https://www.cs.rice.edu/~nakhleh/COMP571/Fall05/DatabaseSearch/blast.pdf

Basic Local Alignment Search Tool

The BLAST programs are widely used tools for searching protein and DNA databases for sequence similarities. For protein comparisons, a variety of definitional, algorithmic and statistical refinements described here permits the execution time of the BLAST programs to be decreased substantially while enhancing their sensitivity to weak similarities. A new criterion for triggering the extension of word hits, combined with a new heuristic for generating gapped alignments, yields a gapped BLAST program that runs at approximately three times the speed of the original. In addition, a method is introduced for automatically combining statistically significant alignments produced by BLAST into a position-specific score matrix, and searching the database using this matrix. The resulting Position-Specific Iterated BLAST (PSIBLAST) program runs at approximately the same speed per iteration as gapped BLAST, but in many cases is much more sensitive to weak but biologically relevant sequence similarities. PSI-BLAST is used to uncover several new and interesting members of the BRCT superfamily.

/pdf/gapped-blast-and-psi-blast-a-new-generation-of-protein-2ybl67x5yx.pdf

Gapped BLAST and PSI-BLAST: a new generation of protein database search programs.

There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This tutorial provides an overview of the basic theory of hidden Markov models (HMMs) as originated by L.E. Baum and T. Petrie (1966) and gives practical details on methods of implementation of the theory along with a description of selected applications of the theory to distinct problems in speech recognition. Results from a number of original sources are combined to provide a single source of acquiring the background required to pursue further this area of research. The author first reviews the theory of discrete Markov chains and shows how the concept of hidden states, where the observation is a probabilistic function of the state, can be used effectively. The theory is illustrated with two simple examples, namely coin-tossing, and the classic balls-in-urns system. Three fundamental problems of HMMs are noted and several practical techniques for solving these problems are given. The various types of HMMs that have been studied, including ergodic as well as left-right models, are described. >

/pdf/a-tutorial-on-hidden-markov-models-and-selected-applications-1vlzbxbk68.pdf

A tutorial on hidden Markov models and selected applications in speech recognition

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The usual assumption that the available partial information is noiseless is replaced by a more realistic statistical model which compensates for the presence of noise. The signal reconstruction is thus viewed as a parameter estimation problem, for which the EM iterative algorithm of A.P. Dempster, N.M. Laird, and D.B. Rubin (J. Roy. Stat. Soc., B, vol.39, p.1-37, 1977) is especially suitable. The posterior probability of the signal increases from iteration to iteration, till the signal converges to a stationary point of the posterior distribution. Each iteration involves one transformation followed by an inverse transformation (usually discrete Fourier transformation (DFT) and inverse DFT). Algorithms for reconstruction of both one- and two-dimensional signals from their spectral magnitude, spectral phase, or modified short time Fourier transform are typical examples of the proposed scheme. >

Signal reconstruction from noisy partial information of its transform

The optimal filtering error for a class of nonlinear cone-bounded filtering problems is analyzed for the case of observation noise intensity tending to zero. A sufficient condition for the resulting optimal error covariance matrix to tend to zero is derived by relating the nonlinear filtering problems to linear ones. The results are specialized for the case of nonlinear autoregressive moving-average (ARMA) processes. >

On the maximal achievable accuracy in nonlinear filtering problems

Self-normalized moderate deviations and lils

Consider an advancing ‘front’ $${R(t) \in \mathbb{Z}_{\geqq 0}}$$
 and particles performing independent continuous time random walks on $${ (R(t), \infty) \cap \mathbb{Z}}$$
 . Starting at $${R(0)=0}$$
 , whenever a particle attempts to jump into $${R(t)}$$
 the latter instantaneously moves $${k \ge 1}$$
 steps to the right, absorbing all particles along its path. We take k to be the minimal random integer such that exactly k particles are absorbed by the move of R, and view the particle system as a discrete version of the Stefan problem

 $$\begin{aligned}&\partial_{t} u_{*}(t,\xi) = \frac{1}{2} \partial^{2}_{\xi} u_{*}(t,\xi), \quad \xi >r(t), \\
&u_{*}(t,r(t))=0, \\
&\frac{{\rm d} }{{\rm d}t}r(t) = \frac{1}{2} \partial_\xi u_{*}(t,r(t)), \\
&t \mapsto r(t)\ \hbox{non-decreasing}, \quad r(0):=0.\end{aligned}
$$
 For a constant initial particle density $${u_{*}(0,\xi)=\rho {\bf 1}_{\{\xi > 0\}}}$$
 , at $${\rho < 1}$$
 the particle system and the PDE exhibit the same diffusive behavior at large time, whereas at $${\rho \ge 1}$$
 the PDE explodes instantaneously. Focusing on the critical density $${ \rho=1 }$$
 , we analyze the large time behavior of the front R(t) for the particle system, and obtain both the scaling exponent of R(t) and an explicit description of its random scaling limit. Our result unveils a rarely seen phenomenon where the macroscopic scaling exponent is sensitive to the amount of initial local fluctuations. Further, the scaling limit demonstrates an interesting oscillation between instantaneous super- and sub-critical phases. Our method is based on a novel monotonicity as well as PDE-type estimates.

Amir Dembo

Papers

Signal reconstruction from noisy partial information of its transform

On the maximal achievable accuracy in nonlinear filtering problems

Self-normalized moderate deviations and lils

Criticality of a Randomly-Driven Front

Large deviations for subsampling from individual sequences