# 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  
Wavelet analysis has been investigated in several financial and economics time series. This is due to its capacity of studying a number of concepts such as non-stationarity, multiresolution and approximate decorrelation emerge from wavelet filters [17; 18; 79; 80]. In fact, several applications were developed in order to study the such as the co-movement among international stock market, analysis of the evolution of the oil, commodity price, etc. In the state of the art, several works were developed in order to study the interaction and co-movement in the international stock market. For example, [PERSON] et al. [59] used the DWT, MODWT and cross wavelet power in order to study eleven stock indices. In addition, [PERSON] [56] applicated the DWT and MRA in order the investigate the dynamics and the protentional interaction in the international stock market. Two basic time series were used: the daily stock market market indices of the KOSPI (Korea Composite Stock Price Index) and the DJIA (Dow Jones industrial average), which cover the period January 1995 to August 2000. Principally, the Haar wavelet was applied as a basic wavelet function. Finally, to study the spill over effects and the interaction between the USA and Korea market, the regression and slope coefficient were calculated. Furthermore, the MODWT was applied in [21] in order to study the relationship between real stock returns and outputs growth. In the first step, the scaling properties of the DJIA stock price index and the industrial production index for the US over the period 1961-2006 were used. Then, the lead/lag relationship between stock returns and real economic activity was analyzed applying wavelet correlation and cross-correlation analyzes to wavelet coefficient. [PERSON] and [PERSON] [57] analyzed the major developed economies, namely, Germany, Japan, United Kingdom and the United States over the last four decades using wavelet squared coherency. First, they calculated the standard deviation, skewness and kurtosis to get information for the different data. Then, the wavelet transform was applied for the analysis of the time- and frequency varying co-movement. [PERSON] et al. [81] used the CWT in order to analyze the level of co-movements, contagion and rolling correlation between the stock markets of the PIIGS (Portugal, Italy, Ireland, Greece and Spain) and of the largest European stock market (UK) and Germany. First, using the wavelet power spectrum, the evolution of the stock market returns series, their volatility and jumps, which appear in different crisis moments were analyzed. Then, the wavelet coherence was applied to analyze the stock markets' co-movements, and the phase-differences in order to identify the lead-lag situations. Additionally, the wavelet transform was applicated to analyze the evolution of the oil. [PERSON] and [PERSON] [60] used cross wavelet analysis to decompose the time-frequency effects of oil price changes on the macroeconomy. They applicated the Morlet wavelet in order to analyze the relation between oil prices and industrial production. [PERSON] [61] used the MRA wavelet in order to study the linear and nonlinear Granger causality between the real oil price and the real effective U.S. Dollar exchange rate. In the first step, the data into macroeconomic variables were decomposed at various scales. Then, the relationships among the decomposed series on a scale by scale basis were studied. More recently, [PERSON] et al. [82] studied the evolution feature of the world crude oil market integration and diversification from the perspective of the interdependent structural relationship of global oil prices. The wavelet based complex network was developed. In fact, this latter is based on the MODWT analysis to extract multi-period characteristics of original oil prices, the grey relation analysis (GRA analysis) to measure the interdependent relationship of oil prices and constructing the multi-period network model of the global oil price co-movement.