Convergence analysis of a simple minor component analysis algorithm

Principal component analysis (PCA) and minor component analysis (MCA) are two important statistical tools which have many applications in the fields of signal processing and data analysis. PCA and MCA neural networks (NNs) can be used to online extract principal component and minor component from input data. It is interesting to develop generalized learning algorithms of PCA and MCA NNs. Some novel generalized PCA and MCA learning algorithms are proposed in this paper. Convergence of PCA and MCA learning algorithms is an essential issue in practical applications. Traditionally, the convergence is studied via deterministic continuous-time (DCT) method. The DCT method requires the learning rate of the algorithms to approach to zero, which is not realistic in many practical applications. In this paper, deterministic discrete-time (DDT) method is used to study the dynamical behaviors of the proposed algorithms. The DDT method is more reasonable for the convergence analysis since it does not require constraints as that of the DCT method. It is proven that under some mild conditions, the weight vector in these proposed algorithms will converge exponentially to principal or minor component. Simulation results are further used to illustrate the theoretical results.

Extracting multiple minor components from the in- put signal is quite useful for many practical applications. In this paper, a globally convergent MCA algorithm that can extract multiple minor components sequentially is proposed. Conver- gence of this MCA algorithm is analyzed via the deterministic discrete time (DDT) method. Sufficient conditions are obtained to guarantee the convergence of this MCA algorithm. Simula- tions are carried out to further illustrate the theoretical results achieved.

A unified learning algorithm to extract principal and minor components

This brief deals with the problem of minor component analysis (MCA). Artificial neural networks can be exploited to achieve the task of MCA. Recent research works show that convergence of neural networks based MCA algorithms can be guaranteed if the learning rates are less than certain thresholds. However, the computation of these thresholds needs information about the eigenvalues of the autocorrelation matrix of data set, which is unavailable in online extraction of minor component from input data stream. In this correspondence, we introduce an adaptive learning rate into the OJAn MCA algorithm, such that its convergence condition does not depend on any unobtainable information, and can be easily satisfied in practical applications.

The minor component analysis is widely used in many fields, such as signal processing and data analysis, so it has very important theoretical significance and practical values for the convergence analysis of these algorithms. In this paper we seek the convergence condition for Feng’s MCA learning algorithm in deterministic discrete time system. Finally numerical experiments show the correctness of our theory.KeywordsMCA learning algorithmDDT systemeigenvalueeigenvectorinvariant sets

Algorithms that extract the principal or minor components of a signal are widely used in signal processing and control applications. This paper explores new frameworks for generating learning rules for iteratively computing the principal and minor components (or subspaces) of a given matrix. Stability analysis using Lyapunov theory and La Salle invariance principle is provided to determine regions of attraction of these learning rules. Among many derivations, it is specifically shown that Oja's rule and many variations of it are asymptotically globally stable. Lyapunov stability theory is also applied to weighted learning rules. Some of the essential features for the proposed MCA/PCA learning rules are that they are self normalized and can be applied to non-symmetric matrices. Exact solutions for some nonlinear dynamical systems are also provided

Edible oil analysis by high-resolution nuclear magnetic resonance spectroscopy: recent advances and future perspectives

X-ray fluorescence analysis is applied in the cement industry to quantitative analysis of major and minor components of raw materials and of cement. The elements Mg, Al, Si, Ca, and Fe may be determined directly in finely ground powders. Better accuracies are obtained, however, when the samples are first decomposed with sodium borate or lithium borate. Linear calibration curves universally applicable to all raw materials are desired, and occasionally certain correction factors are applied in order to obtain a higher accuracy. With recent advances in the efficiencies of commercially available apparatus for the analysis of light elements such as Na, Mg, and Al, it is now possible to determine these elements with the same high accuracy. Minor components such as Sr, Mn, Ti, P, and S can also be determined with comparative ease. In silicate and rock analyses, x-ray fluorescence is applied to the determination of the major components Na2O, MgO, A12O3, SiO2, P2O5, K2O, CaO, TiO2, MnO, and Fe2O3. Depending upon compositional variations of the rocks, the finely ground powders may either be analyzed directly or they are first decomposed with sodium borate or lithium borate, thus eliminating mineralogical and textural variations. Occasionally, addition of an absorbing component such as La2Os makes possible analysis of rocks of considerable variation in composition with uniformly linear calibration curves.

A self-stabilizing MSA algorithm in high-dimension data stream

Ion chromatography: A powerful and versatile analytical technique for environmental chemical characterization

The objective of the present work was the geographical discrimination of the most common honeydew honeys produced in Greece, namely pine and fir, on the basis of sugar, free amino acid and organic acid content, determined by nuclear magnetic resonance spectroscopy (1H NMR) and high-performance liquid chromatography (HPLC), along with moisture content, sugar ratios, or sugars to moisture ratio, using chemometrics. For this purpose, 39 pine and 31 fir honey samples were collected from professional beekeepers from eight different regions in Greece. Data were subjected to multivariate analysis and modeled using supervised statistical methods. The combination of 1H NMR and HPLC based on metabolites along with the aforementioned physicochemical data resulted in the geographical discrimination of pine and fir honeys. Respective prediction rates were 76.9 and 80.6%, using a model validation technique: the cross-validation method. Present results support the combined use of instrumental and conventional methods for honey geographical origin differentiation.

Computation of invariant sets via immersion for discrete-time nonlinear systems

Master the basic concepts and methodologies of digital signal processing with this systematic introduction, without the need for an extensive mathematical background. The authors lead the reader through the fundamental mathematical principles underlying the operation of key signal processing techniques, providing simple arguments and cases rather than detailed general proofs. Coverage of practical implementation, discussion of the limitations of particular methods and plentiful MATLAB illustrations allow readers to better connect theory and practice. A focus on algorithms that are of theoretical importance or useful in real-world applications ensures that students cover material relevant to engineering practice, and equips students and practitioners alike with the basic principles necessary to apply DSP techniques to a variety of applications. Chapters include worked examples, problems and computer experiments, helping students to absorb the material they have just read. Lecture slides for all figures and solutions to the numerous problems are available to instructors.

In the development of adaptive systems in control theory and signal processing, it frequently occurs that the problem of orthonormalization must be addressed. This paper explored the underlying mathematical framework of developing orthonormalization methods that are free of computing matrix square roots. These algorithms are easily modified so that minor and principal component analysis methods are developed. The proposed methods have several important features: 1) higher order convergence can be achieved by choosing a specific stepsize, 2) the methods can be used to compute square root of positive definite matrices.

Since the pioneering work of Amari and Oja, principal component neural networks and their extensions have become an active adaptive signal processing research field. One of such extensions is minor component analysis (MCA), that proves to be effective in tasks such as robust curve/surface fitting and noise reduction. The aims of the paper are to give a detailed and homogeneous review of one-unit first minor/principal component analysis and to propose an application to robust constrained beamforming. In particular, after a careful presentation of first/minor component analysis algorithms based on a single adaptive neuron, along with relevant convergence/steady-state theorems, it is shown how the adaptive robust constrained beamforming theory by Cox et al. may be advantageously recast into an MCA setting. Experimental results obtained with a triangular array of microphones introduced in a teleconference context help to assess the usefulness of the proposed theory.