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Classe « RandomState »

Méthode numpy.random.RandomState.multivariate_normal

Signature de la méthode multivariate_normal

def multivariate_normal(self, mean, cov, size=None, check_valid='warn', tol=1e-08) 

Description

help(RandomState.multivariate_normal)

        multivariate_normal(mean, cov, size=None, check_valid='warn', tol=1e-8)

        Draw random samples from a multivariate normal distribution.

        The multivariate normal, multinormal or Gaussian distribution is a
        generalization of the one-dimensional normal distribution to higher
        dimensions.  Such a distribution is specified by its mean and
        covariance matrix.  These parameters are analogous to the mean
        (average or "center") and variance (standard deviation, or "width,"
        squared) of the one-dimensional normal distribution.

        .. note::
            New code should use the
            `~numpy.random.Generator.multivariate_normal`
            method of a `~numpy.random.Generator` instance instead;
            please see the :ref:`random-quick-start`.

        Parameters
        ----------
        mean : 1-D array_like, of length N
            Mean of the N-dimensional distribution.
        cov : 2-D array_like, of shape (N, N)
            Covariance matrix of the distribution. It must be symmetric and
            positive-semidefinite for proper sampling.
        size : int or tuple of ints, optional
            Given a shape of, for example, ``(m,n,k)``, ``m*n*k`` samples are
            generated, and packed in an `m`-by-`n`-by-`k` arrangement.  Because
            each sample is `N`-dimensional, the output shape is ``(m,n,k,N)``.
            If no shape is specified, a single (`N`-D) sample is returned.
        check_valid : { 'warn', 'raise', 'ignore' }, optional
            Behavior when the covariance matrix is not positive semidefinite.
        tol : float, optional
            Tolerance when checking the singular values in covariance matrix.
            cov is cast to double before the check.

        Returns
        -------
        out : ndarray
            The drawn samples, of shape *size*, if that was provided.  If not,
            the shape is ``(N,)``.

            In other words, each entry ``out[i,j,...,:]`` is an N-dimensional
            value drawn from the distribution.

        See Also
        --------
        random.Generator.multivariate_normal: which should be used for new code.

        Notes
        -----
        The mean is a coordinate in N-dimensional space, which represents the
        location where samples are most likely to be generated.  This is
        analogous to the peak of the bell curve for the one-dimensional or
        univariate normal distribution.

        Covariance indicates the level to which two variables vary together.
        From the multivariate normal distribution, we draw N-dimensional
        samples, :math:`X = [x_1, x_2, ... x_N]`.  The covariance matrix
        element :math:`C_{ij}` is the covariance of :math:`x_i` and :math:`x_j`.
        The element :math:`C_{ii}` is the variance of :math:`x_i` (i.e. its
        "spread").

        Instead of specifying the full covariance matrix, popular
        approximations include:

        - Spherical covariance (`cov` is a multiple of the identity matrix)
        - Diagonal covariance (`cov` has non-negative elements, and only on
          the diagonal)

        This geometrical property can be seen in two dimensions by plotting
        generated data-points:

        >>> mean = [0, 0]
        >>> cov = [[1, 0], [0, 100]]  # diagonal covariance

        Diagonal covariance means that points are oriented along x or y-axis:

        >>> import matplotlib.pyplot as plt
        >>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
        >>> plt.plot(x, y, 'x')
        >>> plt.axis('equal')
        >>> plt.show()

        Note that the covariance matrix must be positive semidefinite (a.k.a.
        nonnegative-definite). Otherwise, the behavior of this method is
        undefined and backwards compatibility is not guaranteed.

        References
        ----------
        .. [1] Papoulis, A., "Probability, Random Variables, and Stochastic
               Processes," 3rd ed., New York: McGraw-Hill, 1991.
        .. [2] Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern
               Classification," 2nd ed., New York: Wiley, 2001.

        Examples
        --------
        >>> mean = (1, 2)
        >>> cov = [[1, 0], [0, 1]]
        >>> x = np.random.multivariate_normal(mean, cov, (3, 3))
        >>> x.shape
        (3, 3, 2)

        Here we generate 800 samples from the bivariate normal distribution
        with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]].  The
        expected variances of the first and second components of the sample
        are 6 and 3.5, respectively, and the expected correlation
        coefficient is -3/sqrt(6*3.5) ≈ -0.65465.

        >>> cov = np.array([[6, -3], [-3, 3.5]])
        >>> pts = np.random.multivariate_normal([0, 0], cov, size=800)

        Check that the mean, covariance, and correlation coefficient of the
        sample are close to the expected values:

        >>> pts.mean(axis=0)
        array([ 0.0326911 , -0.01280782])  # may vary
        >>> np.cov(pts.T)
        array([[ 5.96202397, -2.85602287],
               [-2.85602287,  3.47613949]])  # may vary
        >>> np.corrcoef(pts.T)[0, 1]
        -0.6273591314603949  # may vary

        We can visualize this data with a scatter plot.  The orientation
        of the point cloud illustrates the negative correlation of the
        components of this sample.

        >>> import matplotlib.pyplot as plt
        >>> plt.plot(pts[:, 0], pts[:, 1], '.', alpha=0.5)
        >>> plt.axis('equal')
        >>> plt.grid()
        >>> plt.show()
        

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