By L. Pachter, B. Sturmfels

The quantitative research of organic series facts relies on equipment from facts coupled with effective algorithms from laptop technology. Algebra offers a framework for unifying a number of the doubtless disparate options utilized by computational biologists. This e-book deals an advent to this mathematical framework and describes instruments from computational algebra for designing new algorithms for distinct, actual effects. those algorithms should be utilized to organic difficulties comparable to aligning genomes, discovering genes and developing phylogenies. the 1st a part of this ebook contains 4 chapters at the topics of information, Computation, Algebra and Biology, supplying fast, self-contained introductions to the rising box of algebraic facts and its purposes to genomics. within the moment half, the 4 subject matters are mixed and built to take on actual difficulties in computational genomics. because the first e-book within the fascinating and dynamic quarter, will probably 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**

**Example text**

For a numerical approach see Chapter 20. 4 Markov models We now introduce Markov chains, hidden Markov models and Markov models on trees, using the algebraic notation of the previous sections. , 1998] or other text books. A natural point of departure is the following toric model. 1 Toric Markov chains We ﬁx an alphabet Σ with l letters, and we ﬁx a positive integer n. We shall deﬁne a toric model whose state space is the set Σn of all words of length n. The model is parameterized by the set Θ of positive l × l matrices.

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.

By a Markov chain we mean any point p in the model fl,n (Θ1 ). , 1998, Chapter 3], except that here we require the initial distribution at the ﬁrst state to be uniform. This assumption is made to keep the exposition simple. For instance, if l = 2 then the parameter space Θ1 is a square. Namely, Θ1 is the set of all pairs (θ0 , θ1 ) ∈ R2 such that the following matrix is positive: θ = 1 − θ0 θ0 1 − θ1 θ1 The Markov chain model is the image of the square under the map f2,n . A Markov chain of length n = 4 is any probability distribution p of the form 1 1 1 p0000 = θ03 , p0001 = θ02 (1−θ0 ), p0010 = p1001 = p0100 = θ0 (1−θ0 )(1−θ1 ), 2 2 2 1 p0011 = θ0 (1 − θ0 )θ1 , 2 p0101 = 1 (1 − θ0 )2 (1 − θ1 ) , 2 1 p0110 = p1011 = p1101 = (1 − θ0 )θ1 (1 − θ1 ) , 2 p1000 = 1 p0111 = (1 − θ0 )θ12 , 2 1 p1010 = (1 − θ1 )2 (1 − θ0 ), 2 1 1 1 1 (1−θ1 )θ02 , p1100 = θ1 (1−θ1 )θ0 , p1110 = θ12 (1−θ1 ), p1111 = θ13 .