Entropy-Based Variability and Support Vector Regression-Based Forecast of Drought Index

ABSTRACT

Droughts which are categorized as meteorological or climatogical, agricultural, hydrological and socioeconomic by American Meteorological Society (1997) are expressed as numerically with some index thanks to historical climate records with temperature and precipitation data. Determination of drought index' variability and short-term forecast of these index provide taking necessary precautions, planning water resources, organizing some economic goods. In this study, Palmer Drought Severity Index (PDSI), Palmer Hydrological Drought Index (PHDI), Palmer Z-Index (ZNDX), Modified Palmer Drought Severity Index (PMDI), Standard Precipitation Index (SPI) monthly data between the years 1895 and 2014 which they belongs to NOAA (National Oceanic and Atmospheric Administration) are used for 3-year forecast.

This study aims to determine the entropy-based variability of drought index and forecast these index by using Support Vector Regression (SVR). Variability can be expressed as the difference between the possible maximum entropy and the calculated entropy of the series and the difference is referred as a Disorder Index (DI). The higher the disorder index, the higher the variability.

SV (Support Vector) algorithm is developed as a nonlinear generalization algorithm (Vapnik and Lerner, 1963; Vapnik and Chervonenkis, 1964). SVR is a state of the art method which is a specific implementation of Support Vector Machines (SVMs). SVR transforms input space which is formed from the original observations into high dimensional feature space by way of a kernel function and performs a linear regression in this space.

In the application of SVR, selection of appropriate kernel function is of great importance. Kernel functions enable the SVR to solve the regression problems that require nonlinear approach. There are different kernel functions such as linear, polynomial, sigmoid and radial basis function (RBF). RBF is a local kernel function with a strong learning ability and is able to reduce computational complexity of the training process and improve the generalization performance of SVR (Li and Xu, 2005). Because RBF is the most commonly used kernel function in hydrometeorological studies (see Debnath et al., 2004; Cheng-Ping et al., 2011; Kisi and Cimen, 2011; Li-ying and Wei-guo, 2010; Choy and Chan, 2003; Baydaroğlu and Koçak, 2014; Kalteh, 2013) and it provides good results, therefore it is used in the study.

To prepare the input data for SVR, Chaotic Approach (CA) is employed. In CA, a phase space should be reconstructed. From one dimensional time series, a phase space can be constructed via Embedding Theorem (Takens, 1981). According to Embedding Theorem, embedding parameters (embedding dimension and time delay) are determined from the time series. In this study, False Nearest Neighbour (FNN) (Kennel et al., 1992) method and Mutual Information Function (MIF) (Fraser and Swinney, 1986) algorithm are implemented in order to determine the embedding dimension and the time delay, respectively.

According to calculations, the highest DI belongs to PI. Besides, the results show that forecasts of drought index by using SVR is very accurate. Almost all performance criteria of forecasts are higher than 90%.