Showing papers on "Piecewise linear function published in 1973"

Piecewise Regression Using Cubic Splines

Abstract: Spline theory and piecewise regression theory are integrated to provide a framework in which structural change is viewed as occurring in a smooth fashion. Specifically, structural change occurs at given points through jump discontinuities in the third derivative of a continuous piecewise cubic estimating function. Testing procedures are developed for detecting structural change as well as linear or quadratic segments. Finally, the techniques developed are illustrated empirically in a learning-by-doing model.

Waveform Segmentation Through Functional Approximation

Abstract: Waveform segmentation is treated as a problem of piecewise linear uniform (minmax) approximation. Various algorithms are reviewed and a new one is proposed based on discrete optimization. Examples of its applications are shown on terrain profiles, scanning electron microscope data, and electrocardiograms. The processing is sufficiently fast to allow its use on-line. The results of the segmentation can be used for pattern recognition, data compression, and nonlinear filtering not only for waveforms but also for pictures and maps. In the latter case some additional preprocessing is required and it is described in [19].

Multidimensional Stefan Problems

Abstract: An implicit finite difference method for the multidimensional Stefan problem is discussed. The classical problem with discontinuous enthalpy is replaced by an approximate Stefan problem with continuous piecewise linear enthalpy. An implicit time approximation reduces this formulation to a sequence of monotone elliptic problems which are solved by finite difference techniques. It is shown that the resulting nonlinear algebraic equations are solvable with a Gauss-Seidel method and that the discretized solution converges to the unique weak solution of the Stefan problem as the time and space mesh size approaches zero.

Finite element convergence for singular data

Abstract: Convergence of the finite element solutionu h of the Dirichlet problem Δu=? is proved, where ? is the Dirac ?-function (unit impulse). In two dimensions, the Green's function (fundamental solution)u lies outsideH 1, but we are able to prove that $$\parallel u - u^h \parallel _{L^2 } = O (h)$$ . Since the singularity ofu is logarithmic, we conclude that in two dimensions the function log ? can be approximated inL 2 near the origin by piecewise linear functions with an errorO (h). We also consider the Dirichlet problem Δu=f, wheref is piecewise smooth but discontinuous along some curve. In this case,u just fails to be inH 5/2, but as with the approximation to the Green's function, we prove the full rate of convergence:?u?u h ?1=O (h 8/2) with, say, piecewise quadratics.

Point Estimation and Risk Preferences

Abstract: The decision-theoretic approach to point estimation involves the choice of an estimate to minimize the expected loss associated with the estimate. The purpose of this paper is to indicate the influence of risk aversion on point estimates for classes of payoff functions including the piecewise linear and quadratic payoff functions. Increased risk aversion results in a point estimate closer to zero for a quadratic payoff function and a lower estimate with a piecewise linear payoff function, for example.

Steady-state analysis of piecewise-linear systems with periodic inputs

Abstract: A specific form of a recently proposed algorithm is derived for the rapid determination of the steady-state response of large classes of piecewise-linear systems subjected to periodic inputs. It can be directly incorporated at marginal cost into a computer-oriented state space approach for generating the time domain response of such systems. The algorithm has been successfully applied to several circuits.

Sectional curvature in piecewise linear manifolds

Abstract: A metric complex M is a connected, locally-finite simplicial complex linearly embedded in some Euclidean space R. Metric complexes M and M' are isometric if they have subdivisions L and L and if there is a simplicial isomorphism h:L -• L such that for every a e L, the linear map determined by h\\a -• h(a) is an isometry (that is, it extends to an isometry of the affine spaces generated by these simplexes). This note is concerned with certain characteristics of a metric complex M which are intrinsic, i.e., which depend only on the isometry class of M. The basic such characteristic is the intrinsic metric, which is best described in the piecewise linear context by H. Gluck [3]; for a more general treatment see W. Rinow [8]. Let M ç R be a metric complex and let p be a point of M. Then the tangent cone TPM of M at p is defined to be the infinite cone with vertex p generated by link(/?, M). The isometry class of TPM is intrinsic to M, for each p. An infinite ray px in TpM will be called a tangent direction at p to M. Let NPM be a subcone of TPM and let j be a nonnegative integer. Let R x NPM be given the metric in which its factors are orthogonal. For various choices of NpM and j \\ R j x NPM will be isometric to TPM. For example if p is in the interior of a /-simplex of M, such a factoring exists. Consider those factorings of TPM for which j is maximal; then the corresponding NPM are all isometric. Such an NPM will be called the normal geometry of p in M, and denoted vpM. For example, if M is an «-manifold and p is in the interior of an (n — 1)or «-simplex, then vpM = {/?}. If M is a 2-manifold, then vpM = {p} unless/? is a vertex of nonzero curvature, when vpM — TPM. Clearly j and vpM determine the metric geometry of M near p. For any pe M and any tangent direction px at p lying in vpM I have defined numbers k+(px) and k_(px), with k+(px) ^ k-(px), called the maximum and minimum curvatures of M at /? in the direction /?3c. The definitions are too long to give here. Roughly speaking, k-(px) equals: 2n minus twice the maximum \"angle\" that can occur between px and any other py £ vpM as y varies; k+{px) is defined similarly, using a

Special spines of piecewise linear manifolds

Abstract: A class of so-called special polyhedra is defined for each The following theorems are proved:1 Every piecewise linear manifold with boundary can be collapsed to some -dimensional special polyhedron2 The manifold is uniquely determined by this special polyhedron3 If , then any special polyhedron can be thickened to an -dimensional manifoldThe author also gives applications of the results obtained to a series of questions connected with the Zeeman conjecture about the collapsibility of , where is a contractible polyhedronFigures: 4 Bibliography: 6 items

Pattern Recognition by Piecewise Linear Discriminant Functions

Abstract: A piecewise linear function is represented in terms of a set of linear functions through the use of the maximum and minimum functions. A procedure for finding piecewise linear discriminant functions for pattern recognition is described. The procedure iteratively uses the accelerated relaxation method to find every linear function in a piecewise linear function. The procedure was implemented by a Fortran program. Experimental results with the program showed that the procedure is promising for obtaining piecewise linear discriminant functions.

Analysis of Limit Cycles in a Two-Transistor Saturable-Core Parallel Inverter

Abstract: A familiar two-transistor saturable-core parallel inverter is modeled as a nonlinear negative resistance in parallel with energy-storage elements. The techniques of singular-point analysis are combined with piecewise linear techniques to permit determination of solution trajectories on the phase plane. Clear insight is provided, not only into steady-state oscillation, but also into transient behavior of the circuit. Experimental results confirming the analytical model are included.

MIS array potential calculation

Abstract: A numerical Green's function technique is used to obtain the depletion layer profile of a two-dimensional MIS array. The depletion layer profile is approximated by a piecewise linear curve. Once the depletion layer profile is obtained, the surface potential is calculated by use of Green's function.

Discrete-time models:Bilinear transform and ramp approximation equivalence

Abstract: The problem of developing discrete-time models for continuous systems from integral approximations is considered. In particular, the model obtained by approximating the input by a piecewise linear function is developed. By an appropriate approximation of this model, an equivalence between it and the bilinear transform model is established.

Bayesian Acceptance-Sampling Schemes for Two-Sided Tests of the Mean of a Normal Distribution of Known Variance

Abstract: This article is concerned with the problem of deciding whether the mean θ of a normal distribution of known variance lies in a specified finite interval θ-, θ+). Consideration is given to prior information on θ and to quadratic and piecewise linear utility structures. Computer-aided methods are described for obtaining the optimal decision rule given a sample of observations and for obtaining the optimal sample size when the sampling cost is a linear function of the sample size. Some simpler approximate methods are also described.

On-line learning optimal control using successive approximation techniques

Abstract: The application of learning theory to on-line optimization of unknown or poorly defined plants is discussed An on-line optimization procedure is achieved by means of a learning algorithm which alters a trainable controller on the basis of an instantaneous performance criterion or subgoal The subgoal is related to the over-all goal, the integral cost, by means of successive approximations to the Hamilton-Jacobi equation The resulting piecewise linear controller is implemented by means of an encoder consisting of threshold logic units and a classifier consisting of a set of logic switching functions The classifier is determined by means of an algorithm developed by Arkadev and Braverman Features of the learning algorithm are illustrated by minimum-time and minimum-time-fuel problems

Least squares piecewise cubic curve fitting

Abstract: The matrices involved in a linear least squares formulation are determined for the problem of fitting piecewise cubic functions, those possessing a continuous derivative, to arrays of planar data.

Error Rates on a Data Link with Nonlinear Regeneration

Abstract: A model of nonlinear data wave regeneration without retiming is analyzed. Attention is focused on performance after regeneration and subsequent detection. The communication model consists of a pseudoternary signal that is regenerated at one data station, but not retimed prior to retransmission to another station. The signal regeneration is accomplished by a three-level slicer, and this nonlinear transformation degrades the overall system performance in the presence of noise. The extent of this degradation is our subject matter. The formulation and analysis are constructed so that one can put to effective use the fact that, for a certain Gaussian process, one can give definite expressions for the chance that its sample paths lie below a piecewise linear curve on an appropriately restricted time interval. We are able to use events of this type to give upper and lower bounds for the error rate. The bounds are compared with the error rate that would have been obtained had detection been accomplished before the nonlinear regeneration. We hope that the techniques employed here may be of use in other nonlinear communication problems.

Linear-harmonic generation

Abstract: A linear-harmonic generator is a nonlinear device in which an applied cosine wave of amplitude A causes some harmonic(s) of the input to be generated with amplitude(s) proportional to A. Such devices find application in phase-envelope processing in some types of modulation and speech-transmission systems. The present paper provides a theoretical study of the ability some characteristics have to generate linear harmonics. Linear-harmonic generation by single-valued characteristics which are representable by piecewise linearisation and those representable by a power series are investigated. It is shown that piecewise linear characteristics cannot generate linearly odd components. Also, it is shown that a characteristic expanded by a power series cannot generate linear harmonics, for either even or odd components. A particular single-valued characteristic, which precisely generates odd-harmonic components, is found, although it cannot be practically realised without approximation.

Structural Inelasticity. VII. Elastic-Plastic Analysis of a Wheel Rolling on a Rigid Track,

Abstract: : A consistent finite element model for a circular wheel is developed based on triangular and quasi-triangular domains and a piecewise linear displacement field. The minimum stress-rate principle of plasticity is used to obtain solutions of two-dimensional continuum problems with internal unloading. A piecewise approximation of the Tresca yield condition is used. Elastic-plastic solutions of a wheel rolling on a rigid track under its own weight and a hub load are obtained for the first few revolutions until a steady state condition is reached. Shakedown conditions for the wheel are demonstrated. (Author)

Median and linear loss-functions

Abstract: In this note the relationship between a median and absolute linear loss-functions is studied. The loss-function is assumed to be piecewise linear, but the slope is allowed to vary in the two directions. No restrictions are either made concerning the distribution-function of the random variable considered. The general ralationship is established and proved. Moreover one error arising in literature is corrected.

Piecewise-linear-signal representation

Abstract: A new method for representing a signal is provided. An exact representation of a piecewise linear approximation of the signal with a finite number of basis functions is presented.