Spectral measures of PARMA sequences

This paper presents a constructive neural network model for seasonal streamflow forecasting. This surface water hydrology is basic to the design and operation of the reservoir. A good example is the operation of a reservoir with an uncontrolled inflow but having a means of regulating the outflow. If information on the nature of the inflow is determinable in advance, then the reservoir can be operated by some decision rule to minimize downstream flood damage. For this reasons, several companies in the Brazilian Electrical Sector use the linear time-series models such as PARMA (Periodic Auto regressive Moving Average) models developed by Box-Jenkins. This paper provides for river flow prediction a numerical comparison between neural networks, called nonlinear sigmoidal regression blocks networks (NSRBN) and PARMA models. The model was implemented to forecast weekly average inflow on an step-ahead basis. It was tested on four hydroelectric plants located in different river basins in Brazil. The results obtained in the evaluation of the performance of NSRBN were better than the results obtained with PARMA models.

Lower order periodic stochastic models such as periodic autoregressive (PAR) and periodic autoregressive moving average (PARMA) models with and without harmonics are fitted to the monthly streamflows into Krishnarajasagara (KRS) reservoir located at the upper reaches of the river Cauvery in South India and a comparison is done between these models in terms of preservation of periodic statistics as well as over-year drought characteristics. Further, the comparison has been extended to the Thomas-Fiering lognormal 3-parameter (TFLN3) model which is supposed to be quite simple in structure and at the same time efficient in preservation of low flows. It is found that TFLN3 is able to reproduce the first two moments very well but, not the skewness or periodic correlations. On the other hand, PAR(2) and PARMA(1,1) models are able to preserve the periodic correlations. The periodic models which are not subjected to Fourier fitting, report lower relative root mean square error in overall standard deviation and correlations, compared to the ones which are subjected to Fourier fitting. From the over-year drought analysis, it is observed that all the stochastic models fitted underestimate severe drought durations corresponding to higher truncation levels. The over-year drought characteristics are computed using the method suggested by Wang and Salas (1989), based on one hundred periodic synthetic flow sequences generated from the different models. A comparative study of the preservation of drought characteristics corresponding to various truncation levels and return periods is presented.

ABSTRACTResults involving correlation properties and parameter estimation for autoregressive‐moving average models with periodic parameters are presented. A multivariate representation of the PARMA model is used to derive parameter space restrictions and difference equations for the periodic autocorrelations. Close approximation to the likelihood function for Gaussian PARMA processes results in efficient maximum‐likelihood estimation procedures. Terms in the Fourier expansion of the parameters are sequentially included, and a selection criterion is given for determining the optimal number of harmonics to be included. Application of the techniques is demonstrated through analysis of a monthly streamflow time series.

This paper studies correlation and partial autocorrelation properties of periodic autoregressive moving‐average (PARMA) time series models. An efficient algorithm to compute PARMA autocovariances is first derived. An innovations based algorithm to compute partial autocorrelations for a general periodic series is then developed. Finally, periodic moving averages and autoregressions are characterized as periodically stationary series whose autocovariances and partial autocorrelations, respectively, are zero at all lags that exceed some periodically varying threshold.

In this paper we study the weighted least absolute deviations estimator (WLADE) for Periodic ARMA (PARMA) models with unspecified noises, asymptotic normality of the estimator is derived. The random weighting approach is proposed to estimate the asymptotic covariance matrices of WLADE. Simulation study is carried out to examine the performance of the proposed procedure.

A useful class of models for seasonal time series that cannot be filtered or standardized to achieve second-order stationarity is that of periodic autoregressive moving average (PARMA) models, which are extensions of ARMA models that allow periodic (seasonal) parameters. An approximation to the exact likelihood for Gaussian PARMA processes is developed, and a straightforward algorithm for its maximization is presented. The algorithm is tested on several periodic ARMA(l, 1) models through simulation studies and is compared to moment estimation via the seasonal Yule–Walker equations. Applicability of the technique is demonstrated through an analysis of a seasonal stream-flow series from the Rio Caroni River in Venezuela.

For analysis and design of water resource systems, it is sometimes useful to generate high-resolution (e.g., weekly) synthetic river flows. Periodic autoregressive moving average (PARMA) time series models provide a powerful tool for generating synthetic flows. Periodically stationary models are indicated when the basic statistics (mean, variance, and autocorrelation) of the time series exhibit significant seasonal variations. Parameter estimation for high-resolution PARMA models involves numerous parameters, which can lead to overfitting. Thus, this paper develops a parsimonious method of parameter fitting for high-resolution PARMA models, using discrete Fourier transforms to represent the set of periodic autoregressive and moving average model coefficients. Model parameters are computed via the innovations algorithm, and the asymptotic distributions of the discrete Fourier transform coefficients are obtained. Those asymptotic results are useful to determine the statistically significant Fourier coeffici...

In this article, we derive the asymptotic distributions of residual and normalized residual empirical autocovariances and autocorrelations of (parsimonious) periodic autoregressive moving‐average (PARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. We then deduce the modified portmanteau statistics. We establish the asymptotic behavior of the proposed statistics. It is shown that the asymptotic distribution of the modified portmanteau tests is that of a weighted sum of independent chi‐squared random variables, which can be different from the usual chi‐squared approximation used under independent and identically distributed assumption on the noise. We also propose another test based on a self‐normalization approach to check the adequacy of PARMA models. A set of Monte Carlo experiments and an application to financial data are presented.

Existence of positive periodic solutions of several types of biological models with periodic coefficients

Projections on Convex Sets in Hilbert Space and Spectral Theory: Part I. Projections on Convex Sets: Part II. Spectral Theory

By using the semi-discretization technique of differential equations, the discrete analogue of a kind of cellular neural networks with stochastic perturbations and fuzzy operations is formulated, which gives a more accurate characterization for continuous-time models than that by Euler scheme. Firstly, the existence of at least one p-th mean almost periodic sequence solution of the semi-discrete stochastic models with almost periodic coefficients is investigated by using Minkowski inequality, Hölder inequality and Krasnoselskii’s fixed point theorem. Secondly, the p-th moment global exponential stability of the semi-discrete stochastic models is also studied by using some analytical skills and the proof of contradiction. Finally, a problem of stochastic stabilization for discrete cellular neural networks is studied.

This paper studies the existence of subharmonics of arbitrary order in a generalized class of non-autonomous predator-prey systems of Volterra type with periodic coefficients. When the model is non-degenerate it is shown that the Poincaré–Birkhoff twist theorem can be applied to get the existence of subharmonics of arbitrary order. However, in the degenerate models, whether or not the twist theorem can be applied to get subharmonics of a given order might depend on the particular nodal behavior of the several weight function-coefficients involved in the setting of the model. Finally, in order to analyze how the subharmonics might be lost as the model degenerates, the exact point-wise behavior of the $ T $-periodic solutions of a non-degenerate model is ascertained as a perturbation parameter makes it degenerate.

\n Existence and stability of periodic solutions are studied for a system of delay\n differential equations with two delays, with periodic coefficients. It models the\n evolution of hematopoietic stem cells and mature neutrophil cells in chronic myelogenous\n leukemia under a periodic treatment that acts only on mature cells. Existence of a guiding\n function leads to the proof of the existence of a strictly positive periodic solution by a\n theorem of Krasnoselskii. The stability of this solution is analysed.\n

The paper is devoted to studying the behavior of solutions of the Cauchy problem for large values of time—more precisely, obtaining an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients for large values of the time parameter t. To obtain this asymptotic expansion as t→∞, methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of a non-positive Hill operator with periodic coefficients.

Exponential periodic attractor of impulsive BAM networks with finite distributed delays