By L. Pachter, B. Sturmfels

The quantitative research of organic series information is predicated on tools from information coupled with effective algorithms from desktop technology. Algebra presents a framework for unifying some of the doubtless disparate concepts utilized by computational biologists. This booklet deals an creation to this mathematical framework and describes instruments from computational algebra for designing new algorithms for distinct, exact effects. those algorithms should be utilized to organic difficulties equivalent to aligning genomes, discovering genes and developing phylogenies. the 1st a part of this publication comprises 4 chapters at the issues of facts, Computation, Algebra and Biology, supplying fast, self-contained introductions to the rising box of algebraic information and its purposes to genomics. within the moment half, the 4 issues are mixed and built to take on genuine difficulties in computational genomics. because the first publication within the intriguing and dynamic region, it is going to be welcomed as a textual content for self-study or for complex undergraduate and starting graduate classes.

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**Example text**

N. 40) uij · (θ ) = ∗ fij (θ ) ∂θk i=1 ∂ obs (θ ∗ ) fij (θ ∗ ) = . ∂θk∗ This means that θ ∗ is a critical point of j=1 ∂fij ∗ ui · (θ ) ∗ fi (θ ) ∂θk obs . The remainder of this section is devoted to a simple example which will illustrate the EM algorithm and the issue of multiple local maxima for (θ). 41) We wish to test the hypothesis that these two sequences were generated by DiaNA using one biased coin and four tetrahedral dice, each with four faces labeled by the letters A, C, G and T.

We assume that each random variable Xi takes its values in a ﬁnite alphabet Σi . 59) i=1 and the number of states is m = ni=1 |Σi |. This number is ﬁxed throughout this section. 59) corresponds to an n-dimensional table (pi1 i2 ···in ). We think of pi1 i2 ···in as an unknown which represents the probability of the event X1 = i1 , X2 = i2 , . . , Xn = in . A conditional independence statement about X1 , X2 , . . 60) where A, B, C are pairwise disjoint subsets of {X1 , X2 , . . , Xn }. 60) reads “A is independent of B” and is denoted by A ⊥⊥ B.

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. , 2003]. 46) indicates that the set of optimal solutions to the maximum likelihood problem is the disjoint union of three “surfaces of explanations”.

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