Set Of Random Vectors Research Articles

Finite-horizon Anisotropy-based Estimation with Packet Dropouts

In this paper, the estimation problem is studied for a class of linear discrete time-varying system with packet dropout in the framework of anisotropy-based theory. The extended vector of fragment of the disturbance sequence is from the set of random vectors with bounded anisotropy. The packet dropout effect is considered to be random and described by a binary switching sequence with Bernoulli distribution. The input-to-error dynamics is obtained for multiplicative noise system with mutually independent noises and input disturbance. By using anisotropy-based approach, the estimation problem is reduced to optimization one with convex constraints. The developed method provides the (sub)optimal estimator ensuring the boundedness of anisotropic norm for input-to-output error system. Numerical example is provided to demonstrate efficiency of proposed approach.

Randomized estimation of spectral densities of large matrices made accurate

For a large Hermitian matrix $A\in \mathbb{C}^{N\times N}$, it is often the case that the only affordable operation is matrix-vector multiplication. In such case, randomized method is a powerful way to estimate the spectral density (or density of states) of $A$. However, randomized methods developed so far for estimating spectral densities only extract information from different random vectors independently, and the accuracy is therefore inherently limited to $\mathcal{O}(1/\sqrt{N_{v}})$ where $N_{v}$ is the number of random vectors. In this paper we demonstrate that the "$\mathcal{O}(1/\sqrt{N_{v}})$ barrier" can be overcome by taking advantage of the correlated information of random vectors when properly filtered by polynomials of $A$. Our method uses the fact that the estimation of the spectral density essentially requires the computation of the trace of a series of matrix functions that are numerically low rank. By repeatedly applying $A$ to the same set of random vectors and taking different linear combination of the results, we can sweep through the entire spectrum of $A$ by building such low rank decomposition at different parts of the spectrum. Under some assumptions, we demonstrate that a robust and efficient implementation of such spectrum sweeping method can compute the spectral density accurately with $\mathcal{O}(N^2)$ computational cost and $\mathcal{O}(N)$ memory cost. Numerical results indicate that the new method can significantly outperform existing randomized methods in terms of accuracy. As an application, we demonstrate a way to accurately compute a trace of a smooth matrix function, by carefully balancing the smoothness of the integrand and the regularized density of states using a deconvolution procedure.

Coding ECG beats using multiscale compressed sensing based processing

Compressed sensing recovers a sparse signal from a small set of linear, nonadaptive measurements. A sparse signal can be represented by compressed measurements with a reduced number of projections on a set of random vectors. In this paper, a multiscale compressed sensing based processing is investigated for an electrocardiogram signal which yields coded measurements. In case of an electrocardiogram (ECG) signal, the coded measurements are expected to retain the clinical information. To achieve this, compressed sensing based processing is applied at each wavelet scale and measurements are coded using Huffman coder. The measurements at each scale use random sensing matrix with independent identically distributed (i.i.d.) entries formed by sampling a Gaussian distribution. The proposed method is evaluated using pathological ECG signals from the CSE database, synthetic and normal ECGs. It helps preserve the pathological information and clinical components in compressed signal. The compressed signal quality is evaluated using standard distortion measures and mean opinion score (MOS). The MOS values for the signals range from 5% to 8.3% with a wavelet energy based diagnostic distortion (WEDD) value of 9.46% which falls under the excellent category.

Random Projections of Residuals for Digital Image Steganalysis

The traditional way to represent digital images for feature based steganalysis is to compute a noise residual from the image using a pixel predictor and then form the feature as a sample joint probability distribution of neighboring quantized residual samples-the so - called co-occurrence matrix. In this paper, we propose an alternative statistical representation - instead of forming the co-occurrence matrix, we project neighboring residual samples onto a set of random vectors and take the first-order statistic (histogram) of the projections as the feature. When multiple residuals are used, this representation is called the projection spatial rich model (PSRM). On selected modern steganographic algorithms embedding in the spatial, JPEG, and side-informed JPEG domains, we demonstrate that the PSRM can achieve a more accurate detection as well as a substantially improved performance versus dimensionality trade-off than state-of-the-art feature sets.

Meta-heuristic approach to proportional fairness

Proportional fairness is a concept from resource sharing tasks among n users, where each user receives at least 1/n of her or his total value of the infinitely divisible resource. Here we provide an approach to proportional fairness that allows its extension to discrete domains, as well as for the direct application of evolutionary computation to approximate proportional fair states. We employ the concept of relational optimization, where the optimization task becomes the finding of extreme elements of a binary relation, and define a proportional fairness relation correspondingly. By using a rank-ordered version of proportional fairness, the so-called ordered proportional fairness, we can improve the active finding of maximal proportional fair elements by evolutionary meta-heuristic algorithms. This is demonstrated by using modified versions of the strength pareto evolutionary algorithm (version 2, SPEA2) and multi-objective particle swarm optimization. In comparison between proportional and ordered proportional fairness, and by using relational SPEA2, the evolved maximum sets of ordered proportional fairness achieve 10 % more dominance cases against a set of random vectors than proportional fairness.

Multivariate Association and Dimension Reduction: A Generalization of Canonical Correlation Analysis

In this article, we propose a new generalized index to recover relationships between two sets of random vectors by finding the vector projections that minimize an L(2) distance between each projected vector and an unknown function of the other. The unknown functions are estimated using the Nadaraya-Watson smoother. Extensions to multiple sets and groups of multiple sets are also discussed, and a bootstrap procedure is developed to detect the number of significant relationships. All the proposed methods are assessed through extensive simulations and real data analyses. In particular, for environmental data from Los Angeles County, we apply our multiple-set methodology to study relationships between mortality, weather, and pollutants vectors. Here, we detect existence of both linear and nonlinear relationships between the dimension-reduced vectors, which are then used to build nonlinear time-series regression models for the dimension-reduced mortality vector. These findings also illustrate potential use of our method in many other applications. A comprehensive assessment of our methodologies along with their theoretical properties are given in a Web Appendix.

Random Pixel Purity Index

Endmember extraction has received increasing interest in hyperspectral image analysis. One widely used endmember extraction algorithm is pixel purity index (PPI), which finds endmembers via a set of random vectors, called skewers. Several issues arise in its implementation. One is the prior knowledge of the number of skewers <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> required to be used. Second, due to random nature in skewers, the final results are inconsistent and unreproducible. Third, it needs to know the number of dimensions to be retained after dimensionality reduction. Fourth, it needs to preset a cutoff threshold to extract potential endmembers. Finally, it involves human intervention to manually select final endmembers. This letter derives a random PPI (RPPI) to resolve the aforementioned issues. It considers the result produced by PPI using a random set of initial vectors as skewers as a realization of a random algorithm. From a statistical signal processing view point, if endmembers are crucial in terms of information, they should occur in realizations produced by PPI regardless of what set is chosen for skewers. By virtue of this assumption, the proposed RPPI is developed and validated by experiments.

Accurate Estimation of ICA Weight Matrix by Implicit Constraint Imposition Using Lie Group

This paper presents a new stochastic algorithm to optimize the independence criterion-mutual information-among multivariate data using local, global, and hybrid optimizers, in conjunction with techniques involving a Lie group and its corresponding Lie algebra, for implicit imposition of the orthonormality constraint among the estimated sources. The major advantage of the proposed algorithm is the increased accuracy with which the weight matrix in the independent component analysis (ICA) model is estimated, compared to conventional schemes. When the local optimizer with Lie group techniques and the fast fixed-point (fastICA) algorithm were experimented by inputting the same set of random vectors, the former method superseded the conventional one by producing accurate weight matrix estimates in a majority of the test cases. Importantly, in our approach, the use of a Lie group to "lock" the weight matrix estimates onto the constraint surface enabled easy realization of the hybrid optimizers to yield near-global-optimum solutions consistently in most of the test cases, compared to well-known global optimizers. The inherent computational overhead in the hybrid optimizers was lowered by preprocessing the input data and periodically integrating the local optimizers with the global one. The proposed algorithms were applied to six-dimensional multispectral satellite image data to emphasize their usefulness in terms of accurate ICA weight matrix estimation.

An Informational Measure of Association and Dimension Reduction for Multiple Sets and Groups With Applications in Morphometric Analysis

In this article we propose a new general index that measures relationships between multiple sets of random vectors. This index is based on Kullback–Leibler (KL) information, which measures linear or nonlinear dependence between multiple sets using joint and marginal densities of affine transformations of the random vectors. Estimates of the matrixes are obtained by maximizing the KL information and are shown to be consistent. The motivation for introducing such an index comes from morphological integration studies, a topic in biological science. As a special case of this index, we define an overall measure of association and two other measures for dimension reduction. The use of these measures is illustrated through real data analysis in morphometric studies and extensive simulations, and their performance is compared with that of approaches based on canonical correlation analysis. Extensions of the aforementioned measures to multiple groups are also discussed. In contrast to canonical correlation analysis, our general index not only provides an overall measure of association, but also determines nonlinear relationships, thereby making it useful in many other applications.

Discounting and Risk Neutrality

Consider a binary relation on a real vector space of vector-valued discrete-time stochastic processes. If the binary relation satisfies four axioms, then there are unique discount factors such that preferences regarding stochastic processes induce prefences among present value random vectors. Three of the axioms are familiar: weak ordering, continuity, and non-triviality. The fourth axiom is decomposition. Also, if preferences satisfy the four axioms then the following properties are equivalent: the converse of decomposition, the existence of a felicity function on the set of random vectors, and risk neutrality. In this limited sense, discounting implies risk neutrality.

Analysis and Measurement of Fault Coverage in a Combined ATE and BIST Environment

This paper analyzes an environment which utilizes built-in self-test (BIST) and automatic test equipment (ATE), and presents closed-form expressions for fault coverage as a function of the number of BIST and ATE test vectors. This requires incorporating the time to switch from BIST to ATE (referred to as switchover time), and utilizing ATE generated vectors to finally achieve the desired level of fault coverage. For this environment, we model fault coverage as a function of the testability of the circuit under test and the numbers of vectors which are supplied by the BIST circuitry and the ATE. A novel approach is proposed; this approach is initially based on fault simulation using a small set of random vectors; an estimate of the so-called detection profile of the circuit under test is established as the basis of the test model. This analytical model effectively relates the testable features of the circuit under test to detection using both BIST and ATE as related testing processes.

On samples of independent random vectors in spaces of infinitely increasing dimension

We study the asymptotic behavior of a set of random vectors ξi, i = 1,..., m, whose coordinates are independent and identically distributed in a space of infinitely increasing dimension. We investigate the asymptotics of the distribution of the random vectors, the consistency of the sets Mm(n) = ξ1,..., ξm and Xnλ = x ∈ Xn: ρ(x) ≤ λn, and the mutual location of pairs of vectors.

Normality of a non-linear transformation of AR parameters: Application to reflection and cepstrum coefficients

Two sets of random vectors cannot both be Gaussian if they are nonlinearly related. Thus, autoregressive (AR) parameters and reflection coefficient (resp. cepstrum coefficient) estimators cannot both be Gaussian for a finite number of samples. However, most estimators of AR parameters and reflection coefficients (resp. cepstrum coefficients) are Gaussian asymptotically. Thus, the distribution of AR parameter and reflection coefficient (resp. cepstrum coefficient) estimates are close to Gaussian for large samples. This paper studies the ‘closeness’ between the Gaussian distribution and the ‘non-linear transformation of Gaussian AR parameters’ distribution. A new distance is defined which is based on the Taylor expansion of the non-linear transformation. This ‘Taylor’ distance called M-distance is used to measure the deviations from the Gaussian distribution of reflection coefficient and cepstrum coefficient statistics. A comparison is presented between this distance and Kullback's divergence. The main advantage of the M-distance with respect to other distances is a very simple closed-form expression of the deviations from normality. This closed-form expression shows that the convergence of the reflection and cepstrum coefficient distribution to its asymptotic Gaussian distribution (when the number of samples tends to infinity) depends on the position of AR model poles in the unit circle.

The receptor-like neural network for modeling corticosteroid and testosterone binding globulins.

A neural-net method for simulation of corticosteroid and testosterone binding globulin (CBG, TBG)-ligand interactions is presented. Molecular modeling provides the geometry and partial atomic charges of 31 steroid molecules. The atomic coordinates within the molecule of the compound of the highest affinity are then used to train a self-organizing map (SOM) that forms a template for the comparison to other molecules. Comparison is done using a series of normalized patterns produced by the SOM. The template SOM, after overlaying on the set of random vectors, mimics the topology of the receptor site and is used to train unsupervisedly a neuron capable of recognizing the degree of similarity between the reference and tested patterns. A good correlation is observed for signals generated by the neuron plotted against the experimental CBG affinities. For TBG affinity modeling a modified procedure is designed which is capable of separating electrostatic and shape effects. The high predictive power of the model is achieved by keeping close analogy to the processes taking place at the real receptor sites.