CONSTRUCTION AND APPLICATION BASED ON COMPRESSING DEPICTION IN PROFILE HIDDEN MARKOV MODEL

Transcription

1 CONSRUCON AND APPCAON BASED ON COPRESSNG DEPCON N PROFE HDDEN ARKOV ODE Zhijian Zhou, 1, Daoiang i, 3, 3, i i,3,*, Zetian Fu 1 Coege of Science, China Agricuture University, Beijing, China, Key aboratory of odern Precision Agricuture System ntegration, Research inistry of Education, Beijing, China, Coege of nformation and Eectrica Engineering, China Agricutura University, Beijing, China, * Corresponding author, Address: P.O.Box 09#, China Agricutura University, East Campus, Beijing , P. R. China, e: , Emai:fzt@cau.edu.cn Abstract: Key words: A method to express Profie Hidden arkov ode (Profie H) parameters with compressing matrix is presented, which is obtained by imposing the characteristics of both the state transfer and the character output in the Profie H. Profie H, utipe sequence aignment, Bioinformatics 1. NRODUCON he Profie H is composed by a arkov chain incuding matching, insertion, deete states, and an observabe stochastic process namey as observation chain. he state chain depicts the transfer reationship among different state. he observation chain represents the statistica association between the state and observation. Generay, the state in arkov process cannot be observed directy. t can ony be understood through the observabe process. he Profie H mode was introduced to bioinformatics by Krogh (1994), and now was widey used in the Aignment of bioogica sequence.

2 130 Zhijian Zhou, Daoiang i, i i, Zetian Fu A one grade Profie HH with matching range as is iustrated as Figure 1. his is a inear mode. t progress ony one direction from eft to right. here are 3+1 system states in the system namey as matching (), insertion (), and deete (D) respectivey. For convenience in coding the program, two extra states, say as beginning ( 0 ) and ending ( +1 ), are invoved in the mode. hey do not output any character to infuence the mode activity. he character set depends on the concerning object. For exampe, there are four characters in DNA sequence denoted by A, G, C, and, and twenty characters for amino acid sequence. n the Figure 1, rectange, diamond, and circe denote matching, insertion and deete state respectivey. he arrow connected different states indicates the state transfer reation and direction. n an ascertain mode, the transfer probabiity and character reease between different state is whoy determinate. Figure 1 ode of one grade Profie HH with matching range as. COPRESSON FORS OF PROFE H PARAEERS Based on the stochastic processes theory, a Profie HH can be totay λ = λ S, Ω, ABπ,, is known. Where S denotes as state set, Ω ascertained if ( ) is observation set, A is state transfer probabiity matrix, B is character output probabiity matrix, and π is initia state probabiity distribution function, respectivey. Suppose a Profie H has matching range as. hen the number of eements for S is N = f two extra state, beginning and ending stats, are aso incuded, the number of eements for S become as N = 3( +. So the order of state transfer matrix A is 3( + 3( +. t can be found from Figure 1, at any state such as, three front states as inputs of the state are at the most. hey ocate at same ayer, say as -1. his means ony three occasions, 1, 1and D 1, can turn up when the state transfer from 1 to. Simiary, ony three occasions,,, D and 1D, 1D, D 1D for and D state occur respectivey. Remembering none of character output for deete state, the

3 Construction and Appication Based on Compressing Depiction in Profie Hidden arkov ode 1303 state transfer probabiity matrix as order as 3( + 3( + can be compressed to 9 (+ without ost any information: A a [ 0 1] a [ 0 1] 0 a [ 1 1] = a [ ] ad [ 1 1] a [ 0D1] ad [ 0 1] 0 a [ ] a [ ] ad [ ] a [ ] a [ ] 9( ad [ ] ad [ ] ad [ ] add [ ] a [ 1] a [ 1] a [ + 1] a [ 1] a [ 1] a [ + 1] ad [ 1] ad [ 1] ad [ 1] + a [ ] a [ ] 0 a [ ] a [ ] 0 ad [ ] ad [ ] 0 a [ 1D] a [ 1D] 0 a [ 1D] a [ 1D] 0 a[ D D ] ad [ D] ( + Where S D D D + 1 = {,,,,,,,,,,, } are state set, and axy [ i j] represents the one step transfer probabiity from X i toy j. he eements sign as 0 means no corresponding state transfer, and eements in coumn denote the probabiity of one step state transfer ending at ayer. et A ( a ) a = a[ X Y ], 1 i 9 1 j + 1, then foowing 9 ( + ij 9 ( + reations can be obtained ij d j 1, X = ( i mod 3 =, X = 0, X = D (- ( j 1, ) if 1 i 3 () ( dy, ) = ( j, ) if4 i 6 ( j 1, D) if 7 i 9 (-) (3) a31 = a91 = a4( = = a9( = 0 (-3) + + (4) a + a( 3)( + a( 6) = 1, i= 1,, 3; j =,3, 4 (-4) ij i+ j i+ j Simiary, et Ω = { ω1, ω,, ω K } is observation character set, then character output probabiity matrix B (3 + can be compressed as B K (+ = K b0[ ω1] b1 [ ω1] b1[ ω1] b [ ω1] b[ ω1] b [ ω1] b[ ω1] b0[ ωk] b1 [ ωk] b1[ ωk] b [ ωk] b[ ωk] b [ ωk] b[ ωk] K (+ (-5)

4 1304 Zhijian Zhou, Daoiang i, i i, Zetian Fu X Where bj [ ω k] is the probabiity of output ω k at the state X j. et B ( + ( b ) ( + b X ij = bd ( ωi ) 1 i K 1 j + 1, K ij K foowing reations can be obtained j 1 1, ( X, d) = (, ) j mod = (- j 0, ( X, d) = (, ) 6) herefore a profie H can be simpified by the compression state transfer probabiity matrix with order of 9 (+ and the compression character output probabiity matrix with order of K (+. 3. FORWARD AGORH et us consider the observation sequence O= ( o1, o,, o ). he matching range is in Profie H ( λ ). Based on compress state transfer probabiity matrix A 9 ( + and character output probabiity matrix B K (+, forward agorithm can be obtained. X Definition 1 et α () t = P( o1, o,, ot,endof X λ), X =,, D be the probabiity when part sequence Ot = ( o1, o, ot) output in X state at (1 ) ayer. hen ( α ( ), ( ), D ( )) t α t α t is the probabiity vector for ayer, denoted as α () t. Definition et ϕ( Xr) = ( a[ r 1Xr], a[ r 1Xr], a[ Dr 1Xr]) X =, D be a coumn vector composed by one step transfer probabiity from state X r 1 to X r, and ϕ ( q) = ( a[ qq], a[ qq], a[ Dqq]) aso be a coumn vector composed by one step transfer probabiity from state q 1 to q. husϕ ( ) = ( a, a, a ), ϕ ( ) = ( a, a, a ) and ϕ ( D ) = ( a, a, a ) p 1p p 3p q 4q 5q 6q r 7r 8r 9r compress state transfer probabiity matrix is A 9 ( + and character output probabiity matrix is B K (+. Now the forward agorithm for Profie H can be express as, nitiation α (0) = (1,0,0) 0 0 () t = (0,0,0), t = 1,,, + 1 α (3-, if

5 Construction and Appication Based on Compressing Depiction in Profie Hidden arkov ode 1305 ) α (0) = (0,0,0), = 1,,, + 1 ) Recursion cacuation α α ( t ϕ( ) b ( o ) 1 t () t = α ( t ϕ( ) b ( ot) α 1() t ϕ( D) where bd ( ot) bi( d), b ( ot) b i(d+ = =, when ot = wi. t = 1,,, = 1,,, (3-3) Ending hus α ( + = α ( ) a + α ( ) a D + α ( ) a ( ( + ( + 3( + 3) and the probabiity is PO ( λ) = α + 1( + (3-4) 4. APPCAON OF PROFE H An exampe of compress state transfer probabiity matrix and character output probabiity matrix for known mutipe sequence comparison is shown beow. Suppose DAN sequences have be aiment as abe 1. atching states ocate in first, second, and sixth coumn. hus m is 5, is 6 and is 3. he corresponding Profie H framework is depicted in Figure. abe 7 Aignment of mutipe DNA sequence bat A G rat A cat A G gnat goat A G C A G C A A A A A C C n order to avoid zero in probabiity cacuation, pseudo counting is adapted. For exampe, character A appears 4 times in first coumn, the 4+ 1 probabiity to output A for this state is b1 ( A) = = Character G does not show up and the probabiity to output G is b 1 ( G) = =

6 1306 Zhijian Zhou, Daoiang i, i i, Zetian Fu abe ist character output probabiity at the state in which state transfer rea occurs. As the same rue used, the compress state transfer probabiity matrix can be obtained as matrix (4-. D 1 D D Figure 7 Profie H frameworks for the case of three states abe. Frequency of character output ( ) b1 ω b k ( ωk ) ( ) b ω b k 3 ( ω k ) A G C A = (4- Suppose O= AGC is a new observation sequence. hus the probabiity that the sequence can be observed by using the mode as shown above is PO ( λ ) n practica appication, the observation sequence can be cassed according to its probabiity for different mode λi that estabished by corresponding training data. ACKNOWEDGEENS his work supported by the grant ( ) from the Nationa Natura Science Foundation of China.

7 Construction and Appication Based on Compressing Depiction in Profie Hidden arkov ode 1307 REFERENCES Gong Guang-u, Qian in-ping 004 Appication stochastic process course and stochastic mode in inteigent cacuation, Qinghua Pubisher, Beijing. Katoh A ian S Hausser D A hidden arkov mode that finds genes in E.coi DNA. Nuceic Acids Research : Krogh A Brown ian S Sjoander K and Hausser D Hidden arkov modes in computationa bioogy: appications to protein modeing Journa of oecuar Bioogy 35: Wang Yu-Fei, Shi Ding-Hua 006 Bioinformatiic inteigent arithmetic and its appication, Chemistry industry Pubisher, Beijing.