Table 1 Notations and Meanings

\({x}_{{ij}}\)

Average expression value of the gene i in cluster j

\(\widetilde{{x}_{{ij}}}\)

Maximum-scaled average expression value of the gene i in cluster j

\({S}_{1}\)

Threshold for dividing clusters into high or low-expression groups

\({S}_{2}\)

Percentage of genes to be excluded from each cluster

\({M}_{j}\)

An index set which indicates the rows from the original matrix that the maximum-scaled values for cluster j equal to 1

L

The number of all the input genes

K

The number of all clusters

\({E}_{1}^{i}\)

Mean value of the \({x}_{{ij}}\) (average expression values of gene i) in all the clusters

\({E}_{2}^{i}\)

Mean value of \(\widetilde{{x}_{{ij}}}\) (maximum-scaled average expression values of gene i) which exceed threshold \({S}_{1}\)

\({E}_{3}^{i}\)

Mean value of \(\widetilde{{x}_{{ij}}}\) (maximum-scaled average expression values of gene i) which is inferior to \({S}_{1}\)

\({\rho }_{i}\)

The potential to be a positive marker of gene i

\({\eta }_{i}\)

The potential to be a negative marker of gene i

\({p}_{i}\)

An index set of the clusters with \(\widetilde{{x}_{{ij}}}\) (maximum-scaled average expression) greater than \({S}_{1}\)

\({q}_{i}\)

An index set of the clusters with \(\widetilde{{x}_{{ij}}}\) no bigger than \({S}_{1}\)

\({G}_{i}\)

The set of cells expressing gene i

\({G}_{{{\rm {clust}}}}\)

The set of cells where gene i acts as the marker gene

\({{{\rm {PMI}}}}_{i}\)

Positive molecular index for gene i

\({{{\rm {NMI}}}}_{i}\)

Negative molecular index for gene i

\({{{\rm {MI}}}}_{i}\)

Molecular index for gene i

\({T}_{i}\)

The proportion of cells with gene i as the marker gene to cells expressing gene i