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 Cμ, for μ ∈ 0[1]. (Here we show the space Ω3,2,3). (Right) Normalizing these rays, we obtain the parameter <a onClick="popup('http://genomebiology.com/2013/14/7/R80/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://genomebiology.com/2013/14/7/R80/mathml/M53">View MathML</a>, used in the multinomial likelihood. These parameters are embedded in the simplex Δm - l(gray triangle with a black outline) because their entries sum to one. (This is the space <a onClick="popup('http://genomebiology.com/2013/14/7/R80/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://genomebiology.com/2013/14/7/R80/mathml/M54">View MathML</a>.) For a fixed interval count matrix C = (c1, c2) a blue ray (left) defined by Cμ is mapped to the corresponding red/green ray (right) connecting <a onClick="popup('http://genomebiology.com/2013/14/7/R80/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://genomebiology.com/2013/14/7/R80/mathml/M55">View MathML</a> to <a onClick="popup('http://genomebiology.com/2013/14/7/R80/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://genomebiology.com/2013/14/7/R80/mathml/M56">View MathML</a> (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 <a onClick="popup('http://genomebiology.com/2013/14/7/R80/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://genomebiology.com/2013/14/7/R80/mathml/M57">View MathML</a>, 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

Oesper et al. Genome Biology 2013 14:R80   doi:10.1186/gb-2013-14-7-r80
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