Journal of Climate | Fischer 
The Earth’s ice-ocean-atmosphere system exhibits nonlinear responses, such as the difference in the magnitude of the atmospheric response to positive or negative ocean or sea-ice anomalies. Two classes of methods that have previously been used to investigate the nonlinear dependence between climate fields are kernel methods and neural network methods. In this paper, a third methodology is introduced: gradient-based kernel dimension reduction. Gradient-based kernel methods are an extension of conventional kernel methods, but gradient-based methods focus on the directional derivatives of the regression surface between two fields. Specifically, a new gradient-based method is developed here: gradient kernel canonical correlation analysis (gKCCA). In gKCCA, the canonical directions maximize the directional derivatives between the predictor field and the response field, while the canonical components of the response field maximize the correlation with a nonlinear augmentation of the predictor canonical components. Gradient-based kernel methods have several advantages: their components can be directly related to the original fields (unlike in conventional kernel methods), and the projection vectors are determined by analytical solution (unlike in neural networks). Here gKCCA is applied to the question of nonlinear coupling between high-frequency (2-3 years) and low-frequency (4-6 years) modes in the Pacific Ocean. The leading gKCCA subspace shows a significant nonlinear coupling between the low-pass and high-pass fields. The paper also shows that the results of gKCCA are robust to different levels of noise and different kernel functions.
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