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Nonlinea principal component analysis of climate data Monahan, Adam Hugh
Abstract
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 5layer feedforward 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 Pdimensional 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 Pdimensional NLPCA approximation is determined as a nonlinear nonadditive 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 monthlyaveraged 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 IndoPacific 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 midtroposphere is alternating amplification and attenuation of the climatological ridge over Northern Europe. The second and more episodic regime describes midtropospheric splitflow 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 preindustrial concentrations, the occupation statistics of these preferred modes of variability change, such that the episodic splitflow regime occurs less frequently while the standing oscillation regime occurs more frequently. Finally, a generalisation of Kramer’s NLPCA using a 7layer autoassociative neural network is introduced to address the inability of Kramer’s original network to find Pdimensional 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 7layer network is a substantial improvement over that produced by the 5layer network. [Scientific formulae used in this abstract could not be reproduced.]
Item Metadata
Title 
Nonlinea principal component analysis of climate data

Creator  
Publisher 
University of British Columbia

Date Issued 
2000

Description 
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 5layer feedforward 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 Pdimensional 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 Pdimensional
NLPCA approximation is determined as a nonlinear nonadditive 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 monthlyaveraged 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 IndoPacific 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 midtroposphere is alternating amplification and attenuation
of the climatological ridge over Northern Europe. The second and more episodic
regime describes midtropospheric splitflow 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 preindustrial concentrations,
the occupation statistics of these preferred modes of variability change, such that the
episodic splitflow regime occurs less frequently while the standing oscillation regime
occurs more frequently.
Finally, a generalisation of Kramer’s NLPCA using a 7layer autoassociative neural
network is introduced to address the inability of Kramer’s original network to find Pdimensional
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 7layer
network is a substantial improvement over that produced by the 5layer network. [Scientific formulae used in this abstract could not be reproduced.]

Extent 
8824754 bytes

Genre  
Type  
File Format 
application/pdf

Language 
eng

Date Available 
20090715

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0089632

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
200005

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.