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Stability of nonlinear principal components analysis: an empirical study using the balanced bootstrap
Mariëlle Linting1, Jacqueline J Meulman, Patrick J F Groenen
1Data Theory Group, Leiden University, The Netherlands. linting@fsw.leidenuniv.nl
Psychological Methods
|September 6, 2007
Summary
This study introduces the nonparametric bootstrap procedure to assess the stability of nonlinear principal components analysis (PCA). Researchers recommend using at least 1,000 bootstrap samples for reliable nonlinear PCA results.
Area of Science:
- Statistics
- Data Analysis
Background:
- Principal Components Analysis (PCA) is a technique for exploring data structures with linearly related variables.
- Nonlinear PCA extends this to nonlinearly related numeric and nonnumeric variables.
- Establishing the stability of nonlinear PCA solutions is challenging due to a lack of standard methods.
Purpose of the Study:
- To assess the stability of nonlinear PCA results using the nonparametric bootstrap procedure.
- To provide a benchmark by applying the same bootstrap procedure to linear PCA.
- To offer practical recommendations for improving the reliability of bootstrap results in PCA.
Main Methods:
- Application of the nonparametric bootstrap procedure to empirical data for nonlinear PCA.
- Utilizing confidence intervals for variable transformations and confidence ellipses for eigenvalues, loadings, and scores.
- Comparison with linear PCA using the same bootstrap methodology.
Main Results:
- The bootstrap procedure provides confidence intervals and ellipses for assessing the stability of nonlinear PCA components.
- Procrustes rotation is discussed as a method to align bootstrap results.
- The balanced bootstrap and bias estimation are explored within this context.
Conclusions:
- The study recommends using at least 1,000 bootstrap samples for stable nonlinear PCA.
- Examining bootstrap distributions and confidence regions is crucial.
- Merging categories with small frequencies can reduce bootstrap result variance.
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