Download PDF by L. Pachter, B. Sturmfels: Algebraic Statistics for Computational Biology

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.

Show description

Read or Download Algebraic Statistics for Computational Biology PDF

Best computational mathematicsematics books

Tian-Xiao He's Wavelet analysis and multiresolution methods: proceedings of PDF

Number of papers offered on the Wavelet research and Multiresolution equipment consultation of the yank Mathematical Society assembly held on the collage of Illinois at Urbana-Champaign. specializes in using wavelet research to unravel a vast variety of sign, time sequence, and picture difficulties. Softcover.

Read e-book online Adaptive Atmospheric Modeling: Key Techniques in Grid PDF

This publication offers an summary and counsel within the improvement of adaptive thoughts for atmospheric modeling. Written in an academic variety and that includes an exhaustive record of references, it's a start line for everybody who's drawn to adaptive modeling, now not limited to atmospheric sciences.

Additional resources for Algebraic Statistics for Computational Biology

Sample text

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.

Download PDF sample

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

by Daniel

Rated 4.66 of 5 – based on 50 votes