Distance-based classifier by data transformation for high-dimension, strongly spiked eigenvalue models
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We consider classifiers for high-dimensional data under the strongly spiked eigenvalue (SSE) model. We first show that high-dimensional data often have the SSE model. We consider a distance-based classifier using eigenstructures for the SSE model. We apply the noise-reduction methodology to estimation of the eigenvalues and eigenvectors in the SSE model. We create a new distance-based classifier by transforming data from the SSE model to the non-SSE model. We give simulation studies and discuss the performance of the new classifier. Finally, we demonstrate the new classifier by using microarray data sets.
KeywordsAsymptotic normality Data transformation Discriminant analysis Large p small n Noise-reduction methodology Spiked model
We would like to thank two anonymous referees for their constructive comments.
- Ahn, J., Marron, J. S. (2010). The maximal data piling direction for discrimination. Biometrika, 97, 254–259.Google Scholar
- Alon, U., Barkai, N., Notterman, D. A., Gish, K., Ybarra, S., Mack, D., et al. (1999). Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays. Proceedings of the National Academy of Sciences of the United States of America, 96, 6745–6750.Google Scholar
- Aoshima, M., Yata, K. (2011). Two-stage procedures for high-dimensional data. Sequential Analysis (Editor’s special invited paper), 30, 356–399.Google Scholar
- Aoshima, M., Yata, K. (2014). A distance-based, misclassification rate adjusted classifier for multiclass, high-dimensional data. Annals of the Institute of Statistical Mathematics, 66, 983–1010.Google Scholar
- Aoshima, M., Yata, K. (2015a). Geometric classifier for multiclass, high-dimensional data. Sequential Analysis, 34, 279–294.Google Scholar
- Aoshima, M., Yata, K. (2015b). High-dimensional quadratic classifiers in non-sparse settings. arXiv preprint. arXiv:1503.04549.
- Aoshima, M., Yata, K. (2018). Two-sample tests for high-dimension, strongly spiked eigenvalue models. Statistica Sinica, 28, 43–62.Google Scholar
- Bai, Z., Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem. Statistica Sinica, 6, 311–329.Google Scholar
- Bickel, P. J., Levina, E. (2004). Some theory for Fisher’s linear discriminant function, “naive Bayes”, and some alternatives when there are many more variables than observations. Bernoulli, 10, 989–1010.Google Scholar
- Cai, T. T., Liu, W. (2011). A direct estimation approach to sparse linear discriminant analysis. Journal of the American Statistical Association, 106, 1566–1577.Google Scholar
- Chan, Y.-B., Hall, P. (2009). Scale adjustments for classifiers in high-dimensional, low sample size settings. Biometrika, 96, 469–478.Google Scholar
- Chen, S. X., Qin, Y.-L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. The Annals of Statistics, 38, 808–835.Google Scholar
- Dudoit, S., Fridlyand, J., Speed, T. P. (2002). Comparison of discrimination methods for the classification of tumors using gene expression data. Journal of the American Statistical Association, 97, 77–87.Google Scholar
- Fan, J., Fan, Y. (2008). High-dimensional classification using features annealed independence rules. The Annals of Statistics, 36, 2605–2637.Google Scholar
- Glaab, E., Bacardit, J., Garibaldi, J. M., Krasnogor, N. (2012). Using rule-based machine learning for candidate disease gene prioritization and sample classification of cancer gene expression data. PLoS ONE, 7, e39932.Google Scholar
- Hall, P., Marron, J. S., Neeman, A. (2005). Geometric representation of high dimension, low sample size data. Journal of the Royal Statistical Society, Series B, 67, 427–444.Google Scholar
- Hall, P., Pittelkow, Y., Ghosh, M. (2008). Theoretical measures of relative performance of classifiers for high dimensional data with small sample sizes. Journal of the Royal Statistical Society, Series B, 70, 159–173.Google Scholar
- Jeffery, I. B., Higgins, D. G., Culhane, A. C. (2006). Comparison and evaluation of methods for generating differentially expressed gene lists from microarray data. BMC Bioinformatics, 7, 359.Google Scholar
- Li, Q., Shao, J. (2015). Sparse quadratic discriminant analysis for high dimensional data. Statistica Sinica, 25, 457–473.Google Scholar
- Marron, J. S., Todd, M. J., Ahn, J. (2007). Distance-weighted discrimination. Journal of the American Statistical Association, 102, 1267–1271.Google Scholar
- Nakayama, Y., Yata, K., Aoshima, M. (2017). Support vector machine and its bias correction in high-dimension, low-sample-size settings. Journal of Statistical Planning and Inference, 191, 88–100.Google Scholar
- Ramey J. A. (2016). Datamicroarray: collection of data sets for classification. https://github.com/ramhiser/datamicroarray.
- Shao, J., Wang, Y., Deng, X., Wang, S. (2011). Sparse linear discriminant analysis by thresholding for high dimensional data. The Annals of Statistics, 39, 1241–1265.Google Scholar
- Watanabe, H., Hyodo, M., Seo, T., Pavlenko, T. (2015). Asymptotic properties of the misclassification rates for Euclidean distance discriminant rule in high-dimensional data. Journal of Multivariate Analysis, 140, 234–244.Google Scholar
- Yata, K., Aoshima, M. (2010). Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix. Journal of Multivariate Analysis, 101, 2060–2077.Google Scholar
- Yata, K., Aoshima, M. (2012). Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations. Journal of Multivariate Analysis, 105, 193–215.Google Scholar
- Yata, K., Aoshima, M. (2013). PCA consistency for the power spiked model in high-dimensional settings. Journal of Multivariate Analysis, 122, 334–354.Google Scholar
- Yata, K., Aoshima, M. (2015). Principal component analysis based clustering for high-dimension, low-sample-size data. arXiv preprint. arXiv:1503.04525.