# Wavelet Transform Application for/in Non-Stationary Time-Series Analysis: A Review
## 1 Introduction  
The time series (TS) analysis is the study of a sequence of data collected through time. It is an effective approach for investigating the variability of variables, and thus widely used in many areas of science and engineering. Mathematically, TS (\(y_{t}\)) is stationary if, for all \(t\)[1]:  
\[E(y_{t})=E[(y_{t-1})]=\mu, \tag{1}\]  
\[Var(y_{t})=\gamma_{0}<\infty, \tag{2}\]  
\[Cov(y_{t},y_{t-k})=y_{t}, \tag{3}\]  
where \(E(. )\) is the expected value defined as the ensemble average of the quantity, and \(Var(. )\) and \(Cov(. )\) are, respectively, the variance and the covariance functions. Stationarity in the wide sense is also known as weak stationarity, covariance stationarity or second-order stationarity [2]. If these constraint conditions are violated, the TS would exhibit non-stationary characteristics, which is a major challenge for several fields (e.g., remote sensing, finance, medicine, engineering and hydrology). For this reason, several approaches are developed in order to analyze the non-stationary characteristics. These approaches can be classified into two main types: time methods (e.g., Auto-Correlation Analysis method, Regression method, Seasonal Auto-Regressive Integrated Moving Average, Break for Additive Trend and Season) and Spectro-Temporal methods [3]. Spectro-Temporal methods allow for the characterization of frequency variations. A brief survey of the methods available for processing non-stationary data, such as Short Time-Frequency Transformer (STFT), Wigner-Ville Distribution (WVD), Empirical Mode Decomposition (EMD) and Wavelet Transformation (WT), will be presented. For time-frequency analysis, [PERSON] [4] introduced the STFT, also called a windowing technique, which maps the signal into a two-dimensional function of time and frequency. This technique uses a fixed amount of time for all frequencies which restrict flexibility. However, it allows for obtaining information with a limited precision. This is why [PERSON] et al. [5] introduced the WVD approach as a quadratic time-frequency representation. In fact, WVD is defined as a Fourier transform (FT) of the central covariance. The limitation of this method lies in the cross-term interference problem indicated by the existence of negative power for some frequency ranges. Additionally, [PERSON] et al. [1] developed the EMD method which is suitable for processing non-stationary TS. This method decomposes the data'modes', i.e., Intrinsic Mode Functions (IMFs), without leaving the time domain. However, it is purely empirical and lacks a theoretical basis. Comparatively, WT becomes a widely used tool for signal analysis. One of the main advantages of WT is that it may decompose a signal directly according to the frequency and represents it in the frequency domain distribution state in the time domain. As for the wavelet transformation, both time and frequency information of the signal are retained. It is thus a more powerful transformation for time-frequency analysis. This paper reviews the related studies on the development and applications of WT for different fields over the period between 2004 and 2018. The different references, the used data, the objective behind using WT, the mother wavelet and its performance are all summarized. The purpose of this paper is to provide researchers and engineers with a deeper comprehensive knowledge of the theory and application of the WT method for non-stationary TS. In fact, the primary research questions considered in this survey are:  
* Q1: What are the different types of WT for non-stationary time-series analysis? * Q2: What are the uses of WT in different fields? * Q3: Considering those different fields, what are the limits of the WT method? * Q4: What are the future perspectives of WT for non-stationary time-series analysis? The remainder of this paper is organized into five parts, starting with the Introduction section (Figure 1). Section 2 presents a brief introduction of the theoretical background of WT, including the mostly used wavelets.