By L. Pachter, B. Sturmfels
The quantitative research of organic series information relies on tools from information coupled with effective algorithms from desktop technological know-how. Algebra presents a framework for unifying a few of the doubtless disparate suggestions utilized by computational biologists. This booklet deals an advent to this mathematical framework and describes instruments from computational algebra for designing new algorithms for certain, exact effects. those algorithms should be utilized to organic difficulties akin to aligning genomes, discovering genes and developing phylogenies. the 1st a part of this booklet comprises 4 chapters at the topics of records, Computation, Algebra and Biology, delivering fast, self-contained introductions to the rising box of algebraic records and its functions 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 ebook within the intriguing and dynamic sector, will probably be welcomed as a textual content for self-study or for complicated undergraduate and starting graduate classes.
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Extra resources for Algebraic Statistics for Computational Biology
Thus every algebraic statistical model has a Newton polytope, and it is the structure of this polytope which governs dynamic programming related to that model. Computing the entire polytope is what we call parametric inference. This computation can be done eﬃciently in the polytope algebra which is a natural generalization of tropical arithmetic. , phylogenetic trees, with an emphasis on the neighbor-joining algorithm. 43 44 L. Pachter and B. 1 Tropical arithmetic and dynamic programming Dynamic programming was introduced by Bellman in the 1950s to solve sequential decision problems with a compositional cost structure.
25) . The parameter vector θ is a stationary point of the EM algorithm, so after one step we output θ = θ. 903548889592 . . 16 1 1 1 1 1 1 1 1 Here θ is a critical point of the log-likelihood function obs (θ) but it is not a Statistics 23 local maximum. The Hessian matrix of obs (θ) evaluated at θ has both positive and negative eigenvalues. 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.
Then we make an estimate, given θ, of what we expect the hidden data U might be. This latter step is called the expectation step (or E-step for short). Note that the expected values for the hidden data vector do not have to be integers. 35) to optimality, using the easy and reliable subroutine which we assumed is available for the hidden model F . This step is called the maximization step (or M-step for short). Let θ ∗ be the optimal solution found in the M-step. We then replace the old parameter guess θ by the new and improved parameter guess θ ∗ , and we iterate the process E → M → E → M → E → M → · · · until we are satisﬁed.
Algebraic Statistics for Computational Biology by L. Pachter, B. Sturmfels