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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TL;DR: It is demonstrated that a single, a priori selected spline model recovers a variety of patterns of changes in hazard ratio and fits better than other models, especially when the changes are non-monotonic, as in the case of cancer stages.
Abstract: The authors compare the performance of different regression models for censored survival data in modeling the impact of prognostic factors on all-cause mortality in colon cancer. The data were for 1,951 patients, who were diagnosed in 1977-1991, recorded by the Registry of Digestive Tumors of Cote d'Or, France, and followed for up to 15 years. Models include the Cox proportional hazards model and its three generalizations that allow for hazard ratio to change over time: 1) the piecewise model where hazard ratio is a step function; 2) the model with interaction between a predictor and a parametric function of time; and 3) the non-parametric regression spline model. Results illustrate the importance of accounting for non-proportionality of hazards, and some advantages of flexible non-parametric modeling of time-dependent effects. The authors provide empirical evidence for the dependence of the results of piecewise and parametric models on arbitrary a priori choices, regarding the number of time intervals and specific parametric function, which may lead to biased estimates and low statistical power. The authors demonstrate that a single, a priori selected spline model recovers a variety of patterns of changes in hazard ratio and fits better than other models, especially when the changes are non-monotonic, as in the case of cancer stages.
102 citations
TL;DR: This paper investigates the neural networks with a class of nondecreasing piecewise linear activation functions with 2r corner points with the proposed model of n-neuron dynamical systems, which can have and only have (2r+1)(n) equilibria under some conditions, which are locally exponentially stable and others are unstable.
102 citations
TL;DR: In this paper, a nonlinear calculus of variations problem on time scales with variable endpoints is considered, and the space of functions employed is that of piecewise rd-continuously Δ-differentiable functions (C1prd).
102 citations
TL;DR: Under mild conditions, the Pareto front of a continuous m-objective optimization problem forms an (m - 1)-dimensional piecewise continuous manifold and a self-organizing multiobjective evolutionary algorithm is proposed based on this property.
Abstract: Under mild conditions, the Pareto front (Pareto set) of a continuous $\boldsymbol m$ -objective optimization problem forms an ( $\boldsymbol {m-1}$ )-dimensional piecewise continuous manifold. Based on this property, this paper proposes a self-organizing multiobjective evolutionary algorithm. At each generation, a self-organizing mapping method with ( $\boldsymbol {m-1}$ ) latent variables is applied to establish the neighborhood relationship among current solutions. A solution is only allowed to mate with its neighboring solutions to generate a new solution. To reduce the computational overhead, the self-organizing training step and the evolution step are conducted in an alternative manner. In other words, the self-organizing training is performed only one single step at each generation. The proposed algorithm has been applied to a number of test instances and compared with some state-of-the-art multiobjective evolutionary methods. The results have demonstrated its advantages over other approaches.
102 citations
TL;DR: In this article, the collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous, and a satisfactory sense of point evaluation is given for elements of the integral equation.
Abstract: The collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous. Difficulties with the usual function space setting of $L_\infty (D)$ are discussed, and a satisfactory sense of point evaluation is given for elements of $L_\infty (D)$. Other approaches which are discussed include (i) reformulation as a degenerate kernel method, (ii) the prolongation-restriction framework of Noble, (iii) other function space settings, and (iv) reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation.
102 citations