scipy.stats.multivariate_normal raising `LinAlgError: singular matrix` even though my covariance matrix is invertible

Question

I am having trouble trying to use scipy.stats.multivariate_normal, hopefully one of you might be able to help.

I have a 2x2 matrix which is possible to find the inverse of using numpy.linalg.inv(), however when I attempt to use it as the covariance matrix in multivariate_normal I receive a LinAlgError stating that it is a singular matrix:

In [89]: cov = np.array([[3.2e5**2, 3.2e5*0.103*-0.459],[3.2e5*0.103*-0.459, 0.103**2]])

In [90]: np.linalg.inv(cov)
Out[90]:
array([[  1.23722158e-11,   1.76430200e-05],
       [  1.76430200e-05,   1.19418880e+02]])

In [91]: multivariate_normal([0,0], cov)
---------------------------------------------------------------------------
LinAlgError                               Traceback (most recent call last)
<ipython-input-91-44a6625beda5> in <module>()
----> 1 multivariate_normal([0,0], cov)

/mnt/ssd/Enthought_jli199/Canopy_64bit/User/lib/python2.7/site-packages/scipy/stats/_multivariate.pyc in __call__(self, mean, cov, allow_singular, seed)
    421         return multivariate_normal_frozen(mean, cov,
    422                                           allow_singular=allow_singular,
--> 423                                           seed=seed)
    424
    425     def _logpdf(self, x, mean, prec_U, log_det_cov, rank):

/mnt/ssd/Enthought_jli199/Canopy_64bit/User/lib/python2.7/site-packages/scipy/stats/_multivariate.pyc in __init__(self, mean, cov, allow_singular, seed)
    591         """
    592         self.dim, self.mean, self.cov = _process_parameters(None, mean, cov)
--> 593         self.cov_info = _PSD(self.cov, allow_singular=allow_singular)
    594         self._dist = multivariate_normal_gen(seed)
    595

/mnt/ssd/Enthought_jli199/Canopy_64bit/User/lib/python2.7/site-packages/scipy/stats/_multivariate.pyc in __init__(self, M, cond, rcond, lower, check_finite, allow_singular)
    217         d = s[s > eps]
    218         if len(d) < len(s) and not allow_singular:
--> 219             raise np.linalg.LinAlgError('singular matrix')
    220         s_pinv = _pinv_1d(s, eps)
    221         U = np.multiply(u, np.sqrt(s_pinv))

LinAlgError: singular matrix

Answer 1

By default multivariate_normal checks whether any of the eigenvalues of the covariance matrix are less than some tolerance chosen based on its dtype and the magnitude of its largest eigenvalue (take a look at the source code for scipy.stats._multivariate._PSD and scipy.stats._multivariate._eigvalsh_to_eps for the full details).

As @kazemakase mentioned above, whilst your covariance matrix may be invertible according to the criteria used by np.linalg.inv, it is still very ill-conditioned and fails the more stringent test used by multivariate_normal.

You could pass allow_singular=True to multivariate_normal to skip this test, but in general it would be better to rescale your data to avoid passing such an ill-conditioned covariance matrix in the first place.