Table 1 Synthetic data generation.

Let \({\mathcal{D}}\) be an annotated single-cell data set;

Let Z be the set of all types found in \({\mathcal{D}}\);

Let Idx(z) be the indices of cells belonging to type \(z\in {\mathcal{D}}\);

For s in 1… S

 C_s ~ Unif(lb, ub);

 ∣Z_s∣ ~ Unif(1, ∣Z∣);

 Let Z_s be a subset of Z, consisting of ∣Z_s∣ types formed by uniform sampling without replacement

 \({p}_{sz^{\prime} } \sim {\rm{Dir}}({{\boldsymbol{1}}}_{s}),\quad {{\boldsymbol{1}}}_{s}\in {{\mathbb{R}}}^{| {Z}_{s}| },\quad z^{\prime} \in {Z}_{s}\);

 \({n}_{sz^{\prime} }=\lceil {p}_{sz^{\prime} }\cdot {C}_{s}\rfloor\);

 \({w}_{sz^{\prime} }={n}_{sz^{\prime} }/\mathop{\sum }\limits_{k}^{S}{n}_{kz^{\prime} }\);

 Let \({I}_{sz^{\prime} }\) be \({n}_{sz^{\prime} }\) samples taken from \({\rm{Idx}}(z^{\prime} )\) with equal probability and without replacement;

 \({x}_{sg}={\mathop{\sum}\limits_{z^{\prime} \in {Z}_{s}}}{\mathop{\sum}\limits_{c\in {I}_{sz^{\prime} }}}\lceil \alpha \cdot {y}_{cg}\rfloor\)