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

On large deviations of empirical measures for stationary Gaussian processes

We show that the empirical eigenvalue measure for sum of $d$ independent Haar distributed $n$-dimensional unitary matrices, converge for $n \rightarrow \infty$ to the Brown measure of the free sum of $d$ Haar unitary operators. The same applies for independent Haar distributed $n$-dimensional orthogonal matrices. As a byproduct of our approach, we relax the requirement of uniformly bounded imaginary part of Stieltjes transform of $T_n$ that is made in [Guionnet, Krishnapur, Zeitouni; Theorem 1].

https://projecteuclid.org/download/pdf_1/euclid.ecp/1465315608

Limiting spectral distribution of sum of unitary and orthogonal matrices

Let $A_1,A_2,\ldots,A_n$ be generated governed by an $r$-state irreducible Markov chain and suppose $(X_i,U_i)$ are real valued independently distributed given the sequence $A_1,A_2,\ldots,A_n$, where the joint distribution of $(X_i,U_i)$ depends only on the values of $A_{i-1}$ and $A_i$ and is of bounded support Where $A_0$ is started with its stationary distribution, $E\lbrack X_1\rbrack 0, m = 1,\ldots,k; C\} > 0$ is assumed For the partial sum realizations where $\sum^l_{i=k}X_i \rightarrow \infty$, strong laws are derived for the sums $\sum^l_{i=k}U_i$ Examples with $r = 2, X \in \{-1, 1\}$ and the cases of Brownian motion and Poisson process with negative drift are worked out

/pdf/strong-limit-theorems-of-empirical-distributions-for-large-4c972bei4l.pdf

Strong Limit Theorems of Empirical Distributions for Large Segmental Exceedances of Partial Sums of Markov Variables

A statistical model is used for the optimal design of analysis/synthesis systems that include quantization of the signals in separate bands. Two types of quantization approaches are analyzed: fine quantization modelled by additive noise, and matrix quantization. With proper modification, the model applies to other types of quantization approaches. Assuming that the system operates as a waveform coder, two error measures are used: a generalization of the usual statistical mean-square error (MSE) to time-varying systems, and the MSE between the outputs of the analysis of the original and reconstructed signals. The measures are shown to be equivalent. A method that is applicable to the design of optimal synthesis filters, given the analysis filters, is presented. For fine quantization, an iterative algorithm for the design of an optimal analysis/synthesis system is presented, together with its convergence properties. If no quantization is applied, the results obtained with the presented method coincide with previously reported results. >

Statistical design of analysis/synthesis systems with quantization

Let Θ(x,r) denote the occupation measure of the ball of radius r centered at x for Brownian motion {Wt}0≤t≤1 in R d , d≥2 . We prove that for any analytic set E in [0,1], we have inf t∈E lim inf r→0 Θ(W t ,r)/(r 2 /| log r|)=1/ dim P (E) , where dimP(E) is the packing dimension of E. We deduce that for any a≥1, the Hausdorff dimension of the set of “thin points” x for which lim inf r→0 Θ(x,r)/(r 2 /| log r|)=a , is almost surely 2−2/a; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for the “thin” part of Brownian occupation measure. The methods of this paper differ considerably from those of our work on Brownian thick points, due to the high degree of correlation in the present case. To prove our results, we establish general criteria for determining which deterministic sets are hit by random fractals of `limsup type' in the presence of long-range correlations. The hitting criteria then yield lower bounds on Hausdorff dimension. This refines previous work of Khoshnevisan, Xiao and the second author, that required decay of correlations.

/pdf/thin-points-for-brownian-motion-2g4pxa1ond.pdf

Amir Dembo

Papers

On large deviations of empirical measures for stationary Gaussian processes

Limiting spectral distribution of sum of unitary and orthogonal matrices

Strong Limit Theorems of Empirical Distributions for Large Segmental Exceedances of Partial Sums of Markov Variables

Statistical design of analysis/synthesis systems with quantization

Thin points for Brownian motion