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A Probabilistic Approach to Nonlinear Principal Component Analysis With Applications

Seppo Pulkkinen, A Probabilistic Approach to Nonlinear Principal Component Analysis With Applications. TUCS Technical Reports 1091, TUCS, 2013.

Abstract:

Principal component analysis (PCA) is a well-established tool for identifying the main sources of variation in multivariate data. However, as a linear method it cannot describe complex nonlinear structures. To overcome this limitation, a novel nonlinear generalization of PCA is developed in this paper. The method obtains the nonlinear principal components from ridges of the underlying density of the data. The density is estimated by using Gaussian kernels. Projection onto a ridge of such a density estimate is formulated as a solution to a differential equation, and a predictor-corrector method is developed for this purpose. The method is further extended to time series data by applying it to the phase space representation of the time series. This extension can be viewed as a nonlinear generalization of singular spectrum analysis (SSA). Ability of the nonlinear PCA to capture complex nonlinear shapes and its SSA-based extension to identify periodic patterns from time series are demonstrated on climate data.

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BibTeX entry:

@TECHREPORT{tPulkkinen_Seppo13c,
  title = {A Probabilistic Approach to Nonlinear Principal Component Analysis With Applications},
  author = {Pulkkinen, Seppo},
  number = {1091},
  series = {TUCS Technical Reports},
  publisher = {TUCS},
  year = {2013},
  keywords = {principal component analysis, singular spectrum analysis, nonlinear, kernel density, nonparametric methods, ridge, predictor-corrector method},
  ISBN = {978-952-12-2970-1},
}

Belongs to TUCS Research Unit(s): Turku Optimization Group (TOpGroup)

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