Single-sided asymptotic regression. The figure indicates a typical plot relevant to single-sided asymptotic regression, with the transient regime on the left, the asymptotic regime on the right, and an ill-defined change point separating the two regimes. Once in the asymptotic regime, the data point approximates a horizontal line y = β. The rest of this caption is pertinent only to the Results section, as follows. This figure displays estimates y_i for β = λ_n−5,n from Eq. (2) against sequence length t = n as small open circles (○) with error bars. (Some error bars are small and are not readily visible). The error bars indicate standard errors from 1 000 000 random sequence pairs. The solid horizontal line shows the present best estimate β = λ ≈ 0.2666 ± 0.0003 [32]. For single-sided asymptotic regression, data in the asymptotic regime are indicated by large open circles (○); the estimated β̂ is indicated by a thick dotted horizontal line (indistinguishable from the solid horizontal line). For robust regression with Andrews’ wave function, data in the asymptotic regime are indicated by closed circles (●); the estimated β̂ is indicated by a dashed horizontal line (barely distinguishable from the solid horizontal line). The values c = 2 and a = 0.5 were chosen for single-sided asymptotic regression and robust regression with Andrews’ wave function, respectively, because they minimized the absolute difference between the estimated β and the present best estimate β = λ ≈ 0.2666 ± 0.0003. (Table I summarizes the different estimates of β from different cutoff values c and parameters a.)

Local alignment. Gapped local alignment scores and the corresponding directed paths for two subsequences A(0,10] = TATATGCCTT and B(0,10] = CCATTGCACA of sequences drawn from the nucleotide alphabet {A,C,G,T}. The nucleotide scoring matrix in the figure is commonly used in applications: S(a,b) = 4 if a = b and −5 otherwise and the affine gap penalty Δ(g) = 2 + g. The vertex (i,j) = (10,10) is in the northeast corner of the figure with the origin (0,0) at the southwest corner. The cell (i,j) corresponding to the letter pair (A_i,B_j) displays the corresponding local score Ŝ_i,j. A local score of 0 indicates that no path of positive weight ends at the corresponding point. The optimal local path, which ends at point (8,9), e.g., consists of eight edges, in order: 2 northeast, 1 east, 3 northeast, 1 north, and 1 northeast. It corresponds to the optimal local sequence alignment of A(0,10] and B(0,10], The local score Ŝ_8,9 = 4 + 4 − 3 + 4 + 4 + 4 − 3 + 4 = 18 is the sum of the corresponding edges and represents the path of greatest weight ending at point (8,9). The score is also the greatest weight of any path within the rectangle displayed, so it also corresponds to the local maximum score M̂_10,10 = 18.

Double-sided asymptotic regression. The figure indicates a typical plot relevant to double-sided asymptotic regression, with the transient regime on the left, the asymptotic regime on the right (which is eventually overtaken by a noncompliant regime where the statistical model fails), and two ill-defined change points separating the three regimes. The rest of this caption is pertinent only to the Results section, as follows. This figure displays estimates y_i for β = K from Eq. (3) against global score t = s as small open circles (○) with error bars. The error bars indicate standard errors from 1 000 000 random sequence pairs. The solid horizontal line shows the present best estimate β = K ≈ 0.0409 ± 0.0003 [32]. For double-sided asymptotic regression, data in the asymptotic regime are indicated by large open circles (○); the estimated β̂ is indicated by a thick dotted horizontal line (barely distinguishable from the solid horizontal line). For robust regression with Andrews’ wave function, data are indicated by closed circles (●); the estimated β̂ is represented by a dashed horizontal line. (Note that, in fact, robust regression outliers break up the single asymptotic interval indicated by the dashed line.) For each method, c = 0.5 and a = 0.5 minimized the absolute difference between the estimated K and the present best estimate β = K ≈ 0.0409 ± 0.0003. (Table II summarizes the different estimates of K from different cutoff values c and parameters a.)