Investigation of Temporal-Spatial Characteristics of Precipitation Using Discrete Maximal Overlap Wavelet Transform (MODWT) and Spatial Clustering Tools

Precipitation is one of the important climatic elements and is one of the factors affecting the hydrological cycle. Temporal-spatial variations of precipitation in a watershed can have numerous effects on the engineering, management and planning of water resources. Many researchers have been studied precipitation variations. Most hydrological time series are non-stationary, trendy, or with seasonal fluctuations. Wavelet analysis is one the commonly applied approaches by researchers. It has been utilized as a common tool to break down and excavate complex, periodic, and irregular hydrological and geophysical time series, especially in recent years. On the other hand, clustering techniques can be used to identify structure in an unlabeled precipitation data set by objectively organizing data into homogeneous groups where the within-group-object dissimilarity is minimized and the between-group-object dissimilarity is maximized. Clustering analysis is similar to the homogeneity test. Considering the dynamic characteristics and non-uniform distribution of precipitation data and due to the need for identifying of homogeneous precipitation regions in water resources management, a temporal-spatial model is proposed to investigate the characteristics of precipitation. The time series of the precipitation were decomposed using MODWT mothod and the energy of sub-series was culculated. MODWT is a mathematical technique which transforms a signal into multilevel wavelet and scaling coefficients. Maximal overlap discrete wavelet transform (MODWT) is similar to the discrete wavelet transform (DWT) in that low-pass and high-pass filters are applied to the input signal at each level. However, the MODWT does not decimate the coefficients and the number of wavelet and scaling coefficients is same as the number of sample observation at every level of the transform. In other words, MODWT coefficients consider the result of a simple changing in the pyramid algorithm used in computing DWT coefficients through not down sampling the output at each scale and inserting zeros among coefficients in the scaling and wavelet filters. Clustering is the process of partitioning or grouping a given set of patterns into disjoint clusters. This is done such that patterns in the same cluster are alike and patterns belonging to two different clusters are different. k-means clustering is a method of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining. k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. RBF methods have their origins in techniques for performing the exact interpolation of a set of data points in a multi-dimensional space. RBF mappings provide an interpolating function which passes exactly through every data point. In the present study, the classical and proposed methods were used to investigate the monthly rainfall characteristics of 30 stations in the southeastern of United States during 1968- 2018. In the proposed method, the time series pre-processing method including MODWT discrete wavelet transform and K-means clustering were used. At first the monthly precipitation time series of the stations were districted into several sub-series using MODWT method and db mother wavelet. Then, the energy of sub-series was calculated and used as input for K-means and RBF methods. The optimum number of clusters for the stations in both classical and proposed methods was five clusters. In order to use the data as the input of the RBF method, the correlation of the data was evaluated by variogram and covariance graphs. Then Spline with Tension method was selected in RBF model and zoning maps were drawn. In order to evaluate the temporal-spatial characteristics of monthly rainfall of 30 selected stations, two classical and proposed methods were used. At first the monthly precipitation time series of the stations were districted into several sub-series using MODWT method and db mother wavelet. Then, the energy of sub-series was calculated and used as input for K-means and RBF methods. The optimum number of clusters for the stations in both classical and proposed methods was five clusters. In order to use the data as the input of the RBF method, the correlation of the data was evaluated by variagram and covariance graphs. Based on the results of clustering and in accordance with the latitude and longitude variations of the stations, it was found that with increasing the energy of the clusters, the amount of precipitation in the stations decreased and vice versa. Silhouette coefficient of clustering in classical method was 0.3 and in proposed method was 0.8, which indicates better clustering of studied stations in the proposed method.