Abstract: . The determination of the inverse autocorrelation function of a weakly stationary autoregressive process using the autocorrelation function is considered. Usually this is carried out either by using frequency domain methods or by solving first the parameters of the process and then using them. In this paper we give a simple formula by which the inverse autocorrelation function can be determined directly from the autocorrelation function.

Abstract: In a linear regression model even when the errors are autocorrelated and non-normal the ordinary least squares (OLS) estimator of the regression coefficients ( ) converges in probability to β. But the effects of autocorrelation among errors on this rate of convergence are unknown. In this paper, we investigate these effects for the case of a linear trend model. It is shown that the rate of convergence becomes faster if the autocorrelation is negative while it becomes slower if the autocorrelation is positive as compared to when the errors are independent. Thus, if the autocorrelation among errors is negative the consistency of is achieved by the same sample size (even less) as needed when the errors are independent. But if the autocorrelation is positive the sample size needed to achieve the consistency of increases with the intensity of autocorrelation and it can be extremely large for high positive autocorrelations.

Abstract: For noisy signals it is not enough to achieve a match between the first p+1 correlations of the measured data and that of the AR model. The higher order correlations begin to diverge as the level of data noise increases. In an effort to perform extended correlation matching we have found the need to use an Autocorrelation Distortion Function (ADF), defined as the difference between the sample acf of the noisy-data and the statistical acf of the true signal. Using the model bα|n|for the ADF we develop a procedure for improved AR modeling of a stationary time series. In this procedure we estimate the parameters of the ADF in such a way that while the autocorrelation of the signal model plus noise matches the data acf exactly for the first p+1 lags, simultaneously, the mismatch at certain higher-order lags is minimized. These higher-order lags are taken to be p+1, .... ,p+q or, alternatively, they are chosen according to a peak-picking scheme described in the paper.

Abstract: (1987). Charts for autocorrelation and partial-autocorrelation functions for the ar(2) and ma(2) processes. Communications in Statistics - Theory and Methods: Vol. 16, No. 9, pp. 2717-2725.

Abstract: In this paper, we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process. The standing assumption is the availability of an (incomplete) partial sequence of covariance samples of the process. We study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples. The classical theory of orthogonal polynomials (with respect to the unit circle) and the theory of moments have been utilized extensively in a variety of engineering problems, including the one we are dealing with. These have been known to provide a unifying framework for a variety of current spectral estimation techniques (maximum entropy method, Pisarenko's harmonic decomposition, etc.). In this work, we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence (and equal to the dimension of the maximum entropy realization). The ME solution is a point in this set. Other points correspond to "pole-zero" models. A general formula is obtained for recursively updated "pole-zero" models of dimension increasing with the data record. Information about the "zeros" is obtained from the asymptotic behavior of the "partial autocorrelation coefficients." Our approach combines techniques from the theory of orthogonal polynomials on the unit circle, the theory of moments, and also techniques from degree theory/topology. Our objective is to develop a theory that will provide a frame for constructing recursively "pole-zero" realizations of increasing dimension.

Abstract: The application of time series analysis methods to load forecasting is reviewed. It is shown than Box and Jenkins time series models, in particular, are well suited to this application. The logical and organized procedures for model development using the autocorrelation function and the partial autocorrelation function make these models particularly attractive. One of the drawbacks of these models is the inability to accurately represent the nonlinear relationship between load and temperature. A simple procedure for overcoming this difficulty is introduced, and several Box and Jenkins models are compared with a forecasting procedure currently used by a utility company.