Robust tests for time series comparison based on Laplace periodograms

Abstract

Statistical comparison of time series is useful for the detection of mechanical damage and many other real-world applications. New methods have been proposed to check whether two semi-stationary time series have the same normalized dynamics. The proposed methods differ from traditional methods in that they are based on the Laplace periodogram, which is a robust tool to analyze the serial dependence of time series. Via the method of estimating equations, a generalized score statistic and an order selected statistic are developed for the comparison. Their asymptotic distributions under the null are obtained. The proposed methods are applicable to compare two semi-stationary time series which may be dependent on each other. They also can be used to compare two time series whose traditional spectral densities or autocovariance structures may not exist. A Monte Carlo simulation study illustrates the validity of the asymptotic results and the finite sample performance. The proposed methods have been applied to an analysis of non-stationary vibration signals for mechanical damage detection.

Introduction

Mechanical systems such as turbo pumps, bridges, and power systems play a critical role in the world. However, damage occurs to these systems over time due to various reasons. Unexpected damage may cause the equipment breakdown, leading to financial losses or even personnel casualties. Early damage detection of mechanical systems has gained wide attention for the prevention of accidents. Usually, mechanical systems under abnormal situations are accompanied by changes in the dynamics of the vibration signals (Cempel and Tabaszewski, 2007). Due to their non-invasive characteristics and effectiveness, vibration-based damage detection methods are popular and useful. They are generally based on a comparison between vibration signals and a reference signal collected when the system is healthy. Many vibration-based damage detection algorithms are only valid for stationary signals. For example, Nair et al. (2006) used autoregressive moving average (ARMA) time series to model the vibration signals obtained from sensors for damage detection by checking the differences in the ARMA coefficients. In practice, the excitation condition such as input power can be time-varying for a mechanical system during regular operations, producing non-stationary vibration signals. However, Bartelmus and Zimroz (2009) mentioned that the spectral features of the vibration and the excitation condition are often linearly related in a specific range of operating conditions. One may transform a time-varying vibration signal to be a stationary one via normalizing its amplitude. However, the time-varying amplitude of a non-stationary signal is usually unknown, and it is not easy to obtain a good estimate. In this paper, new methods are proposed to compare the normalized dynamics of semi-stationary time series naturally, which can be useful for damage detection of some mechanical systems.

The comparison of time series has been widely studied for different purposes in various applications. Most of the existing methods have been developed to compare different time series through their spectral densities or the autocovariance functions. To compare two stationary time series which are independent of each other, the methods include Coates and Diggle (1986), Diggle and Fisher (1991), Fokianos and Savvides (2008), Lund et al. (2009), Dette and Paparoditis (2009), Decowski and Li (2015), Jin and Wang (2016), Grant and Quinn (2017) and others. For comparing stationary time series that are dependent on each other, the methods include Alonso and Maharaj (2006), Jin (2011), Jentsch and Pauly (2015) and Jin et al. (2019). In addition, methods such as Maharaj (2002), Salcedo et al. (2012) and Zhang and Tu (2018) can be used to compare non-stationary time series. Because it is easy to obtain a consistent variance estimate when a time series is stationary, there are no major differences between methods to compare the spectral densities or the normalized spectral densities for stationary time series. However, the situation could be different when the processes are non-stationary because the local normalization is complicated in general.

The periodogram of a time series plays an essential role in spectral analysis. It has been widely used to develop various statistical methods in the frequency domain (see e.g. Brockwell and Davis, 1991; Shao and Wu, 2007). Recently, with the periodogram, Huang et al., 2016, Huang et al., 2019 have also proposed some semi-parametric Whittle likelihood approaches for complex statistical models with dependent errors. The traditional periodograms are related to $L_2$ least-squares projections of the time series data on a harmonic basis. According to Kley et al. (2016), they have optimal properties for Gaussian processes, but they are not robust against outliers and cannot capture important dynamic features. To overcome these weaknesses, Li (2008), Hagemann (2011), Li (2012) and Kley et al. (2016) studied the Laplace periodogram, or the quantile periodogram, both of which are robust tools to analyze serial dependence. The least absolute deviations estimate is the same as the maximum likelihood estimate if the errors have a Laplace distribution in a regression setting. The Laplace periodogram can be derived via the robust least absolute deviations method in the harmonic regression procedure.

In this paper, new methods based on the Laplace periodogram are proposed to compare two semi-stationary time series in terms of their normalized dynamics. The proposed methods can be useful for damage detection, especially when the vibration signals may not be stationary. If the variance of a process is not finite, the existing methods do not work because the traditional spectral density cannot be defined. By inheriting some of the nice properties of the Laplace periodogram, the proposed methods are capable of dealing with heavy-tailed processes whose variances may not exist. According to Li (2008), the Laplace periodogram is connected to the zero-crossing information of a process. Therefore, the proposed methods naturally work for semi-stationary processes whose nonstationarity is only due to the time-varying magnitude. When two processes are not independent, it can be relatively difficult to obtain the likelihood functions. The proposed methods rely on the method of estimating equations and some empirical variance estimation so that they can be used to compare two dependent processes without the likelihood information.

The paper is organized as follows. Section 2 describes some notations, introduces the proposed statistics, and then studies their statistical properties. A simulation study is given in Section 3. In Section 4, the proposed methods are applied for damage detection of a mechanical system based on vibration data. Concluding remarks are given in Section 5.