Detecting local deviations. Optimisation and applications to RNA-gene searching.

Transcription

1 Detecting Local Deviations Detecting local deviations. Optimisation and applications to R-gene searching. iels Richard ansen niversity of openhagen Department of pplied Mathematics and Statistics p. 1/20

2 My scientific grandfather Ole E. Barndorff-ielsen Michael Sørensen (1986) iels Richard ansen (2004) ollaboration Distance (Barndorff-ielsen number): 6 iels Richard ansen coauthored with Ernst ansen Ernst ansen coauthored with Søren Fiig Jarner Søren Fiig Jarner coauthored with areth O. Roberts areth O. Roberts coauthored with Jesper Møller Jesper Møller coauthored with Michael Sørensen Michael Sørensen coauthored with Ole E. Barndorff-ielsen p. 2/20

3 Ribonucleic acid(r) ytosine 's T's ytosine 2 2 O O uanine uanine O O itrogenous Bases denine denine 2 Base pair 2 Sugar phosphate backbone racil T Thymine O O 3 O replaces Thymine in R itrogenous Bases R Ribonucleic acid D Deoxyribonucleic acid itrogenous Bases O ational Institutes of ealth ational uman enome Research Institute Division of Intramural Research

4 R molecular structure Let-7 (pre-cursor) from. Elegans. Member of the family of micro Rs that terminate or inhibit the translation of mr to protein. The pre-cursor is embedded as a gene in the D we want to find genes with similar structure. p. 4/20

5 Epidemic change point problem Two change points the epidemic alternative. The null model: (X n ) n 1 are i.i.d. The alternative: For some, unknown k 1 k 2, (X n ) n {k1,...,k 2 } are still i.i.d. but (X n ) n {k1,...,k 2 } has another distribution. Objectives: Test the null against the alternative. Estimate k 1 and k 2. Example: X n (µ n, 1), (X n ) n 1 independent. ull model: µ n = µ for all n 1 and some µ R. lternative: µ n = µ for all n {k 1,...,k 2 }, µ n = µ for all n {k 1,...,k 2 } with µ, µ R. p. 5/20

6 SM test The SM test statistic ( µ > µ) is M n = max 1 k 1 k 2 n { k2 k=k 1 Z k, 0 } = max 1 k 2 n max 1 k 1 k 2 { k2 } Z k, 0 k=k 1 T k2, where e.g. Z k = X k µ + µ 2. T k2 = max 1 k 1 k 2 { k2 k=k 1 Z k, 0 } = max{t k2 1 + Z k2, 0}. p. 6/20

7 SM test example X n S n T µ = 0 and µ = 1 n p. 7/20

8 SM test example X n S n W ˆk 1 = 37 (True k 1 = 38) ˆk2 = 50 (True k 2 = 50) n p. 8/20

9 Stem-loop alternative X 1...X i 1 5 -stem hairpin-loop 3 -stem X i...x i+δ δ+1 X i+δ+1...x j δ 1 j i 2δ 1 X j δ...x j δ+1 X j+1...x n. ull: lternative: (X n ) n 1 i.i.d. π 0 -distributed. (X i+k, X j k ) k=0,...,δ i.i.d. π-distributed and independent of everything else. Define Z i,j = f(x i, X j ), f(x, y) = log π(x, y) π 0 (x)π 0 (y) for i < j, x, y {,,, }. p. 9/20

10 Test statistic - without loop penalty X 1...X i 1 5 -stem hairpin-loop 3 -stem X i...x i+δ δ+1 X i+δ+1...x j δ 1 j i 2δ 1 X j δ...x j δ+1 X j+1...x n. Directly generalising the SM test gives M n = max i,j,δ { δ k=0 Z i+k,j k }= max i,j { δ } max Z i+k,j k δ k=0 T i,j. p. 10/20

11 Test statistics with loop penalty X 1...X i 1 5 -stem hairpin-loop 3 -stem X i...x i+δ δ+1 X i+δ+1...x j δ 1 j i 2δ 1 X j δ...x j δ+1 X j+1...x n. Large loops are not favourable. We introduce a penalty function g : (, 0] and M n = max i,j,δ = max i,j max δ { δ } Z i+k,j k + g(j i 2δ 1) k=0 { δ } Z i+k,j k + g(j i 2δ 1). k=0 } {{ } T i,j p. 11/20

12 recursion Initialisation: Score g(2) T k,k+1 = g(2) (T k,k = g(1)) g(4) g(4) + 1 g(4) + 2 g(4) max{g(4) 2, g(12)} T i,j = max{t i+1,j 1 + Z i,j, g(j i + 1)} Index k k 1 k 2 k 3 k 4 k 5 i R Struct. X i Z i,j = f(x i, X j ) X j Index k + 1 k + 2 k + 3 k + 4 k + 5 k + 6 j p. 12/20

13 The reflected random walk The process T n = max{t n 1 + Z n, g(n)}, T 0 = 0 for i.i.d. (Z n ) n 0 and g : (, 0] (g(0) = 0) is a random walk reflected at the barrier g. It fulfills that where S n = n k=1 Z k, S 0 = 0. T n = S n + max 0 k n {g(k) S k} L n M n := max 0 k n T k M := sup k 0 T k. p. 13/20

14 Three examples g(n)=0 g(n) = 15 log(n) g(n) = n p. 14/

15 Results Let θ > 0 solve E exp(θz 1 ) = 1, and let P have local Radon-ikodym derivative exp(θ S n ) w.r.t. P, then with L = sup {g(n) S n } = lim L n n 0 n we have P(M < ) = 1 E exp(θ L) <. Moreover (with L 1 = ), E exp(θ L) = exp(θ g(n))e(1 exp(θ (L n 1 L n ))) n=0 exp(θ g(n)) n=0 p. 15/20

16 Results Theorem: If E exp(θ L) < and if Z 1 is non-arithmetic P(M > u) exp( θ u) E exp(θ 1 P(τ + < ) L) θ E S τ+ K g K for u where and τ + = inf{n 0 S n > 0} L = sup{g(n) S n }. n 0 p. 16/20

17 Levy process detour With (S t ) t 0 a Lévy process and g : [0, ) (, 0] (cadlag) the reflection at g is defined by T t = S t + max {g(s) S s}. 0 s t L t onjecture: With θ > 0 solving E exp(θs 1 ) = 1, P having local Radon-ikodym derivative exp(θ S t ) w.r.t. P, and L = lim t L t, then P(sup t 0 T t < ) = 1 E exp(θ L) <, and ( E exp(θ L) = θ E 0 ) exp(θ g(t))dl t. p. 17/20

18 Results with loop penalty onsider T i,j = max{t i+1,j 1 + f(x i, X j ), g(j i + 1)} and let n (t) be the number of diagonals in (T i,j ) exceeding t. Define g i (n) = g(2n + i), i = 0, 1. Theorem: If K gi < then with t n = log {nk(k g 0 + K g1 )} + x θ it holds that D( n (t n )) Poi(exp( x)) 0 for n. p. 18/20

19 Results without loop penalty onsider T i,j = max{t i+1,j 1 + f(x i, X j ), 0} and let n (t) be the number of excursions in (T i,j ) exceeding t. Define K = KP (τ = ). and let π = exp(θ f) π 0 π 0. Theorem: If (π π 0 π 0 ) > 2 max{(π 1 π 0 ), (π 2 π 0 )} then with t n = log { n 2 K /2 } + x θ it holds that D( n (t n )) Poi(exp( x)) 0 for n. (Essentially Karlin et al. 1994). p. 19/20

20 onclusions and outlook We have dealt with a central test problem in bioinformatics identified as a kind of change point detection problem for a few simple test statistics. Realistic tests statistics have a more complicated combinatorial structure. Empirical evidence support that the results generalise qualitatively. urrent project: Estimate the asymptotic parameters from data/simulations. Investigate the space of score functions f and more elaborated score systems using the theory in particular, optimisation of the score system based on a dataset from the alternative (some R-structures). p. 20/20