By L. Pachter, B. Sturmfels

ISBN-10: 0521857007

ISBN-13: 9780521857000

The quantitative research of organic series information relies on tools from statistics coupled with effective algorithms from machine technological know-how. Algebra offers a framework for unifying a number of the probably disparate thoughts utilized by computational biologists. This ebook deals an advent to this mathematical framework and describes instruments from computational algebra for designing new algorithms for specified, exact effects. those algorithms will be utilized to organic difficulties equivalent to aligning genomes, discovering genes and developing phylogenies. the 1st a part of this booklet includes 4 chapters at the subject matters of records, Computation, Algebra and Biology, providing fast, self-contained introductions to the rising box of algebraic records 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 e-book within the interesting and dynamic zone, it will likely be welcomed as a textual content for self-study or for complex undergraduate and starting graduate classes.

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**Extra resources for Algebraic Statistics for Computational Biology**

**Sample text**

The observable states in this model are pairs (σ, τ ) ∈ Σn × (Σ )n of words of length n. A sequence of N observations in this model is summarized n n in a matrix u ∈ Nl ×(l ) where u(σ,τ ) is the number of times the pair (σ, τ ) was observed. Hence, in this model, m = (l · l )n . The fully observed Markov model is parameterized by a pair of matrices (θ, θ ) where θ is an l × l matrix and θ is an l × l matrix. The matrix θ encodes a Markov chain as before: the entry θij represents the probability of transitioning from state i ∈ Σ to j ∈ Σ.

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.

The characteristic polynomial of the Hessian equals z(z − 64)(z − 16)2 (z + 16)2 (z + 64)(z + 80)4 (z + 320)2 . 25) . 152332481077 . . 3 4 4 4 54 3 4 4 4 The Hessian of obs (θ) at θ has rank 11, and all eleven non-zero eigenvalues are distinct and negative. 25) . 223952742410 . . 3 3 3 3 48 3 3 3 3 The Hessian of obs (θ) at θ has rank 11, with nine eigenvalues negative. 25) . The EM algorithm converges to a probability distribution which is a local maximum of the likelihood function. This maximum is greater than the local maximum found previously in Experiment 2.

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

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