2.2 Background Correction / Signal Adjustment Methods

Transcription

1 7 It is importnt to note tht this definition is somewht roder thn is often used in the wider community. Mny times only methods deling with the first prolem hve een referred to s ckground correction methods. Unlike other rry systems, such s cdna microrrys, where pixels surrounding spot cn e used to compute the ckground djustment, the proe intensities themselves must e used to determine ny djustment for Affymetrix Genechips. This is ecuse proe loctions re very densely spced on the rry.. Bckground Correction / Signl Adjustment Methods.. RMA Convolution Model The RMA convolution model ckground correction method is motivted y looking t the distriution of proe intensities. Figure. shows the proe intensity distriution for group of typicl rrys. We model the oserved intensity s the sum of signl nd ckground component. In prticulr, our model is tht we oserve S X +Y, where X is signl nd Y is ckground. Assume tht X is distriuted expα nd tht Y is distriuted N μ,, with X nd Y independent. Furthermore, ssume tht Y 0 to void producing negtive vlues. Thus, Y is normlly distriuted with trunction t 0. This model is motivted y the oserved proe densities in Figure.. Under this model the ckground corrected proe intensities will e given y EX S s. A formul for this quntity is derived elow. We define z nd φz s the stndrd norml distriution function nd density function respectively. More specificlly z z exp w dw nd φz exp z. Rememering tht we oserve only S X +Y, under the conditions of this model, the density of the joint distriution of X nd Y is given y f X,Y x,yαexp αx y μ φ when y > 0, x > 0

2 8 density x Figure.: Smoothed histogrms of the proe intensities for numer of rrys from the HGU95A spike-in dtset.

3 9 Then, we get the joint distriution of X nd S from f X,S x,s f X,Y x,s x J where J is the Jcoin of the trnsformtion. Now, J nd so the joint distriution of X nd S is f X,S x,sαexp αx s x μ φ α exp αx x s + μ φ The conditionl distriution of X given S is f X,S x,s f X S x s 0 f X,S x,sdx where the denomintor the mrginl pdf of S is α exp αx x s + μ 0 φ dx Let w x so tht dw dx nd x w+s μ. Mking the sustitution, the integrl ecomes α exp αw + s μφ wdw α exp αs μ exp αw exp w dw Now, we consider the integrl on the right hnd side exp αwexp w dw exp α exp α exp w + αw dw exp w + αw + α dw exp w + α dw Let z w + α nd then the integrl ecomes +α exp s μ α +α z dz nd the denomintor is s μ α α exp α αs μ μ + α μ + α ]

4 0 thus, f X S x s f X,S x,s 0 f X,S x,sdx α exp α αs μ α exp αx φ x s μ α exp αx + αs μ α exp ] s μ α μ+α μ+α ] x s + μ exp x xs μ+s μ + αx αs μ+α 4 ] s μ α μ+α exp x xs μ α +s μ s μ α + α 4 ] s μ α μ+α exp x s μ α s μ α μ+α ] Let s μ α nd Therefore, the conditionl distriution of x given S is f x s φ x s ] nd so s E x s x x s 0 φ dx Let z x so dz dx. Thus x x φ 0 dx / / z + φzdz φzdz+ s / zφzdz ] + φ φ ] s nd so E X S s + φ φ s s

5 In most Affymetrix micorrry pplictions φ s is neglile nd s is close to one. So in prctice, it is only necessry to compute the first term in the numertor nd the first term in the denomintor to mke the djustment. It is somewht troulesome to estimte the prmeters μ, nd α. Some pproches re either pinfully slow the EM lgorithm or numericlly unstle Newton methods. An d-hoc pproch is used to estimte the prmeters. First, non-prmetric density estimte of the oserved proe intensities on n rry is tken, the mode of which is used s the estimte of μ. Then, the vriilty of the lower til out μ is used for nd n exponentil is fitted to the right til to estimte α. In this thesis, we hve elected to only djust PM proe intensities ecuse we focus on expression mesures which use only PM proes, ut in principle we could djust MM proe intensities using this method, either seprtely or together with the PM proe intensities... Methods Proposed y Affymetrix There re two seprte djustment steps tht hve een proposed y Affymetrix 00. For our nlysis, they re considered oth seprtely nd in the sequence in which they re used in the MAS 5.0 softwre Affymetrix, 00, which is the loction specific correction followed y the idel mismtch djustment. It should e noted tht we creted our own implementtions of these methods sed upon the ville documenttion nd there my e some slight differences from the Affymetrix softwre. Loction Specific Correction The gol of this step is to remove overll ckground noise. Ech rry is divided into set of regions, then ckground vlue for tht is grid estimted. Then ech proe intensity is djusted sed upon weighted verge of ech of the ckground vlues. The weights re dependent on the distnce from the centroid of ech of the grids. In prticulr, the weights re w k x,y d k x,y+smooth where d k x,y is the eucliden distnce from loction x,y to the centroid of region k. The defult vlue for smooth is 00. Specil cre is tken to void negtive vlues or other numericl prolems