Abstract
Signals are often destroyed by various kinds of noises. A common way to statistically assess the significance of a broad spectral peak in signals and the synchronization between signals is to compare with simple noise processes. At present, wavelet analysis of red noise is studied limitedly and there is no general formula on the distribution of the wavelet power spectrum of red noise. Moreover, the distribution of the wavelet phase of red noise is also unknown. In this paper, for any given real/analytic wavelet, we will use a rigorous statistical framework to obtain the distribution of the wavelet power spectrum and wavelet phase of red noise and apply these formulas in climate diagnosis.
1. Introduction
Signals are often destroyed by various kinds of noises during the process of generation, transportation, and processing [1, 2]. A common way to statistically assess the significance of a broad Fourier/wavelet spectral peak in signals [3, 4] and the synchronization between signals is to compare with a white/red noise process [4, 5]. White noise has zero mean, has constant variance, and is uncorrelated in time. As its name suggests, white noise has a power spectrum which is uniformly spread across all allowable frequencies. Different from white noise, red noise has a power spectrum weighted toward low frequencies and is serially correlated in time. Red noise can describe climatic background noise with relatively enhanced low-frequency fluctuations arising from the interaction of white noise forcing with the slow-response components in the earth system (e.g., the thermal inertia of the oceans provides memory, effectively integrating atmospheric “weather” forcing [6]). In practice, the red noise model has always provided a reasonable description of the noise spectra for a variety of climatic and hydrological time series [5–9]. The red noise is also an important noise model in outputs of the feedback system [4], the neural network coupled with genetic algorithm [10], the optimization system in random scenarios [11, 12], and the big data processing system [5, 13].
Wavelet analysis is a very popular analysis tool for a wide range of applications, including time-frequency analysis, feature extraction, statistical estimation, and denoising. A wavelet is a waveform-like function which has zero mean and is localized in both time and frequency space. The wavelet transform is the set of inner products of all dilated and translated wavelets with a signal; in detail, the wavelet transform of a discrete signal with time step is defined as follows [14, 15]:where is the conjugate of . is called the wavelet power spectrum, and is called the wavelet phase. Due to time and frequency localization of the wavelet, the wavelet transform can extract localized intermittent periodicity of any signal very well.
Wavelet analysis of white noise has been widely studied (e.g., [16–19]), while wavelet analysis of red noise is studied limitedly, only using some specific wavelets (Morlet/Paul/DOG wavelets [15] and modulated Haar wavelet [20]). Since 1980s, a large family of wavelets with nice properties has been constructed [14]. However, there is no general formula on the distribution of the wavelet power spectrum of red noise. On the other hand, the wavelet phase can be used to test the synchronization between signals, but the distribution of the wavelet phase of red noise is also unknown. In this paper, for any given real/analytic wavelet, we will use a rigorous statistical framework to derive a general formula for the distribution of the wavelet power spectrum and wavelet phase of red noise.
2. Background
The simplest red noise model is the lag-1 autoregressive (AR (1)) red noise process [1, 4–6, 15]: Let with time step be an AR (1) red noise with parameters and ; then,where is a Gaussian white noise with mean 0 and variance and . The discrete Fourier transform of an AR (1) red noise iswhere . Again, by equation (1), the wavelet transform of an AR (1) red noise can be expressed as [15]where , and (here, we have fixed some typos according to Torrence and Compo’s Matlab codes [15]).
Torrence and Compo [15] gave an empirical distribution of the Morlet wavelet power spectrum of an AR (1) red noise:where is the chi-square distribution with two degrees of freedom and is the variance of the AR (1) red noise . For Paul/DOG wavelet power spectrum of an AR (1) red noise, Torrence and Compo [15] empirically gave the following:
Formulas (5) and (6) are empirically obtained by assuming that “the wavelet spectrum of a red noise is distributed as its Fourier spectrum” [15].
Zhang and Jorgensen [20] showed that under the rigorous statistical framework, the modulated Haar wavelet power spectra of AR (1) red noise is distributed aswhere and are two independent standard Gaussian distributions. Also, and
Until now, wavelet analysis of red noise is studied limitedly, only using some specific wavelets. There is no general formula on the distribution of the wavelet power spectrum and wavelet phase of an AR (1) red noise.
3. Main Results
We will use a rigorous statistical framework to establish the distribution of the wavelet power spectrum and wavelet phase of an AR (1) red noise with time step , length , and parameters and . Here, we assume a very weak condition: and . In practice, all climatic background noises satisfy this condition, e.g., if and , then and .
A wavelet is real if it is a real-valued function. All of spline wavelets and most of compactly supported wavelets are real wavelets.
Theorem 1. For any real wavelet, the wavelet power spectrum of an AR (1) red noise is distributed asA wavelet is said to be analytic if its Fourier transform satisfies . Different from real wavelets, analytic wavelets can extract not only spectral information but also phase information from any given signal.
Theorem 2. For any analytic wavelet, the wavelet power spectrum of an AR (1) red noise is distributed as
Theorem 3. For any analytic wavelet, the wavelet phase of an AR (1) red noise is distributed uniformly on .
4. Proofs
The AR (1) red noise with parameters and can be expressed as
By ,and so the real part of the wavelet transform is
Since is a linear combination of , is a Gaussian random variable and
Similarly, is a Gaussian random variable and
Denote . Then, , where . By equation (4), we have
Let
Then, the variances of and and their correlation are
From these, we see that the computations of variances and the correlation of and are reduced to the computations of , , and .
4.1. Computation of
By equation (3), we havewhere . Furthermore,
By equation (11), it follows thatwhere
Hence, we have for ,
Similarly, for ,
From these, we obtainwhere
We first compute . By using Euler formula , the inner summation becomeswhere and
From this and equation (25), we obtain
For , when ,
For , we havewhere
Noticing that , we consider four cases to compute and then estimate :(i)For and , by and , we have and then .(ii)For , we obtain and then Therefore,(iii)For and , we obtain Therefore,(iv)For , we obtain and so
Now, we compute in equation (25). When , it follows that
By equation (25), we obtainwhere
Furthermore,
Similar to the estimation process of , we have
Combining this and equations (29)–(42), noticing that , it follows that
Noticing and , we have
Again, by equation (25), we obtainwhere is a constant.
Finally, we obtain the following.
Lemma 1. When and , the series satisfy
4.2. Computation of
By equation (3), we have
Similar to the deducing process of equation (25), we have
Noticing thatwe obtainwhere
We estimate in equation (51):
Notice that
For and , we haveand so .
For , we have
For and , we have
For , .
We estimate in equation (51). Since , it follows that
Furthermore,
By equation (55), it follows that .
For ,
From all of the above, it follows that
Therefore, we have the following.
Lemma 2. When , the series satisfy
4.3. Computation of
By equation (3), we have
Similar to the deducing process of equation (25), we obtain
Sincewherewe have
Furthermore,
Noticing that , we obtain
For , we obtainwhere
Sincewe have the following.
Lemma 3. When , the series satisfy
4.4. Proof of Theorem 1
Without loss of generalization, we assume that is an odd number. By equation (4), we obtainwhere
Since is real, we have . This implies that
Since is real-valued, ; by , and , we obtain
Therefore, by equation (77),
Hence, . From this, , and equation (76), it follows that
Since is a Gaussian random variable with mean 0, we obtain Theorem 1.
4.5. Proof of Theorems 2 and 3
Without loss of generalization, we assume that is an old number. By equation (4), for , we have
Since is an analytic wavelet, . Notice that . The second term on the right-hand side of equation (83) is vanishing. Therefore,
Furthermore,
However, by Lemma 3 and equation (77), it follows that
Similarly, we have
By Lemma 3,
This implies that
Therefore, we have
Now, we prove that and are independent.
Since both and are Gaussian random variables with mean 0 andwherewe obtain Theorem 2.
Consider the phase . Notice that the p.d.f.s of and are, respectively,and and are independent, the joint p.d.f. of and is
The probabilitywhere the domain . Hence,i.e., Theorem 3 holds.
5. Climate Diagnosis
Climatic background noises are often modeled as red noise [4–9]. In order to extract instinct features of climatic time series, we can use Theorems 1 and 2 to compare the wavelet power spectrum of climatic time series with that of red noise. If the values of the wavelet power spectrum of climate time series at some regions are all outside the 95% confidence interval for the distribution for the wavelet power spectrum of the red noise, then the wavelet power spectrum at this region contain instinct features of climatic time series. Moreover, we can use Theorem 3 to test the synchronization between the two climatic time series. If the difference of the wavelet phase of the two climatic time series does not change in some significant regions of the wavelet power spectrum, noticing that Theorem 3 indicates that this cannot be caused by climatic background noise effects, these two climatic time series demonstrate synchronization.
Data Availability
No data were used to support this research.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This research was partially supported by the European Commissions Horizon 2020 Framework Program under No. 861584 and Taishan Distinguished Professor Fund.