Approximation factor of the piecewise linear functions in Mamdani fuzzy system and its realization process1

2021 ◽  
pp. 1-15
Author(s):  
Yujie Tao ◽  
Chunfeng Suo ◽  
Guijun Wang
Keyword(s):  
Continuous Function ◽  
Linear Function ◽  
Hypothesis Test ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Two Dimensions ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Approximation Accuracy ◽  
Approximation Factor

Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.

UNIVERSAL CNN CELLS

1999 ◽  
Vol 09 (01) ◽  
pp. 1-48 ◽  
Author(s):  
RADU DOGARU ◽  
LEON O. CHUA
Keyword(s):  
Linear Function ◽  
Boolean Functions ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Linear Functions ◽  
Compact Representation ◽  
Value Functions ◽  
Linear Discriminant ◽  
Absolute Value ◽  
Parity Function

A cellular neural/nonlinear network (CNN) [Chua, 1998] is a biologically inspired system where computation emerges from a collection of simple nonlinear locally coupled cells. This paper reviews our recent research results beginning from the standard uncoupled CNN cell which can realize only linearly separable local Boolean functions, to a generalized universal CNN cell capable of realizing arbitrary Boolean functions. The key element in this evolutionary process is the replacement of the linear discriminant (offset) function w(σ)=σ in the "standard" CNN cell in [Chua, 1998] by a piecewise-linear function defined in terms of only absolute value functions. As in the case of the standard CNN cells, the excitation σ evaluates the correlation between a given input vector u formed by the outputs of the neighboring cells, and a template vector b, which is interpreted in this paper as an orientation vector. Using the theory of canonical piecewise-linear functions [Chua & Kang, 1977], the discriminant function [Formula: see text] is found to guarantee universality and its parameters can be easily determined. In this case, the number of additional parameters and absolute value functions m is bounded by m<2n-1, where n is the number of all inputs (n=9 for a 3×3 template). An even more compact representation where m<n-1 is also presented which is based on a special form of a piecewise-linear function; namely, a multi-nested discriminant: w (σ) =s (zm +| zm -1 +⋯ | z1 +| z0 +σ |||). Using this formula, the "benchmark" Parity function with an arbitrary number of inputs n is found to have an analytical solution with a complexity of only m =O ( log 2 (n)).

scholarly journals Models and Algorithms for Optimal Piecewise-Linear Function Approximation

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Eduardo Camponogara ◽  
Luiz Fernando Nazari
Keyword(s):  
Function Approximation ◽  
Linear Models ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Linear Function Approximation ◽  
Data Points ◽  
Stochastic Functions ◽  
Piecewise Linear Models

Piecewise-linear functions can approximate nonlinear and unknown functions for which only sample points are available. This paper presents a range of piecewise-linear models and algorithms to aid engineers to find an approximation that fits best their applications. The models include piecewise-linear functions with a fixed and maximum number of linear segments, lower and upper envelopes, strategies to ensure continuity, and a generalization of these models for stochastic functions whose data points are random variables. Derived from recursive formulations, the algorithms are applied to the approximation of the production function of gas-lifted oil wells.

scholarly journals Osiris wavelets and Set wavelets

2004 ◽  
Vol 2004 (6) ◽  
pp. 495-528
Author(s):  
Guy Battle
Keyword(s):  
Lower Bound ◽  
Piecewise Linear ◽  
Building Blocks ◽  
Two Dimensions ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Wavelet System ◽  
Equilateral Triangles ◽  
Overlap Matrix

An alternative to Osiris wavelet systems is introduced in two dimensions. The basic building blocks are continuous piecewise linear functions supported on equilateral triangles instead of on squares. We refer to wavelets generated in this way as Set wavelets. We introduce a Set wavelet system whose homogeneous mode density is2/5. The system is not orthonormal, but we derive a positive lower bound on the overlap matrix.

scholarly journals MINLP formulations for continuous piecewise linear function fitting

2021 ◽  
Author(s):  
Noam Goldberg ◽  
Steffen Rebennack ◽  
Youngdae Kim ◽  
Vitaliy Krasko ◽  
Sven Leyffer
Keyword(s):  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Mixed Integer ◽  
Function Fitting ◽  
Computational Performance ◽  
Integer Nonlinear Programming ◽  
Minlp Model ◽  
Necessary Constraints ◽  
Continuous Piecewise Linear Function

AbstractWe consider a nonconvex mixed-integer nonlinear programming (MINLP) model proposed by Goldberg et al. (Comput Optim Appl 58:523–541, 2014. 10.1007/s10589-014-9647-y) for piecewise linear function fitting. We show that this MINLP model is incomplete and can result in a piecewise linear curve that is not the graph of a function, because it misses a set of necessary constraints. We provide two counterexamples to illustrate this effect, and propose three alternative models that correct this behavior. We investigate the theoretical relationship between these models and evaluate their computational performance.

scholarly journals DESIGN OF TIME DELAYED CHAOTIC CIRCUIT WITH THRESHOLD CONTROLLER

2011 ◽  
Vol 21 (03) ◽  
pp. 725-735 ◽  
Author(s):  
K. SRINIVASAN ◽  
I. RAJA MOHAMED ◽  
K. MURALI ◽  
M. LAKSHMANAN ◽  
SUDESHNA SINHA
Keyword(s):  
Numerical Simulations ◽  
Linear Function ◽  
Piecewise Linear ◽  
Piecewise Linear Function ◽  
Operational Amplifiers ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Chaotic Oscillator ◽  
Threshold Values ◽  
Chaotic Circuit

A novel time delayed chaotic oscillator exhibiting mono- and double scroll complex chaotic attractors is designed. This circuit consists of only a few operational amplifiers and diodes and employs a threshold controller for flexibility. It efficiently implements a piecewise linear function. The control of piecewise linear function facilitates controlling the shape of the attractors. This is demonstrated by constructing the phase portraits of the attractors through numerical simulations and hardware experiments. Based on these studies, we find that this circuit can produce multi-scroll chaotic attractors by just introducing more number of threshold values.

Modelling the single-electron transistor with piecewise linear functions

2009 ◽  
Author(s):  
Arturo Sarmiento-Reyes ◽  
Luis Hernandez-Martinez ◽  
Miguel Angel Gutierrez de Anda ◽  
Francisco Javier Castro Gonzalez
Keyword(s):  
Piecewise Linear ◽  
Linear Functions ◽  
Single Electron ◽  
Single Electron Transistor ◽  
Piecewise Linear Functions

Mesh duality and Legendre duality

1990 ◽  
Vol 428 (1875) ◽  
pp. 351-377 ◽  
Keyword(s):  
Piecewise Linear ◽  
Voronoi Tessellation ◽  
Tangent Plane ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Dual Transformation ◽  
Delaunay Mesh ◽  
Definition Of

We describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.

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