DOCSLIB.ORG

Finite Element Function Approximation 1 Piecewise Linear Interpolation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Finite Element Function Approximation 1 Piecewise Linear Interpolation THE UNIVERSITY OF WESTERN ONTARIO LONDON ONTARIO Paul Klein Office: SSC 4028 Phone: 661-2111 ext. 85227 Email: [email protected] URL: www.ssc.uwo.ca/economics/faculty/klein/ Finite element function approximation In these notes we consider function approximation where the basis functions do not have full support (are zero on most of the domain). 1 Piecewise linear interpolation Let the grid be given by fx1; x2; : : : ; xng. Then we define the following ”tent” functions. 8 x − xi−1 > if x 2 [xi−1; xi] > xi − xi−1 < xi+1 − x i(x) = if x 2 [xi; xi+1] (1) > xi+1 − xi > :> 0 elsewhere: These functions are illustrated in Figure1 with a grid given by f0; 1=4; 2=3; 1g. Notice that the points needn’t be evenly spaced. One may verify that if n ^ X f(x) = f(xi) i(x) i=1 1 1 φ φ φ φ 0.9 1 2 3 4 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1: Basis functions for piecewise linear interpolation. 2 ^ then f(x) = f(x) for each x 2 fx1; x2; : : : ; xng. In the notation of the handout on multi-dimensional interpolation, Ψ = I. In several dimensions, life is a little bit more complicated. Then the usual approach is to let f^ be piecewise multilinear, i.e. on each element (rectangle), f^ is linear in each of the dimensions separately, keeping the others constant.1 We will define a piecewise bilinear approximation for the two-dimensional case. For arbitrarily many dimensions, see the handout on multi-dimensional interpolation. Let f be a function of two variables x1 and x2, and let the grids be given by 1 1 1 2 2 2 j x1; x2; : : : ; xn1 and x1; x2; : : : ; xn2 . Write i for the ith tent function associ- ated with the jth grid. Then n1 n2 ^ 1 2 X X 1 2 1 1 2 2 f(x ; x ) = f(xi ; xj ) i (x ) j (x ): i=1 j=1 1 1 2 2 Notice that this is not a linear function on a rectangle [xi ; xi+1] × [xj ; xj+1] but that it is bilinear on such a rectangle. A smart way of calculating the relevant sum is based on the fact that, in any given dimension, only two of the basis functions are non-zero, and these are consecutive; call them k and k+1. To find k given x you use a locating algo- rithm that essentially proceeds by bisection. 1It is possible, and for some purposes preferable, to define elementwise linear functions on multi-dimensional domains. But we won’t do that here. 3 2 Cubic splines Piecewise linear interpolation is nice, but it yields an interpolating function that is not differentiable at the grid points. An interpolation technique that does yield differentiable approximating functions, and has other nice proper- ties as well, is the cubic spline. Like linear interpolation, cubic spline interpolation begins with a grid of x- values fx1; x2; : : : ; xng and the value of the approximating function fb at those points: fy1; y2; : : : ; yng. A cubic spline specifies fb to be a separate cubic function on each segment [xi; xi+1). Each cubic function has four parameters. This means that there are 4(n − 1) parameters to determine. To do this, we impose the following conditions: 1. fb(xi) = yi; i = 1; 2; : : : ; n (n conditions) 2. Continuity of fb at x2; x3; : : : ; xn−1 (n − 2 conditions) 0 3. Continuity of fb at x2; x3; : : : ; xn−1 (n − 2 conditions) 00 4. Continuity of fb at x2; x3; : : : ; xn−1 (n − 2 conditions) 00 00 5. fb (x1) = fb (xn−1) = 0 (2 conditions). The final two conditions can of course easily be modified to something else depending on what is suitable in the particular case at hand. In principle, one could impose all of these conditions at once, and the result 4 would be a large linear system. However, this problem is best approached in 00 two steps. First, we impose fb(xi) = yi, continuity of fb and continuity of fb at the gridpoints. It turns out that these requirements force us to write the cubic on each interval [xi; xi+1) as xi+1 − x x − xi fb(x) = · yi + · yi+1+ xi+1 − xi xi+1 − xi ( " 3 # " 3 #) 1 2 xi+1 − x xi+1 − x x − xi x − xi + (xi+1−xi) ai − + ai+1 − 6 xi+1 − xi xi+1 − xi xi+1 − xi xi+1 − xi where ai; i = 1; 2; : : : ; n remain to be determined. With this specification, it 00 turns out that fb (xi) = ai. We now proceed to determine these second deriva- 0 00 00 tives. We do it by imposing continuity of fb and fb (x1) = fb (xn−1) = 0. Conti- nuity of fb0 at the gridpoints boils down to yi+1 − yi 1 1 + (xi+1 − xi)ai + (xi+1 − xi)ai+1 = xi+1 − xi 6 3 yi+2 − yi+1 1 1 = − (xi+2 − xi+1)ai+1 − (xi+2 − xi+1)ai+2 xi+2 − xi+1 3 6 for i = 1; 2; : : : ; n − 2. These equations, together with a1 = an = 0 (this choice for the endpoints is called a “natural” spline), together constitute a tridiagonal linear system of equations. Specifically, we have Ma = b where 2 3 0 6 7 6 q2 q1 7 6 − 7 6 h2 h1 7 6 q q 7 6 3 − 2 7 6 7 b = 6 h3 h2 7 6 . 7 6 . 7 6 7 6 qn−1 qn−2 7 6 − 7 6 hn−1 hn−2 7 4 5 0 5 and 2 3 1 0 0 0 0 ··· 0 6 7 6 7 6 h1 2(h1 + h2) h2 0 0 ··· 0 7 6 7 6 7 1 6 0 h2 2(h2 + h3) h3 0 ··· 0 7 M = 6 7 6 6 . .. .. .. .. 7 6 . 7 6 7 6 7 6 0 0 0 0 hn−2 2(hn−2 + hn−1) hn−1 7 4 5 0 0 0 0 0 0 1 where qi = yi+1 − yi and hi = xi+1 − xi. Setting a1 = an = 0 is of course a bit arbitrary. Another, equally arbitrary choice would be a1 = a2 and that an = an−1. 2.1 Not-a-knot cubic splines n So far we have worked with nodes (or gridpoints) fxigi=1 and until now that concept has had two distinct connotations. One is that we have data (function values) for these points, and the other is that the interpolating function fbis a polynomial on each subinterval [xi; xi+1]. Now we will change the terminology somewhat and distinguish between a node, for which we have a given function value, and a knot, where we switch from using one polynomial to another. Notice that a knot may or may not be a node and a node may or may not be a knot. The not-a-knot approach to cubic splines is an alternative to setting a1 = an = 0 or a1 = a2 and an = an−1. It imposes continuity of the third derivative at the nodes x2 and xn−1. This of course amounts to insisting that these nodes are 6 not really knots in the sense that the first cubic polynomial, i.e. the one valid for the interval [x1; x2), is in fact identical to the second cubic polynomial, i.e. the one valid for the interval [x2; x3), and similarly for the (n − 1)th and nth cubic polynomials. Equating the relevant third derivatives, we have −a + a −a + a 1 2 = 2 3 x2 − x1 x3 − x2 and −a + a −a + a n−2 n−1 = n−1 n xn−1 − xn−2 xn − xn−1 This modification leads to the following modified coefficient matrix of our lin- ear system Ma = b. The modified b is left as an exercise for the reader. 2 3 −h2 h1 + h2 −h1 0 0 ··· 0 6 7 6 7 6 h1 2(h1 + h2) h2 0 0 ··· 0 7 6 7 6 7 1 6 0 h2 2(h2 + h3) h3 0 ··· 0 7 M = 6 7 6 6 . .. .. .. .. 7 6 . 7 6 7 6 7 6 0 0 0 0 hn−2 2(hn−2 + hn−1) hn−1 7 4 5 0 0 0 0 −hn−1 hn−2 + hn−1 −hn−2 3 Shape-preserving splines Sometimes it is important that the interpolating function preserve the “shape” of the true function. More precisely, it should be increasing (decreasing) wher- ever the true function is increasing (decreasing) and convex (concave) wherever the true function is convex (concave). Use Schumaker(1983) with the slope ap- proximation of Butland(1980), a combination suggested by Iqbal(1992), who 7 shows that Butland’s choice of slope approximations implies that Schumaker’s conditions 3.3 for co-monotonicity are satisfied. Here is a cook-book description of Iqbal’s Butland-Schumaker algorithm. What is given is the following: Nodes xi; i = 1; 2;:::;N and function values at these nodes yi; i = 1; 2;:::;N. What we want to compute are knots zj; j = 1; 2;:::;K and coefficients Aj, Bj and Cj; j = 1; 2;:::;K so that the following function defines a shape-preserving quadratic interpolating spline, defined on the interval [x1; xN ]: 2 fb(x) = Aj + Bj(x − xj) + Cj(x − xj) if x 2 [zj; zj+1) Except by fluke, we have that K = 2N − 2 because exactly one extra knot is inserted inside each interval between adjacent nodes.
Load more

Recommended publications
  • Kriging Prediction with Isotropic Matérn Correlations: Robustness
    Journal of Machine Learning Research 21 (2020) 1-38 Submitted 12/19; Revised 7/20; Published 9/20 Kriging Prediction with Isotropic Mat´ernCorrelations: Robustness and Experimental Designs Rui Tuo∗y [email protected] Wm Michael Barnes '64 Department of Industrial and Systems Engineering Texas A&M University College Station, TX 77843, USA Wenjia Wang∗ [email protected] The Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong Editor: Philipp Hennig Abstract This work investigates the prediction performance of the kriging predictors. We derive some error bounds for the prediction error in terms of non-asymptotic probability under the uniform metric and Lp metrics when the spectral densities of both the true and the imposed correlation functions decay algebraically. The Mat´ernfamily is a prominent class of correlation functions of this kind. Our analysis shows that, when the smoothness of the imposed correlation function exceeds that of the true correlation function, the prediction error becomes more sensitive to the space-filling property of the design points. In particular, the kriging predictor can still reach the optimal rate of convergence, if the experimental design scheme is quasi-uniform. Lower bounds of the kriging prediction error are also derived under the uniform metric and Lp metrics. An accurate characterization of this error is obtained, when an oversmoothed correlation function and a space-filling design is used. Keywords: Computer Experiments, Uncertainty Quantification, Scattered Data Approx- imation, Space-filling Designs, Bayesian Machine Learning 1. Introduction In contemporary mathematical modeling and data analysis, we often face the challenge of reconstructing smooth functions from scattered observations.
    [Show full text]
  • On Prediction Properties of Kriging: Uniform Error Bounds and Robustness
    On Prediction Properties of Kriging: Uniform Error Bounds and Robustness Wenjia Wang∗1, Rui Tuoy2 and C. F. Jeff Wuz3 1The Statistical and Applied Mathematical Sciences Institute, Durham, NC 27709, USA 2Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843, USA 3The H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA Abstract Kriging based on Gaussian random fields is widely used in reconstructing unknown functions. The kriging method has pointwise predictive distributions which are com- putationally simple. However, in many applications one would like to predict for a range of untried points simultaneously. In this work we obtain some error bounds for the simple and universal kriging predictor under the uniform metric. It works for a scattered set of input points in an arbitrary dimension, and also covers the case where the covariance function of the Gaussian process is misspecified. These results lead to a better understanding of the rate of convergence of kriging under the Gaussian or the Matérn correlation functions, the relationship between space-filling designs and kriging models, and the robustness of the Matérn correlation functions. Keywords: Gaussian Process modeling; Uniform convergence; Space-filling designs; Radial arXiv:1710.06959v4 [math.ST] 19 Mar 2019 basis functions; Spatial statistics. ∗Wenjia Wang is a postdoctoral fellow in the Statistical and Applied Mathematical Sciences Institute, Durham, NC 27709, USA (Email: [email protected]); yRui Tuo is Assistant Professor in Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX 77843, USA (Email: [email protected]); Tuo’s work is supported by NSF grant DMS 1564438 and NSFC grants 11501551, 11271355 and 11671386.
    [Show full text]
  • Rational Function Approximation
    Rational function approximation Rational function of degree N = n + m is written as p(x) p + p x + + p xn r(x) = = 0 1 ··· n q(x) q + q x + + q xm 0 1 ··· m Now we try to approximate a function f on an interval containing 0 using r(x). WLOG, we set q0 = 1, and will need to determine the N + 1 unknowns p0,..., pn, q1,..., qm. Numerical Analysis I – Xiaojing Ye, Math & Stat, Georgia State University 240 Pad´eapproximation The idea of Pad´eapproximation is to find r(x) such that f (k)(0) = r (k)(0), k = 0, 1,..., N This is an extension of Taylor series but in the rational form. i Denote the Maclaurin series expansion f (x) = i∞=0 ai x . Then i m i n i ∞ a x q x P p x f (x) r(x) = i=0 i i=0 i − i=0 i − q(x) P P P If we want f (k)(0) r (k)(0) = 0 for k = 0,..., N, we need the − numerator to have 0 as a root of multiplicity N + 1. Numerical Analysis I – Xiaojing Ye, Math & Stat, Georgia State University 241 Pad´eapproximation This turns out to be equivalent to k ai qk i = pk , k = 0, 1,..., N − Xi=0 for convenience we used convention p = = p = 0 and n+1 ··· N q = = q = 0. m+1 ··· N From these N + 1 equations, we can determine the N + 1 unknowns: p0, p1,..., pn, q1,..., qm Numerical Analysis I – Xiaojing Ye, Math & Stat, Georgia State University 242 Pad´eapproximation Example x Find the Pad´eapproximation to e− of degree 5 with n = 3 and m = 2.
    [Show full text]
  • Piecewise Linear Approximation of Streaming Time Series Data with Max-Error Guarantees
    Piecewise Linear Approximation of Streaming Time Series Data with Max-error Guarantees Ge Luo Ke Yi Siu-Wing Cheng Zhenguo Li Wei Fan Cheng He Yadong Mu HKUST Huawei Noah’s Ark Lab AT&T Labs Abstract—Given a time series S = ((x1; y1); (x2; y2);::: ) and f(xi) yi " for all i. This is because the `1=`2-error is a prescribed error bound ", the piecewise linear approximation ill-suitedj − forj ≤ online algorithms as it is a sum of errors over (PLA) problem with max-error guarantees is to construct a the entire time series. When the algorithm has no knowledge piecewise linear function f such that jf(xi) − yij ≤ " for all i. In addition, we would like to have an online algorithm that takes about the future, in particular the length of the time series n, the time series as the records arrive in a streaming fashion, and it is impossible to properly allocate the allowed error budget outputs the pieces of f on-the-fly. This problem has applications over time. Another advantage of the ` -error is that it gives wherever time series data is being continuously collected, but us a guarantee on any data record in1 the time series, while the data collection device has limited local buffer space and the ` =` -error only ensures that the “average” error is good communication bandwidth, so that the data has to be compressed 1 2 without a bound on any particular record. Admittedly, the ` - and sent back during the collection process. 1 Prior work addressed two versions of the problem, where error is sensitive to outliers, but one could remove them before either f consists of disjoint segments, or f is required to be feeding the stream to the PLA algorithm, and there is abundant a continuous piecewise linear function.
    [Show full text]
  • Rethinking Statistical Learning Theory: Learning Using Statistical Invariants
    Machine Learning (2019) 108:381–423 https://doi.org/10.1007/s10994-018-5742-0 Rethinking statistical learning theory: learning using statistical invariants Vladimir Vapnik1,2 · Rauf Izmailov3 Received: 2 April 2018 / Accepted: 25 June 2018 / Published online: 18 July 2018 © The Author(s) 2018 Abstract This paper introduces a new learning paradigm, called Learning Using Statistical Invariants (LUSI), which is different from the classical one. In a classical paradigm, the learning machine constructs a classification rule that minimizes the probability of expected error; it is data- driven model of learning. In the LUSI paradigm, in order to construct the desired classification function, a learning machine computes statistical invariants that are specific for the problem, and then minimizes the expected error in a way that preserves these invariants; it is thus both data- and invariant-driven learning. From a mathematical point of view, methods of the classical paradigm employ mechanisms of strong convergence of approximations to the desired function, whereas methods of the new paradigm employ both strong and weak convergence mechanisms. This can significantly increase the rate of convergence. Keywords Intelligent teacher · Privileged information · Support vector machine · Neural network · Classification · Learning theory · Regression · Conditional probability · Kernel function · Ill-Posed problem · Reproducing Kernel Hilbert space · Weak convergence Mathematics Subject Classification 68Q32 · 68T05 · 68T30 · 83C32 1 Introduction It is known that Teacher–Student interactions play an important role in human learning. An old Japanese proverb says “Better than thousand days of diligent study is one day with a great teacher.” What is it exactly that great Teachers do? This question remains unanswered.
    [Show full text]
  • Function Approximation with Mlps, Radial Basis Functions, and Support Vector Machines
    Table of Contents CHAPTER V- FUNCTION APPROXIMATION WITH MLPS, RADIAL BASIS FUNCTIONS, AND SUPPORT VECTOR MACHINES ..........................................................................................................................................3 1. INTRODUCTION................................................................................................................................4 2. FUNCTION APPROXIMATION ...........................................................................................................7 3. CHOICES FOR THE ELEMENTARY FUNCTIONS...................................................................................12 4. PROBABILISTIC INTERPRETATION OF THE MAPPINGS-NONLINEAR REGRESSION .................................23 5. TRAINING NEURAL NETWORKS FOR FUNCTION APPROXIMATION ......................................................24 6. HOW TO SELECT THE NUMBER OF BASES ........................................................................................28 7. APPLICATIONS OF RADIAL BASIS FUNCTIONS................................................................................38 8. SUPPORT VECTOR MACHINES........................................................................................................42 9. PROJECT: APPLICATIONS OF NEURAL NETWORKS AS FUNCTION APPROXIMATORS ...........................52 10. CONCLUSION ..............................................................................................................................59 CALCULATION OF THE ORTHONORMAL WEIGHTS ..................................................................................63
    [Show full text]
  • Uhm Ms 3812 R.Pdf
    UNIVERSITY OF HAWAI'I LIBRARY ADVANCED MARINE VEIDCLE PRODUCTS DATABASE -A PRELIMINARY DESIGN TOOL A THESIS SUBMITTED TO THE GRADUATE DMSION OF THE UNIVERSITY OF HAWAI'I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN OCEAN ENGINEERING AUGUST 2003 By Kristen A.L.G. Woo Thesis Committee: Kwok Fai Cheung, ChaiIperson Hans-Jurgen Krock John C. Wiltshire ACKNOWLEDGEMENT I would like to thank my advisor Prof. Kwok Fai Cheung and the other committee members Prof. Hans-Jiirgen Krock and Dr. John Wiltshire for the time and effort they spent with me on this project. I would also like to thank the MHPCC staff, in particular, Mr. Scott Splean for their advice and comments on the advanced-marine-vehicle products database. Thanks are also due to Drs. Woei-Min Lin and JUll Li of SAIC for their advice on the neural network and preliminary ship design tools. I would also like to thank Mr. Yann Douyere for his help with MatLab. The work described in this thesis is a subset of the project "Environment for Design of Advanced Marine Vehicles and Operations Research" supported by the Office of Naval Research, Grant No. NOOOI4-02-1-0903. iii ABSTRACT The term advanced marine vehicle encompasses a broad category of ship designs typically referring to multihull ships such as catamarans, trimarans and SWATH (small waterplane area twin hull) ships, but also includes hovercrafts, SES (surface effect ships), hydrofoils, and advanced monohulls. This study develops an early stage design tool for advanced marine vehicles that provides principal particulars and additional parameters such as fuel capacity and propulsive power based on input ship requirements.
    [Show full text]
  • Rational Function Approximation
    Approximation Theory Pad´eApproximation Numerical Analysis and Computing Lecture Notes #13 — Approximation Theory — Rational Function Approximation Joe Mahaffy, [email protected] Department of Mathematics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182-7720 http://www-rohan.sdsu.edu/∼jmahaffy Spring 2010 Joe Mahaffy, [email protected] Rational Function Approximation — (1/21) Approximation Theory Pad´eApproximation Outline 1 Approximation Theory Pros and Cons of Polynomial Approximation New Bag-of-Tricks: Rational Approximation Pad´eApproximation: Example #1 2 Pad´eApproximation Example #2 Finding the Optimal Pad´eApproximation Joe Mahaffy, [email protected] Rational Function Approximation — (2/21) Pros and Cons of Polynomial Approximation Approximation Theory New Bag-of-Tricks: Rational Approximation Pad´eApproximation Pad´eApproximation: Example #1 Polynomial Approximation: Pros and Cons. Advantages of Polynomial Approximation: [1] We can approximate any continuous function on a closed inter- val to within arbitrary tolerance. (Weierstrass approximation theorem) Joe Mahaffy, [email protected] Rational Function Approximation — (3/21) Pros and Cons of Polynomial Approximation Approximation Theory New Bag-of-Tricks: Rational Approximation Pad´eApproximation Pad´eApproximation: Example #1 Polynomial Approximation: Pros and Cons. Advantages of Polynomial Approximation: [1] We can approximate any continuous function on a closed inter- val to within arbitrary tolerance. (Weierstrass approximation theorem) [2] Easily evaluated at arbitrary values. (e.g. Horner’s method) Joe Mahaffy, [email protected] Rational Function Approximation — (3/21) Pros and Cons of Polynomial Approximation Approximation Theory New Bag-of-Tricks: Rational Approximation Pad´eApproximation Pad´eApproximation: Example #1 Polynomial Approximation: Pros and Cons. Advantages of Polynomial Approximation: [1] We can approximate any continuous function on a closed inter- val to within arbitrary tolerance.
    [Show full text]
  • Simplified Neural Networks Algorithms for Function Approximation and Regression
    SIMPLIFIED NEURAL NETWORKS ALGORITHMS FOR FUNCTION APPROXIMATION AND REGRESSION BOOSTING ON DISCRETE INPUT SPACES A THESIS SUBMITTED TO THE UNIVERSITY OF MANCHESTER FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE FACULTY OF ENGINEERING AND PHYSICAL SCIENCES 2010 By Syed Shabbir Haider School of Computer Science Table of Contents Abstract 9 Declaration 11 Copyright 12 Acknowledgements 13 1 Introduction 14 1.1 Neural Networks-A Brief Overview 14 1.2 Basic Terminology and Architectural Considerations 15 1.3 Real World Applications Of Neural Networks 22 1.4 The Problem Statement 23 1.5 Motivations Behind Initiation Of This Research 23 1.6 Research Objectives To Be Met 25 1.7 Main Contributions 26 1.8 Structure Of Thesis 27 1.9 Summary 28 2 Learning in Feedforward Neural Networks (FNN) 29 2.1 The Learning Process 29 2.1.1 Supervised Learning 30 2.1.2 Un-Supervised Learning 30 2.2.1 Graded (Reinforcement) Learning 31 2.2 Supervised Learning Laws for MLPs 32 2 2.2.1 The Perceptron Training Rule 32 2.2.2 The Widrow-Hoff Learning (Delta) Rule 34 2.3 Backpropagation Algorithm for MLPs 37 2.4 Special Issues in BP Learning and MLPs 39 2.4.1 Convergence, Stability and Plasticity 39 2.4.2 Selection of Hidden Layer Units (Activation Function) 40 2.4.3 When To Stop Training? 40 2.4.4 Local Minima 41 2.4.5 Number of Hidden Layers 41 2.4.6 Number of Hidden Units 42 2.4.7 Learning Rate and Momentum 43 2.4.8 The Training Style 43 2.4.9 Test, Training and Validation sets 44 2.5 Variants of BP Learning 45 2.6 Summary 47 3 Approximation Capabilities of FNNs and
    [Show full text]
  • Time Series Compression: a Survey
    1 Time series compression: a survey Giacomo Chiarot, Claudio Silvestri Abstract—The presence of smart objects is increasingly widespread and their ecosystem, also known as Internet of Things, is relevant in many different application scenarios. The huge amount of temporally annotated data produced by these smart devices demand for efficient techniques for transfer and storage of time series data. Compression techniques play an important role toward this goal and, despite the fact that standard compression methods could be used with some benefit, there exist several ones that specifically address the case of time series by exploiting their peculiarities to achieve a more effective compression and a more accurate decompression in the case of lossy compression techniques. This paper provides a state-of-the-art survey of the principal time series compression techniques, proposing a taxonomy to classify them considering their overall approach and their characteristics. Furthermore, we analyze the performances of the selected algorithms by discussing and comparing the experimental results that where provided in the original articles. The goal of this paper is to provide a comprehensive and homogeneous reconstruction of the state-of-the-art which is currently fragmented across many papers that use different notations and where the proposed methods are not organized according to a classification. Index Terms—Time series, Data Compaction and Compression, Internet of Things, State-of-the-art Survey, Sensor Data, Approximation, and Data Abstraction. F 1 INTRODUCTION IME series are relevant in several different contexts in the IoT). Further, these algorithms take advantage of the T and Internet of Things ecosystem (IoT) is among the peculiarities of time series produced by sensors, such as: most pervasive ones.
    [Show full text]
  • Deep Neural Network Approximation Theory
    Deep Neural Network Approximation Theory Dennis Elbrachter,¨ Dmytro Perekrestenko, Philipp Grohs, and Helmut Bolcskei¨ Abstract This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the complexity of the function (class) to be approximated and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the associated quantized weights. The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces. In addition, deep networks provide exponential approximation accuracy—i.e., the approximation error decays exponentially in the number of nonzero weights in the network—of the multiplication operation, polynomials, sinusoidal functions, and certain smooth functions. Moreover, this holds true even for one-dimensional oscillatory textures and the Weierstrass function— a fractal function, neither of which has previously known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks. I. INTRODUCTION Triggered by the availability of vast amounts of training data and drastic improvements in computing power, deep neural networks have become state-of-the-art technology for a wide range of practical machine learning tasks such as image classification [1], handwritten digit recognition [2], speech recognition [3], or game intelligence [4].
    [Show full text]
  • Modeling Periodic Functions for Time Series Analysis
    From: AAAI Technical Report WS-98-07. Compilation copyright © 1998, AAAI (www.aaai.org). All rights reserved. Modeling periodic functions for time series analysis Anish Biswas Department of Mathematics and Computer Science University of Tulsa 600 South College Avenue Tulsa, OK 74014 E-mail: [email protected] Abstract the model can be improved. Wehave also provided the convergence proof of the algorithm, which says that un- Timeseries problemsinvolve analysis of periodic functions for predicting the future. Aflexible re- der infinite sampling this method is guaranteed to give gression method should be able to dynamically accurate model and the level of accuracy is limited only select the appropriate model to fit the available by error due to truncation of the series. data. In this paper, we present a function ap- proximation scheme that can be used for model- Chebychev polynomials ing periodic functions using a series of orthogonal polynomials, named Chebychev polynomials. In Chebychev polynomials are a family of orthogonal poly- our approach, we obtain an estimate of the error nomials (Geronimus 1961). Any function f(x) may due to neglecting higher order polynomials and approximated by a weighted sum of these polynomial thus can flexibly select a polynomialmodel of the functions with an appropriate selection of the coeffi- proper order. Wealso showthat this approxima- cients. OO tion approachis stable in the presence of noise. a0 f(x) = -~ + E ai * 7](x) Keywords: Function approximation,Orthogonal poly- i=1 nomials where Introduction Analysis of time series data plays an important role and in finance and marketing. A model derived from past 2 F f(x) * 7](x) data can help to predict future behavior of the phe- ai=-zrJ_l ~-1---’-’~ dx.
    [Show full text]