# 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 Remote Sensing in Vegetation Analysis  
In the remote sensing field, WT has been widely used to analyze vegetation TS. However, several applications are developed to ameliorate the quality of forecasting and to detect the land change as well as crop phenology. On the one hand, WT has been used as a filter for denoising TS in several research works. For example, [PERSON] et al. [31] used the EVI from the MODIS TS to detect the phenological stages of paddy rice. First, the abnormal data affected by clouds was removed. Then, the DWT (Equation (9)) was applied as a filter to denoise the EVI TS using several types of mother wavelets, such as the Coiflets, Daubechies and Symlet. Finally, the phenological stages were specified. To prove the effectiveness of the proposed approach, the RMSE of the estimated phenological date and growing period against the statistical data (e.g., growing period, planning date, heading date and harvesting date) was calculated. Moreover, [PERSON] et al. [32] drew a comparison between wavelet-based and Fourier-based features. The authors proved that the result of a classification was better than Fourier when using a 3-level Haar wavelet as a denoising method in the original data. Different methods for denoising, such as median filtering, moving average filtering and wavelet denoising, were applied. Then, two classifiers were used, namely Nearest Mean (NM) and Nearest Neighbour (NN). Additionally, [PERSON] et al. [33] analyzed three types of vegetation from MODIS data (e.g., 8-day NDVI/LAI and 16-day MODIS albedo): the deciduous broadleaf forest, evergreen broadleaf forest and double-cropping rice. As a first step, the control information was used. The blue band of MODIS data was exploited as an indicator to detect the small cloud and to overcome the drawbacks using the inverse weighted interpolation. Finally, the Daubechies SWT was applied to remove the noise in TS. To prove the effectiveness of the proposed approach in comparison with the Fourier-based fitting method, the Savitzky-Golay filter and Best Index Slope Extraction (BISE) algorithm were both applied [33]. On the other hand, the WT has been widely used for multi-temporal scale analysis. For example, [PERSON] et al. [7] applied the WT to analyze the NDVI data for four vegetation groups (boreal, temperate, tundra and frost) based on the modified Koppen classification system. In addition, the authors studied the variance across different scales as well as the correlation structure of TS, and applied the Haar MODWT (Section 2.5) to analyze the multi-temporal scale. [PERSON] and [PERSON] [34] used the Meyer wavelet into the MRA (introduced in Section 2.3) to capture and describe both inter- and intra-annual vegetation changes of AVHRR NDVI TS. The intra-annual change was associated with the seasonal change to define several key features in relation to the vegetation phenology, such as the minimum NDVI and the amplitude of the phenological cycle. The inter-annual change was calculated to detect the abrupt and subtle changes by trend analysis where the Mann-Kendall test was applied. This approach was tested on AVHR/NDVI data in Spain over the period between 1989 and 2002 and used the precipitation data to analyze the climatic influences in some regions. Moreover, [PERSON] et al. [35] investigated the Meyer wavelet along with the MRA to examine the land condition of a dry land region in Senegal from 2001 to 2009 using 1-km AGC SPOT vegetation data. The vegetation seasonality was studied from the denoised TS, whereas the temporal variability of the inter-annual component was studied using the Mann-Kendall test. [PERSON] and [PERSON] [36] applied the Meyer WT-MRA to analyze the land cover change. Four MODIS TS from 2000 to 2010 were used. These TS expressed different changes, such as inundation in Cambodia and Vietnam, fire in Spain, farming in Argentina and urbanization in China. In addition, two Landsat-TM images were analyzed to characterize different change levels in the spatial dimension. Afterwards, [PERSON] et al.