameters
----------
x : array_like
Input data that is to be histogrammed, trimmed to range. May not
be empty.
Returns
-------
h : An estimate of the optimal bin width for the given data.
See Also
--------
_hist_bin_fd, _hist_bin_sturges
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Determine the outer bin edges to use, from either the data or the range
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Function to calculate only the edges of the bins used by the `histogram`
function.
Parameters
----------
a : array_like
Input data. The histogram is computed over the flattened array.
bins : int or sequence of scalars or str, 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.
If `bins` is a string from the list below, `histogram_bin_edges` will use
the method chosen to calculate the optimal bin width and
consequently the number of bins (see `Notes` for more detail on
the estimators) from the data that falls within the requested
range. While the bin width will be optimal for the actual data
in the range, the number of bins will be computed to fill the
entire range, including the empty portions. For visualisation,
using the 'auto' option is suggested. Weighted data is not
supported for automated bin size selection.
'auto'
Maximum of the 'sturges' and 'fd' estimators. Provides good
all around performance.
'fd' (Freedman Diaconis Estimator)
Robust (resilient to outliers) estimator that takes into
account data variability and data size.
'doane'
An improved version of Sturges' estimator that works better
with non-normal datasets.
'scott'
Less robust estimator that takes into account data variability
and data size.
'stone'
Estimator based on leave-one-out cross-validation estimate of
the integrated squared error. Can be regarded as a generalization
of Scott's rule.
'rice'
Estimator does not take variability into account, only data
size. Commonly overestimates number of bins required.
'sturges'
R's default method, only accounts for data size. Only
optimal for gaussian data and underestimates number of bins
for large non-gaussian datasets.
'sqrt'
Square root (of data size) estimator, used by Excel and
other programs for its speed and simplicity.
range : (float, float), optional
The lower and upper range of the bins. If not provided, range
is simply ``(a.min(), a.max())``. Values outside the range are
ignored. The first element of the range must be less than or
equal to the second. `range` affects the automatic bin
computation as well. While bin width is computed to be optimal
based on the actual data within `range`, the bin count will fill
the entire range including portions containing no data.
weights : array_like, optional
An array of weights, of the same shape as `a`. Each value in
`a` only contributes its associated weight towards the bin count
(instead of 1). This is currently not used by any of the bin estimators,
but may be in the future.
Returns
-------
bin_edges : array of dtype float
The edges to pass into `histogram`
See Also
--------
histogram
Notes
-----
The methods to estimate the optimal number of bins are well founded
in literature, and are inspired by the choices R provides for
histogram visualisation. Note that having the number of bins
proportional to :math:`n^{1/3}` is asymptotically optimal, which is
why it appears in most estimators. These are simply plug-in methods
that give good starting points for number of bins. In the equations
below, :math:`h` is the binwidth and :math:`n_h` is the number of
bins. All estimators that compute bin counts are recast to bin width
using the `ptp` of the data. The final bin count is obtained from
``np.round(np.ceil(range / h))``. The final bin width is often less
than what is returned by the estimators below.
'auto' (maximum of the 'sturges' and 'fd' estimators)
A compromise to get a good value. For small datasets the Sturges
value will usually be chosen, while larger datasets will usually
default to FD. Avoids the overly conservative behaviour of FD
and Sturges for small and large datasets respectively.
Switchover point is usually :math:`a.size \approx 1000`.
'fd' (Freedman Diaconis Estimator)
.. math:: h = 2 \frac{IQR}{n^{1/3}}
The binwidth is proportional to the interquartile range (IQR)
and inversely proportional to cube root of a.size. Can be too
conservative for small datasets, but is quite good for large
datasets. The IQR is very robust to outliers.
'scott'
.. math:: h = \sigma \sqrt[3]{\frac{24 \sqrt{\pi}}{n}}
The binwidth is proportional to the standard deviation of the
data and inversely proportional to cube root of ``x.size``. Can
be too conservative for small datasets, but is quite good for
large datasets. The standard deviation is not very robust to
outliers. Values are very similar to the Freedman-Diaconis
estimator in the absence of outliers.
'rice'
.. math:: n_h = 2n^{1/3}
The number of bins is only proportional to cube root of
``a.size``. It tends to overestimate the number of bins and it
does not take into account data variability.
'sturges'
.. math:: n_h = \log _{2}(n) + 1
The number of bins is the base 2 log of ``a.size``. This
estimator assumes normality of data and is too conservative for
larger, non-normal datasets. This is the default method in R's
``hist`` method.
'doane'
.. math:: n_h = 1 + \log_{2}(n) +
\log_{2}\left(1 + \frac{|g_1|}{\sigma_{g_1}}\right)
g_1 = mean\left[\left(\frac{x - \mu}{\sigma}\right)^3\right]
\sigma_{g_1} = \sqrt{\frac{6(n - 2)}{(n + 1)(n + 3)}}
An improved version of Sturges' formula that produces better
estimates for non-normal datasets. This estimator attempts to
account for the skew of the data.
'sqrt'
.. math:: n_h = \sqrt n
The simplest and fastest estimator. Only takes into account the
data size.
Examples
--------
>>> arr = np.array([0, 0, 0, 1, 2, 3, 3, 4, 5])
>>> np.histogram_bin_edges(arr, bins='auto', range=(0, 1))
array([0. , 0.25, 0.5 , 0.75, 1. ])
>>> np.histogram_bin_edges(arr, bins=2)
array([0. , 2.5, 5. ])
For consistency with histogram, an array of pre-computed bins is
passed through unmodified:
>>> np.histogram_bin_edges(arr, [1, 2])
array([1, 2])
This function allows one set of bins to be computed, and reused across
multiple histograms:
>>> shared_bins = np.histogram_bin_edges(arr, bins='auto')
>>> shared_bins
array([0., 1., 2., 3., 4., 5.])
>>> group_id = np.array([0, 1, 1, 0, 1, 1, 0, 1, 1])
>>> hist_0, _ = np.histogram(arr[group_id == 0], bins=shared_bins)
>>> hist_1, _ = np.histogram(arr[group_id == 1], bins=shared_bins)
>>> hist_0; hist_1
array([1, 1, 0, 1, 0])
array([2, 0, 1, 1, 2])
Which gives more easily comparable results than using separate bins for
each histogram:
>>> hist_0, bins_0 = np.histogram(arr[group_id == 0], bins='auto')
>>> hist_1, bins_1 = np.histogram(arr[group_id == 1], bins='auto')
>>> hist_0; hist_1
array([1, 1, 1])
array([2, 1, 1, 2])
>>> bins_0; bins_1
array([0., 1., 2., 3.])
array([0. , 1.25, 2.5 , 3.75, 5. ])
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n>> np.histogram([1, 2, 1], bins=[0, 1, 2, 3])
(array([0, 2, 1]), array([0, 1, 2, 3]))
>>> np.histogram(np.arange(4), bins=np.arange(5), density=True)
(array([0.25, 0.25, 0.25, 0.25]), array([0, 1, 2, 3, 4]))
>>> np.histogram([[1, 2, 1], [1, 0, 1]], bins=[0,1,2,3])
(array([1, 4, 1]), array([0, 1, 2, 3]))
>>> a = np.arange(5)
>>> hist, bin_edges = np.histogram(a, density=True)
>>> hist
array([0.5, 0. , 0.5, 0. , 0. , 0.5, 0. , 0.5, 0. , 0.5])
>>> hist.sum()
2.4999999999999996
>>> np.sum(hist * np.diff(bin_edges))
1.0
.. versionadded:: 1.11.0
Automated Bin Selection Methods example, using 2 peak random data
with 2000 points:
>>> import matplotlib.pyplot as plt
>>> rng = np.random.RandomState(10) # deterministic random data
>>> a = np.hstack((rng.normal(size=1000),
... rng.normal(loc=5, scale=2, size=1000)))
>>> _ = plt.hist(a, bins='auto') # arguments are passed to np.histogram
>>> plt.title("Histogram with 'auto' bins")
Text(0.5, 1.0, "Histogram with 'auto' bins")
>>> plt.show()
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Compute the multidimensional histogram of some data.
Parameters
----------
sample : (N, D) array, or (N, D) array_like
The data to be histogrammed.
Note the unusual interpretation of sample when an array_like:
* When an array, each row is a coordinate in a D-dimensional space -
such as ``histogramdd(np.array([p1, p2, p3]))``.
* When an array_like, each element is the list of values for single
coordinate - such as ``histogramdd((X, Y, Z))``.
The first form should be preferred.
bins : sequence or int, optional
The bin specification:
* A sequence of arrays describing the monotonically increasing 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 length D, each an optional (lower, upper) tuple giving
the outer bin edges to be used if the edges are not given explicitly in
`bins`.
An entry of None in the sequence results in the minimum and maximum
values being used for the corresponding dimension.
The default, None, is equivalent to passing a tuple of D None values.
density : bool, optional
If False, the default, returns the number of samples in each bin.
If True, returns the probability *density* function at the bin,
``bin_count / sample_count / bin_volume``.
weights : (N,) array_like, optional
An array of values `w_i` weighing each sample `(x_i, y_i, z_i, ...)`.
Weights are normalized to 1 if density is True. If density is False,
the values of the returned histogram are equal to the sum of the
weights belonging to the samples falling into each bin.
Returns
-------
H : ndarray
The multidimensional histogram of sample x. See density and weights
for the different possible semantics.
edges : list
A list of D arrays describing the bin edges for each dimension.
See Also
--------
histogram: 1-D histogram
histogram2d: 2-D histogram
Examples
--------
>>> r = np.random.randn(100,3)
>>> H, edges = np.histogramdd(r, bins = (5, 8, 4))
>>> H.shape, edges[0].size, edges[1].size, edges[2].size
((5, 8, 4), 6, 9, 5)
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