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# scipy.stats.Covariance.from_diagonal

static Covariance.from_diagonal(diagonal)

  Return a representation of a covariance matrix from its diagonal.

  Parameters:
    diagonal array_like

      The diagonal elements of a diagonal matrix.

  Notes

  Let the diagonal elements of a diagonal covariance matrix \(D\) be stored in the vector \(d\).

  When all elements of \(d\) are strictly positive, whitening of a data point \(x\) is performed by computing \(x \cdot d^{-1/2}\), where the inverse square root can be taken element-wise. \(\log\det{D}\) is calculated as \(-2 \sum(\log{d})\), where the \(\log\) operation is performed element-wise.

  This `Covariance` class supports singular covariance matrices. When computing `_log_pdet`, non-positive elements of \(d\) are ignored. Whitening is not well defined when the point to be whitened does not lie in the span of the columns of the covariance matrix. The convention taken here is to treat the inverse square root of non-positive elements of \(d\) as zeros.

  Examples

  Prepare a symmetric positive definite covariance matrix `A` and a data point `x`.

      >>> import numpy as np
      >>> from scipy import stats
      >>> rng = np.random.default_rng()
      >>> n = 5
      >>> A = np.diag(rng.random(n))
      >>> x = rng.random(size=n)

  Extract the diagonal from `A` and create the `Covariance` object.

      >>> d = np.diag(A)
      >>> cov = stats.Covariance.from_diagonal(d)

  Compare the functionality of the `Covariance` object against a reference implementations.

      >>> res = cov.whiten(x)
      >>> ref = np.diag(d**-0.5) @ x
      >>> np.allclose(res, ref)
      True
      >>> res = cov.log_pdet
      >>> ref = np.linalg.slogdet(A)[-1]
      >>> np.allclose(res, ref)
      True