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Machine Learning Basics
Published in Peter Wlodarczak, Machine Learning and its Applications, 2019
Function approximation (FA) is sometimes used interchangeably with regression. Regression is a way to approximate a given data set. Function approximation can be considered a more general concept since there are many different methods to approximate data or functions. The goal of function approximation is to find a function f that maps an input to an output vector. The function is selected among a defined class, e.g., quadratic functions or polynomials. For instance, equation 4.14 is a first degree polynomial equation. Contrary to function fitting, where a curve is fitted to a set of data points, function approximation aims to find a function f that approximates a target function. Given one set of samples (x, y), we try to find a function f: () f:X→Y
Function Approximation
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
Function approximation is a branch of mathematics that seeks to find, among a family of functions, the function that best approximates a given function f. Function approximation is necessary in many applied settings, including numerical computation of functions, and is of broad interest to applied mathematicians and computer scientists. In this Extended Exploration, we will examine a number of ways we can carefully and systematically approximate functions, beginning with Taylor polynomials (which are well-known to calculus students) and Lagrange Interpolation polynomials (which are less so).
Data Interpolation and Functional Approximation Problems
Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018
The so-called function approximation problem is to extract the function representation from the measured sample points. Polynomial approximations and the least squares method for nonlinear function approximation will be studied. Furthermore, Pad´e approximations and continued fraction approximations for given functions are explored in Section 8.3. In Section 8.6, the correlation analysis of signals and experimental data are introduced. Fast Fourier transforms and filter-based de-noising and other signal processing problems are introduced.
The Reconstruction Approach: From Interpolation to Regression
Published in Technometrics, 2021
Interpolation is an important technique for function approximation and has been intensively studied by mathematicians (Wendland 2004). It can be viewed as the limit of a regression problem as noises go to zeros, and iterative regression techniques have been used to approximate an interpolator (Friedman 2001; Kang and Joseph 2016). Also, many techniques used in regression are applicable to interpolation such as basis representation. In statistics, interpolation is commonly used to model some spatial data (Cressie 2015), functional data (Ramsay, Hooker, and Graves 2009), and computer experiments (Santner, Williams, and Notz 2018), which do not contain any random noise. For noisy data, applications of interpolation are very limited. It is sometimes served as an auxiliary technique in nonparametric regression (Hall and Turlach 1997).
Research on efficient and accurate computing strategies for complex optimal control problems
Published in International Journal of Control, 2022
Aipeng Jiang, Qiuyun Huang, Yun Chen, Yudong Xia, Xiaoqing Zheng, Qiang Ding
Here is the value of control variable in time interval , and is the unit switching function as follows: According to function approximation theory (Sun & Fang, 1900), the control variables in each subinterval can be approximated by a series of linear combinations of basis functions: where is an Rij order basis function, and is a linear combination coefficient of the Rij function called the control parameter. Based on the basis functions selected, the control variable can be approximated differently. Piecewise constant approximation, piecewise linear approximation, piecewise parabolic approximation, and piecewise smooth spline function approximation can be selected as the functions according to different requirements. Piecewise constant approximation is the most commonly used function and is thus, used for simplicity in the current study. Under this condition, Rij is set to zero and the value of uj(t) in time interval can be reduced to and written as follows: Then the above infinite-dimensional optimal control problem is transformed into a finite-dimensional nonlinear programming (NLP) problem with variable control parameters to be optimised:
A new approximate dynamic programming algorithm based on an actor–critic framework for optimal control of alkali–surfactant–polymer flooding
Published in Engineering Optimization, 2019
Shurong Li, Lu Han, Yulei Ge, Yuhuan Shi
The value function approximation in ADP has a variety of methods, including the parametric function approximation method and the non-parametric function approximation method. The parametric approximation method covers linear function approximation and nonlinear function approximation. Since the linear function approximation structure has many advantages, such as faster convergence speed, fewer adjustable parameters and a simple form, therefore in this article the linear structure is used to approximate the optimal value function. The detailed process is shown as follows: where , , denotes weight value of each basis function and denotes the basis function. The selections of basis function are generally RBF basis functions, polynomial basis functions, Gaussian basis functions; Fourier basis functions, etc. However, from the mechanism model of ASP flooding in Section 2 it can be shown that ASP flooding is a complex system with partial differential equations. The general basis function has a low approximation accuracy when approximating the value function of the complex system and it cannot reflect the system's features fully. As a consequence, some scholars have suggested a method of constructing basis functions from system features (Wen et al.2011). Using this method can effectively improve the approximation precision and generalization in approximating complex system. This method uses system features to construct new basis functions; the number of basis functions can be chosen according to actual need. Compared with general basis function constructors, this method has the following advantages: the basis function can fully reflect the features of the entire system states and can improve the approximation accuracy;it has a relatively simple mathematical form and it can improve convergence effectively. In view of the above advantages, the method of constructing basis functions from system features is adopted to construct basis functions in this article. The detailed structure of basis function construction is shown in Figure 1.