Table 2 Overview of symbols used in the description of the scregclust algorithm

n

Total number of cells in the input matrix

p_t

Number of target genes

p_r

Number of regulators

Z_t, Z_r

target gene and regulator expression (n × p_t resp. n × p_r matrices)

K

Desired number of clusters

Π

K × p_t cluster membership matrix for target genes with Π^(i, j) ∈ {0, 1} and \({\sum }_{i=1}^{K}{{{{\mathbf{\Pi }}}}}^{(i,j)}\le 1\)

J

p_t × p_t indicator matrix describing prior knowledge of biological relationships between target genes (e.g. pathway co-occurence)

n_d

Number of cells in the d-th data split

Z_t,d

target gene expression used in the d-th data split (n_d × p_t matrix)

Z_r,d

regulator expression used in the d-th data split (n_d × p_r matrix)

C_i

Set of target genes in cluster i

R_i

Set of regulators associated with cluster i

N_i

Maximum number of regulators associated with cluster i

s_i

Sign vectors containing one sign for each regulator in R_i

B_i

non-negative regression coefficients (∣R_i∣ × p_t matrix)

\({\sigma }_{i,j}^{2}\)

positive variance parameter for each target gene and cluster

λ

Positive penalty parameter used in coop-Lasso

w_i

Positive weight vector of length p_r for each cluster i in coop-Lasso

B_i,OLS

Ordinary least squares estimates of the regression coefficients in cluster i (p_r × ∣C_i∣ matrix)

B_i,CL

Coop-lasso estimates of the regression coefficients in cluster i (p_r × ∣C_i∣ matrix)

τ

Non-negative threshold for rag-bag clustering

p_i,j

Prior probability of target gene j being in cluster i

L_j

K × n₂ likelihood matrix for target gene j

v_j

Vector of n₂ votes for the cluster assignment of target gene j

μ

Prior strength in [0, 1] for trade-off between likelihood and prior in cluster allocation

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

Predictive R² for cluster i

\({R}_{i,-k}^{2}\)

Predictive R² for cluster i with regulator k omitted

I_i,k

Importance of regulator k in cluster i

\({R}_{i,j}^{2}\)

Predictive R² for target gene j predicted by regulators in R_i

S_j

Silhouette score for target gene j

T

Regulatory table providing a summary of regulator strength in each cluster (p_r × K matrix)