Wavelet Analysis of Red Noise and Its Application in Climate Diagnosis

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.


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][6][7][8][9]. e 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. e 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 X � x k k�0,...,N− 1 with time step δt is defined as follows [14,15]: where ψ is the conjugate of ψ. |W n (s)| 2 is called the wavelet power spectrum, and Arg(W n (s)) 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][17][18][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.
Torrence and Compo [15] gave an empirical distribution of the Morlet wavelet power spectrum of an AR (1) red noise: where χ 2 2 is the chi-square distribution with two degrees of freedom and σ 2 is the variance of the AR (1) red noise (σ 2 ≈ σ 2 /(1 − λ 2 )). 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 as where χ 1 and χ 2 are two independent standard Gaussian distributions. Also, Until now, wavelet analysis of red noise is studied limitedly, only using some specific wavelets.
ere is no general formula on the distribution of the wavelet power spectrum and wavelet phase of an AR (1) red noise.

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 δt, length N, and parameters λ and σ 2 . Here, we assume a very weak condition: λ N ≈ 0 and (1/N 2 ) ≈ 0. In practice, all climatic background noises satisfy this condition, e.g., if λ � 0.5 and N � 50, then λ N ≈ 10 − 8 and (1/N 2 ) � 0.0004.
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 |W n (s)| 2 of an AR (1) red noise is distributed as
A wavelet ψ is said to be analytic if its Fourier transform satisfies ψ(ω) � 0, ω < 0. Different from real wavelets, analytic wavelets can extract not only spectral information but also phase information from any given signal.

Proofs
e AR (1) red noise with parameters λ, |λ| < 1 and σ 2 can be expressed as and so the real part of the wavelet transform is Since Re(W n (s)) is a linear combination of x m , Re(W n (s)) is a Gaussian random variable and Similarly, Im(W n (s)) is a Gaussian random variable and en, the variances of Re(W n (s)) and Im(W n (s)) and their correlation are ψ sω j ψ sω j α k,j cos η n,k cos η n,j − β k,j sin η n,k cos η n,j − β j,k cos η n,k sin η n,j + c k,j sin η n,k sin η n,j , ψ sω j ψ sω j α k,j sin η n,k sin η n,j + β k,j cos η n,k sin η n,j + β j,k sin η n,k sin η n,j + c k,j cos η n,k cos η n,j , ψ sω k ψ sω j α k,j cos η n,k sin η n,j + β k,j sin η n,k sin η n,j − β j,k cos η n,k cos η n,j − c k,j sin η n,k cos η n,j .

(18)
From these, we see that the computations of variances and the correlation of Re(W n (s)) and Im(W n (s)) are reduced to the computations of α k,j , β k,j , and c k,j .

Computation of α k,j . By equation (3), we have
Mathematical Problems in Engineering By equation (11), it follows that where Hence, we have for l > p, Similarly, for l ≤ p, From these, we obtain where We first compute Σ 1 . By using Euler formula cos(pθ k ) � Re(e ipθ k ), the inner summation becomes where From this and equation (25), we obtain For Σ 11 , when λ N ≈ 0, For Σ 13 , we have where 4

Computation of c k,j . By equation (3), we have
Similar to the deducing process of equation (25), we have (51) Noticing that

Proof of eorems 2 and 3.
Without loss of generalization, we assume that N is an old number. By equation (4) x k ψ sω k e inθ k .

(83)
Since ψ is an analytic wavelet, ψ(ω) � 0, ω < 0. Notice that ω k < 0, k > (N/2). e second term on the right-hand side of equation (83)  r k r j E Im x k e iη n,k Im x j e iη n,j . (88) By Lemma 3, E Im x k e iη n,k Im x j e iη n,j � E Re x k Re x j sin η n,k sin η n,j + E Im x k Im x j cos η n,k cos η n,j .
is implies that Now, we prove that Re(W n (s)) and Im(W n (s)) are independent.
By Lemmas 1-3, r k r j E Re x k Re x j cos η n,k sin η n,j − E Im x k Im x j sin η n,k cos η n,j Since both Re(W n (s)) and Im(W n (s)) are Gaussian random variables with mean 0 and Var Re W n (s) � Var Im W n (s) ≕ σ 2 (s), where we obtain eorem 2. Consider the phase Φ � Arg(W n (s)). Notice that the p.d.f.s of Re(W n (s)) and Im(W n (s)) are, respectively, and Re(W n (s)) and Im(W n (s)) are independent, the joint p.d.f. of Re(W n (s)) and Im(W n (s)) is p(x, y) � 1 2πσ 2 (s) e − x 2 +y 2 ( )/2σ 2 (s) ( ) .

Climate Diagnosis
Climatic background noises are often modeled as red noise [4][5][6][7][8][9]. In order to extract instinct features of climatic time series, we can use eorems 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 eorem 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 eorem 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
e author declares no conflicts of interest.