Copyright © 2013 Osamu Komori et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper discusses mathematical and statistical aspects in analysis methods applied to microarray gene expressions. We focus on pattern recognition to extract informative features embedded in the data for prediction of phenotypes. It has been pointed out that there are severely difficult problems due to the unbalance in the number of observed genes compared with the number of observed subjects. We make a reanalysis of microarray gene expression published data to detect many other gene sets with almost the same performance. We conclude in the current stage that it is not possible to extract only informative genes with high performance in the all observed genes. We investigate the reason why this difficulty still exists even though there are actively proposed analysis methods and learning algorithms in statistical machine learning approaches. We focus on the mutual coherence or the absolute value of the Pearson correlations between two genes and describe the distributions of the correlation for the selected set of genes and the total set. We show that the problem of finding informative genes in high dimensional data is ill-posed and that the difficulty is closely related with the mutual coherence.