# Wavelet Transform Application for/in Non-Stationary Time-Series Analysis: A Review
## 3 Wavelet Transformation for Non-Stationary Time-Series: State-of-the-Art
### Application of WT in Finance
Finally, two critical reference indexes, namely the reference decision-making cycle and the target regional market, were proposed for decision makers to better adjust their strategies. Furthermore, a novel multivariate, dynamic approach--the wavelet local multiple correlation (WLMC)--was proposed recently by [83] and this tool was firstly applied to analyze the relationship between oil time series in the time-scale domain [84]. This approach is suitable for use with energy data of any kind that change over time and involve heterogeneous agents who make decisions across different time horizons and operate on different time scales. The different approach to study the crude oil energy resources worldwide were reviewed in [85]. In fact, the different work that analyzes the behavior of oil spot and future prices and their determinants during periods of market uncertainty, particularly in the context of economic and financial crises, were studied. [PERSON] et al. [58] investigated the cross-wavelet in order to study the monetary policy variables (money aggregates and interest rates) and macroeconomic variables (industrial production and inflation). The Morlet wavelet was chosen as the optimal joint time-frequency concentration. [PERSON] [86] used the MODWT in order to test whether contagion occurred during the US subprime crisis of 2007. In fact, the wavelet was applied to decompose macroeconomic time series, and data in general, into their time-scale components, and to provide an alternative representation of the variability and association structure of certain stochastic processes on a scale-by-scale basis. To handle multivariate time series, [PERSON] [19] used the MODWT in order to analyze a daily Eurozone stock market returns during a recent period. In fact, two statistical tools were studied to determine the overall correlation for the whole multivariate set on a scale-by-scale basis. [PERSON] [87] investigated the MODWT to examine contagion among the major world markets during the last 25 years. In addition, the wavelet coherency was used in order to analyze the correlation structure between co-movements at different time scales. Recently, [PERSON] et al. [20] analyzed EU peripheral (so-called PIIGS) stock market indices and the S&P Europe 350 index (SPEURO), as a European benchmark market, over the pre-crisis (2004-2007) and crisis (2008-2011) periods. In the first step, they decomposed the daily log returns for the two time intervals (pre-crisis and 90 crisis periods) applyingthe MODWT with a Daubechies least asymmetric (LA) wavelet. Then, they computed the MODWT wavelet correlation in order to analyze the relationships among the five daily log returns of the PIIGS. Finally, they applied a nonlinear Granger causality test to the wavelet decomposition coefficients of these stock market returns. In addition, [PERSON] and [PERSON] [52] studied the financial time-series using DWT. First, the unit root test was used to check the stationarity of the daily price of the Dow Jones Index (DJIA30) series from 2004 to 2012. Therefore, the original and return data were decomposed into different components based on five wavelet families, namely Haar, Daubechies, Symmlet, Coiflets and a discrete approximation of the Meyer wavelet. In addition, the ACF and the ANOVA table were applied to compare the five wavelets. Finally, the authors determined the best mother wavelet for decomposition in order to reduce noise from the original and return data. Moreover, a bootstrap wavelet-based hypothesis test was proposed by [PERSON] [53]. The proposed approach was tested using the yearly U.S Gross Domestic Product (GPD) TS from 1929 to 2014. The Haar MODWT was also applied (Section 2.5). First, the robust filtering and modified cross-validation were used to smooth the squared MODWT wavelet in order to study the variability of the series. Then, several statistical tests, such as the standard deviation, were performed to get more information on the variability and non-stationarity of the series.