基于小波和差分的时间序列时变一阶和二阶结构建模
Modelling Time-Varying First and Second-Order Structure of Time Series via Wavelets and Differencing
摘要：
在实践中观察到的大多数时间序列表现出随时间变化的趋势（一阶）和自协方差（二阶）的行为。差分是一种常用的技术，以消除这种序列中的趋势，以估计（差分序列的）时变二阶结构。然而，我们通常需要对原始序列的二阶行为进行推断，例如，在进行趋势估计时。在这篇文章中，我们提出了一种方法，使用差分，联合估计的时变趋势和二阶结构的非平稳时间序列，在局部平稳小波建模框架。我们开发了一个基于小波的估计的二阶结构的原始时间序列的基础上的差分估计，并展示了如何将其纳入估计的趋势的时间序列。我们进行了模拟研究，以调查的方法的性能，并通过分析环境和生物医学科学的数据例子，证明该方法的实用性。
英文摘要：
Most time series observed in practice exhibit time-varying trend (first-order) and autocovariance (second-order) behaviour. Differencing is a commonly-used technique to remove the trend in such series, in order to estimate the time-varying second-order structure (of the differenced series). However, often we require inference on the second-order behaviour of the original series, for example, when performing trend estimation. In this article, we propose a method, using differencing, to jointly estimate the time-varying trend and second-order structure of a nonstationary time series, within the locally stationary wavelet modelling framework. We develop a wavelet-based estimator of the second-order structure of the original time series based on the differenced estimate, and show how this can be incorporated into the estimation of the trend of the time series. We perform a simulation study to investigate the performance of the methodology, and demonstrate the utility of the method by analysing data examples from environmental and biomedical science.
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