Bootstrapping the Local Periodogram of Locally Stationary Processes

Abstract. Locally stationary processes are non‐stationary stochastic processes the second‐order structure of which varies smoothly over time. In this paper, we develop a method to bootstrap the local periodogram of a locally stationary process. Our method generates pseudo local periodogram ordinates by combining a parametric time and non‐parametric frequency domain bootstrap approach. We first fit locally a time varying autoregressive model so as to capture the essential characteristics of the underlying process. A locally calculated non‐parametric correction in the frequency domain is then used so as to improve upon the locally parametric autoregressive fit. As an application, we investigate theoretically the asymptotic properties of the bootstrap method proposed applied to the class of local spectral means, local ratio statistics and local spectral density estimators. Some simulations demonstrate the ability of our method to give accurate estimates of the quantities of interest in finite sample situations and an application to a real‐life data‐set is presented.     

Spectral estimation of video signals

Spectrum analyzers are ubiquitous in laboratory work concerning one dimensional signals. This is because linear operators are best examined in the frequency domain. Linear operators, such as linear filters, DCT coders, line shufflers, etc., dominate also the video systems scenario. Their frequency domain study is as appropriate and informative as it is in the case of their one-dimensional counterparts. This paper considers the problems associated with the introduction of two well-known spectral estimation techniques, the periodogram and AR estimates, to the context of television signals. The potential for application of spectral estimation to video problems is exemplified by a number of applications related to the fields of enhanced quality television and HDTV. Special attention is paid to the computational aspects, whose effective solution conditions the practical applicability of the proposed spectral estimation techniques.R. Rinaldo is currently at the Department of Electrical and Computer Engineering of the University of California, Berkeley.     

PEAK-INSENSITIVE NON-PARAMETRIC SPECTRUM ESTIMATION

Abstract. We study the problem of non-parametric spectrum estimation of a stationary time series that might contain periodic components. In this case the periodogram ordinates have a significant amplitude at frequencies near the frequencies of the periodic components. These can be regarded as outliers in an asymptotically exponential sample. We develop a non-parametric estimator for the spectral density that is insensitive to these outliers in the frequency domain. This is done by robustifying the usual kernel estimator (smoothed periodogram) by means of M-estimation in the frequency domain. We propose to use data-tapered periodograms, which yield a drastic improvement of the procedure, typically for the contaminated situation. This is both shown theoretically and supported by means of simulation. We show consistency of the resulting estimator in the general case, and asymptotic normality in the special case of a Gaussian time series, whether contamination is present or not. Finally we illustrate the finite sample performance of the estimating procedure by some simulation results and by application to the Canadian lynx trappings data.     

Fuzzy clustering of time series in the frequency domain

Traditional and fuzzy cluster analyses are applicable to variables whose values are uncorrelated. Hence, in order to cluster time series data which are usually serially correlated, one needs to extract features from the time series, the values of which are uncorrelated. The periodogram which is an estimator of the spectral density function of a time series is a feature that can be used in the cluster analysis of time series because its ordinates are uncorrelated. Additionally, the normalized periodogram and the logarithm of the normalized periodogram are also features that can be used. In this paper, we consider a fuzzy clustering approach for time series based on the estimated cepstrum. The cepstrum is the spectrum of the logarithm of the spectral density function. We show in our simulation studies for the typical generating processes that have been considered, fuzzy clustering based on the cepstral coefficients performs very well compared to when it is based on other features.     

非高斯背景下的频域广义匹配检测

采用广义匹配滤波器在功率谱未知的非高斯噪声中检测宽带信号。在估计功率谱时,为了减小周期图法谱估计产生的误差,采用平滑周期图法来估计非高斯噪声的功率谱,并在此基础上对信号进行恒虚警概率损失下的广义匹配检测。实验表明,用平滑周期图法估计功率谱并进行的广义匹配检测,其检测性能稍弱于功率谱取真值时的检测,但远远优于传统的匹配滤波器。     

一种基于平均周期图的频域信噪比估计算法

在信噪比估计算法中,频域估计法是一种经典算法。为了降低算法复杂度并在低信噪比条件下准确估计,在频域估计法的基础上,结合平均周期图的思想,通过分段运算、累加合并的方法,给出一种改进的频域信噪比估计法。仿真结果表明,在实际信噪比为[-16 dB,-6 dB]时,估计均方误差不超过0.4 dB;通过与最大似然法仿真对比可知,该算法不受多普勒频移影响,应用范围更广。同时,该算法运算量少,复杂度低,易于在工程中实现。     

SPECTRAL ANALYSIS OF STATIONARY POINT PROCESSES USING THE FAST FOURIER TRANSFORM ALGORITHM

Abstract. In this paper we consider techniques of spectral analysis for stationary point processes in order to study the behaviour of a complex physiological system. The estimates of the power spectrum are obtained by smoothing the periodogram which is computed very rapidly with the help of the fast Fourier transform algorithm. In the computation of the estimates we can use either the whole record of the data or a number of disjoint records.     

AN ALTERNATIVE CONSISTENT PROCEDURE FOR DETECTING HIDDEN FREQUENCIES

Abstract. An alternative procedure is developed to detect the hidden frequencies in linear processes which differs from the procedure proposed earlier by the author. The advantage of the new procedure is that it enables us to detect a hidden frequency in the noise when the spectral density possesses high peaks. In such cases the earlier procedure often fails in practice. We also prove the convergence of a spectral estimate designed to reduce the influence of hidden frequencies.     

ON AN OPTIMALITY PROPERTY OF WHITTLE'S GAUSSIAN ESTIMATE OF THE PARAMETER OF THE SPECTRUM OF A TIME SERIES

Abstract. Whittle has obtained an optimality property of a method of estimation of the parameter of the spectrum. In this paper we present a proof in the vector parameter case. Gaussian estimation is a natural method to consider, based on finite Fourier transforms and periodograms.     

ON THE ROBUSTNESS TO SMALL TRENDS OF ESTIMATION BASED ON THE SMOOTHED PERIODOGRAM

Abstract. In this paper we are concerned with the robustness of inferences, carried out on a stationary process contaminated by a small trend, to this departure from stationarity. It is shown that a smoothed periodogram approach to model fining and parameter estimation is highly robust to the presence of a small trend if the underlying stationary process is short-range dependent. If the underlying process is long-range dependent the robustness properties are still good but now depend on the Hurst index of the process and deteriorate with increasing Hurst index.