Sparse Sequence Research Articles

Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences

An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavy-tailed density γ, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried out using the posterior median, this is a random thresholding procedure. Other thresholding rules employing the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n, allowing for sparsity of various kinds and degrees. The signal classes considered are “nearly black” sequences where only a proportion η is allowed to be nonzero, and sequences with normalized ℓp norm bounded by η, for η>0 and 0<p≤2. Estimation error is measured by mean qth power loss, for 0<q≤2. For all the classes considered, and for all q in (0,2], the method achieves the optimal estimation rate as n→∞ and η→0 at various rates, and in this sense adapts automatically to the sparseness or otherwise of the underlying signal. In addition the risk is uniformly bounded over all signals. If the posterior mean is used as the estimator, the results still hold for q>1. Simulations show excellent performance. For appropriately chosen functions γ, the method is computationally tractable and software is available. The extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered.

MRI language dominance assessment in epilepsy patients at 1.0�T: region of interest analysis and comparison with intracarotid amytal testing

The primary goal of this study was to test the reliability of presurgical language lateralization in epilepsy patients with functional magnetic resonance imaging (fMRI) with a 1.0-T MR scanner using a simple word generation paradigm and conventional equipment. In addition, hemispherical fMRI language lateralization analysis and region of interest (ROI) analysis in the frontal and temporo-parietal regions were compared with the intracarotid amytal test (IAT). Twenty epilepsy patients under presurgical evaluation were prospectively examined by both fMRI and IAT. The fMRI experiment consisted of a word chain task (WCT) using the conventional headphone set and a sparse sequence. In 17 of the 20 patients, data were available for comparison between the two procedures. Fifteen of these 17 patients were categorized as left hemispheric dominant, and 2 patients demonstrated bilateral language representation by both fMRI and IAT. The highest reliability for lateralization was obtained using frontal ROI analysis. Hemispherical analysis was less powerful and reliable in all cases but one, while temporo-parietal ROI analysis was unreliable as a stand-alone analysis when compared with IAT. The effect of statistical threshold on language lateralization prompted for the use of t-value-dependent lateralization index plots. This study illustrates that fMRI-determined language lateralization can be performed reliably in a clinical MR setting operating at a low field strength of 1 T without expensive stimulus presentation systems.

Automatic indexing of underwater survey video: algorithm and benchmarking method

It is often the case that only a few sparse sequences of long videos from scientific underwater surveys actually contain important information for the expert. Locating such sequences is time consuming and tedious. A system that automatically detects those critical parts, online or during post-mission tape analysis, would alleviate the expert workload and improve data exploitation. In this paper, a methodology for evaluating the performance of such a system on real data is presented. Interesting sequences are started by changes of visual context. An algorithm to detect significant context changes in benthic videos in real time has been presented by Lebart et al. in 2000. It is used as an illustration for this methodology - its performance is studied and benchmarked on real underwater data, ground truthed by an expert biologist. Various issues relating to the complexity of the problems of automatically analyzing underwater video are also discussed.

Genomic sequencing reveals gene content, genomic organization, and recombination relationships in barley.

Barley (Hordeum vulgare L.) is one of the most important large-genome cereals with extensive genetic resources available in the public sector. Studies of genome organization in barley have been limited primarily to genetic markers and sparse sequence data. Here we report sequence analysis of 417.5 kb DNA from four BAC clones from different genomic locations. Sequences were analyzed with respect to gene content, the arrangement of repetitive sequences and the relationship of gene density to recombination frequencies. Gene densities ranged from 1 gene per 12 kb to 1 gene per 103 kb with an average of 1 gene per 21 kb. In general, genes were organized into islands separated by large blocks of nested retrotransposons. Single genes in apparent isolation were also found. Genes occupied 11% of the total sequence, LTR retrotransposons and other repeated elements accounted for 51.9% and the remaining 37.1% could not be annotated.

Waring's Problem with Polynomial Summands

Let f ( x ) be an integer-valued polynomial with no fixed integer divisor [ges ] 2, that is, for no integer d [ges ] 2 does d [mid ] f ( x ) for all integers x . One generalization of the famous Waring problem is to determine whether, for large enough s , the equation formula here is solvable in positive integers x 1 , …, x s for sufficiently large integers n . The existence of such s for every f was established by Kamke [ 5 ] in 1921. Subsequent authors (Pillai, Hua [ 2–4 ], Vinogradov, Nacaev [ 7 ], and others) have studied the problem of bounding G ( f ), the least s for which (1.1) is solvable for all large n . Questions of local solubility of (1.1), that is, solubility of the congruence formula here play a more important and complicated role in this problem than in the classical Waring problem. Let Γ 0 ( f ) denote the least number s so that (1.2) is solvable for every pair n , q . It is well known that Γ 0 ( x k ) [les ] 4 k for every k , but Hua [ 4 ] found that for every k , the polynomial formula here satisfies Γ 0 ( f k ) [ges ] 2 k − 1 (take s = 2 k −2, q = 2 k and n = (−1) k in (1.2)). Clearly G ( f ) [ges ] Γ 0 ( f ), but one can say more by restricting the values of n under consideration, as has been done by several authors in the case f ( x ) = x 4 (for example [ 1 , 6 ]). The singular series formula here where e ( z ) = e 2π iz , encapsulates the local solubility information. In particular, [Sfr ] s, f ( n ) [ges ] 0 for every n and [Sfr ] s, f ( n ) > 0 if and only if (1.2) is soluble for every q . Define G ( f ) to be the least number s so that for every δ >0 and every n > n 0 (δ) with [Sfr ] s, f ( n ) [ges ] δ, (1.1) is soluble. The reason for taking [Sfr ] s, f ( n ) [ges ] δ instead of [Sfr ] s, f ( n ) > 0 is that we wish to exclude from consideration certain n lying in sparse sequences for which (1.1) is insoluble but [Sfr ] s, f ( n ) > 0. For example, taking f ( x ) = x 4 , s = 15 and n j = 79·16 j ( j = 0, 1, …), it can be shown that (1.1) is not soluble for n = n j , that [Sfr ] s, f ( n j ) > 0 for all j , and that [Sfr ] s, f ( n j ) → 0 as j → ∞. It is known that G ( x 4 ) = 16 (see [ 1 ]) and that G ( x 4 ) [ges ] 11 almost holds (see [ 6 ]).

Bounded and compact multipliers between Bergman and Hardy spaces

This paper studies the boundedness and compactness of the coefficient multiplier operators between various Bergman spacesA p and Hardy spacesH q. Some new characterizations of the multipliers between the spaces with exponents 1 or 2 are derived which, in particular, imply a Bergman space analogue of the Paley-Rudin Theorem on sparse sequences. Hardy and Bergman spaces are shown to be linked using mixed-norm spaces, and this linkage is used to improve a known result on (A p,A 2), 1<p<2.Compact (H 1,H 2) and (A 1,A 2) multipliers are characterized. The essential norms and spectra of some multiplier operators are computed. It is shown that forp>1 there exist bounded non-compact multiplier operators fromA p toA q if and only ifp≤q.

Nonstationary Subdivision Schemes and Multiresolution Analysis

Nonstationary subdivision schemes consist of recursive refinements of an initial sparse sequence with the use of masks that may vary from one scale to the next finer one. This paper is concerned with both the convergence of nonstationary subdivision schemes and the properties of their limit functions. We first establish a general result on the convergence of such schemes to $C^\infty $ compactly supported functions. We show that these limit functions allow us to define a multiresolution analysis that has the property of spectral approximation. Finally, we use these general results to construct $C^\infty $ compactly supported cardinal interpolants and also $C^\infty $ compactly supported orthonormal wavelet bases that constitute Riesz bases for Sobolev spaces of any order.

An adaptive maximum-likelihood deconvolution algorithm

Abstract Kormylo and Mendel proposed a maximum-likelihood deconvolution (MLD) algorithm for estimating a desired sparse spike sequence μ ( k ), modelled as a Bernoulli-Gaussian (BG) sigbal, which was distorted by a linear time-invariant system v ( k ). Then Chi, Mendel and Hampson proposed another MLD algorithm which is a computationally fast MLD algorithm and has been successfully used to process real seismic data. In this paper, we propose an adaptive MLD algorithm, which allows v ( k ) to be a slowly time-varying linear system, for estimating the BG signal μ ( k ) from noisy data. Like the previous MLD algorithms, the proposed adaptive MLD algorithm can also recover the phase of v ( k ) when v ( k ) is time-invariant. Some simulation results are provided to support the proposed algorithm.

The Fractional Fourier Transform and Applications

This paper describes the “fractional Fourier transform,” which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm. Whereas the discrete Fourier transform (DFT) is based on integral roots of unity $e^{{{ - 2\pi i} / n}} $, the fractional Fourier transform is based on fractional roots of unity $e^{ - 2\pi i\alpha } $ where $\alpha $ is arbitrary. The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing DFTs of sparse sequences, analyzing sequences with noninteger periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images, and detecting signals with linearly drifting frequencies. In many cases, the resulting algorithms are faster by arbitrarily large factors than conventional techniques.

On almost symmetric sequences inL p

The case p--1 in Theorem 1 is due to H. P. RosenthaI (unpublished) and the cases p =2, 4, 6 . . . . to W. B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri [91. The notion of almost symmetric sequences comes from functional analysis; we refer to [7] and [9] for applications of almost symmetry and Theorem 1 for the study of the structure of Banach spaces. In [8] S. Guerre and Y. Raynaud showed that for every 1-<p<2 there exists a weakly null sequence in Lp without almost symmetric subsequences. Hence condition (2) in Theorem 1, although it can be considerably weakened (see the Corollary after Theorem 2) cannot be omitted completely. The example in [8] also shows that despite the strong similarity between sparse subsequences of Lp-bounded sequences and exchangeable sequences (see Aldous [1]), not all asymptotic properties of sparse sequences are symmetric. No characterization of sequences (X,) having an almost symmetric subsequence is known.

An interpolation technique for computing the DFT of a sparse sequence

An interpolation technique for the approximate computation of the DFT of a sparse sequence is presented. This new method is more efficient than FFT pruning, particularly for long sequences where the effectiveness of FFT pruning diminishes. This interpolation method for computing the DFT of a sparse sequence is very useful in filter design and spectrum analysis.

Detection of the movements of persons from a sparse sequence of TV images

This paper describes a method for detecting persons walking in a passageway from two consecutive TV images at long intervals. The method consists of the following steps: (1) extracting candidate areas for persons from TV images by image processing; (2) detecting persons in the areas on the basis of their shapes and sizes, and locating them in a passageway; (3) finding the correspondence of persons between two images on the basis of their locations. Special hardware called DIP was employed for high speed image processing.

ALGEBRAIC NUMBER FIELDS WITH LARGE CLASS NUMBER

We prove that almost real quadratic fields of a given type have a large ideal class number. For example, the number of ideal classes of the fields , where is the field of rational numbers, grows unbounded with , as ranges through all natural numbers, except for a very sparse sequence. An analogous fact is established for the fields of Ankeny-Brauer-Chowla, Amer. J. Math. 78 (1965), 51-61.