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# scipy.stats.multivariate_t

scipy.stats.multivariate_t = <scipy.stats._multivariate.multivariate_t_gen object>

  A multivariate t-distributed random variable.

  The loc parameter specifies the location. The shape parameter specifies the positive semidefinite shape matrix. The df parameter specifies the degrees of freedom.

  In addition to calling the methods below, the object itself may be called as a function to fix the location, shape matrix, and degrees of freedom parameters, returning a “frozen” multivariate t-distribution random.

  Parameters:
    loc array_like, optional

      Location of the distribution. (default `0`)

    shape array_like, optional

      Positive semidefinite matrix of the distribution. (default `1`)

    df float, optional

      Degrees of freedom of the distribution; must be greater than zero. If `np.inf` then results are multivariate normal. The default is `1`.

    allow_singular bool, optional

      Whether to allow a singular matrix. (default `False`)

    seed{None, int, np.random.RandomState, np.random.Generator}, optional

      Used for drawing random variates. If seed is None, the RandomState singleton is used. If seed is an int, a new `RandomState` instance is used, seeded with seed. If seed is already a `RandomState` or `Generator` instance, then that object is used. Default is None.

  Methods

  pdf(x, loc=None, shape=1, df=1, allow_singular=False) | Probability density function.
  logpdf(x, loc=None, shape=1, df=1, allow_singular=False) | Log of the probability density function.
  cdf(x, loc=None, shape=1, df=1, allow_singular=False, *, | maxpts=None, lower_limit=None, random_state=None) Cumulative distribution function.
  rvs(loc=None, shape=1, df=1, size=1, random_state=None) | Draw random samples from a multivariate t-distribution.
  entropy(loc=None, shape=1, df=1) | Differential entropy of a multivariate t-distribution.
  marginal(dimensions, loc=None, shape=1, df=1, allow_singular=False) | Return a marginal multivariate t-distribution.

  Notes

  Setting the parameter loc to `None` is equivalent to having loc be the zero-vector. The parameter shape can be a scalar, in which case the shape matrix is the identity times that value, a vector of diagonal entries for the shape matrix, or a two-dimensional array_like. The matrix shape must be a (symmetric) positive semidefinite matrix. The determinant and inverse of shape are computed as the pseudo-determinant and pseudo-inverse, respectively, so that shape does not need to have full rank.

  The probability density function for `multivariate_t` is

  \[f(x) = \frac{\Gamma((\nu + p)/2)}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}|\Sigma|^{1/2}} \left[1 + \frac{1}{\nu} (\mathbf{x} - \boldsymbol{\mu})^{\top} \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right]^{-(\nu + p)/2},\]

  where \(p\) is the dimension of \(\mathbf{x}\), \(\boldsymbol{\mu}\) is the \(p\)-dimensional location, \(\boldsymbol{\Sigma}\) the \(p \times p\)-dimensional shape matrix, and \(\nu\) is the degrees of freedom.

  Added in version 1.6.0.

  References

  [1]

  Arellano-Valle et al. “Shannon Entropy and Mutual Information for Multivariate Skew-Elliptical Distributions”. Scandinavian Journal of Statistics. Vol. 40, issue 1.

  Examples

  The object may be called (as a function) to fix the loc, shape, df, and allow_singular parameters, returning a “frozen” multivariate_t random variable:

      >>> import numpy as np
      >>> from scipy.stats import multivariate_t
      >>> rv = multivariate_t([1.0, -0.5], [[2.1, 0.3], [0.3, 1.5]], df=2)
      >>> # Frozen object with the same methods but holding the given location,
      >>> # scale, and degrees of freedom fixed.

  Create a contour plot of the PDF.

      >>> import matplotlib.pyplot as plt
      >>> x, y = np.mgrid[-1:3:.01, -2:1.5:.01]
      >>> pos = np.dstack((x, y))
      >>> fig, ax = plt.subplots(1, 1)
      >>> ax.set_aspect('equal')
      >>> plt.contourf(x, y, rv.pdf(pos))