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.
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Ahn, J., Marron, J. S. (2010). The maximal data piling direction for discrimination. Biometrika, 97, 254–259.
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.
Aoshima, M., Yata, K. (2011). Two-stage procedures for high-dimensional data. Sequential Analysis (Editor’s special invited paper), 30, 356–399.
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.
Aoshima, M., Yata, K. (2015a). Geometric classifier for multiclass, high-dimensional data. Sequential Analysis, 34, 279–294.
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.
Bai, Z., Saranadasa, H. (1996). Effect of high dimension: By an example of a two sample problem. Statistica Sinica, 6, 311–329.
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.
Cai, T. T., Liu, W. (2011). A direct estimation approach to sparse linear discriminant analysis. Journal of the American Statistical Association, 106, 1566–1577.
Chan, Y.-B., Hall, P. (2009). Scale adjustments for classifiers in high-dimensional, low sample size settings. Biometrika, 96, 469–478.
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.
Christensen, B. C., Houseman, E. A., Marsit, C. J., Zheng, S., Wrensch, M. R., Wiemels, J. L., et al. (2009). Aging and environmental exposures alter tissue-specific DNA methylation dependent upon CpG island context. PLoS Genetics, 5, e1000602.
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.
Fan, J., Fan, Y. (2008). High-dimensional classification using features annealed independence rules. The Annals of Statistics, 36, 2605–2637.
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.
Gravier, E., Pierron, G., Vincent-Salomon, A., Gruel, N., Raynal, V., Savignoni, A., et al. (2010). A prognostic DNA signature for T1T2 node-negative breast cancer patients. Genes, Chromosomes and Cancer, 49, 1125–1134.
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.
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.
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.
Li, Q., Shao, J. (2015). Sparse quadratic discriminant analysis for high dimensional data. Statistica Sinica, 25, 457–473.
Marron, J. S., Todd, M. J., Ahn, J. (2007). Distance-weighted discrimination. Journal of the American Statistical Association, 102, 1267–1271.
McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. The Annals of Probability, 2, 620–628.
Naderi, A., Teschendorff, A. E., Barbosa-Morais, N. L., Pinder, S. E., Green, A. R., Powe, D. G., et al. (2007). A gene-expression signature to predict survival in breast cancer across independent data sets. Oncogene, 26, 1507–1516.
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.
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.
Shipp, M. A., Ross, K. N., Tamayo, P., Weng, A. P., Kutok, J. L., Aguiar, R. C., et al. (2002). Diffuse large B-cell lymphoma outcome prediction by gene-expression profiling and supervised machine learning. Nature Medicine, 8, 68–74.
Tian, E., Zhan, F., Walker, R., Rasmussen, E., Ma, Y., Barlogie, B., et al. (2003). The role of the Wnt-signaling antagonist DKK1 in the development of osteolytic lesions in multiple myeloma. The New England Journal of Medicine, 349, 2483–2494.
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.
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.
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.
Yata, K., Aoshima, M. (2013). PCA consistency for the power spiked model in high-dimensional settings. Journal of Multivariate Analysis, 122, 334–354.
Yata, K., Aoshima, M. (2015). Principal component analysis based clustering for high-dimension, low-sample-size data. arXiv preprint. arXiv:1503.04525.
We would like to thank two anonymous referees for their constructive comments.
Research of the first author was partially supported by Grants-in-Aid for Scientific Research (A) and Challenging Exploratory Research, Japan Society for the Promotion of Science (JSPS), under Contract Numbers 15H01678 and 26540010. Research of the second author was partially supported by Grant-in-Aid for Young Scientists (B), JSPS, under Contract Number 26800078.
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Aoshima, M., Yata, K. Distance-based classifier by data transformation for high-dimension, strongly spiked eigenvalue models. Ann Inst Stat Math 71, 473–503 (2019). https://doi.org/10.1007/s10463-018-0655-z
- Asymptotic normality
- Data transformation
- Discriminant analysis
- Large p small n
- Noise-reduction methodology
- Spiked model