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 Nino/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 RP. 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. [Scientific formulae used in this abstract could not be reproduced.]Science, Faculty ofEarth, Ocean and Atmospheric Sciences, Department ofGraduat