Figure 6

Convex geometry of the MLMDP used in the THetA algorithm. (Left) For a single cancer genome with normal admixture, the interval count vector c₂ of the cancer genome and tumor purity μ define a collection of rays C μ, for μ ∈ 0[1]. (Here we show the space Ω_3,2,3). (Right) Normalizing these rays, we obtain the parameter $p=\widehat{C\mu }$, used in the multinomial likelihood. These parameters are embedded in the simplex Δ _{m - l}(gray triangle with a black outline) because their entries sum to one. (This is the space $P_{{\text{Ω}}_{3,2,3}}$.) For a fixed interval count matrix C = (c₁, c₂) a blue ray (left) defined by C μ is mapped to the corresponding red/green ray (right) connecting $\widehat{c_1}$ to $\widehat{c_2}$ (right), the normalized columns of C, as described in Theorem 1. For n > 2, hyperplanes are mapped to hyperplanes (see Additional file 1, Figure S2). We show $p^{*}=\widehat{r}$, the maximum likelihood solution when interval counts are not constrained to be integers. Note that this point is not on any of the rays defined by interval count matrices. Rays that satisfy the ordering constraint from Theorem 2 are in green. MLMDP: maximum likelihood mixture decomposition problem