Adjacent Extreme Points Research Articles

Nonlinear vibration feature extraction based on power spectrum envelope adaptive empirical Fourier decomposition

Analyzing the vibration features of a shipboard stabilized platform (SSP) is significant to the design or vibration compensation of a marine gravimetric survey vibration isolation system. Empirical Fourier decomposition (EFD) is a recently developed method for nonlinear and non-stationary signal decomposition. However, the spectral segmentation boundary needs to be set in advance according to sophisticated experience, and it is easy to be disturbed by noise, and the decomposition result is inaccurate. In order to accurately extract the nonlinear vibration characteristics of SSP, this paper proposes a new method called power spectrum envelope adaptive empirical Fourier decomposition (PSEEFD). Firstly, the number of selected modal decomposition is determined based on the mutual information to realize adaptive segmentation. Then, the improved power spectrum envelope segmentation method is adopted to effectively diminish the interference of noise since the segmentation boundary is formed by the minimum of the adjacent extreme points enveloped by the maximum value of the power spectrum. The spectrum segments obtained from segmentation contain less interference. Finally, the component signal in each frequency band is reconstructed by inverse fast Fourier transform, and the instantaneous frequency signal component with physical significance is obtained. Through the analysis of vibration simulation signals and measured data of SSP, the proposed method is compared with EMD, AFVMD, EWT and EFD. The results show that PSEEFD has a well suppression of noise interference and can effectively extract the characteristics of nonlinear vibration signals.

Spectral envelope-based adaptive empirical Fourier decomposition method and its application to rolling bearing fault diagnosis

Adaptive empirical Fourier decomposition (AEFD) is a recently developed approach of nonstationary signal mode separation. However, it requires to set the spectrum segmentation boundary relying on the users’ professional experience ahead of time. In this paper, a novel spectral envelope-based adaptive empirical Fourier decomposition (SEAEFD) method is proposed to improve the performance of AEFD for rolling bearing vibration signal analysis. In the proposed SEAEFD approach, fast Fourier transform (FFT) of the raw signal is calculated to obtain the frequency spectrum at first. Then, the spectral envelope processing is implemented on the spectrum signal obtained by FFT to achieve an adaptive segmentation. In the traditional segmentation method, generally, the minima and midpoints between adjacent extreme points are taken as the spectrum segmentation boundary, in which the obtained frequency band contains more interference components. To achieve the effect of denoising and restrain the noise that existed in the collected vibration signal, SEAEFD is proposed to optimize the spectrum segmentation boundary so that the obtained frequency band contains the least noise components. Lastly, the inverse FFT is used to reconstruct the component signal within each frequency band and the gained signals are termed as Fourier intrinsic mode functions (FIMFs). Therefore, SEAEFD enables a nonstationary signal to be decomposed into several single-component signals with instantaneous frequencies of physical significance. The proposed SEAEFD method is compared with recently developed methods, including EAEFD, AEFD, EWT, VMD and EMD methods, by analyzing the simulation signals and the measured data of rolling bearing. The results indicate that SEAEFD is valid in diagnosing rolling bearing faults and gets a better diagnosis performance than the compared methods.

A Novel Median-Point Mode Decomposition Algorithm for Motor Rolling Bearing Fault Recognition

Precise fault recognition of motor rolling bearing fault is playing a significant role in any machinery and equipment. However, conventional decomposition methods fail to completely reveal the fault signal information of motor rolling bearing due to mixed modes problem. To solve the problem, the median-point mode decomposition (MMD) method is presented. The MMD method uses sort-based inversion to sort out each variation of the same time interval for better and specific mode decomposition, with the assistance of the advanced envelope curve formed by the median points between adjacent extreme points. It certainly alleviates the mixed mode during the iteration of intrinsic mode functions (IMFs). Therefore, comparison results are simulated in the proposed MMD method with conventional methods. Experiment of motor rolling bearing fault is operated for fault recognition in order to demonstrate the MMD algorithm.

Extreme point bias compensation: A similarity method of functional clustering and its application to the stock market

Functional clustering is based on functional similarity measures that are adapted to functional data. However, the existing functional similarity measures account either for the position (value) or temporal deviation (bias) of extreme points of the functional curves. This may lead to erroneous conclusions on the similarities of the curves. In this case, most functional clustering measures underperform in (for example) the analysis of stock market data. To address this methodological limitation, a new similarity measure that is based on extreme point bias compensation is proposed in this paper. By penalizing the curves with the temporal deviation of extreme points and rewarding the curves that are close to each other, the new similarity measure better reflects the shape of the curve. In addition, the proposed method overcomes the difficulty of unifying the dimensions of the horizontal and vertical axes (i.e., time and function value) when calculating the distance between two adjacent extreme points. Finally, an empirical example of stock return analysis verifies the validity of this new measure.

Adjacency based method for generating maximal efficient faces in multiobjective linear programming

Multiobjective linear optimization problems (MOLPs) arise when several linear objective functions have to be optimized over a convex polyhedron. In this paper, we propose a new method for generating the entire efficient set for MOLPs in the outcome space. This method is based on the concept of adjacencies between efficient extreme points. It uses a local exploration approach to generate simultaneously efficient extreme points and maximal efficient faces. We therefore define an efficient face as the combination of adjacent efficient extreme points that define its border. We propose to use an iterative simplex pivoting algorithm to find adjacent efficient extreme points. Concurrently, maximal efficient faces are generated by testing relative interior points. The proposed method is constructive such that each extreme point, while searching for incident faces, can transmit some local informations to its adjacent efficient extreme points in order to complete the faces’ construction. The performance of our method is reported and the computational results based on randomly generated MOLPs are discussed.

Analysis of heart rate variability during breath‐holding using fast and continuous calculation of frequency and power

AbstractMany reports show that the dominant frequency of the high‐frequency component (HF: 0.15 to 0.4 Hz) of heart rate time series is synchronized with the respiratory frequency. In this paper, we propose a method for estimating the condition of respiration continuously by using the dominant frequency and power of the HF component (HFP) of the heart rate time series. The dominant frequency and HFP are calculated from the interval between two adjacent extreme points and subtraction of the two neighboring extremals. In the experimental results, high‐frequency components did not disappear completely during breath‐holding. This result differs from results obtained in previous studies. The subjects were divided into two groups. In one group, the dominant frequency of the HF was significantly increased during breath‐holding compared with normal breathing. In the other group, this phenomenon was not observed. On the other hand, the HFP of all subjects decreased significantly during breath‐holding compared with normal breathing. The correct rate during breath‐holding and the error rate during rest and recovery were calculated using the HFP. In the results, the mean and standard deviation of the correct rate during breath‐holding were 65.0 ± 26.3%. The correct rate for 18 subjects was 80.0 ± 14.1% and the correct rate for the other 8 subjects was 31.5 ± 11.9%. Our method is expected to be applicable to the development of respiratory monitors which can measure respiratory conditions continuously in a nonrestrictive manner. © 2011 Wiley Periodicals, Inc. Electron Comm Jpn, 94(6): 24–31, 2011; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/ecj.10324

Extremist vs. centrist decision behavior: quasi-convex utility functions for interactive multi-objective linear programming problems

This paper presents the fundamental theory and algorithms for identifying the most preferred alternative for a decision maker (DM) having a non-centrist (or extremist) preferential behavior. The DM is requested to respond to a set of questions in the form of paired comparison of alternatives. The approach is different than other methods that consider the centrist preferential behavior. In this paper, an interactive approach is presented to solve the multiple objective linear programming (MOLP) problem. The DM's underlying preferential function is represented by a quasi-convex value (utility) function, which is to be maximized. The method presented in this paper solves MOLP problems with quasi-convex value (utility) functions by using paired comparison of alternatives in the objective space. From the mathematical point of view, maximizing a quasi-convex (or a convex) function over a convex set is considered a difficult problem to solve, while solutions for quasi-concave (or concave) functions are currently available. We prove that our proposed approach converges to the most preferred alternative. We demonstrate that the most preferred alternative is an extreme point of the MOLP problem, and we develop an interactive method that guarantees obtaining the global most preferred alternative for the MOLP problem. This method requires only a finite number of pivoting operations using a simplex-based method, and it asks only a limited number of paired comparison questions of alternatives in the objective space. We develop a branch and bound algorithm that extends a tree of solutions at each iteration until the MOLP problem is solved. At each iteration, the decision maker has to identify the most preferred alternatives from a given subset of efficient alternatives that are adjacent extreme points to the current basis. Through the branch and bound algorithm, without asking many questions from the decision maker, all branches of the tree are implicitly enumerated until the most preferred alternative is obtained. An example is provided to show the details of the algorithm. Some computational experiments are also presented. Scope and purpose This paper presents the fundamental theory, algorithm, and examples for identifying the most preferred alternative (solution) for a decision maker (DM) having a non-centrist (or extremist) preferential behavior for Multiple Objective Linear Programming (MOLP) problems. The DM is requested to respond to a set of questions in the form of paired comparison of alternatives. Although widely applied, Linear Programming is limited to a single objective function. In many real world situations, DMs are faced with multiple objective problems in that several competing and conflicting objectives have to be considered. For these problems, there exist many alternatives that are feasible and acceptable. However, the DM is interested in finding “the most preferred alternative”. In the past three decades, many methods have been developed for solving MOLP problems. One class of these methods is called “interactive”, in which the DM responds to a set of questions interactively so that his/her most preferred alternative can be obtained. In most of these methods, the value (utility) function (that presents the DM's preference) is assumed to be linear or additive, concave, pseudo-concave, or quasi-concave. However, for MOLP problems, there has not been any effort to recognize and solve the quasi-convex utility functions, which are among the most difficult class of problems to solve. The quasi-convex class of utility functions represents an extremist preferential behavior, while the other aforementioned methods (such as quasi-concave) represent a conservative behavioral preference. It is shown that the method converges to the optimal (the most preferred) alternative. The approach is computationally feasible for moderately sized problems.

Adjacent Extreme Points Of A Transportation Polytope

The adjacent extreme points of a Transportation Polytope are characterized by the circuit s contained in the support of its vertices. This result generalizes the Balinski and Russakoff [4] result on the Assignment Polytope

The polytope of degree sequences

A nonnegative integer sequence ( d 1, d 2,…, d n ) is called a degree sequence if there exists a simple graph on the vertex set V= {1,2,…, n} such that deg( i)= d i for all i. The degree sequence of a threshold graph is a threshold sequence. Let D n = Convex Hull {( x 1, x 2,…, x n )|( x 1,…, x n ) is a degree sequence}. It is proved that: (1) A degree sequence f is an extreme point of D n if and only if f is a threshold sequence. (2) Two threshold sequences f and g are adjacent extreme points of D n if and only if f can be obtained from g by either adding 1 to two components of g or subtracting 1 from two components of g. (3) D n is determined bythe following system of inequalities: ∑ i∈ x i− ∑ i∈T x i⩽|S|(n−1−|T|) for all sets S, T with ⊘ ≠ S ∪ T ⊆ {1,2,…,n}, S ∩T = ⊘. Moreover, this system is totally dual integral. Furthermore, for n ⩾ 4, ( S, T) determines a facet of D n if and only if either ⋎S ∪ T| = 1 or else S ≠ ⊘,T ≠ ⊘, |S ∪T| ≠n - 1,n - 2. (4) f is a threshold sequence if and only if the only degree sequences majorizing f in the sense of Hardy, Littlewood, and Pólya are the rearrangements of f. Consequently, every degree sequence is a convex combination of isomorphic threshold sequences (i.e., threshold sequences that are rearrangements of each other).

An algorithm for bicriterion linear programming

An algorithm is presented here for bicriterion linear programs which generates either all efficient points or a subset of such efficient points corresponding to a decision-maker's specified space of objective weights. The computational requirements of the algorithm are quite low; in fact only a series of divisions and comparisons are needed for the determination of adjacent efficient extreme points. As a by-product, the range of objective weights corresponding to each efficient extreme point is also generated. This additional information is used to characterize the set of all efficient points as a union of maximal efficient faces.

Some inverse problems involving conditional expectations

Let (Ω, F, P) be a probability space, let H be a sub-σ-algebra of F, and let Y be positive and H-measurable with E[ Y] = 1. We discuss the structure of the convex set CE( Y; H) = { X ∈ p F: Y = E[ X| H]} of random variables whose conditional expectation given H is the prescribed Y. Several characterizations of extreme points of CE( Y; H) are obtained. A necessary and sufficient condition is given in order that CE( Y; H) be the closed, convex hull of its extreme points. For the case of finite F we explicitly calculate the extreme points of CE( Y; H), identify pairs of adjacent extreme points, and characterize extreme points of CE( Y; H) ⌢ CE( Z; G), where G is a second sub-σ-algebra of F and Z ∈ p G. When H = σ( Y) and appropriate topological hypotheses hold, extreme points of CE( Y; H) are shown to be in explicit one-to-one correspondence with certain left inverses of Y. Finally, it is shown how the same approach can be applied to the problem of extremal random measures on R + with a prescribed compensator, to deduce that the number of extreme points is zero or one.

A Heuristic Adjacent Extreme Point Algorithm for the Fixed Charge Problem

An algorithm with three variations is presented for the approximate solution of fixed charge problems. Computational experience shows it to be extremely fast and to yield very good solutions. The basic approach is (1) to obtain a local optimum by using the simplex method with a modification of the rule for selection of the variable to enter the basic solution, and (2) once at a local optimum, to search for a better extreme point by jumping over adjacent extreme points to resume iterating two or three extreme points away. This basic approach is the same as that used by Steinberg [Steinberg, D. I. 1970. The fixed charge problem. Naval Res. Log. Quart. 17 217–236.], Cooper [Cooper, L. 1975. The fixed charge problem—I: A new heuristic method. Comp. & Maths, with Appls. 1 89–95.], and Denzler [Denzler, D. R. 1969. An approximate algorithm for the fixed charge problem. Naval Res. Log. Quart. 16 411–416.] in their algorithms, but is an extension and improvement of all three. The algorithm is being used by the U.S Environmental Protection Agency's Office of Solid Waste Management Programs to decide on the number, type, size, and location of the disposal facilities to operate in a region, and how to allocate the region's wastes to these facilities.