共查询到20条相似文献,搜索用时 15 毫秒
1.
Adaptive Filtering: a Theoretical and Simulation Study of the Convergence of the Parameter Estimates
Keith D. C. Stoodley 《The Journal of the Operational Research Society》1982,33(12):1077-1087
The existence of bias in the final parameter estimates using adaptive filtering is demonstrated theoretically. For observations generated by an autoregressive model of order one, an approximate theoretical expression for the bias is derived which is valid for long series of observations. The validity of the expression is investigated by simulation and comparing the theoretical bias with the simulated bias at the end of one major iteration. By carrying out a number of major iterations, it is shown that the bias reaches a stable value which is a function of the learning constant. The magnitude of the stable bias may be made as small as desired by taking smaller values for the learning constant. However, as the learning constant decreases, the number of major iterations required to achieve stability increases. By means of simulation experiments, the existence of bias in the final parameter estimates is demonstrated for shorter series of observations generated by an AR(1) model, and for long series of observations generated by an AR(2) model. Again the bias appears to increase with the magnitude of the learning constant. It is argued that the presence of this bias need not be a drawback in the practical application of the method since, by systematic reduction of the training constant between major iterations, the bias may also be reduced, while reasonably rapid convergence can still be maintained. 相似文献
2.
Tsogtgerel Gantumur 《Foundations of Computational Mathematics》2017,17(4):917-956
This paper concerns characterizations of approximation classes associated with adaptive finite element methods with isotropic h-refinements. It is known from the seminal work of Binev, Dahmen, DeVore and Petrushev that such classes are related to Besov spaces. The range of parameters for which the inverse embedding results hold is rather limited, and recently, Gaspoz and Morin have shown, among other things, that this limitation disappears if we replace Besov spaces by suitable approximation spaces associated with finite element approximation from uniformly refined triangulations. We call the latter spaces multievel approximation spaces and argue that these spaces are placed naturally halfway between adaptive approximation classes and Besov spaces, in the sense that it is more natural to relate multilevel approximation spaces with either Besov spaces or adaptive approximation classes, than to go directly from adaptive approximation classes to Besov spaces. In particular, we prove embeddings of multilevel approximation spaces into adaptive approximation classes, complementing the inverse embedding theorems of Gaspoz and Morin. Furthermore, in the present paper, we initiate a theoretical study of adaptive approximation classes that are defined using a modified notion of error, the so-called total error, which is the energy error plus an oscillation term. Such approximation classes have recently been shown to arise naturally in the analysis of adaptive algorithms. We first develop a sufficiently general approximation theory framework to handle such modifications, and then apply the abstract theory to second-order elliptic problems discretized by Lagrange finite elements, resulting in characterizations of modified approximation classes in terms of memberships of the problem solution and data into certain approximation spaces, which are in turn related to Besov spaces. Finally, it should be noted that throughout the paper we paid equal attention to both conforming and non-conforming triangulations. 相似文献
3.
John R. Collins 《Annals of the Institute of Statistical Mathematics》2000,52(2):351-366
Asymptotic biases and variances of M-, L- and R-estimators of a location parameter are compared under ε-contamination of the known error distribution F 0 by an unknown (and possibly asymmetric) distribution. For each ε-contamination neighborhood of F 0, the corresponding M-, L- and R-estimators which are asymptotically efficient at the least informative distribution are compared under asymmetric ε-contamination. Three scale-invariant versions of the M-estimator are studied: (i) one using the interquartile range as a preliminary estimator of scale: (ii) another using the median absolute deviation as a preliminary estimator of scale; and (iii) simultaneous M-estimation of location and scale by Huber's Proposal 2. A question considered for each case is: when are the maximal asymptotic biases and variances under asymmetric ε-contamination attained by unit point mass contamination at ∞? Numerical results for the case of the ε-contaminated normal distribution show that the L-estimators have generally better performance (for small to moderate values of ε) than all three of the scale-invariant M-estimators studied. 相似文献
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Petru A. Cioica Stephan Dahlke Nicolas Döhring Ulrich Friedrich Stefan Kinzel Felix Lindner Thorsten Raasch Klaus Ritter René L. Schilling 《Foundations of Computational Mathematics》2014,14(5):863-912
This paper is concerned with the convergence analysis of the horizontal method of lines for evolution equations of the parabolic type. Following a semidiscretization in time by \(S\) -stage one-step methods, the resulting elliptic stage equations per time step are solved with adaptive space discretization schemes. We investigate how the tolerances in each time step must be tuned in order to preserve the asymptotic temporal convergence order of the time stepping also in the presence of spatial discretization errors. In particular, we discuss the case of linearly implicit time integrators and adaptive wavelet discretizations in space. Using concepts from regularity theory for partial differential equations and from nonlinear approximation theory, we determine an upper bound for the degrees of freedom for the overall scheme that are needed to adaptively approximate the solution up to a prescribed tolerance. 相似文献
6.
As main result we prove that certain means of the partial sums of two-parameter Vilenkin-Fourier series are uniformly bounded operators from H P to L p (0 < p ≦ 1). The Hardy space H p (0 < p ≦ 1) will be defined by means of a diagonal maximal function. As a consequence we obtain a so-called strong convergence theorem for the Vilenkin-Fourier partial sums. Some dual inequalities are also verified for BMO spaces. 相似文献
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We establish particular wavelet-based decompositions of Gaussian stationary processes in continuous time. These decompositions
have a multiscale structure, independent Gaussian random variables in high-frequency terms, and the random coefficients of
low-frequency terms approximating the Gaussian stationary process itself. They can also be viewed as extensions of the earlier
wavelet-based decompositions of Zhang and Walter (IEEE Trans. Signal Process. 42(7):1737–1745, [1994]) for stationary processes, and Meyer et al. (J. Fourier Anal. Appl. 5(5):465–494, [1999]) for fractional Brownian motion. Several examples of Gaussian random processes are considered such as the processes with
rational spectral densities. An application to simulation is presented where an associated Fast Wavelet Transform-like algorithm
plays a key role.
The second author was supported in part by the NSF grant DMS-0505628. 相似文献
9.
Carsten Carstensen 《PAMM》2004,4(1):27-30
State of the art simulations in computational mechanics aim reliability and efficiency via adaptive finite element methods (AFEMs) with a posteriori error control. The a priori convergence of finite element methods is justified by the density property of the sequence of finite element spaces which essentially assumes a quasi‐uniform mesh‐refining. The advantage is guaranteed convergence for a large class of data and solutions; the disadvantage is a global mesh refinement everywhere accompanied by large computational costs. AFEMs automatically refine exclusively wherever the refinement indication suggests to do so and so violate the density property on purpose. Then, the a priori convergence of AFEMs is not guaranteed automatically and, in fact, crucially depends on algorithmic details. The advantage of AFEMs is a more effective mesh accompanied by smaller computational costs in many practical examples; the disadvantage is that the desirable error reduction property is not always guaranteed a priori. Efficient error estimators can justify a numerical approximation a posteriori and so achieve reliability. But it is not clear from the start that the adaptive mesh‐refinement will generate an accurate solution at all. This paper discusses particular versions of an AFEMs and their analyses for error reduction, energy reduction, and convergence results for linear and nonlinear problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
10.
The adaptively compressed exchange (ACE) method provides an efficient way to solve Hartree-Fock-like equations in quantum physics, chemistry, and materials science. The key step of the ACE method is to adaptively compress an operator that is possibly dense and full-rank. In this paper, we present a detailed study of the adaptive compression operation and establish rigorous convergence properties of the adaptive compression method in the context of solving linear eigenvalue problems. Our analysis also elucidates the potential use of the adaptive compression method in a wide range of problems. © 2018 Wiley Periodicals, Inc. 相似文献
12.
Liudas Giraitis Peter M. Robinson Alexander Samarov 《Journal of multivariate analysis》2000,72(2):183
In Giraitis, Robinson, and Samarov (1997), we have shown that the optimal rate for memory parameter estimators in semiparametric long memory models with degree of “local smoothness” β is n−r(β), r(β)=β/(2β+1), and that a log-periodogram regression estimator (a modified Geweke and Porter-Hudak (1983) estimator) with maximum frequency m=m(β)n2r(β) is rate optimal. The question which we address in this paper is what is the best obtainable rate when β is unknown, so that estimators cannot depend on β. We obtain a lower bound for the asymptotic quadratic risk of any such adaptive estimator, which turns out to be larger than the optimal nonadaptive rate n−r(β) by a logarithmic factor. We then consider a modified log-periodogram regression estimator based on tapered data and with a data-dependent maximum frequency m=m(β), which depends on an adaptively chosen estimator β of β, and show, using methods proposed by Lepskii (1990) in another context, that this estimator attains the lower bound up to a logarithmic factor. On one hand, this means that this estimator has nearly optimal rate among all adaptive (free from β) estimators, and, on the other hand, it shows near optimality of our data-dependent choice of the rate of the maximum frequency for the modified log-periodogram regression estimator. The proofs contain results which are also of independent interest: one result shows that data tapering gives a significant improvement in asymptotic properties of covariances of discrete Fourier transforms of long memory time series, while another gives an exponential inequality for the modified log-periodogram regression estimator. 相似文献
13.
An adaptive multi-penalty discontinuous Galerkin method (AMPDG) for
the diffusion problem is considered. Convergence and quasi-optimality of the AMPDG
are proved. Compared with the analyses for the adaptive finite element method
or the adaptive interior penalty discontinuous Galerkin method, extra works is done to overcome the difficulties caused by the additional penalty terms. 相似文献
14.
在一元实函数无穷积分定义的基础上,定义了含参量Fuzzy区间值函数的正常积分和无穷积分,给出了含参量无穷积分一致收敛的定义和判定定理. 相似文献
15.
In this article, we study the robustness of strong stability of the homogeneous difference systems via the concept of strong stability radii: complex, real and positive radii under parameter perturbations. We also show that in the case of positive systems, these radii coincide and can be computed by a simple formula. The results generalize those obtained in [5]. 相似文献
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Computational Mathematics and Mathematical Physics - The paper proposes a new method for choosing a regularization parameter when solving an integral equation of convolution type in problems of... 相似文献
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In this article, the problem on the estimation of the convolution model parameters is considered. The recursive algorithm for estimating model parameters is introduced from the orthogonal procedure of the data, the convergence of this algorithm is theoretically discussed, and a sufficient condition for the convergence criterion of the orthogonal procedure is given. According to this condition, the recursive algorithm is convergent to model wavelet A- = (1, α1,..., αq). 相似文献
20.
In this paper, we propose a discrepancy rule-based method to automatically choose the regularization parameters for total variation image restoration problems. The regularization parameters are adjusted dynamically in each iteration. Numerical results are shown to illustrate the performance of the proposed method. 相似文献