Fig. 1

Enviro-geno-pheno state as biomarker, shortly E-GPS biomarker a Each element of \(\mathcal{D}_{g\phi e}\) is generally a \({ d_g}\times { d_e}\times { d_{\phi }}\) data cubic, where \({ d_g}\), \({ d_e}\), and \({ d_{\phi }}\) are the dimensionalities of \(\mathbf{g}\), \(\mathbf{e}\), and \(\mathbf{\pmb {\phi }}\), respectively. b When g, e, and \({ \phi }\) are univariate, the case is illustrated by a scattering map, which is degenerated into an \({ m_g}\times {m_e}\times {m_{\phi }}\) table that represents a discrete distribution when g, e, and \({ \phi }\) take \({ m_g}, {m_e}\), and \({m_{\phi }}\) discrete values, respectively. c A convex set \(R_s\) acts as E-GPS biomarker, with the system status indicated by Type(s) and the boundary condition by COND(s) about genotypes, phenotypes, and environments by the boundary of \(R_s\). d The possible system statuses are featured by E-GPS states that are learned from given samples, by minimising the criterion given by Eq. (1) or (4). e For a finite size of samples, we prefer a simple parametric model, e.g., by one of the two choices given in Eq. (7). f An E-GPS state corresponds to a convex subset with all its elements dedicated to the same status type, e.g., \(s_{11}\) is a biomarker of ‘green’, which maybe relaxed to require a probabilistic dedication, i.e., samples falling in a convex subset are mostly dedicated to a same status type. Contrastingly, a c-state is featured by that two status type compete samples, e.g., \(s_{01}\) and \(s_{10}\). g Prognosis analysis can be made per d-state, as addressed in Table 1 (3)(a). In addition, subtype analysis is made per state, with the top row indicating ‘green’ and ‘red’ samples and other rows indicating subtypes in binary values. The relation between the E-GPS state in consideration and each subtype is examined by their intersection. h We may compare the configuration of states jointly. In addition, the results of phenotype analysis per state can be combined, with help of the weighting probability \(p(j|s_j)\) in accordance with the individual performance of each state. We may further make state transient analysis by estimating the transfer probabilities \(p(s_i|s_j)\)