By L. Pachter, B. Sturmfels

ISBN-10: 0521857007

ISBN-13: 9780521857000

The quantitative research of organic series facts relies on equipment from facts coupled with effective algorithms from laptop technological know-how. Algebra offers a framework for unifying a few of the doubtless disparate suggestions utilized by computational biologists. This ebook deals an advent to this mathematical framework and describes instruments from computational algebra for designing new algorithms for designated, actual effects. those algorithms should be utilized to organic difficulties resembling aligning genomes, discovering genes and developing phylogenies. the 1st a part of this publication includes 4 chapters at the topics of records, Computation, Algebra and Biology, delivering quickly, self-contained introductions to the rising box of algebraic statistics and its purposes to genomics. within the moment half, the 4 subject matters are mixed and constructed to take on genuine difficulties in computational genomics. because the first publication within the intriguing and dynamic quarter, will probably be welcomed as a textual content for self-study or for complex undergraduate and starting graduate classes.

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This diﬀerence is nonnegative because the parameter vector θ ∗ was chosen so as to maximize the log-likelihood function for the hidden model with data (uij ). We next show that the last sum is non-negative as well. The parenthesized expression equals fi (θ ∗ ) − log fi (θ) n j=1 fij (θ ∗ ) uij log ui fij (θ) fi (θ ∗ ) = log + fi (θ) n j=1 fij (θ) fij (θ) log . 38) . This last expression is non-negative. This can be seen as follows. Consider the non-negative quantities πj = fij (θ) fi (θ) and σj = fij (θ ∗ ) fi (θ ∗ ) for j = 1, 2, .

The parameter λij represents the probability that the ith dice in her left pocket comes up with nucleotide j. The parameter ρij represents the probability that the ith dice in her right pocket comes up with nucleotide j. In total there are d = 13 free parameters because λiA + λiC + λiG + λiT = ρiA + ρiC + ρiG + ρiT = 1 for i = 1, 2. More precisely, the parameter space in this example is a product of simplices Θ = ∆1 × ∆3 × ∆3 × ∆3 × ∆3 . Statistics 21 The model is given by the polynomial map f : R13 → R4×4 , θ → (fij ) where fij = π·λ1i ·λ2j + (1−π)·ρ1i ·ρ2j .

Based on these ﬁndings, we would like to conclude that the 24 L. Pachter and B. 0981283 . 46) 4040 Assuming that this conclusion is correct, let us discuss the set of all optimal solutions. Since the data matrix u is invariant under the action of the symmetric group on {A, C, G, T}, that group also acts on the set of optimal solutions. There are three matrices like the one found in Experiment 4: 3 3 2 2 3 2 3 2 3 2 2 3 1 1 1 3 3 2 2 , 2 3 2 3 and 2 3 3 2 . 47) · · · 40 2 2 3 3 40 3 2 3 2 40 2 3 3 2 max Lobs (θ) : θ ∈ Θ 2 2 3 3 = 2 3 2 3 3 2 2 3 The preimage of each of these matrices under the polynomial map f is a surface in the space of parameters θ, namely, it consists of all representations of a rank 2 matrix as a convex combination of two rank 1 matrices.

### Algebraic Statistics for Computational Biology by L. Pachter, B. Sturmfels

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