Blind Identification of Underdetermined Mixtures with Complex Sources Using the Generalized Generating Function

Blind identification of under-determined mixtures based on the characteristic function

In this paper, we propose a novel algorithm for underdetermined blind identification problem in blind source separation. The proposed algorithm is based on the charrelation matrix of observed signals. The charrelation matrix not only can be considered as a generalized second-order covariance matrix, but also incorporates higher-order information. It is significant for blind separation problem based on statistic characteristics to extract statistical information. The problem of underdetermined blind identification is converted as a tensor decomposition model. The mixing matrix can be estimated from the rank-1 terms of the tensor decomposition. Theoretical analysis and simulation results illustrate the proposed algorithm performs competitive estimation performance compared to the underdetermined blind identification algorithm based on second-order covariance and fourth-order cumulant.

The modal identification framework known as Blind Modal Identification (BMID) has recently been developed, drawing on techniques from Blind Source Separation (BSS). Therein, a BSS algorithm known as Second Order Blind Identification (SOBI) was adapted to solve the Modal IDentification (MID) problem. One of the drawbacks of the technique is that the number of modes identified must be less than the number of sensors used to measure the vibration of the equipment or structure. In this paper, an extension of the BMID method is presented for the underdetermined case, where the number of sensors is less than the number of modes to be identified. The analytic signal formed from measured vibration data is formed and the Second Order Blind Identification of Underdetermined Mixtures (SOBIUM) algorithm is applied to estimate the complex-valued modes and modal response autocorrelation functions. The natural frequencies and modal damping ratios are then estimated from the corresponding modal auto spectral density functions using a simple Single Degree Of Freedom (SDOF), frequency-domain method. Theoretical limitations on the number of modes identified given the number of sensors are provided. The method is demonstrated using a simulated six DOF mass-spring-dashpot system excited by white noise, where displacement at four of the six DOF is measured. All six modes are successfully identified using data from only four sensors. The method is also applied to a more realistic simulation of ambient building vibration. Seven modes in the bandwidth of interest are successfully identified using acceleration data from only five DOF. In both examples, the identified modal parameters (natural frequencies, mode shapes, modal damping ratios) are compared to the analytical parameters and are demonstrated to be of good quality.

Algorithm ALESCAF (alternating least squares identification based on the characteristic function) uses the derivatives of the second characteristic function (c.f.) of observations, without any need of sparsity assumption on sources, but assuming their statistical independence. ALESCAF was already proposed by the authors in P. Comon and M. Rajih (2005), where only one derivative order was considered. In this paper, new versions of ALESCAF are proposed, that jointly use derivatives of different orders. We also propose ALESCAS, a new algorithm that uses the knowledge of source c.f.'s. Computer simulations demonstrate that both algorithms accelerate the convergence

A Khatri–Rao subspace approach to blind identification of mixtures of quasi-stationary sources

Analysis of fractional order Bonhoeffer–van der Pol oscillator

In this contribution, we analyze the memory effect associated with fractional derivatives in a Cournot-type oligopoly model. Such an analysis will be made from the comparison of the solutions of a duopoly with fractional derivatives of different orders. We show that the firm that has more memory on the competition, given by the fractional derivative, will have advantages over its competitor in the long run. Moreover, we show that the choices of fractional derivatives αj ∈ (0,3/5) for both firms j = 1,2 will result in advantages in profit compared to the profit of firms without memory.

<p>Two inverse source problems for a space-time fractional differential equation involving bi-fractional Laplacian operators in the spatial variable and Caputo time-fractional derivatives of different orders between 1 and 2 are studied. In the first inverse source problem, the space-dependent term along with the diffusion concentration is recovered, while in the second inverse source problem, the time-dependent term along with the diffusion concentration is identified. Both inverse source problems are ill-posed in the sense of Hadamard. The existence and uniqueness of solutions for both inverse source problems are investigated. Finally, several examples are presented to illustrate the obtained results for the inverse source problems.</p>

This paper presents a novel space-time feature-based human activity analysis system. We detect Space Time Interest Points (STIP) and generate their description based on the facet model. The proposed approach detects interest points in video data using the three-dimensional facet model efficiently. Then we describe each interest point by three-dimensional Haar wavelet transform and time derivatives of different order obtained from said facet model. Here we represent each video clip following the bag-of-words approach by learning feature specific dictionary. Finally, classification is done using non-linear SVM with χ 2 -kernel. We evaluate the performance of our system on standard datasets like Weizmann, KTH, UCF sports, ICD, UCF YouTube, and UCF50 and get better, or at least comparable results compared to other state-of-the-art systems.

In this article the existence as well as the uniqueness (EU) of the solutions for nonlinear multiorder fractional-differential equations (FDE) with local boundary conditions and fractional derivatives of different orders (Caputo and Riemann–Liouville) are covered. The existence result is derived from Krasnoselskii’s fixed point theorem and its uniqueness is shown using the Banach contraction mapping principle. To illustrate the reliability of the results, two examples are given.

In this paper, we investigate the existence of solutions for a nonlinear fractional-order coupled system involving both Caputo and Riemann–Liouville generalized fractional derivatives of different orders equipped with coupled integral boundary conditions. We transform the given system into an equivalent fixed point problem and solve it by applying the standard fixed point theorems. Examples are constructed for the illustration of the obtained results.

Flicker-Noise Spectroscopy (FNS), a general approach to the extraction and parameterization of resonant and stochastic components contained in medical time series, is presented. The basic idea of FNS is to treat the correlation links present in sequences of different irregularities, such as spikes, jumps, and discontinuities in derivatives of different orders, on all levels of the spatiotemporal hierarchy of the system under study as main information carriers. The tools to extract and analyze the information are power spectra and difference moments (structural functions), which complement the information of each other. The structural function stochastic component is formed exclusively by jumps of the dynamic variable while the power spectrum stochastic component is formed by both spikes and jumps on every level of the hierarchy. The information passport characteristics that are determined by fitting the derived expressions to the experimental variations for the stochastic components of power spectra and structural functions are interpreted as the correlation times and parameters that describe the rate of memory loss on these correlation time intervals for different irregularities. The number of the extracted parameters is determined by the requirements of the problem under study. Application of this approach to the analysis of tremor velocity signals for a Parkinsonian patient is discussed.

In addition to the well-known and widely-used adaptive strategies for region subdivision and the choice of local quadrature rules, there still exists a third possibility for doing numerical integration adaptively, when interval computation is considered. Based on the Peano's kernels theorem and interval arithmetic, we are able to estimate the truncation error of a quadrature rule by means of different derivatives of the integrand, where the available orders of the derivatives depend on the degree of smoothness of the integrand and the exactness degree of the underlying quadrature rule. We classify the methods as the adaptive orders strategies, if they make use of the derivatives of different orders to improve each single local error estimation. In this paper, alternatives for adaptive error estimation are discussed. Moreover, a practical way for realization of the optimal error estimation is suggested. Numerical results for integrands of different classes as well as numerical comparisons of different methods are given.

We obtain a characterization of local Besov spaces of periodic functions in terms of trigonometric polynomial operators. We construct a sequence of operators whose values are (global) trigonometric polynomials, and yet their behavior at different points reflects the behavior of the target function near each of these points. In addition to being localized, our operators preserve trigonometric polynomials of degree commensurate with the degree of polynomials given by the operators. Our constructions are “universal;” i.e., they do not require an a priori knowledge about the smoothness of the target functions. Several numerical examples are discussed, including applications to the problem of direction finding in phased array antennas and finding the location of jump discontinuities of derivatives of different order.

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. The study of existence and uniqueness of solution of fractional partial differential equations are then proved by mani methods as: the well-known Lax–Milgram theorem, Energy estimate and fixed point theorem... Among them, In this work we investigate the existence and uniqueness of solutions for a new class of tripled system, which containing of mixed fractional derivatives between the Riemann-Liouville and the Caputo fractional derivatives of different orders with Laplacian operator, by using Leray-Schauder alternative fixed point theorem type in generalized Banach space and Banach contraction principle. The stability of the Ulam type of the proposed coupled system is also studied. Due to the importance of variable-order fractional calculus, both from a theoretical and an applied perspective, we think that the results obtained can be of interest to the research pursuits of many readers. At the end, to conclude this paper we provide an example illustrating our findings and approach.