Further Examinations of SMs—Defect of Revised LP-OLDF and Correlations of Genes

In this chapter, we analyze Alon’s microarray in 2018 and obtain two SMs from the RIP and Revised LP-OLDF. In Sect. 4.2, RIP separates the microarray into a union of 62 SMs (1,968 genes). Six MP-based LDFs find this subspace is LSD and a noise subspace (32 genes) is not LSD. In Sect. 4.3, Revised LP-OLDF separates the microarray into a union of 32 SMs (1,005 genes) and a noise subspace (995 genes). Six MP-based LDFs find both subspaces are LSD. This fact suggests us that a noise subspace includes other SMs in it. We find Revised LP-OLDF cannot find all SMs from the microarray correctly. We guess Problem1 causes the defect of Revised LP-OLDF. Namely, Revised LP-OLDF cannot find other SMs from noise subspace. Section 4.4 analyzes 62 SMs found by the RIP and evaluates 62 SMs by RatioSV and NM. Moreover, the 1,891 correlations of 62 RIP discriminant scores (RipDSs) are computed. At first, we consider each gene set included in SM is cancer genes and a signal subspace. However, standard statistical methods cannot show the linear separable facts. Thus, we conclude that the gene sets included in all SMs are not signals. We recognize the data made by RipDSs is signal data. Two signal data of SM13 with maximum RatioSV and SM62 with minimum RatioSV are validated. Section 4.5 analyzes two signal data made by RipDSs and HsvmDSs obtained by 62 SMs of the RIP. The results are almost the same in Chaps. 2 and 3 . However, these findings can open a new field of cancer gene diagnosis only after verification of the subjects used in the study of Alon et al. (Proc Natl Acad Sci USA, 96(1.1): 6745–6750 1999). Section 4.6 explains the reason why standard statistical methods could not find the linear separable facts. Section 4.9 is the conclusion.