= Global orientation ambiguity in a rank-one covariance fit
{title2=$C_\theta=C_{\theta+\pi}$}
\b[Global orientation ambiguity in a rank-one <covariance> fit] means that <covariance> identifies an axis without identifying its positive direction. For $\mathbf u_\theta=(\cos\theta,\sin\theta)^T$ and $\lambda>0$,
$$
C_\theta=\lambda\mathbf u_\theta\mathbf u_\theta^T
=C_{\theta+\pi},
$$
because $\mathbf u_{\theta+\pi}=-\mathbf u_\theta$. Adding an isotropic transverse-noise term does not remove this equality. The <angular resolution set by transverse spectral noise> can make each local orientation lobe narrow while leaving the two directed orientations indistinguishable.
For an original example, let $\lambda=4$ and $\theta=30^{\circ}$. Both $30^{\circ}$ and $210^{\circ}$ give
$$
C=\begin{pmatrix}3&\sqrt3\\\sqrt3&1\end{pmatrix}.
$$
The eigenvalues are four and zero, and the principal eigenvector can be written as either $(\sqrt3/2,1/2)^T$ or its negative. More samples can improve estimation of this matrix without choosing between those signs.
In a zero-mean Gaussian <covariance> model, equal <covariance> gives equal probability laws, including the supported line in the singular case.<SciPy 1.16.2 documentation>[scipy.stats.multivariate_normal, mean/cov parameters and Notes on singular covariance]{ref} No fitting procedure using only that law can distinguish the two parameter labels. A fitting range $0\leq\theta<\pi$ removes duplicate labels by convention; it does not measure the sign of a physical direction.
For example, local high-likelihood neighborhoods around $30^{\circ}$ and $210^{\circ}$ may each span only a few degrees. Reporting only the first as uncertainty in a directed angle discards an equally supported region unless additional information rules it out. A prior or coordinate convention that selects one sign should be distinguished from an experimental observation.
<Sign-sensitive measurements that resolve a covariance-axis ambiguity> can supply information outside the <covariance>-only model, such as an independently calibrated signed mean or phase reference. If the scientific quantity is genuinely an unoriented axis, an angle modulo $\pi$ is already the appropriate parameter. The interpretation of the parameter determines whether the ambiguity needs resolving.
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