17.9: Extension of the EM Approach
- Page ID
- 41022
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)ZOOPS Model
The approach presented before (OOPS) relies on the assumption that every sequence is characterized by only one motif (e.g., there is exactly one motif occurrence in a given sequence). The ZOOPS model takes into consideration the possibility of sequences not containing motifs.
In this case let i be a sequence that does not contain a motif. This extra information is added to our previous model using another parameter λ to denote the prior probability that any position in a sequence is the start of a motif. Next, the probability of the entire sequence to contain a motif is λ = (L − W + 1) ∗ λ
The E-Step
The E-step of the ZOOPS model calculates the expected value of the missing information–the probability that a motif occurrence starts in position j of sequence Xi. The formulas used for the three types of model are given below.
where λt is the probablity that sequence i has a motif, Prt(Xi|Qi = 0) is the probablity that Xi is generated from a sequence i that does not contain a motif
The M-Step
The M-step of EM in MEME re-estimates the values for λ using the preceding formulas. The math remains the same as for OOPS, we just update the values for λ and γ
The model above takes into consideration sequences that do not have any motifs. The challenge is to also take into consideration the situation in which there is more than one motif per sequence. This can be accomplished with the more general model TCM. TCM (two-component mixture model) is based on the assumption that there can be zero, one, or even two motif occurrences per sequence.
Finding Multiple Motifs
All the above sequence model types model sequences containing a single motif (notice that TCM model can describe sequences with multiple occurences of the same motif). To find multiple, non-overlapping, different motifs in a single dataset, one incorporates information about the motifs already discovered into the current model to avoid rediscovering the same motif. The three sequence model types assume that motif occurrences are equally likely at each position j in sequences xi. This translates into a uniform prior probability distribution on the missing data variables Zij. A new prior on each Zij had to be used during the E-step that takes into account the probability that a new width-W motif occurrence starting at position Xij might overlap occurrences of the motifs previously found. To help compute the new prior on Zij we introduce variables Vij where Vij = 1 if a width-W motif occurrence could start at position j in the sequence Xi without overlapping an occurrence of a motif found on a previous pass. Otherwise Vij = 0.
