J. Earth Syst. Sci. (2021) 130:3 Ó Indian Academy of Sciences https://doi.org/10.1007/s12040-020-01490-1 (0123456789().,-volV)(0123456789().,-volV) Prediction of water shortage loss in situations with small samples based on an improved Gumbel copula 1 2, 3 4 LONGXIA QIAN ,YONG ZHAO *, HONGRUI WANG and SUZHEN DANG 1School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210 023, China. 2State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100 038, China. 3College of Water Sciences, Beijing Normal University, Key Laboratory for Water and Sediment Sciences, Ministry of Education, Beijing 100 875, China. 4Yellow River Institute of Hydraulic Research, Yellow River Conservancy Commission, Zhengzhou 45003, China. *Corresponding author. e-mail: [email protected] MS received 25 May 2020; revised 4 August 2020; accepted 6 August 2020 Prediction of water shortage losses is of great importance for water resources management. A new mathematical expression of water shortage loss was proposed in order to describe the random uncertainty and economic attributes of water resources. Then, Gumbel copula with a new method of parameter estimation was introduced to model the joint probabilistic characteristics for water supply and water use in situations when sufBcient data is unavailable. The new parameter estimation method requires only the minimum and maximum values of two variables. The improved Gumbel copula was proved to be reliable based on the RMSEs (root mean square error) and AICs (Akaike information criterion), statis- tical tests and upper tail dependence tests. The potential water shortage losses for all the districts of Tianjin were predicated. The water shortage loss in the Urban district is highest (7.02 billion CNY), followed by the new district of Binhai and Wuqing district, while those in the Baodi district and Ji County are very small. Keywords. Economic attributes; water shortage loss; insufBcient data; improved Gumbel copula; Tianjin. 1. Introduction predict the occurrence probability of water short- age risk or other risks. For example, the probabil- Water resources have multiple attributes, ities of water shortage were forecasted through a including natural and economic attributes. Many stochastic approach (Yu et al. 2015). The infor- countries in the world are facing water shortage, mation diffusion model was employed to predict which has caused considerable economic losses water shortage risk and compared with some tra- (Zhang et al. 2015;Liet al. 2016; Qian et al. 2016). ditional methods (Feng and Huang 2008). Yang Prediction of water shortage risk is of great et al. (2015) proposed an integrated method for importance for risk prevention and control. systematic quantitative risk analysis. Qian et al. Haimes (2009) pointed out that risk is a measure (2019) put forward a new model to predict water of the probability and severity of consequences. At shortage probability in situations with small sam- present, many researches concentrates on how to ple numbers. The probability of earthquake risk 3 Page 2 of 12 J. Earth Syst. Sci. (2021) 130:3 was forecasted through Markov chains (Fujinawa water shortage losses in all the districts of Tianjin 1991). are predicted by using these methods in section 3. Researches about how to deBne or predict water Some conclusions and future work are given in shortage loss are very scarce. Vulnerability was section 4. Brstly used to express the expected maximum severity of a sojourn into the set of unsatisfactory states (Hashimoto et al. 1982). But it is difBcult to 2. Methods estimate the probability of every unsatisfactory state. Qian et al. (2014)deBned potential economic In this section, some details about the expression of losses due to water shortage in terms of vulnera- water shortage loss and Gumbel copula maximum bility. However, the random uncertainty was not entropy estimation are presented. considered about water supply and water demand in this definition. Qian et al. (2016) constructed the joint distribution of water supply and water 2.1 Expression of water shortage loss demand by supposing that the relationship The amount of water shortage refers to the differ- between water supply and water demand is inde- ence between the amounts of water supply and pendent. However, there is a strong correlation water use in this paper. Water shortage loss is between water supply and water demand. Copulas referred to as the potential economic loss due to have been widely used to build the joint distribu- water shortage. Based on the definition of mathe- tion among variables. Gao et al. (2018) applied matical expectation, water shortage loss can be copulas to construct the joint probabilistic char- calculated quantitatively as follows: acteristics of water shortage. Zhang et al. (2016) Z Z used Student t-copula to build the joint distribu- b d tion of water supply and water demand. Ayantobo EL ¼ q Á fxðÞÁ; y ðÞy À x dxdy; ð1Þ et al. (2019) applied copulas to analyze the joint a c characteristics among three drought variables. where x and y are the amounts of water supply and Zhang et al. (2015) employed copulas to model the water use, and fxðÞ; y is the joint probability den- joint distribution of two major drought character- sity function of x and y; and q denotes the unit istics. Maximum likelihood estimation (MLE) is average economic beneBt of water resources (Qian often applied to estimate the parameters of copula et al. 2014); a and b are the possible range of water which requires a large amount of data (Qian et al. supply, and c and d are the possible range of water 2018). Unfortunately, data is often scarce when use. If the amount of water supply is greater than assessing water resources risk for many developing that of water use, i.e., x [ y, then EL =0. regions (Qian et al. 2019). Copulas with maximum likelihood estimation may be inapplicable in data- 2.2 Gumbel copula with maximum entropy scarce regions. Therefore, Gumbel copula with estimation maximum entropy estimation for small sample situations is introduced to model the joint proba- Gumbel copula, Clayton copula and Frank copula bilistic characteristics of water supply and water are often used to study the joint probabilistic use in situations with small samples. characteristics of random variables. Maximum There are some contributions in this paper. likelihood estimation or maximum pseudo-likeli- First, water shortage losses were deBned in terms of hood method are often applied to compute the mathematical expectation for the purpose of con- unknown parameters of copulas, while both meth- sideration of random uncertainty and economic ods require a great number of samples (Vergni attributes of water resources. Then, Gumbel cop- et al. 2015; Zhang et al. 2016). However, small ula with maximum entropy estimation was intro- samples are often available in water resources risk duced to model the joint distribution between (Qian et al. 2019). Qian et al. (2018) proposed a water supply and water use in situation with new method for estimating the parameter of insufBcient sample. The structure of this paper is Gumbel copula in situations when small samples as follows. Section 2 presents some methods pro- are available. The new method is called as maxi- posed and applied in this paper, including a new mum entropy estimation. The performance of expression about water shortage losses and Gumbel Gumbel copula with maximum entropy estimation copula with maximum entropy estimation. The is satisfying when small samples are used. J. Earth Syst. Sci. (2021) 130:3 Page 3 of 12 3 Moreover, it is still reliable when the true joint parameter p can be obtained by solving the distribution of two variables is not Gumbel copula following equation (Qian et al. 2018). The principle of the maximum oEpðÞ entropy estimation for Gumbel copula is as follows. ¼ 0; ð3Þ Let X and Y be two random variables with op marginal distributions u ¼ G1ðÞx and v ¼ G2ðÞy , where an optimization model can be built based on the Z Z maximum entropy principle (Qian et al. 2018): oEpðÞ a2 b2 ocuðÞ; v; p ¼ Á g ðÞÁx g ðÞy lnðcuðÞ; v; p Z Z op op 1 2 a2 b2 a1 b1 min À EpðÞ¼ cuðÞÁ; v; p g1ðÞÁx g2ðÞy ocuðÞ; v; p Ág ðÞÁx g ðÞÞþy dxdy; a1 b1 1 2 op  lnðÞcuðÞÁ; v; p g1ðÞÁx g2ðÞy dxdy; ð2Þ oðÞcuðÞ; v; p C 0 C ¼ E þ E0 where p is the parameter of Gumbel copula to be op D D estimated, and C 0D À CD0 C ¼ þ E0; D2 D 8 hihi9 > 1 1 p 1 1 > < p p p p = C ðÞÀ log u þÀðÞlog v log ðÞÀ log u þÀðÞlog v Á C 0 ¼ hi: p2 > 1 1 > : ðÞÀ log u p logðÞþÀÀ log u ðÞlog v p logðÞÀ log v À logðÞ log u log v ; 8 hihi9 > 1 1 p 1 1 > < p p p p = À1 ðÞÀ log u þÀðÞlog v log ðÞÀ log u þÀðÞlog v Á D0 ¼ hi p2 > 1 1 > : ðÞÀ log u p logðÞþÀÀ log v ðÞlog v p logðÞÀ log v þ 1 ; cuðÞ; v; p More details about the maximum entropy nohi p estimation can be found in Qian et al. (2018). 1 1 1À1 exp ÀÀðÞlog u pþÀðÞlog v p ðÞlog u Á log v p RMSE (root mean square error) and AIC criteria ¼ hi 1 1 2Àp (Akaike information criterion) are employed to uv ðÞÀ log u pþÀðÞlog v p evaluate the goodness of Bt. They are expressed as hi follows: 1 1 p 1 ÂÀðÞlog u pþÀðÞlog v p þ À 1 ; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p Xn 1 2 RMSE ¼ ðÞEi À Ti ; ð4Þ n i 1 ln cuðÞ; v; p ¼ hi 1 1 p 1 AIC ¼ n lnðÞþMSE 2m; ð5Þ ¼ÀðÞ Àlog u pþÀðÞlog v p þ À 1 logðÞ log u Á log v p hi where Ei and Ti are the joint empirical and theo- 1 1 p 1 þ log ðÞÀ log u pþÀðÞlog v p þ À 1 À logðÞuv retical frequency of the ith sample respectively, n is hip samples size and m is the number of parameters, 1 1 P À ðÞ2 À p log ðÞÀ log u pþÀðÞlog v p : 1 n 2 and MSE ¼ n i¼1 ðÞEi À Ti .