Piecewise linear function

Summary

In mathematics and statistics, a piecewise linear, PL or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments. Definition A piecewise linear function is a function defined on a (possibly unbounded) interval of real numbers, such that there is a collection of intervals on each of which the function is an affine function. (Thus "piecewise linear" is actually defined to mean "piecewise affine".) If the domain of the function is compact, there needs to be a finite collection of such intervals; if the domain is not compact, it may either be required to be finite or to be locally finite in the reals. Examples The function defined by : f(x) = \begin{cases} -x - 3 & \text{if }x \leq -3 \ x + 3 & \text{if }-3 < x < 0 \ -2x + 3 & \text{if }0 \leq x < 3 \ 0.5x - 4.5 & \text{if }x \geq 3 \end{cases} is piecewise linear with four pieces. The graph of this function is shown to the rig

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https://en.wikipedia.org/wiki/Piecewise_linear_function

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A recursive max-linear vector models causal dependence between its components by expressing each node variable as a max-linear function of its parental nodes in a directed acyclic graph and some exogenous innovation. Motivated by extreme value theory, innovations are assumed to have regularly varying distribution tails. We propose a scaling technique in order to determine a causal order of the node variables. All dependence parameters are then estimated from the estimated scalings. Furthermore, we prove asymptotic normality of the estimated scalings and dependence parameters based on asymptotic normality of the empirical spectral measure. Finally, we apply our structure learning and estimation algorithm to financial data and food dietary interview data. (C) 2020 Elsevier Inc. All rights reserved.

ELSEVIER INC2021

CAT(0) Geometry for the Thompson Group

Dan Titus Salajan

We investigate Farley’s CAT(0) cubical model for Thompson’s group F (we adopt the classical language of F, using binary trees and piecewise linear maps, avoiding the one of diagram groups and pictures). Main results include: in general, Thompson’s group elements are parabolic; we find simple, exact formulas for the CAT(0) translation lengths, in particular the elements of F are ballistic and uniformly bounded away from zero; there exist flats of any dimension and we construct explicitly many of them; we reveal large regions in the Tits Boundary, for example the positive part of a non-separable Hilbert sphere , but also more complicated objects. En route, we solve several open problems proposed in Farley’s papers.

EPFL2012

Switch Detection in Genetic Regulatory Networks

Giancarlo Ferrari Trecate

This paper considers piecewise affine models of genetic regulatory networks and focuses on the problem of detecting switches among different modes of operation in gene expression data. This task constitutes the first step of a procedure for the complete identification of the network. We propose two methods and illustrate the application to the reconstruction of switching times in data produced by a piecewise affine model of the network regulating the carbon starvation response in Es- cherichia coli.

Springer-Verlag2007