Topic
Piecewise
About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.
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01 Jan 2004
TL;DR: In this paper, the authors developed spatial adaptive estimates for restoring functions from noisy observations and showed that the traditional least square (piecewise polynomial) estimate equipped with adaptively adjusted window possesses simultaneously many attractive adaptive properties, namely, 1) it is near-optimal within lnn-factor for estimating a function (or its derivative) at a single point; 2) its quality is close to that one which could be achieved if smoothness of the underlying function was known in advance.
Abstract: The paper is devoted to developing spatial adaptive estimates for restoring functions from noisy observations. We show that the traditional least square (piecewise polynomial) estimate equipped with adaptively adjusted window possesses simultaneously many attractive adaptive properties, namely, 1) it is near– optimal within lnn–factor for estimating a function (or its derivative) at a single point; 2) it is spatial adaptive in the sense that its quality is close to that one which could be achieved if smoothness of the underlying function was known in advance; 3) it is optimal in order (in the case of “strong” accuracy measure) or near–optimal within lnn–factor (in the case of “weak” accuracy measure) for estimating whole function (or its derivative) over wide range of the classes and global loss functions. We demonstrate that the “spatial adaptive abilities” of our estimate are, in a sense, the best possible. Besides this, our adaptive estimate is computationally efficient and demonstrates reasonable practical behavior.
153 citations
TL;DR: In this article, an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based density estimators is provided, which is available for densities which are smooth in only a piecewise sense.
Abstract: We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based density estimators. We show that, unlike the analogous situation for kernel density estimators, this MISE formula is relatively unaffected by assumptions of continuity. In particular, it is available for densities which are smooth in only a piecewise sense. Another difference is that in the wavelet case the classical MISE formula is valid only for sufficiently small values of the bandwidth. For larger bandwidths MISE assumes a very different form and hardly varies at all with changing bandwidth. This remarkable property guarantees a high level of robustness against oversmoothing, not encountered in the context of kernel methods. We also use the MISE formula to describe an asymptotically optimal empirical bandwidth selection rule.
152 citations
TL;DR: Results for a set of water network design problems show that the new algorithm can lead to orders of magnitude reduction in the optimality gap compared to commercial solvers.
152 citations
TL;DR: It is shown that the two formulations without additional binary variables have the same LP bounds as those of the corresponding formulations with binary variables and therefore are preferable for efficient computation.
152 citations
TL;DR: In this article, the application of the method of fundamental solutions to the Cauchy problem associated with two-dimensional Helmholtz-type equations is investigated, where the resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularized by employing the first-order Tikhonov functional, while the choice of the regularization parameter is based on the L-curve method.
152 citations