A nonparametric test for slowly-varying nonstationarities
Introduction
Stationarity is a crucial assumption for many statistical models, but many real-world signals turn out to be nonstationary. For instance, rainfall [1] and sunspot [2] signals are real-world processes that commonly exhibit nonstationary behaviors. Assessment of stationarity is thus an important task in signal processing and time series analysis. The goal of this paper is to propose a method suitable for testing slow nonstationary evolutions. As a matter of fact, detecting such nonstationarities is especially challenging and most of traditional tests fail.
Several stationarity tests have been proposed in the last decades, some being rooted in time series modeling [3], [4], [5], spectral analysis [6], or detection of abrupt changes [7]. The emerging alternatives in the literature can be categorized into parametric and nonparametric approaches. The definition of nonparametric technique adopted in this work is that of a method which does assume any a priori functional form or parametric model for the input signal [8]. Therefore, even if the technique makes use of a particular window function to analyze the signal (which could be considered a priori a kind of model), if the methods requires no assumption regarding the distribution (Gaussian, gamma, etc.) or process model (TVAR, ARMA, etc.) of the input signal, then we consider this method as nonparametric. Nonparametric methods may be preferable in real-world applications, as the performance of a parametric method depends on the accuracy of the chosen model, which is hardly assessed for real-world data.
Methods for detecting slow nonstationary evolutions, however, are more common in the class of parametric techniques. Some parametric methods that have been proposed in the past years assume the underlying signal can be modeled by a time-varying autoregressive process (TVAR) [3], [4], [9], [10]. Among the nonparametric methods, the work in [11] has presented a technique for detecting changes in high-order statistics, whereas [12] has proposed a test for second-order stationarity in the TF domain. A common drawback of TF-based methods, however, is the computational load required to estimate full TF representations [13]. Moreover, traditional TF representations often estimate poorly the spectral content at very low frequencies [14]. The latter is an important issue if the goal is to detect slowly-varying nonstationarities. In this regard, the method of [12] has been modified in [15] so as to improve the detection of nonstationary signals with spectral content more concentrated at low frequencies. Some other methods have also been proposed in the literature to improve the resolution of TF transforms at low frequency bands [16], [17]. Unfortunately, these modifications end up increasing even more the computational complexity of the TF-based approaches.
In this paper, we show how a nonparametric test for slow nonstationary evolutions can be built in time domain, by developing a hypothesis test for the presence of trends in the marginal of the time-varying spectrum. A crucial point of the proposed technique is that we compute the time marginal directly in time domain, therefore avoiding the problems mentioned above involving TF representations. We remark that this paper is a modified and extended version of the work presented in [18]. Different from the present paper, the work in [18] has been built in TF domain and has not been developed as a proper hypothesis test. To build the new stationarity test, concepts like bootstrapping and GEV modeling are introduced. Furthermore, this paper develops the mathematics to describe the link between slowly-varying nonstationarities and trend-like structures in the time marginals. Also, the experimental study is extended significantly, by testing more nonstationary processes against a longer list of alternative methods.
This paper is organized as follows. In Section 2, we show how nonstationarities that vary slow in time can appear as trends in the time marginal. We define how to approximate these trends and assess their significance by means of the trend importance estimator. In Section 3, we describe the framework and the resampling method needed to build the hypothesis test. In Section 4, we analyze the behavior of the trend importance estimator in stationary and nonstationary situations. In Section 5, we propose a model for the distribution of this estimator under stationarity. This model allows for characterizing a hypothesis test for stationarity in Section 6. The experimental study and the conclusions are shown in Sections 7 and 8, respectively.
Section snippets
The rationale behind the framework
This paper proposes a method for detecting slowly-varying nonstationarities that can be seen as a trend in the time marginal y(n) of the time-varying spectrum of the signal. As illustration, Fig. 1 shows the time-varying spectra and the time marginals of a stationary Gaussian signal (x1(n)) and a nonstationary one (x2(n)), (Fig. 1a and d, respectively). The mean and variance of the nonstationary process start to increase slowly at . Notice the difference between the two TF representations (
Building the stationarity test with the trend c(n) and time marginal y(n)
The flowchart of the proposed stationarity test is shown in Fig. 3. More specifically, a one-sided hypothesis test is built, where the null hypothesis of stationarity refers to the situation where the approximated trend c(n) cannot be distinguished from spurious ones that could appear by resampling the original signal x(n) (i.e. by approximating virtual realizations of x(n)). These resamples are bootstrap replicates of x(n), which are supposed to be stationary, so trends in their time marginal
Behavior of the trend importance estimator
We analyze the behavior of in two situations. The first one stands for the case of a trendless time marginal (indicating stationarity), where a significant trend cannot be detected in y(n). The second one stands for the case of a trended time marginal (indicating nonstationarity), where a significant trend can be detected in y(n).
The generalized extreme value distribution
The bootstraps are assumed to be stationary. Thus, any resample with a significantly trended time marginal could be interpreted as an extreme event, very unlikely to be found in the stationarized data. Nevertheless, when these rare events occur, we will likely observe large values of as the ratio of energies in (28) will attain a peak value if a trend is detected. A heuristic argument suggests that the distribution under stationarity of could be modeled by a heavy tailed pdf with a
Hypothesis test
Having defined the distribution of for the stationary references, it is now simple to derive a threshold above which the null hypothesis of stationarity is rejected under a given level. Putting it in the form of a one-sided test, we use as test statistic the estimated importance of the trend in the original signal. The hypothesis test is given in (30), where the threshold τ is computed given a false alarm rate of 5%.
Alternative methods considered for evaluation
The proposed approach has been compared with other stationarity tests available in the literature, namely the nonstationarity detector proposed by S. Kay [3], the classical KPSS test [4], [35], and the TF approach of [12] based on surrogate resampling. The approach of [3] assumes the signal follows a time-varying autoregressive (TVAR) model. The rationale behind the technique is to test the nullity of certain TVAR parameters, which should be equal to zero in case of stationarity. The KPSS
Conclusions
We have proposed a nonparametric test for slowly-varying nonstationarities that can be seen as a trend in the time marginal series of the signal. Being nonparametric, this method does not require any a priori knowledge about the functional form of the signal, which makes it better suited to test real-world signals. In the proposed framework, we measure the importance of the trend in the time marginal of the original signal and its stationarized counterparts by using a proper test statistic. We
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