A nonlinear generalisation of Principal Component Analysis (PCA), denoted Nonlinear Principal Component Analysis (NLPCA), is introduced and applied to the analysis of climate data. This method is implemented using a 5-layer feed-forward neural network introduced originally in the chemical engineering literature. The method is described and details of its implementation are addressed. It is found empirically that NLPCA partitions variance in the same fashion as does PCA, that is, that the sum of the total variance of the NLPCA approximation with the total variance of the residual from the original data is equal to the total variance of the original data. An important distinction is drawn between a modal P-dimensional NLPCA analysis, in which P successive 1D approximations are determined iteratively so that the approximation is the sum of P nonlinear functions of one variable, and a nonmodal analysis, in which the P-dimensional NLPCA approximation is determined as a nonlinear non-additive function of P variables.
Nonlinear Principal Component Analysis is first applied to a data set sampled from the Lorenz attractor. It is found that the NLPCA approximations are much more representative of the data than are the corresponding PCA approximations. In particular, the 1D and 2D NLPCA approximations explain 76% and 99.5% of the total variance, respectively, in contrast to 60% and 95% explained by the 1D and 2D PCA approximations.
When applied to a data set consisting of monthly-averaged tropical Pacific Ocean sea surface temperatures (SST), the modal 1D NLPCA approximation describes average variability associated with the El Nin/Southern Oscillation (ENSO) phenomenon, as does the 1D PCA approximation. The NLPCA approximation, however, characterises the asymmetry in spatial pattern of SST anomalies between average warm and cold events (manifested in the skewness of the distribution) in a manner that the PCA approximation cannot. The second NLPCA mode of SST is found to characterise differences in ENSO variability between individual events, and in particular is consistent with the celebrated 1977 ``regime shift''. A 2D nonmodal NLPCA approximation is determined, the interpretation of which is complicated by the fact that a secondary feature extraction problem has to be carried out to interpret the results. It is found that this approximation contains much the same information as that provided by the modal analysis. A modal NLPC analysis of tropical Indo-Pacific sea level pressure (SLP) finds that the first mode describes average ENSO variability in this field, and also characterises an asymmetry in SLP fields between average warm and cold events. No robust nonlinear mode beyond the first could be found.
Nonlinear Principal Component Analysis is used to find the optimal nonlinear approximation to SLP data produced by a 1001 year integration of the Canadian Centre for Climate Modelling and Analysis (CCCma) coupled general circulation model (CGCM1). This approximation's associated time series is strongly bimodal and partitions the data into two distinct regimes. The first and more persistent regime describes a standing oscillation whose signature in the mid-troposphere is alternating amplification and attenuation of the climatological ridge over Northern Europe. The second and more episodic regime describes mid-tropospheric split-flow south of Greenland. Essentially the same structure is found in the 1D NLPCA approximation of the 500mb height field itself. In a 500 year integration with atmospheric CO2 at four times pre-industrial concentrations, the occupation statistics of these preferred modes of variability change, such that the episodic split-flow regime occurs less frequently while the standing oscillation regime occurs more frequently.
Finally, a generalisation of Kramer's NLPCA using a 7-layer autoassociative neural network is introduced to address the inability of Kramer's original network to find P-dimensional structure topologically different from the unit cube in Re(P). The example of an ellipse is considered, and it is shown that the approximation produced by the 7-layer network is a substantial improvement over that produced by the 5-layer network.
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