Topic
Piecewise linear function
About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.
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TL;DR: In this article, a piecewise linearized version of the Lorenz system is proposed, which has an independent total amplitude control parameter, and a chaotic circuit with a single amplitude controller is then implemented using a new switch element, producing a chaotic oscillation that agrees with the numerical calculation for the piece-wise linear diffusionless system.
64 citations
TL;DR: A stable nonparametric adaptive control approach using a piecewise local linear approximator with universal approximation capability and analyses of asymptotic stability of the tracking error and parameter convergence for the proposed adaptive control scheme with the on-line self-organizing structure are presented.
Abstract: Presents a stable nonparametric adaptive control approach using a piecewise local linear approximator. The continuous piecewise linear approximator is developed and its universal approximation capability is proved. The controller architecture is based on adaptive feedback linearization plus sliding mode control. A time varying activation region is introduced for efficient self-organization of the approximator during operation. We modify the adaptive control approach for piecewise linear approximation and self-organizing structures. In addition, we provide analyses of asymptotic stability of the tracking error and parameter convergence for the proposed adaptive control scheme with the online self-organizing structure. The method with a deadzone is also discussed to prevent a high-frequency input which might excite the unmodeled dynamics in practical applications. The application of the piecewise linear adaptive control method is demonstrated by a computational simulation.
64 citations
01 Jan 2007
TL;DR: It is proved that for a universal penalty level, the MC+ has high probability of correct selection under much weaker conditions compared with existing results for the LASSO for large n and p, including the case of p ≫ n.
Abstract: We introduce MC+, a fast, continuous, nearly unbiased, and accurate method of penalized variable selection in high-dimensional linear regression. The LASSO is fast and continuous, but biased. The bias of the LASSO interferes with variable selection. Subset selection is unbiased but computationally costly. The MC+ has two elements: a minimax concave penalty (MCP) and a penalized linear unbiased selection (PLUS) algorithm. The MCP provides the minimum non-convexity of the penalized loss given the level of bias. The PLUS computes multiple local minimizers of a possibly non-convex penalized loss function in certain main branch of the graph of such solutions. Its output is a continuous piecewise linear path encompassing from the origin to an optimal solution for zero penalty. We prove that for a universal penalty level, the MC+ has high probability of correct selection under much weaker conditions compared with existing results for the LASSO for large n and p, including the case of p ≫ n. We provide estimates of the noise level for proper choice of the penalty level. We choose the sparsest solution within the PLUS path for a given penalty level. We derive degrees of freedom and Cp-type risk estimates for general penalized LSE, including the LASSO estimator, and prove their unbiasedness. We provide necessary and sufficient conditions for the continuity of the penalized LSE under general sub-square penalties. Simulation results overwhelmingly support our claim of superior variable selection properties and demonstrate the computational efficiency of the proposed method.
63 citations
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TL;DR: In this article, the authors characterize optimal tax parameters for the cases in which budget sets are convex and nonconvex respectively, and a numerical analysis of a discrete version of the model shows the circumstances under which each of these cases will hold as a global optimum.
Abstract: Given its significance in practice, piecewise linear taxation has received relatively little attention in the literature. This paper offers a simple and transparent analysis of its main characteristics. We fully characterize optimal tax parameters for the cases in which budget sets are convex and nonconvex respectively. A numerical analysis of a discrete version of the model shows the circumstances under which each of these cases will hold as a global optimum. We find that, given plausible parameter values and wage distributions, the globally optimal tax system is convex, and marginal rate progressivity increases with rising inequality.
63 citations
TL;DR: An algorithm to minimize a function of several variables with no convexity nor smoothness assumptions is introduced, using an objective function model which is the difference of two piecewise affine convex functions.
Abstract: We introduce an algorithm to minimize a function of several variables with no convexity nor smoothness assumptions. The main peculiarity of our approach is the use of an objective function model which is the difference of two piecewise affine convex functions. Bundling and trust region concepts are embedded into the algorithm. Convergence of the algorithm to a stationary point is proved and some numerical results are reported.
63 citations