Extended EOFs^*

Propagation may be studied with a technique called 'Extended EOFs' (EEOFs). The math is essentially the same as for ordinary EOFs, and the difference lies in the preprocessing of the data. The EEOFs maximize the variance in a $(n_x \times n_y) \times n_l$ window.

Compute the covariances at all spatial lags and out to time-lag n_l -1. Assuming that we are looking for patterns which are invariant under the sliding-window operation. Let the $\vec{x}$ describe the geographically distributed data at time i, denoted by the subscript. Then ${\bf X} = [\vec{x}_1 .. \vec{x}_T] \rightarrow {\bf X}'= [\{\vec{x}_1 .. \vec{x}_L\}.. \{\vec{x}_{T-L} .. \vec{x}_T\}] = [\vec{x'}_1 .. \vec{x'}_{T-L}]$, where $\vec{x'}_i= \{\vec{x}_i .. \vec{x}_{i+L}\}$.

PCs have rank n_t - n_l +1.

Advantages: i) more averaging $\rightarrow$ smoother patterns and sometimes better S/N; ii) contain lag-relationship information that can help interpretation of the patterns.

Pitfalls: The eigenvectors of the inverse covariance matrix are similar to EOFs of common noise process. Errors: ${\bf\Omega}^{-1} \vec{e} = \vec{e} \lambda$, which is the wave equation. Take care!

Sanity check: i) compare with EEOFs applied to data filtered through a few of the leading conventional EOFs; ii) model each PC as an AR(1) (red noise) process (MC-test): H₀= ``Data consists of mutually independent, non-oscillatory processes''; iii) Compare power in each extended EOF/PC pair with the distribution of power in that from from the surrogates: if all are inside the null-distribution, H₀ can be rejected.