# scipy.stats.vonmises_line

scipy.stats.vonmises_line = <scipy.stats._continuous_distns.vonmises_gen object>

  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.

  See also

  `scipy.stats.vonmises_fisher`

    Von-Mises Fisher distribution on a hypersphere

  Notes

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

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

  for \(-\pi \le x \le \pi\), \(\kappa > 0\). \(I_0\) 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 \([-\pi, \pi]\) 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(loc, kappa, 0)
      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(loc=loc, kappa=kappa, x=x)
      >>> 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.

      >>> number_of_samples = 1000
      >>> samples = vonmises(loc=loc, kappa=kappa).rvs(number_of_samples)

  Plot the von Mises density on a Cartesian and polar grid to emphasize that is 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(loc, kappa, x)
      >>> 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(number_of_samples))
      >>> left.hist(samples, 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(samples, density=True, bins=number_of_bins,
      ...            label="Histogram")
      >>> right.set_title("Polar plot")
      >>> right.legend(bbox_to_anchor=(0.15, 1.06))

  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 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.