Week 3 Statistics for bioinformatics and escience

Transcription

1 Week 3 Statitic for bioinformatic and escience Line Skotte 28. november ) In thi eercie we conider microrna data from Human and Moue. The data et repreent 685 independent realiation of the random vector (X 1, X 2, X 3, X 4 ), with value in the ample pace E = {A,C,G,T} 4. Here (X 1, X 4 ) repreent hydrogen bonding nucleotide from the Moue RNA and (X 2, X 3 ) repreent hydrogen bonding nucleotide from the Human RNA. The RNA are at the ame time aumed to be aligned, uch that (X 1, X 2 ) and (X 3, X 4 ) repreent evolutional alignment pair. Let n( 1, 2, 3, 4 ) denote the number of time each of the 256 poible letter combination occur in the dataet. Let p( 1, 2, 3, 4 ) = n( 1, 2, 3, 4 )/685 be the relative frequencie (e R code for calculation). We hall now ue the relative frequencie p from above a the point probabilitie of a probability meaure P on E = {A, C, G, T } 4. Aume that the ditribution of (X 1, X 2, X 3, X 4 ) i given by P. The point probabilitie p 1 of the marginal ditribution P 1 of the random variable X 1 i found by definition For the nucleotide A we get p 1 (A) = P 1 ({A}) = P(X 1 {A}) = P(X 1 {A}, X 2 {A,C,G,T}, X 3 {A,C,G,T}, X 4 {A,C,G,T}) = p(a, 2, 3, 4 ). 2 {A,C,G,T} 3 {A,C,G,T} 4 {A,C,G,T} The point probabilitie of the other nucleotide i found in the eact ame way. Finding the other marginal ditribution i analog the above calculation (e R code for calculation). To compute the point probabilitie p m of the ditribution P m of the random vector 1

2 (X 1, X 4 ), the needed derivation are imilar. p m (A,A) = P m ({A} {A}) = P((X 1, X 4 ) = (A,A)) = P(X 1 {A}, X 2 {A,C,G,T}, X 3 {A,C,G,T}, X 4 {A}) = p(a, 2, 3, A). 2 {A,C,G,T} 3 {A,C,G,T} To find point probabilitie of the other 15 nucleotide pair, the calculation are the ame. The the point probabilitie p h of the ditribution P h of (X 2, X 3 ) i found in a imilar way (e R code for calculation). The ditribution of thee hydrogen bond pair are of coure probability meaure on the product et {A, C, G, T } 2. Let P 2 denote the probability meaure on E, where (X 1, X 4 ) and (X 2, X 3 ) are independent, with marginal ditribution P m and P h repectively. To compute the point probabilitie of P 2, we ue Definition or Theorem Thi give u that P 2 ({(A,A,A,A)}) = P ((X 1, X 4 ) {(A,A)}, (X 2, X 3 ) {(A,A)}) = P ((X 1, X 4 ) {(A,A)}) P ((X 2, X 3 ) {(A,A)}) = p m (A,A)p h (A,A). The ame for the other 255 combination of 4 nucleotide. Now we conider the four dimenional core array S 1, 2, 3, 4 = log p( 1, 2, 3, 4 ) p m ( 1, 2 )p h ( 2, 3 ). If P = P 2, we have for all ( 1, 2, 3, 4 ) E that p( 1, 2, 3, 4 ) = p m ( 1, 2 )p h ( 2, 3 ). Since log(1) = 0 we have that S 1, 2, 3, 4 = 0 for all ( 1, 2, 3, 4 ) E. Note that log i undefined in zero. Thu S i not well defined when ome of the 256 poible combination of nucleotide are unoberved. Therefore we introduce peudo count. Thi i done by replacing the frequencie n( 1, 2, 3, 4 ) with n( 1, 2, 3, 4 ) + δ, for ome δ > 0. Then the total count i no longer 685, but δ, ince we add the peudo count δ to each of the 256 frequencie. (Se R code for the computation of S 1, 2, 3, 4 with peudo count). Now let (X 1, X 2, X 3, X 4 ) be a random vector with ditribution P. Then S X1,X 2,X 3,X 4 i a tranformation of the random variable, and we can ue eample to find the mean of thi tranformation. Let p denote the point probabilitie of P, then µ = S 1, 2, 3, 4 p( 1, 2, 3, 4 ). ( 1, 2, 3, 4 ) E Similarly, if (X 1, X 2, X 3, X 4 ) ha ditribution P 2. (Se R code for calculation of the mean) 2

3 2.10.1) The ditribution function F of the tandard gumbel ditribution i by Eample F () = ep( ep( )). To find the ditribution function G of the gumbel ditribution with location parameter l and cale parameter, we ue Eample and get ( ) ( ( l G() = F = ep ep l )). Ditribution function G() From Eample we know the denity f of the tandard gumbel ditribution f() = ep( ep( )). Therefore we can find the denity function g of the gumbel ditribution with location parameter l and cale parameter by uing the formula found in Eample (of by differentiating the ditribution function found above) g() = 1 f ( l ) = 1 ( ep l ep ( l )). 3

4 Denity function g() Definition define quantile function. According to Theorem the invere G 1 of the ditribution function G i a quantile function for the gumbel ditribution with location l and cale. To find the invere conider that i equivalent to y = ep ( ep ( l l log( log(y)) =. Thu a poible quantile function Q : (0, 1) R i (e implementation in R code) )) Q(y) = l log( log(y)). 4

5 Inver ditribution function Q() ) In order to imulate from the gumbel ditribution with location parameter l and cale parameter, alle we need i the invere ditribution function found above. Thi follow from Theorem og Algorithm , ince the invere i a generalized invere. (e implementation in R code) 5

6 Simulation from gumbel ditribution Denity ) Let X have the geometric ditribution with ucce probability p. According to Eample thi ditribution i given by the point probabilitie p(n) = p(1 p) n for n N 0. Thi correpond to the probability of having n failure and then one ucce. The ditribution function i F () = P(X ) = 1 P(X > ) = 1 P(X + 1) where the lat equality follow from the fact that X only take value in N 0. Now the probability that X + 1 i the probability of having at leat + 1 failure before the firt ucce. Therefore F () = 1 (1 p) +1. Since i the integer atiefying 1 >, we can conclude that = ma{n Z n }. 6

7 Likewie = min{n Z n }. Note that z, implie that the integer z i an element in the interval (, z). Therefore the mallet integer, which i larger than, mut be maller than z. Thi i the ame a aying z. In the ame way z, implie that the integer i an element in the interval (, z). Therefore the larget integer, which i maller than z, mut be larger than. Thi i the ame a aying z. To how that F (y) = log(1 y) log(1 p) 1 we imply how that the defining relation (2.29) i atified. F () y 1 (1 p) +1 y (1 p) +1 1 y ( + 1) log(1 p) log(1 y) log(1 y)/ log(1 p) 1 log(1 y) log(1 p) 1. Note that log(1 p) i negative, which i the reaon for flipping the inequality ign the econd time. We have indeed found a generalied invere of the ditribution function for the geometric ditribution ) To imulate independent geometrically ditributed random value, alle we need i the generalied inver of the ditribution function found above. Thi follow form Theorem (Algorithm ). (Se R code for implementation) 7

8 Simulation from geometric ditribution