 Global orientation ambiguity in a rank-one covariance fit

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 $\lambda >0$,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 $210^{\circ }$ giveThe eigenvalues are four and zero, and the principal eigenvector can be written as either $(\sqrt{3}/2,1/2{)}^T$ 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.^[ref] No fitting procedure using only that law can distinguish the two parameter labels. A fitting range $0\le \theta <\pi$ 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.