Functional setting

Keyword(s):  
Biometrika ◽  
2021 ◽  
Author(s):  
Juhyun Park ◽  
Jeongyoun Ahn ◽  
Yongho Jeon

Abstract Functional linear discriminant analysis offers a simple yet efficient method for classification, with the possibility of achieving a perfect classification. Several methods are proposed in the literature that mostly address the dimensionality of the problem. On the other hand, there is a growing interest in interpretability of the analysis, which favors a simple and sparse solution. In this work, we propose a new approach that incorporates a type of sparsity that identifies nonzero sub-domains in the functional setting, offering a solution that is easier to interpret without compromising performance. With the need to embed additional constraints in the solution, we reformulate the functional linear discriminant analysis as a regularization problem with an appropriate penalty. Inspired by the success of ℓ1-type regularization at inducing zero coefficients for scalar variables, we develop a new regularization method for functional linear discriminant analysis that incorporates an L1-type penalty, ∫ |f|, to induce zero regions. We demonstrate that our formulation has a well-defined solution that contains zero regions, achieving a functional sparsity in the sense of domain selection. In addition, the misclassification probability of the regularized solution is shown to converge to the Bayes error if the data are Gaussian. Our method does not presume that the underlying function has zero regions in the domain, but produces a sparse estimator that consistently estimates the true function whether or not the latter is sparse. Numerical comparisons with existing methods demonstrate this property in finite samples with both simulated and real data examples.


1996 ◽  
Vol 3 (37) ◽  
Author(s):  
Gerth Stølting Brodal ◽  
Chris Okasaki

Brodal recently introduced the first implementation of imperative priority queues to support findMin, insert, and meld in O(1) worst-case time, and deleteMin in O(log n) worst-case time. These bounds are asymptotically optimal among all comparison-based priority queues. In this paper, we adapt<br />Brodal's data structure to a purely functional setting. In doing so, we both simplify the data structure and clarify its relationship to the binomial queues of Vuillemin, which support all four operations in O(log n) time. Specifically, we derive our implementation from binomial queues in three steps: first, we reduce the running time of insert to O(1) by eliminating the possibility of cascading links; second, we reduce the running time of findMin to O(1) by adding a global root to hold the minimum element; and finally, we reduce the running time of meld to O(1) by allowing priority queues to contain other<br />priority queues. Each of these steps is expressed using ML-style functors. The last transformation, known as data-structural bootstrapping, is an interesting<br />application of higher-order functors and recursive structures.


PeerJ ◽  
2016 ◽  
Vol 4 ◽  
pp. e1978 ◽  
Author(s):  
Gabriela Fontanarrosa ◽  
Virginia Abdala

Grasping is one of a few adaptive mechanisms that, in conjunction with clinging, hooking, arm swinging, adhering, and flying, allowed for incursion into the arboreal eco-space. Little research has been done that addresses grasping as an enhanced manual ability in non-mammalian tetrapods, with the exception of studies comparing the anatomy of muscle and tendon structure. Previous studies showed that grasping abilities allow exploitation for narrow branch habitats and that this adaptation has clear osteological consequences. The objective of this work is to ascertain the existence of morphometric descriptors in the hand skeleton of lizards related to grasping functionality. A morphological matrix was constructed using 51 morphometric variables in 278 specimens, from 24 genera and 13 families of Squamata. To reduce the dimensions of the dataset and to organize the original variables into a simpler system, three PCAs (Principal Component Analyses) were performed using the subsets of (1) carpal variables, (2) metacarpal variables, and (3) phalanges variables. The variables that demonstrated the most significant contributions to the construction of the PCA synthetic variables were then used in subsequent analyses. To explore which morphological variables better explain the variations in the functional setting, we ranGeneralized Linear Modelsfor the three different sets. This method allows us to model the morphology that enables a particular functional trait. Grasping was considered the only response variable, taking the value of 0 or 1, while the original variables retained by the PCAs were considered predictor variables. Our analyses yielded six variables associated with grasping abilities: two belong to the carpal bones, two belong to the metacarpals and two belong to the phalanges. Grasping in lizards can be performed with hands exhibiting at least two different independently originated combinations of bones. The first is a combination of a highly elongated centrale bone, reduced palmar sesamoid, divergence angles above 90°, and slender metacarpal V and phalanges, such as exhibited byAnolissp. andTropidurussp. The second includes an elongated centrale bone, lack of a palmar sesamoid, divergence angles above 90°, and narrow metacarpal V and phalanges, as exhibited by geckos. Our data suggest that the morphological distinction between graspers and non-graspers is demonstrating the existence of ranges along the morphological continuum within which a new ability is generated. Our results support the hypothesis of the nested origin of grasping abilities within arboreality. Thus, the manifestation of grasping abilities as a response to locomotive selective pressure in the context of narrow-branch eco-spaces could also enable other grasping-dependent biological roles, such as prey handling.


2016 ◽  
Vol 2016 ◽  
pp. 1-32 ◽  
Author(s):  
Massimo Tessarotto ◽  
Claudio Cremaschini

A discussion of the functional setting customarily adopted in General Relativity (GR) is proposed. This is based on the introduction of the notion of nonlocal point transformations (NLPTs). While allowing the extension of the traditional concept of GR-reference frame, NLPTs are important because they permit the explicit determination of the map between intrinsically different and generally curved space-times expressed in arbitrary coordinate systems. For this purpose in the paper the mathematical foundations of NLPT-theory are laid down and basic physical implications are considered. In particular, explicit applications of the theory are proposed, which concern(1)a solution to the so-called Einstein teleparallel problem in the framework of NLPT-theory;(2)the determination of the tensor transformation laws holding for the acceleration 4-tensor with respect to the group of NLPTs and the identification of NLPT-acceleration effects, namely, the relationship established via general NLPT between particle 4-acceleration tensors existing in different curved space-times;(3)the construction of the nonlocal transformation law connecting different diagonal metric tensors solution to the Einstein field equations; and(4)the diagonalization of nondiagonal metric tensors.


2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Christian Lamour

Abstract Mass media can represent and help to recompose European spaces. The aim of the current article is to ascertain whether the journalistic representation of space within a European cross-border region is related to the economy driven functional integration favored by the EU’s new regionalism policies. Based on a content analysis of two interconnected newspapers located in the trans-frontier area centered around the Luxembourg economy, the objective is to explore the spatial arrangements orienting the mediatization of cross-border regionalization. The results show that new regionalism—meaning the definition of a de-bordered and urban regional unit based on economic competitive advantages—is not necessarily the most important geographical set of ideas leading to the production of images in a trans-frontier and European region. Each of the chosen newspapers has developed a cross-border agenda assigning specific values to state borders and places within this regional and functional setting.


1999 ◽  
Vol 9 (4) ◽  
pp. 471-477 ◽  
Author(s):  
CHRIS OKASAKI

Everybody learns about balanced binary search trees in their introductory computer science classes, but even the stouthearted tremble at the thought of actually implementing such a beast. The details surrounding rebalancing are usually just too messy. To show that this need not be the case, we present an algorithm for insertion into red-black trees (Guibas and Sedgewick, 1978) that any competent programmer should be able to implement in fifteen minutes or less.


2012 ◽  
Vol 163 (7) ◽  
pp. 918-934 ◽  
Author(s):  
Antonio Bucciarelli ◽  
Thomas Ehrhard ◽  
Giulio Manzonetto

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