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Module « scipy.stats »

Fonction vonmises_line - module scipy.stats

Signature de la fonction vonmises_line

def vonmises_line(*args, **kwds) 

Description

help(scipy.stats.vonmises_line)

A Von Mises continuous random variable.

As an instance of the `rv_continuous` class, `vonmises_line` object inherits from it
a collection of generic methods (see below for the full list),
and completes them with details specific for this particular distribution.

Methods
-------
rvs(kappa, loc=0, scale=1, size=1, random_state=None)
    Random variates.
pdf(x, kappa, loc=0, scale=1)
    Probability density function.
logpdf(x, kappa, loc=0, scale=1)
    Log of the probability density function.
cdf(x, kappa, loc=0, scale=1)
    Cumulative distribution function.
logcdf(x, kappa, loc=0, scale=1)
    Log of the cumulative distribution function.
sf(x, kappa, loc=0, scale=1)
    Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(x, kappa, loc=0, scale=1)
    Log of the survival function.
ppf(q, kappa, loc=0, scale=1)
    Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, kappa, loc=0, scale=1)
    Inverse survival function (inverse of ``sf``).
moment(order, kappa, loc=0, scale=1)
    Non-central moment of the specified order.
stats(kappa, loc=0, scale=1, moments='mv')
    Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(kappa, loc=0, scale=1)
    (Differential) entropy of the RV.
fit(data)
    Parameter estimates for generic data.
    See `scipy.stats.rv_continuous.fit <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#scipy.stats.rv_continuous.fit>`__ for detailed documentation of the
    keyword arguments.
expect(func, args=(kappa,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)
    Expected value of a function (of one argument) with respect to the distribution.
median(kappa, loc=0, scale=1)
    Median of the distribution.
mean(kappa, loc=0, scale=1)
    Mean of the distribution.
var(kappa, loc=0, scale=1)
    Variance of the distribution.
std(kappa, loc=0, scale=1)
    Standard deviation of the distribution.
interval(confidence, kappa, loc=0, scale=1)
    Confidence interval with equal areas around the median.

See Also
--------
scipy.stats.vonmises_fisher : Von-Mises Fisher distribution on a
                              hypersphere

Notes
-----
The probability density function for `vonmises` and `vonmises_line` is:

.. math::

    f(x, \kappa) = \frac{ \exp(\kappa \cos(x)) }{ 2 \pi I_0(\kappa) }

for :math:`-\pi \le x \le \pi`, :math:`\kappa \ge 0`. :math:`I_0` is the
modified Bessel function of order zero (`scipy.special.i0`).

`vonmises` is a circular distribution which does not restrict the
distribution to a fixed interval. Currently, there is no circular
distribution framework in SciPy. The ``cdf`` is implemented such that
``cdf(x + 2*np.pi) == cdf(x) + 1``.

`vonmises_line` is the same distribution, defined on :math:`[-\pi, \pi]`
on the real line. This is a regular (i.e. non-circular) distribution.

Note about distribution parameters: `vonmises` and `vonmises_line` take
``kappa`` as a shape parameter (concentration) and ``loc`` as the location
(circular mean). A ``scale`` parameter is accepted but does not have any
effect.

Examples
--------
Import the necessary modules.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.stats import vonmises

Define distribution parameters.

>>> loc = 0.5 * np.pi  # circular mean
>>> kappa = 1  # concentration

Compute the probability density at ``x=0`` via the ``pdf`` method.

>>> vonmises.pdf(0, loc=loc, kappa=kappa)
0.12570826359722018

Verify that the percentile function ``ppf`` inverts the cumulative
distribution function ``cdf`` up to floating point accuracy.

>>> x = 1
>>> cdf_value = vonmises.cdf(x, loc=loc, kappa=kappa)
>>> ppf_value = vonmises.ppf(cdf_value, loc=loc, kappa=kappa)
>>> x, cdf_value, ppf_value
(1, 0.31489339900904967, 1.0000000000000004)

Draw 1000 random variates by calling the ``rvs`` method.

>>> sample_size = 1000
>>> sample = vonmises(loc=loc, kappa=kappa).rvs(sample_size)

Plot the von Mises density on a Cartesian and polar grid to emphasize
that it is a circular distribution.

>>> fig = plt.figure(figsize=(12, 6))
>>> left = plt.subplot(121)
>>> right = plt.subplot(122, projection='polar')
>>> x = np.linspace(-np.pi, np.pi, 500)
>>> vonmises_pdf = vonmises.pdf(x, loc=loc, kappa=kappa)
>>> ticks = [0, 0.15, 0.3]

The left image contains the Cartesian plot.

>>> left.plot(x, vonmises_pdf)
>>> left.set_yticks(ticks)
>>> number_of_bins = int(np.sqrt(sample_size))
>>> left.hist(sample, density=True, bins=number_of_bins)
>>> left.set_title("Cartesian plot")
>>> left.set_xlim(-np.pi, np.pi)
>>> left.grid(True)

The right image contains the polar plot.

>>> right.plot(x, vonmises_pdf, label="PDF")
>>> right.set_yticks(ticks)
>>> right.hist(sample, density=True, bins=number_of_bins,
...            label="Histogram")
>>> right.set_title("Polar plot")
>>> right.legend(bbox_to_anchor=(0.15, 1.06))

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