Ambiguity and errors in inferring segment copy number (SCN) profiles for a heterogeneous sample S = (G₁, G₂) under different assumptions about the sample composition. (A) A two-genome proper sample S = (G₁, G₂): each genome G_i ∈ S is depicted as collections of adjacent blocks (top), and the corresponding sequences of signed blocks (bottom). (B) The copy number profile c = [c₁, c₂, c₃, c₄] inferred under the assumption that the sample is homogeneous (i.e., comprised of a single derived genome) and the reference genome is haploid (i.e., each segment has only a single haplotype in the reference). Each value c_j is the weighted average of the sums of haplotype-specific (or allele-specific) copy numbers $a_{i,j}+b_{i,j}={\hat{c}}_{i,j}+{č}_{i,j}$ over the genomes G_i ∈ S. (C) Allele-specific copy number profiles $\hat{\mathbf{c}}=\left[{\hat{c}}_1,{\hat{c}}_2,{\hat{c}}_3,{\hat{c}}_4\right]$ and $\mathbf{č}=[{č}_1,{č}_2,{č}_3,{č}_4]$ inferred under the assumption that the sample is homogeneous and the reference genome is diploid (i.e., each segment has two haplotypes labeled A and B). Here, the entries ${\hat{c}}_j$ and ${č}_j$ for segment j are averages $\left({\hat{c}}_{1,j}+{\hat{c}}_{2,j}\right)/2$ and $\left({č}_{1,j}+{č}_{2,j}\right)/2$ of genome- and allele-specific copy number values. Note that the vectors $\hat{\mathbf{c}}$ and č do not preserve the true A/B label of each allele: dark blue are true counts of allele A and light blue are true counts of allele B. Here, segments 2 and 4 are flipped. (D) Genome-specific copy number profiles c₁ = [c_1,1, c_1,2, c_1,3, c_1,4] and c₂ = [c_2,1, c_2,2, c_2,3, c_2,4] inferred under the assumption that the sample is heterogeneous, but the reference genome is haploid. Here, the entry c_i,j for a segment j and genome G_i is the sum ${\hat{c}}_{i,j}+{č}_{i,j}$ of allele-specific copy number values in a genome G_i. (E) Allele- and genome-specific copy number matrices $\stackrel{\sim }{\mathbf{C}}=\left(\hat{\mathbf{C}}={\left[{\hat{\mathbf{c}}}_1,{\hat{\mathbf{c}}}_2,\dots ,{\hat{\mathbf{c}}}_n\right]}^T,\mathbf{Č}={\left[{\mathbf{č}}_1,{\mathbf{č}}_2,\dots ,{\mathbf{č}}_n\right]}^T\right)$ inferred under the assumption that the sample is heterogeneous and the reference genome is diploid. Segments 2 and 4 are flipped alleles: $\left({č}_{1,2},{\hat{c}}_{2,2}\right)=\left(a_{1,2},b_{2,2}\right)$ and $\left({č}_{1,4},{\hat{c}}_{2,4}\right)=\left(a_{1,4},b_{2,4}\right)$.