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By L. Pachter, B. Sturmfels
The quantitative research of organic series info is predicated on tools from facts coupled with effective algorithms from desktop technological know-how. Algebra presents a framework for unifying some of the probably disparate recommendations utilized by computational biologists. This ebook bargains an creation to this mathematical framework and describes instruments from computational algebra for designing new algorithms for certain, exact effects. those algorithms could be utilized to organic difficulties equivalent to aligning genomes, discovering genes and developing phylogenies. the 1st a part of this ebook comprises 4 chapters at the subject matters of records, Computation, Algebra and Biology, providing quickly, self-contained introductions to the rising box of algebraic information and its purposes to genomics. within the moment half, the 4 topics are mixed and constructed to take on genuine difficulties in computational genomics. because the first ebook within the fascinating and dynamic region, it will likely be welcomed as a textual content for self-study or for complicated undergraduate and starting graduate classes.
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Additional resources for Algebraic Statistics for Computational Biology
Our tree models are parameterized by a collection of matrices θ kl , one for each edge kl ∈ E(T ). The rows of the matrix θ kl are indexed by Σk , and the columns are indexed by Σl . As before, we restrict ourselves to positive matrices whose rows sum to one. Let Θ1 denote the collection of tuples θ kl kl∈E(T ) of such matrices. The dimension of the parameter space Θ1 is therefore d = kl∈E(T ) |Σk |(|Σl | − 1). The fully observed tree model is the restriction to Θ1 of the monomial map FT : Rd → Rm , θ = θ kl pσ = 1 · |Σr | kl∈E(T ) θσklk σl .
Suppose we observe the game N times. These observations are our data. The suﬃcient statistic is the vector (uτ ) ∈ N1296 , where uτ = uτ1 τ2 τ3 τ4 counts the number of times the output sequence τ = τ1 τ2 τ3 τ4 was observed. Hence τ ∈(Σ )4 uτ = N . The goal of EM is to maximize the log-likelihood function x, y, f1 , . . , f5 , l1 , . . , l5 uτ1 τ2 τ3 τ4 · log(pτ1 τ2 τ3 τ4 ), = τ ∈Σ 4 Statistics 31 where (x, y) ranges over a square, (f1 , . . , f5 ) runs over a 5-simplex, and so does (l1 , . .
The cardinality of the state space is m = |Σ1 | · |Σ2 | · · · |Σn |. There is a natural linear marginalization 32 L. Pachter and B. Sturmfels map ρT : Rm → Rm which takes real-valued functions on real-valued functions on ni=1 Σi . We now have fT = ρT ◦ FT . 22 The hidden tree model fT : Rd → Rm is a multilinear polynomial map. Each of its coordinates has total degree |E(T )|, but is linear when regarded as a function of the entries of each matrix θ kl separately. The model fT described here is also known as the general Markov model on the tree T , relative to the given alphabets Σi .
Algebraic Statistics for Computational Biology by L. Pachter, B. Sturmfels