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 desktop technological know-how. Algebra offers a framework for unifying the various probably disparate concepts utilized by computational biologists. This e-book deals an creation to this mathematical framework and describes instruments from computational algebra for designing new algorithms for special, exact effects. those algorithms should be utilized to organic difficulties corresponding to aligning genomes, discovering genes and developing phylogenies. the 1st a part of this e-book comprises 4 chapters at the subject matters of records, Computation, Algebra and Biology, providing quickly, 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 constructed to take on genuine difficulties in computational genomics. because the first publication within the intriguing and dynamic sector, it is going to be welcomed as a textual content for self-study or for complicated undergraduate and starting graduate classes.

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**Example text**

X1 X2 X3 X4 Fig. 3. Graph of the 4-chain Markov random ﬁeld. There are 3 pairs of nodes not connected by an edge, so that MG = X1 ⊥⊥ X3 | {X2 , X4 } , X1 ⊥⊥ X4 | {X2 , X3 } , X2 ⊥⊥ X4 | {X1 , X3 } . Statistics 37 For binary alphabets Σi the set QMG consists of the twelve quadratic forms p0010 p1000 − p0000 p1010 , p0001 p1000 − p0000 p1001 , p0001 p0100 − p0000 p0101 , p0011 p1001 − p0001 p1011 , p0011 p1010 − p0010 p1011 , p0011 p0110 − p0010 p0111 , p0110 p1100 − p0100 p1110 , p0101 p1100 − p0100 p1101 , p1001 p1100 − p1000 p1101 , p0111 p1101 − p0101 p1111 , p0111 p1110 − p0110 p1111 , p1011 p1110 − p1010 p1111 .

The entry vσ1 i2 in row σ1 and column i2 of the matrix v equals the number of occurrences of σ1 i2 ∈ Σ2 as a consecutive pair in any of the N observed sequences. 28 L. Pachter and B. 17 The maximum likelihood estimate of the data u ∈ Nl in the Markov chain model is the l × l matrix θ = θij in Θ1 with coordinates θij vij = where s∈Σ vis v = Al,n · u. Proof The likelihood function for the toric Markov chain model equals L(θ) = θ Al,n ·u = θv v θijij . = ij∈Σ2 The log-likelihood function can be written as follows: vi1 · log(θi1 ) + vi2 · log(θi2 ) + · · · + vi,l−1 · log(θi,l−1 ) + vil · log(θil ) .

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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