Let's consider more general nonlinear approaches to study teleconnections of climate variables
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The recent work by (Rieger et al 2021) is concerned with the problem of extracting features from spatio-temporal geophysical signals. The authors introduce the complex rotated MCA (xMCA) to deal with lagged effects and non-orthogonality of the feature representation. This method essentially (1) transforms the signals to a complex plane with the Hilbert transform; (2) applies an oblique (Varimax and Promax) rotation to remove the orthogonality constraint; and (3) performs the eigendecomposition in this complex space (Horel et al, 1984). We argue that this method is essentially a particular case of the method called rotated complex kernel principal component analysis (ROCK-PCA) introduced in (Bueso et al., 2019, 2020), where we proposed the same approach: first transform the data to the complex plane with the Hilbert transform and then apply the varimax rotation, with the only difference that the eigendecomposition is performed in the dual (kernel) Hilbert space. The latter allows us to generalize the xMCA solution by extracting nonlinear (curvilinear) features when nonlinear kernel functions are used. Hence, the solution of xMCA boils down to ROCK-PCA when the inner product is computed in the input data space instead of in the high-dimensional (possibly infinite) kernel Hilbert space to which data has been mapped. In this short correspondence we show theoretical proof that xMCA is a special case of ROCK-PCA and provide quantitative evidence that more expressive and informative features can be extracted when working with kernels; results of the decomposition of global sea surface temperature (SST) fields are shown to illustrate the capabilities of ROCK-PCA to cope with nonlinear processes, unlike xMCA.
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