First Theory of Cancer Gene Data Analysis by 169 Microarrays—Four Universal Data Structures of Discriminant Data

The new discriminant theory (Theory1) analyzed six microarrays having two classes. The Revised IP Optimal LDF (RIP) finds the minimum number of misclassifications (MNMs) and shows that all are zero, indicating that the six microarrays are linearly separable data (LSD). Program3 and Program4 coded in LINGO, can split microarrays into many exclusive Small Matryoshkas (SMs) and Basic Gene Sets (BGSs). SMs and BGSs allow us to discriminate between normal and cancer patients (or two cancer classes) with just a few genes. We confirmed these facts by analyzing 163 2nd-generation microarrays registered on the GSE database after 2007. The tenfold CV (Program2) finds SMs and BGSs and shows that many average error rates (ERs) for 10 validation samples (M2) are zero. Therefore, we consider these gene sets valuable for cancer gene diagnosis. Moreover, we find four universal data structures of microarrays (Fact3). Thus, we can establish the first cancer gene data analysis theory in 2020 (Theory2). This chapter explains the details of the SM decomposition of Golub data in Program3. Surprisingly, Program3 can find 723 constant-value genes in Golub’s data and omit them from the analysis. This result shows a convenient ability when other high-dimensional measurement values include meaningless constant values.