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: A flexible optimization-based framework for intentional islanding is presented that provides islands that are balanced in real and reactive power, satisfy AC power flow laws, and have a healthy voltage profile.
Abstract: In this paper, a flexible optimization-based framework for intentional islanding is presented. The decision is made of which transmission lines to switch in order to split the network while minimizing disruption, the amount of load shed, or grouping coherent generators. The approach uses a piecewise linear model of AC power flow, which allows the voltage and reactive power to be considered directly when designing the islands. Demonstrations on standard test networks show that solution of the problem provides islands that are balanced in real and reactive power, satisfy AC power flow laws, and have a healthy voltage profile.
147 citations
TL;DR: Wang et al. as mentioned in this paper proposed a Regularized Optimal Affine Discriminant (ROAD) algorithm to select an increasing number of features as the regularization relaxes, and further benefits can be achieved when a screening method is employed to narrow the feature pool before hitting the ROAD.
Abstract: For high-dimensional classification, it is well known that naively performing the Fisher discriminant rule leads to poor results due to diverging spectra and noise accumulation. Therefore, researchers proposed independence rules to circumvent the diverging spectra, and sparse independence rules to mitigate the issue of noise accumulation. However, in biological applications, there are often a group of correlated genes responsible for clinical outcomes, and the use of the covariance information can significantly reduce misclassification rates. In theory the extent of such error rate reductions is unveiled by comparing the misclassification rates of the Fisher discriminant rule and the independence rule. To materialize the gain based on finite samples, a Regularized Optimal Affine Discriminant (ROAD) is proposed. ROAD selects an increasing number of features as the regularization relaxes. Further benefits can be achieved when a screening method is employed to narrow the feature pool before hitting the ROAD. An efficient Constrained Coordinate Descent algorithm (CCD) is also developed to solve the associated optimization problems. Sampling properties of oracle type are established. Simulation studies and real data analysis support our theoretical results and demonstrate the advantages of the new classification procedure under a variety of correlation structures. A delicate result on continuous piecewise linear solution path for the ROAD optimization problem at the population level justifies the linear interpolation of the CCD algorithm.
147 citations
TL;DR: Two novel IV methods, a fractional polynomial method and a piecewise linear method, were used to investigate the shape of relationship of body mass index with systolic blood pressure and diastolicBlood pressure.
Abstract: Mendelian randomization, the use of genetic variants as instrumental variables (IV), can test for and estimate the causal effect of an exposure on an outcome. Most IV methods assume that the function relating the exposure to the expected value of the outcome (the exposure-outcome relationship) is linear. However, in practice, this assumption may not hold. Indeed, often the primary question of interest is to assess the shape of this relationship. We present two novel IV methods for investigating the shape of the exposure-outcome relationship: a fractional polynomial method and a piecewise linear method. We divide the population into strata using the exposure distribution, and estimate a causal effect, referred to as a localized average causal effect (LACE), in each stratum of population. The fractional polynomial method performs metaregression on these LACE estimates. The piecewise linear method estimates a continuous piecewise linear function, the gradient of which is the LACE estimate in each stratum. Both methods were demonstrated in a simulation study to estimate the true exposure-outcome relationship well, particularly when the relationship was a fractional polynomial (for the fractional polynomial method) or was piecewise linear (for the piecewise linear method). The methods were used to investigate the shape of relationship of body mass index with systolic blood pressure and diastolic blood pressure.
146 citations
TL;DR: A nonlinear orientation model for the representation of the orientation matrix of a fingerprint image shows substantial decrease in the orientation error and is explored for efficient identity verification based on the ridge orientation.
146 citations
TL;DR: In this article, the problem of computing the dimension of the space of continuous piecewise linear functions over nonsimplicial decompositions of a triangulated d-dimensional region in road was studied.
Abstract: For A a triangulated d-dimensional region in Rd, let Sr (A) denote the vector space of all cr functions F on A that, restricted to any simplex in A, are given by polynomials of degree at most m. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds A in the plane, getting lower bounds on the dimension of Sr (A) for all r. For r = 1, we prove a conjecture of Strang concerning the generic dimension of the space of Cl splines over a triangulated manifold in R2. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.
146 citations