Figure 11

Evaluation with dynamic programming. The simple DP scheme computes all probability vectors Q₀, Q₁, Q₂ completely within the green marked area, corresponding to score ranges of prefix PSSMs M _k . In contrast to the simple scheme, the restricted probability computation method computes only the upper end of the probability distribution until the given p-value threshold is exceeded, omitting parts of the green area. In this example we show how to compute the score threshold $Tmi{n}_{\mathcal{P}}$(π, M) for PSSM M of length m = 3 and a score range of [4,11] corresponding to a given p-value threshold of π = $\frac{1}{8}$. For simplicity we assume a uniform character distribution of f(A) = f(C) = f(G) = f(T) = $\frac{1}{4}$. Cells of the matrix that are computed in the step actually under consideration are marked red. In step d = 0, see (A), the algorithm computes Q₂(11) recursively for all paths through M that achieve a score of 11, i.e. Q₂(11) = Q₁(8)·f(G), Q₁(8) = Q₀(4)·f(G), Q₀(4) = Q_-1(0)·f(A) = 1·$\frac{1}{4}$, since AGG is the only path achieving score 11. It follows Q₂(11) = $\frac{1}{64}$. In step d = 1 all paths achieving a score of 11 - d = 10 to determine Q₂(10) are computed, see (B). We conclude Q₂(10) = $\frac{1}{16}$. In this step, DP allows to reuse value Q₁(8) without recomputation. In step d = 2, see (C) values Q₁(7) and Q₀(3) can be reused to compute Q₂(9) = $\frac{5}{64}$. In step d = 2 the cumulated probability Q₂(11) + Q₂(10) + Q₂(9) = $\frac{5}{32}$ exceeds the given p-value threshold of π = $\frac{1}{8}$, and the restricted probability computation method skips the rest of the computation. We obtain a score threshold of th = 10 correponding to π.