scipy.stats.gaussian_kde

class scipy.stats.gaussian_kde(dataset, bw_method=None)

Representation of a kernel-density estimate using Gaussian kernels.

Kernel density estimation is a way to estimate the probability density function (PDF) of a random variable in a non-parametric way. gaussian_kde works for both uni-variate and multi-variate data. It includes automatic bandwidth determination. The estimation works best for a unimodal distribution; bimodal or multi-modal distributions tend to be oversmoothed.

Parameters:

• dataset (array_like) -- Datapoints to estimate from. In case of univariate data this is a 1-D array, otherwise a 2-D array with shape (# of dims, # of data).
• bw_method (str, scalar or callable, optional) -- The method used to calculate the estimator bandwidth. This can be 'scott', 'silverman', a scalar constant or a callable. If a scalar, this will be used directly as kde.factor. If a callable, it should take a gaussian_kde instance as only parameter and return a scalar. If None (default), 'scott' is used. See Notes for more details.

dataset

ndarray -- The dataset with which gaussian_kde was initialized.

d

int -- Number of dimensions.

n

int -- Number of datapoints.

factor

float -- The bandwidth factor, obtained from kde.covariance_factor, with which the covariance matrix is multiplied.

covariance

ndarray -- The covariance matrix of dataset, scaled by the calculated bandwidth (kde.factor).

inv_cov

ndarray -- The inverse of covariance.

kde.evaluate(points) : ndarray

Evaluate the estimated pdf on a provided set of points.

kde(points) : ndarray

Same as kde.evaluate(points)

kde.integrate_gaussian(mean, cov) : float

Multiply pdf with a specified Gaussian and integrate over the whole domain.

kde.integrate_box_1d(low, high) : float

Integrate pdf (1D only) between two bounds.

kde.integrate_box(low_bounds, high_bounds) : float

Integrate pdf over a rectangular space between low_bounds and high_bounds.

kde.integrate_kde(other_kde) : float

Integrate two kernel density estimates multiplied together.

kde.resample(size=None) : ndarray

Randomly sample a dataset from the estimated pdf.

kde.set_bandwidth(bw_method='scott') : None

Computes the bandwidth, i.e. the coefficient that multiplies the data covariance matrix to obtain the kernel covariance matrix. .. versionadded:: 0.11.0

kde.covariance_factor : float

Computes the coefficient (kde.factor) that multiplies the data covariance matrix to obtain the kernel covariance matrix. The default is scotts_factor. A subclass can overwrite this method to provide a different method, or set it through a call to kde.set_bandwidth.

Notes

Bandwidth selection strongly influences the estimate obtained from the KDE (much more so than the actual shape of the kernel). Bandwidth selection can be done by a "rule of thumb", by cross-validation, by "plug-in methods" or by other means; see [3], [4] for reviews. gaussian_kde uses a rule of thumb, the default is Scott's Rule.

Scott's Rule [1], implemented as scotts_factor, is:

n**(-1./(d+4)),

with n the number of data points and d the number of dimensions. Silverman's Rule [2], implemented as silverman_factor, is:

n * (d + 2) / 4.)**(-1. / (d + 4)).

Good general descriptions of kernel density estimation can be found in [1] and [2], the mathematics for this multi-dimensional implementation can be found in [1].

References

[1]	(1, 2, 3) D.W. Scott, "Multivariate Density Estimation: Theory, Practice, and Visualization", John Wiley & Sons, New York, Chicester, 1992.

[2]	(1, 2) B.W. Silverman, "Density Estimation for Statistics and Data Analysis", Vol. 26, Monographs on Statistics and Applied Probability, Chapman and Hall, London, 1986.

[3]	B.A. Turlach, "Bandwidth Selection in Kernel Density Estimation: A Review", CORE and Institut de Statistique, Vol. 19, pp. 1-33, 1993.

[4]	D.M. Bashtannyk and R.J. Hyndman, "Bandwidth selection for kernel conditional density estimation", Computational Statistics & Data Analysis, Vol. 36, pp. 279-298, 2001.

Examples

Generate some random two-dimensional data:

>>> from scipy import stats
>>> def measure(n):
>>>     "Measurement model, return two coupled measurements."
>>>     m1 = np.random.normal(size=n)
>>>     m2 = np.random.normal(scale=0.5, size=n)
>>>     return m1+m2, m1-m2

>>> m1, m2 = measure(2000)
>>> xmin = m1.min()
>>> xmax = m1.max()
>>> ymin = m2.min()
>>> ymax = m2.max()

Perform a kernel density estimate on the data:

>>> X, Y = np.mgrid[xmin:xmax:100j, ymin:ymax:100j]
>>> positions = np.vstack([X.ravel(), Y.ravel()])
>>> values = np.vstack([m1, m2])
>>> kernel = stats.gaussian_kde(values)
>>> Z = np.reshape(kernel(positions).T, X.shape)

Plot the results:

>>> import matplotlib.pyplot as plt
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.imshow(np.rot90(Z), cmap=plt.cm.gist_earth_r,
...           extent=[xmin, xmax, ymin, ymax])
>>> ax.plot(m1, m2, 'k.', markersize=2)
>>> ax.set_xlim([xmin, xmax])
>>> ax.set_ylim([ymin, ymax])
>>> plt.show()

__init__(dataset, bw_method=None)

Methods

__init__(dataset[, bw_method])
covariance_factor()
evaluate(points)	Evaluate the estimated pdf on a set of points.
integrate_box(low_bounds, high_bounds[, maxpts])	Computes the integral of a pdf over a rectangular interval.
integrate_box_1d(low, high)	Computes the integral of a 1D pdf between two bounds.
integrate_gaussian(mean, cov)	Multiply estimated density by a multivariate Gaussian and integrate over the whole space.
integrate_kde(other)	Computes the integral of the product of this kernel density estimate with another.
resample([size])	Randomly sample a dataset from the estimated pdf.
scotts_factor()
set_bandwidth([bw_method])	Compute the estimator bandwidth with given method.
silverman_factor()