Spectral Density Of Time Series Research Articles

A maximum likelihood estimator for spectral models of acoustic noise processes

The power spectral density of time series from many acoustic phenomena can vary rapidly through several decades of magnitude, particularly for noise processes. This property can complicate evaluation of spectral density models with respect to a non-parametric estimate of the spectral density, also known as the periodogram. Typical measures, such as the sum of squares of the residuals, can suffer from bias errors and oversensitivity to large values. A log-likelihood function is here presented for a periodogram derived from a sequence of independent, circularly-symmetric complex normal Fourier components. The bias-corrected Akaike’s information criterion of an expected-value spectral model is shown to be a useful measure for multi-model selection. A maximum-likelihood estimator is formed for spectral model parameters with respect to a known periodogram, along with approximate confidence intervals for the parameter estimates. Results from the maximum-likelihood estimator are compared with weighted and unweig...

Power spectral based detection of brain activation from fMR images

An efficient method for detecting activation on single and multiple epoch functional MRI (fMRI) data based on power spectral density of time-series and hidden Markov model is presented. Conventional methods of analysis of fMRI data are generally based on time-domain correlation analysis concentrating mainly on the multiple epoch data and generally do not provide good results for single epoch data. The main focus of this study is the analysis of single epoch data, constrained by certain experiments such as pain response, sleep, administration of pharmacological agents, which can only have a single or very few stimulus cycles. Further, our method obviates the need to exclusively model the hemodynamic response function and correctly identifies the voxels with delayed activation. We demonstrate the efficacy of our method in detecting brain activation by using both synthetic and real fMRI data.

Bayesian Estimation of the Spectral Density of a Time Series

This article describes a Bayesian approach to estimating the spectral density of a stationary time series. A nonparametric prior on the spectral density is described through Bernstein polynomials. Because the actual likelihood is very complicated, a pseudoposterior distribution is obtained by updating the prior using the Whittle likelihood. A Markov chain Monte Carlo algorithm for sampling from this posterior distribution is described that is used for computing the posterior mean, variance, and other statistics. A consistency result is established for this pseudoposterior distribution that holds for a short-memory Gaussian time series and under some conditions on the prior. To prove this asymptotic result, a general consistency theorem of Schwartz is extended for a triangular array of independent, nonidentically distributed observations. This extension is also of independent interest. A simulation study is conducted to compare the proposed method with some existing methods. The method is illustrated with the well-studied sunspot dataset.

Data Driven Order Selection for Projection Estimator of the Spectral Density of Time Series with Long Range Dependence

Fractional exponential (FEXP) models have been introduced by Robinson (1991) and Beran (1993) to model the spectral density of a covariance stationary long‐range dependent process. In this class of models, the spectral density f(x) of the process is decomposed as f(x) = |1 − exp(ix)|−2df*(x), where f*(x) accounts for the short‐memory component. In this contribution, FEXP models are used to construct semi‐parametric estimates of the fractional differencing coefficient and of the spectral density, by considering an infinite Fourier series expansion of log f*(x). A data‐driven order selection procedure, adapted from the Mallows' Cp procedure, is proposed to determine the order of truncation. The optimality of the data‐driven procedure is established, under mild assumptions on the short‐memory component f*(x). A limited Monte‐Carlo experiment is presented to support our claims.

MODEL SELECTION AND ORDER DETERMINATION FOR TIME SERIES BY INFORMATION BETWEEN THE PAST AND THE FUTURE

Abstract. In this paper, the information between the past and the future of a Gaussian stationary sequence is calculated either by its spectral density or by its autocovariances, and is related to the problem of model fitting. It is demonstrated that the criterion of minimum mutual information is the generalization of that of maximum entropy. By employing the above information quantity, we propose a procedure, which is called LIC for simplicity, to obtain consistent estimate of the order of the Bloomfield model or the autoregressive model. In Monte Carlo studies, we illustrate the LIC procedure by several examples, and also estimate the spectral density of time series by the Bloomfield model and LIC method.

On a generalization of maximum entropy method for processing of nonstationary multichannel complex valued data

Maximum entropy method for estimating power spectral density of time series has been generalized to incorporate the multichannel, complex, and nonstationary characteristics. A relation between the prediction-error-filter and the output has been derived and the prediction-error-filter are obtained by designing Wiener filter for multichannel nonstationary complex valued data. The computational procedure for a two point long prediction-error-filter has been described as an example.