numpy.lib.histograms

Parameters

a : ndarray

Ravelled data array

bins, range

Forwarded arguments from histogram.

weights : ndarray, optional

Ravelled weights array, or None

Returns

bin_edges : ndarray

Array of bin edges

uniform_bins : (Number, Number, int):

The upper bound, lowerbound, and number of bins, used in the optimized implementation of histogram that works on uniform bins.

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

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.

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.

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.

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.

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.

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.

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.

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 a monotonically increasing array of bin edges, including the rightmost edge, allowing for non-uniform bin widths.

New in version 1.11.0.

If bins is a string, it defines the method used to calculate the optimal bin width, as defined by histogram_bin_edges.

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.

normed : bool, optional

Deprecated since version 1.6.0.

This is equivalent to the density argument, but produces incorrect results for unequal bin widths. It should not be used.

Changed in version 1.15.0: DeprecationWarnings are actually emitted.

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). If density is True, the weights are normalized, so that the integral of the density over the range remains 1.

density : bool, optional

If False, the result will contain the number of samples in each bin. If True, the result is the value of the probability density function at the bin, normalized such that the integral over the range is 1. Note that the sum of the histogram values will not be equal to 1 unless bins of unity width are chosen; it is not a probability mass function.

Overrides the normed keyword if given.

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

histogramdd, bincount, searchsorted, digitize, histogram_bin_edges

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.

Examples

>>> 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

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()

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 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 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, h is the binwidth and 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 a.size ≈ 1000.

'fd' (Freedman Diaconis Estimator)

h = 2(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'

h = σ³√((24*√(π))/(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'

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'

n_h = log₂n + 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'

n_h = 1 + log₂(n) + log₂(1 + (|g₁|)/(σ_g1))

g₁ = mean[((x − μ)/(σ))³]

σ_g1 = √((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'

n_h = √(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.  ])

Parameters

sample : (N, D) array, or (D, N) 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.

normed : bool, optional

An alias for the density argument that behaves identically. To avoid confusion with the broken normed argument to histogram, density should be preferred.

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 normed is True. If normed 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 normed 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)