Wavelet estimation of long-range dependent processes

具有长期相关性的时间序列是近年来研究的热点。它们出现在许多应用中，例如，在计算机网络流量分析中。在有限方差时间序列中，远程相关性的特征是协方差函数，随着滞后的增加，协方差函数缓慢地减小到0。这种下降是如此缓慢，以至于相应的频谱密度在非常低的频率上爆炸，这种现象也被称为“长距离依赖”、“长记忆”或“1/f噪声”。有许多经验程序本质上是图形化的，但严格的估计程序很少。较好的方法是使用半参数方法，因为远程依赖不涉及短期依赖结构。最成功的方法是基于傅里叶的。一种方法是对周期图的对数进行回归。另一种是当地的惠特尔方法。它是拟似然方法，仅在低频时使用周期图。研究者建议使用伪似然方法，而不是基于傅立叶谱，而是基于小波。傅里叶小波的优点是，如果时间序列不是平稳的，就不需要对它们进行差分，而且一般来说，它们对偏离平稳的鲁棒性要强得多。由于小波与尺度相关，因此，尝试使用小波来估计远程依赖的强度是很自然的。小波已经被成功地用作周期图对数回归的替代方法，但小波还没有被用作基于傅立叶的局部惠特尔方法的替代方法。调查员建议这样做。这将提供一种鲁棒的基于半参数伪似然的方法来估计远程依赖的强度。在过去的几十年里，时间序列分析的主题引发了相当多的研究活动。从本质上讲，它关注的是随着时间推移对各种现象的测量——从年收入、汇率到河流的水位——其目标是开发合适的模型，并为这些测量获得准确的预测，其核心成分是时间依赖性的概念。Benoit Mandelbrot在60年代建议使用涉及长期依赖的模型。简而言之，长期依赖影响的现象是，现在和过去之间的相关性随着时间的推移而缓慢衰减，因此不能轻易忽略。具有长期相关性的时间序列及其变化已被用于水文学、地球物理学和生物物理学，以及最近的金融和计算机网络流量分析。因此，能够有效地估计远程依赖的强度是很重要的。本研究的目的是开发新的方法来实现这一目标。它是基于小波的，因此对趋势和其他偏离模型的影响不敏感。

Time series with long-range dependence have been recently the focus ofmuch attention. They appear in a number of applications, for example,in the analysis of traffic in computer networks. In finite variancetime series, long-range dependence is characterized by a covariancefunction which decreases slowly to 0 as the lag increases. Thedecrease is so slow that the corresponding spectral density blows upat very low frequencies, a phenomenon also known as ''long-rangedependence'', ``long memory'' or ``1/f noise''. There are manyempirical procedures which are graphical in nature but rigorousestimation procedures are few. The better ones use a semi-parametricapproach because long-range dependence does not involve the shortrange dependence structure. The most successful methods have beenFourier-based. One approach involves a regression on the logarithm ofthe periodogram. Another is the local Whittle approach. It is apseudo-likelihood approach which uses the periodogram at lowfrequencies only. The investigator proposes to use a pseudo-likelihoodapproach based not on the Fourier spectrum but on wavelets. Theadvantage of wavelets on Fourier is that there is no need todifference the time series if these are not stationary and, ingeneral, they are much more robust against departure fromstationarity. Since wavelets are associated with scaling it is,moreover, natural to attempt to use wavelets in order to estimate theintensity of long-range dependence. Wavelets have been successfullyused as an alternative to the regression on the logarithm of theperiodogram but wavelets have not beenused as an alternative to the Fourier-based local Whittle approach.The investigator proposes to do so. This would provide a robustsemi-parametic pseudo-likelihood based method to estimate theintensity of long-range dependence.The subject of time series analysis has sparked considerable researchactivity over the past several decades. Essentially concerned withmeasurements over time of various kinds of phenomena --- from yearlyincome, exchange rates, to the level of a river --- the goal is todevelop suitable models and obtain accurate predictions for suchmeasurements, the core ingredient being the notion of time dependence.Benoit Mandelbrot in the sixties suggested using models involvinglong-range dependence. Long-range dependence, to put it concisely,affects phenomena in which correlations between the present and thepast decay slowly with time and thus they cannot be easily ignored.Time series with long-range dependence and their variations have beenused in hydrology, geophysics and biophysics, and more recently, infinance and in analyzing traffic in computer networks. It is thusimportant to be able to estimate effectively the intensity oflong-range dependence. The purpose of this research is to develop anew methodology to achieve this goal. It is wavelet-based and henceinsensitive to the effects of trends and other deviations from themodel.