estimating the frequency of a periodic function summary the problem of estimating the frequency and other parameters of a cyclical oscillation is considered where the data consists of a periodic function observed subject to stationary additive noise an estimation procedure is proposed and the asymptotic properties of the estimators established tests for unknown frequencies to be harmonics of a fundamental frequency are also developed and their asymptotic properties investigated the procedures are applied to observations on the variable star s carinae

nonparametric inference for periodic sequences this article proposes a nonparametric method for estimating the period and values of a periodic sequence when the data are evenly spaced in time the period is estimated by a leave out one cycle version of cross validation cv and complements the periodogram a widely used tool for period estimation the cv method is computationally simple and implicitly penalizes multiples of the smallest period leading to a virtually consistent estimator of integer periods this estimator is investigated both theoretically and by simulation we also propose a nonparametric test of the null hypothesis that the data have constant mean against the alternative that the sequence of means is periodic finally our methodology is demonstrated on three well known time series the sunspots and lynx trapping data and the el nino series of sea surface temperatures

an indirect boundary integral equation method for the biharmonic equation an integral equation method for the dirichlet problem for the biharmonic equation is proposed it leads to a 2 times 2 matrix integral equation system by taking suitable norms on the spaces of density functions the fredholm operator theory can be used to prove the solvability the kernels in this system are relatively complicated therefore especially when a high order polynomial approximation is used for a numerical purpose it is costly to evaluate the integrals that appear in the numerical system a discrete galerkin method that has shown superb convergence is proposed here as in k atkinson j integral equations appl 1 1988 pp 343 363 and elsewhere when the boundary functions are smooth exponential convergence is observed

new indirect scalar boundary integral equation formulas for the biharmonic equation we derive scalar boundary integral equation formulas for both interior and exterior biharmonic equations with the dirichlet boundary data they are based on indirect boundary integral equation formulas so called the chakrabarty and almansi formulas the scalar formulas are derived through an unconventional variational approach the unique solvability results of the formulas are also obtained

stability analysis of neural net controllers using fuzzy neural networks neural networks can only be trained with a crisp and finite data set therefore the approximation quality of a trained network is hard to verify so a common way in proving stability of a trained neural net controller is to demonstrate the existence of a lyapunov function in this article we propose a new method how stability of a neural net controller used as a closed loop feedback controller can be proven instead of finding a lyapunov function conditions for a fuzzy training set are developed if a fuzzy neural net is trained using this training set special properties of fuzzy neural nets can be used for estimating the generalization error after defuzzification of the fuzzy net finite stability of the process can be concluded

semiparametric estimation of the frequency of unknown periodic functions and its application to laser vibrometry signals we propose a semiparametric approach to fundamental frequency estimation of an unknown periodic signal in additive white noise based on model selection our estimator maximizes a penalized version of the cumulated periodogram and is proved to be consistent and asymptotically efficient under very general conditions when the number of observations is fixed an implementation of this estimation method is proposed and illustrated on specific synthetic signals which arise in laser vibrometry we extend this method for estimating the fundamental frequencies of two periodic functions having different fundamental frequencies when the data consist of their sum and additive white noise we also compare the performances of our procedure with the so called microdoppler technique which is commonly used for laser vibrometry signals analysis we show on simulated data that the penalized cumulated periodogram yields an accurate estimation of the frequencies at very low signal to noise ratios

nonparametric estimation of a periodic function summary motivated by applications to brightness data on periodic variable stars we study nonparametric methods for estimating both the period and the amplitude function from noisy observations of a periodic function made at irregularly spaced times it is shown that nonparametric estimators of period converge at parametric rates and attain a semiparametric lower bound which is the same if the shape of the periodic function is unknown as if it were known also first order properties of nonparametric estimators of the amplitude function are identical to those that would obtain if the period were known numerical simulations and applications to real data show the method to work well in practice

nonparametric methods for deconvolving multiperiodic functions summary multiperiodic functions or functions that can be represented as finite additive mixtures of periodic functions arise in problems related to stellar radiation there they represent the overall variation in radiation intensity with time the individual periodic components generally correspond to different sources of radiation and have intrinsic physical meaning provided that they can be deconvolved from the mixture we suggest a combination of kernel and orthogonal series methods for performing the deconvolution and we show how to estimate both the sequence of periods and the periodic functions themselves we pay particular attention to the issue of identifiability in a nonparametric sense of the components this aspect of the problem is shown to exhibit particularly unusual features and to have connections to number theory the matter of rates of convergence of estimators also has links there although we show that the rate of convergence problem can be treated from a relatively conventional viewpoint by considering an appropriate prior distribution for the periods

stability analysis of dynamical neural networks in this paper we use the matrix measure technique to study the stability of dynamical neural networks testable conditions for global exponential stability of nonlinear dynamical systems and dynamical neural networks are given it shows how a few well known results can be unified and generalized in a straightforward way local exponential stability of a class of dynamical neural networks is also studied we point out that the local exponential stability of any equilibrium point of dynamical neural networks is equivalent to the stability of the linearized system around that equilibrium point from this some well known and new sufficient conditions for local exponential stability of neural networks are obtained

solution of elliptic partial differential equations by an optimization based domain decomposition method an optimization based non overlapping domain decomposition method for the solution of elliptic partial differential equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the partial differential equations the method is reformulated as a linear least squares problem the latter is examined and a conjugate gradient method for its solution is presented and analyzed the results of some numerical experiments are also given

nonparametric estimation of a periodic sequence in the presence of a smooth trend in this paper we study a nonparametric regression model including a periodic component a smooth trend function and a stochastic error term we propose a procedure to estimate the unknown period and the function values of the periodic component as well as the nonparametric trend function the theoretical part of the paper establishes the asymptotic properties of our estimators in particular we show that our estimator of the period is consistent in addition we derive the convergence rates as well as the limiting distributions of our estimators of the periodic component and the trend function the asymptotic results are complemented with a simulation study that investigates the small sample behaviour of our procedure finally we illustrate our method by applying it to a series of global temperature anomalies

can neural nets be universal approximators for fuzzy functions abstract we first argue that the extension principle is too computationally involved to be an efficient way for a computer to evaluate fuzzy functions we then suggest using cuts and interval arithmetic to compute the valued of fuzzy functions using the method of computing fuzzy functions we then show that neural nets are universal approximators for computable fuzzy functions when we only input non negative or non positive fuzzy numbers

a learning algorithm of fuzzy neural networks with triangular fuzzy weights abstract in this paper first we propose an architecture of fuzzy neural networks with triangular fuzzy weights the proposed fuzzy neural network can handle fuzzy input vectors as well as real input vectors in both cases outputs from the fuzzy neural network are fuzzy vectors the input output relation of each unit of the fuzzy neural network is defined by the extension principle of zadeh next we define a cost function for the level sets i e cuts of fuzzy outputs and fuzzy targets then we derive a learning algorithm from the cost function for adjusting three parameters of each triangular fuzzy weight finally we illustrate our approach by computer simulations on numerical examples

on the existence of positive solutions for nonlinear differential equations with periodic boundary conditions we prove the existence of positive solutions of second order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions obtained results yield positive periodic solutions of the equation on the whole real axis provided that the coefficients are periodic

detecting periodic components in a white gaussian time series abstract a family of tests for periodic components in a white gaussian series is proposed the test is based on a statistic which is proportional to the ratio of the maximum periodogram to the trimmed mean of the periodograms the asymptotic distribution of the statistic is obtained it is shown that the test proposed and fisher s test have the same asymptotic powers at the alternative hypotheses that the series contains a single periodic component at a non zero fourier frequency the tests are applied to detect the eigenfrequencies of the earth the proposed test detects some peaks that fisher s test fails to detect

two dimensional multiple non homogeneous harmonic equation and its boundary integral equations the paper discusses 2 dimensional multiple non homogeneous harmonic equation and its boundary integral equation and gives recurrence formulae of fundamental solution under the assumption that non homogeneous term is m degree harmonic the integral term in domain is shifted into boundary integral therefore the boundary integral equations are obtained that the integral in domain is not included and the special solution can be expressed by boundary integrals furthermore the error and convergence analysis are given by taylor s polynomial approximating non homogeneous term f x and analytic expressions of boundary integral for 2 dimensional harmonic equation are given in local right angle coordinate system

semiparametric bayesian inference for time series with mixed spectra summary a bayesian analysis is presented of a time series which is the sum of a stationary component with a smooth spectral density and a deterministic component consisting of a linear combination of a trend and periodic terms the periodic terms may have known or unknown frequencies the advantage of our approach is that different features of the data such as the regression parameters the spectral density unknown frequencies and missing observations are combined in a hierarchical bayesian framework and estimated simultaneously a bayesian test to detect deterministic components in the data is also constructed by using an asymptotic approximation to the likelihood the computation is carried out efficiently using the markov chain monte carlo method in o mn operations where n is the sample size and m is the number of iterations we show empirically that our approach works well on real and simulated samples

scalar boundary integral equation formulas for the biharmonic equation numerical experiments we consider numerical methods of the four scalar integral equation formulas for the biharmonic equation suggested by the author in an earlier paper the numerical methods are fully discrete collocation methods based on technique of singularity subtraction and the rectangular quadrature rule the numerical methods are computationally effective for all four formulas when the domain has a smooth boundary when the boundary has corners mesh grading of sufficient order yields desired convergences

using the periodogram to estimate period in nonparametric regression summary properties of the periodogram are seldom studied in the setting of nonparametric regression although that is the context in which the periodogram is widely applied in astronomy there it is a competitor with more recent least squares methods the periodogram has the advantage of providing significant graphical insight into statistical and numerical aspects of the problem however as we show in the present paper it also has drawbacks the estimator that it produces has somewhat higher variance than its least squares counterpart and a periodogram based approach is more prone to suffer difficulties caused by periodicity of the observation schedule while the periodogram remains a very attractive tool the information provided in this paper allows users to assess more readily the extent to which it can be relied upon in a nonparametric setting this aspect of our contributions is discussed theoretically and illustrated by numerical studies involving a real dataset

stability and the fuzzy controller abstract this paper corrects a part of the main result theorem 1 in buckley fuzzy sets and systems 77 1996 167 173