Mixed‐phase deconvolution

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 637-647 ◽  
Author(s):  
Milton J. Porsani ◽  
Bjorn Ursin
Keyword(s):  
Unit Circle ◽  
Amplitude Spectrum ◽  
Real Data ◽  
Coefficient Matrix ◽  
Mixed Phase ◽  
Time Signal ◽  
Lp Norm ◽  
Optimal Value ◽  
Minimum Delay ◽  
Delay Component

We describe a new algorithm for mixed‐phase deconvolution. It is valid only for pulses whose Z-transform has no zeros on the unit circle. That is, the amplitude spectrum cannot be zero for any frequency. Using the Z-transform of a discrete‐time signal, and assuming that the signal has α zeros inside the unit circle, the inverse of its minimum‐delay component may be estimated by solving the extended Yule‐Walker (EYW) system of equations with the lag α of the autocorrelation function (ACF) on diagonal of the coefficient matrix. This property of the solution of EYW equations is exploited to derive mixed‐phase inverse filters and their corresponding mixed‐phase pulses. For different values of α, a suite of inverse filters is generated using the same ACF. To choose the best decomposition and its corresponding mixed‐phase inverse filter, we have used the value of α which gives the maximum value of the Lp norm of the filtered signal. The optimal value of α does not seem to be very sensitive to the choice of norm as long as p > 2. In the numerical examples, we have used p = 5. The mixed‐phase deconvolution filter performs better than minimum‐phase deconvolution on the synthetic and real data examples shown.

scholarly journals Sparsity-Constrained NMF Algorithm Based on Evolution Strategy for Hyperspectral Unmixing

2018 ◽  
Vol 232 ◽  
pp. 04019
Author(s):  
ShangBin Ning ◽  
FengChao Zuo
Keyword(s):  
Optimization Technique ◽  
Differential Evolution Algorithm ◽  
Real Data ◽  
Coefficient Matrix ◽  
Selection Strategy ◽  
Blind Separation ◽  
Basis Matrix ◽  
Hyperspectral Unmixing ◽  
Update Rules ◽  
Multiplicative Update

As a powerful and explainable blind separation tool, non-negative matrix factorization (NMF) is attracting increasing attention in Hyperspectral Unmixing(HU). By effectively utilizing the sparsity priori of data, sparsity-constrained NMF has become a representative method to improve the precision of unmixing. However, the optimization technique based on simple multiplicative update rules makes its unmixing results easy to fall into local minimum and lack of robustness. To solve these problems, this paper proposes a new hybrid algorithm for sparsity constrained NMF by intergrating evolutionary computing and multiplicative update rules (MURs). To find the superior solution in each iteration,the proposed algorithm effectively combines the MURs based on alternate optimization technique, the coefficient matrix selection strategy with sparsity measure, as well as the global optimization technique for basis matrix via the differential evolution algorithm .The effectiveness of the proposed method is demonstrated via the experimental results on real data and comparison with representative algorithms.

scholarly journals Multiple Haplotype Reconstruction from Allele Frequency Data

2020 ◽  
Author(s):  
Marta Pelizzola ◽  
Merle Behr ◽  
Housen Li ◽  
Axel Munk ◽  
Andreas Futschik
Keyword(s):  
Allele Frequency ◽  
Real Data ◽  
Coefficient Matrix ◽  
Point Of View ◽  
Design Matrix ◽  
Frequency Data ◽  
Computationally Efficient ◽  
Regression Problem ◽  
Allele Frequency Data ◽  
Multiple Samples

AbstractSince haplotype information is of widespread interest in biomedical applications, effort has been put into their reconstruction. Here, we propose a new, computationally efficient method, called haploSep, that is able to accurately infer major haplotypes and their frequencies just from multiple samples of allele frequency data. Our approach seems to be the first that is able to estimate more than one haplotype given such data. Even the accuracy of experimentally obtained allele frequencies can be improved by re-estimating them from our reconstructed haplotypes. From a methodological point of view, we model our problem as a multivariate regression problem where both the design matrix and the coefficient matrix are unknown. The design matrix, with 0/1 entries, models haplotypes and the columns of the coefficient matrix represent the frequencies of haplotypes, which are non-negative and sum up to one. We illustrate our method on simulated and real data focusing on experimental evolution and microbial data.

Sensitivity of ℓ1 minimization to parameter choice

2020 ◽  
Author(s):  
Aaron Berk ◽  
Yaniv Plan ◽  
Özgür Yilmaz
Keyword(s):  
Real Data ◽  
Optimal Choice ◽  
High Dimensional ◽  
Parameter Choice ◽  
Optimal Error ◽  
Governing Parameter ◽  
ℓ1 Minimization ◽  
Optimal Value ◽  
Generalized Lasso ◽  
Common Technique

Abstract The use of generalized Lasso is a common technique for recovery of structured high-dimensional signals. There are three common formulations of generalized Lasso; each program has a governing parameter whose optimal value depends on properties of the data. At this optimal value, compressed sensing theory explains why Lasso programs recover structured high-dimensional signals with minimax order-optimal error. Unfortunately in practice, the optimal choice is generally unknown and must be estimated. Thus, we investigate stability of each of the three Lasso programs with respect to its governing parameter. Our goal is to aid the practitioner in answering the following question: given real data, which Lasso program should be used? We take a step towards answering this by analysing the case where the measurement matrix is identity (the so-called proximal denoising setup) and we use $\ell _{1}$ regularization. For each Lasso program, we specify settings in which that program is provably unstable with respect to its governing parameter. We support our analysis with detailed numerical simulations. For example, there are settings where a 0.1% underestimate of a Lasso parameter can increase the error significantly and a 50% underestimate can cause the error to increase by a factor of $10^{9}$.

Matrix Perturbation and Optimal Partition Invariancy in Linear Optimization

2015 ◽  
Vol 32 (03) ◽  
pp. 1550013 ◽  
Author(s):  
Alireza Ghaffari-Hadigheh ◽  
Nayyer Mehanfar
Keyword(s):  
Linear Optimization ◽  
Optimal Solution ◽  
Singular Values ◽  
Coefficient Matrix ◽  
Optimal Partition ◽  
Matrix Perturbation ◽  
Optimal Value ◽  
Special Perturbation ◽  
Transition Points ◽  
The Given

Understanding the effect of variation of the coefficient matrix in linear optimization problem on the optimal solution and the optimal value function has its own importance in practice. However, most of the published results are on the effect of this variation when the current optimal solution is a basic one. There is only a study of the problem for special perturbation on the coefficient matrix, when the given optimal solution is strictly complementary and the optimal partition (in some sense) is known. Here, we consider an arbitrary direction for perturbation of the coefficient matrix and present an effective method based on generalized inverse and singular values to detect invariancy intervals and corresponding transition points.

Nonlinear impedance inversion for attenuating media

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. R111-R117 ◽  
Author(s):  
Sérgio Oliveira ◽  
Luiz Loures ◽  
Fernando Moraes ◽  
Carlos Theodoro
Keyword(s):  
Amplitude Spectrum ◽  
Real Data ◽  
Alternative Formulation ◽  
Nonlinear Inversion ◽  
Inversion Algorithm ◽  
Impedance Inversion ◽  
Reflectivity Function ◽  
Analytical Formulas ◽  
Zero Offset ◽  
Internal Multiples

Applications of seismic impedance inversion normally assume the data are free of multiples and transmission effects, requiring knowledge of the seismic pulse that is assumed to be stationary. An alternative formulation for impedance inversion is based on an exact frequency-domain, zero-offset reflectivity function for a 1D medium. Analytical formulas for the Fréchet derivatives are derived for efficient implementation of an iterative nonlinear inversion. The exact zero-offset reflectivity accounts for internal multiples and transmission effects in the data. Absorption and dispersion are also conveniently handled if a reasonable estimate for the quality [Formula: see text] factor of the medium is available. A series of convenient features are included in the inversion algorithm: an automatic estimation of the amplitude spectrum of the seismic pulse, an impedance transform that makes the inversion independent from the initial smooth model, and a practical approach to estimate the regularization weight. Numerical tests using synthetic and real data show that the method is stable and needs only a few iterations to converge.

scholarly journals Optimal forecasting accuracy using Lp-norm combination

METRON ◽  
2021 ◽  
Author(s):  
Massimiliano Giacalone
Keyword(s):  
Real Data ◽  
Optimization Techniques ◽  
Combination Method ◽  
Forecast Combination ◽  
Forecasting Accuracy ◽  
Lp Norm ◽  
Combination Methods ◽  
Historical Series ◽  
Non Gaussian ◽  
Standard Regression

AbstractA well-known result in statistics is that a linear combination of two-point forecasts has a smaller Mean Square Error (MSE) than the two competing forecasts themselves (Bates and Granger in J Oper Res Soc 20(4):451–468, 1969). The only case in which no improvements are possible is when one of the single forecasts is already the optimal one in terms of MSE. The kinds of combination methods are various, ranging from the simple average (SA) to more robust methods such as the one based on median or Trimmed Average (TA) or Least Absolute Deviations or optimization techniques (Stock and Watson in J Forecast 23(6):405–430, 2004). Standard regression-based combination approaches may fail to get a realistic result if the forecasts show high collinearity in several situations or the data distribution is not Gaussian. Therefore, we propose a forecast combination method based on Lp-norm estimators. These estimators are based on the Generalized Error Distribution, which is a generalization of the Gaussian distribution, and they can be used to solve the cases of multicollinearity and non-Gaussianity. In order to demonstrate the potential of Lp-norms, we conducted a simulated and an empirical study, comparing its performance with other standard-regression combination approaches. We carried out the simulation study with different values of the autoregressive parameter, by alternating heteroskedasticity and homoskedasticity. On the other hand, the real data application is based on the daily Bitfinex historical series of bitcoins (2014–2020) and the 25 historical series relating to companies included in the Dow Jonson, were subsequently considered. We showed that, by combining different GARCH and the ARIMA models, assuming both Gaussian and non-Gaussian distributions, the Lp-norm scheme improves the forecasting accuracy with respect to other regression-based combination procedures.

scholarly journals A Feasible Approach to Determine the Optimal Relaxation \\[6pt]Parameters in Each Iteration for the SOR Method

2021 ◽  
Vol 13 (1) ◽  
pp. 1
Author(s):  
Chein-Shan Liu
Keyword(s):  
Large Scale ◽  
Coefficient Matrix ◽  
Merit Function ◽  
Relaxation Parameter ◽  
Projection Technique ◽  
Sor Method ◽  
Optimal Value ◽  
Alternative Choice ◽  
Successive Over Relaxation ◽  
Previous Step

The paper presents a dynamic and feasible approach to the successive over-relaxation (SOR) method for solving large scale linear system through iteration. Based on the maximal orthogonal projection technique, the optimal relaxation parameter is obtained by minimizing a derived merit function in terms of right-hand side vector, the coefficient matrix and the previous step values of unknown variables. At each iterative step, we can quickly determine the optimal relaxation value in a preferred interval. When the theoretical optimal value is hard to be achieved, the new method provides an alternative choice of the relaxation parameter at each iteration. Numerical examples confirm that the dynamic optimal successive over-relaxation (DOSOR) method outperforms the classical SOR method.

SEISMIC WAVE PROPAGATION IN LAYERED MEDIA IN TERMS OF COMMUNICATION THEORY

Geophysics ◽  
1966 ◽  
Vol 31 (1) ◽  
pp. 17-32 ◽  
Author(s):  
S. Treitel ◽  
E. A. Robinson
Keyword(s):  
Impulse Response ◽  
Communication Theory ◽  
Amplitude Spectrum ◽  
P Wave ◽  
Earth Model ◽  
Phase Lag ◽  
Elastic Plates ◽  
Pass System ◽  
Maximum Delay ◽  
Minimum Delay

The problem of a normally incident plane P wave propagating in a system of horizontally layered homogeneous perfectly elastic plates is reformulated in terms of concepts drawn from communication theory. We show how both the reflected and transmitted responses of such a system can be expressed as a z transform which is the ratio of two polynomials in z. Since this response must be stable, the denominators of the z transforms describing the reflected and transmitted motion are minimum delay (i.e., minimum‐phase lag). If the layered medium is bounded at depth by a perfect reflector, then the reflected impulse response recorded at the surface is in the form of a dispersive all‐pass z transform. A dispersive all‐pass system is one whose z transform is the ratio of the z transform of a maximum‐delay wavelet to that of its corresponding minimum‐delay wavelet; hence, the amplitude spectrum of a dispersive all‐pass system is unity for all frequencies. This means that the amplitude spectrum of the reflected response is identical to the amplitude spectrum of the input wavelet used to excite the system. More specifically, all the energy put in is returned with the same frequency content, but is differentially delayed. The phase‐lag spectrum of the reflected response lies everywhere above the phase‐lag spectrum of the input wavelet. Thus, the all‐pass situation implies that the layered earth model considered here, while not able to alter the amplitude of the frequency components of the input wavelet, will introduce differential time delays with certain properties into each such component. Finally, since the reflected impulse response is an all‐pass wavelet, its autocorrelation is a spike of unit magnitude at τ=0, and zero for all other lags.

Could the processed seismic wavelet be simpler than we think?

Geophysics ◽  
1991 ◽  
Vol 56 (5) ◽  
pp. 681-690 ◽  
Author(s):  
N. S. Neidell
Keyword(s):  
Unit Circle ◽  
Frequency Distribution ◽  
Amplitude Spectrum ◽  
Root Structure ◽  
Symmetric Form ◽  
Central Frequency ◽  
Seismic Wavelet ◽  
Phase Property ◽  
Short Piece ◽  
Zero Phase

J. P. Lindsey, (1988) in a clearly written short piece, opens an old question which concerns the analytic properties of seismic wavelets. This well conceived study concludes that most of the roots of a seismic wavelet as expressed by its z transform representation lie on or are very near the unit circle. The present discussion does not seek to characterize the form of all seismic wavelets, but only many if not most of those which have been processed with deconvolutions or “inversion” type operators to have reduced length, broadened bandwidth, and some desirable phase property. For such wavelets, despite the diversity by which they are obtained, remarkably simple operations having very few parameters can be extremely effective. As a case in point, constant‐phase rotations appear to carry such wavelets to zero‐phase symmetric form to a very good approximation. I start with empirical attributes which appear to characterize most processed seismic wavelets. Such wavelets tend to be of 40–100 ms duration with a smooth and unimodal amplitude spectrum of “peak” or “central” frequency between 15 and 30 Hz. The amplitude spectrum itself is further largely concentrated at frequencies between 5 and 55 Hz. A z transform root structure having essentially all of its roots only on the unit circle and on the real axis seems able to characterize all of the observed attributes rather well. This structure will be termed the band‐limiting root approximation (BLRA) and describes the attributes I seek to explain which are not as readily understood from alternative descriptions of the wavelets. Since the class of wavelets we address is obtained by a variety of means, and because the differences in character are at best subtle according to interpretive criteria, my justification is heuristic. The BLRA wavelet structure can be represented with remarkably few parameters (typically fewer than five). Of these few parameters, two relate to the frequency distribution. Such a formalism should be exceptionally useful for designing seismic techniques which seek to extract interpretive information based on properties of the wavelet.

scholarly journals Hyperspectral Image Denoising Based on Nonconvex Low-Rank Tensor Approximation and lp Norm Regularization

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Li Bo ◽  
Luo Xuegang ◽  
Lv Junrui
Keyword(s):  
Image Denoising ◽  
Hyperspectral Image ◽  
Structural Information ◽  
Real Data ◽  
Optimization Method ◽  
Nuclear Norm ◽  
Low Rank ◽  
Phase Congruency ◽  
Structure Information ◽  
Lp Norm

A new nonconvex smooth rank approximation model is proposed to deal with HSI mixed noise in this paper. The low-rank matrix with Laplace function regularization is used to approximate the nuclear norm, and its performance is superior to the nuclear norm regularization. A new phase congruency lp norm model is proposed to constrain the spatial structure information of hyperspectral images, to solve the phenomenon of “artificial artifact” in the process of hyperspectral image denoising. This model not only makes use of the low-rank characteristic of the hyperspectral image accurately, but also combines the structural information of all bands and the local information of the neighborhood, and then based on the Alternating Direction Method of Multipliers (ADMM), an optimization method for solving the model is proposed. The results of simulation and real data experiments show that the proposed method is more effective than the competcing state-of-the-art denoising methods.

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