A New Parameter Estimation Method for a Logistic Regression Model of Water Shortage Risk in the Case of Small Sample Num
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A New Parameter Estimation Method for a Logistic Regression Model of Water Shortage Risk in the Case of Small Sample Numbers Longxia Qian1 Caiyun Deng2
· Hongrui Wang2 · Chengzu Bai3 ·
Received: 17 June 2019 / Accepted: 19 August 2019 © International Association for Mathematical Geosciences 2019
Abstract The ability to predict the risk of water shortage is critical, and therefore it is important to develop methods of parameter estimation for statistical models in situations when insufficient data are available. Based on the maximum entropy principle, this paper proposes an alternative method of parameter estimation for a logistic regression model in the case of small sample numbers. The new method requires very little data about risk factors, whereas the maximum likelihood estimation requires a high quantity of data regarding risk and risk factors. In addition, the paper applies a new formula for normalized information flow (information flow is a physical notion logically associated with causality, which can be used to quantify the cause–effect relation between dynamic events) to select important risk factors. Five experiments are performed based on predictions of water shortage risk in the Beijing–Tianjin–Tangshan region to validate the performance of the new method with different small sample sizes. The results show that the new method is generally reliable and performs much better than the maximum likelihood estimation when only small samples are used. Specifically, an improvement of between 87.9 and 95.3% is observed when the number of samples is more than 15 and less than 30. The new method still generates an acceptable result using only 10 samples, while the maximum likelihood estimation is unreliable in such situations.
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Longxia Qian [email protected]
1
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
2
College of Water Sciences, Beijing Normal University, Key Laboratory for Water and Sediment Sciences, Ministry of Education, Beijing 100875, China
3
Beijing Institute of Applied Meteorology, Beijing 100029, China
123
Math Geosci
Keywords Small samples · Maximum entropy principle · Maximum likelihood estimation · Normalized information flow · Water shortage risk
1 Introduction As a result of climate change, dramatic changes are occurring in the frequency and intensity of extreme weather events, which in turn strongly affect water resources (Vanwindekens et al. 2018; Adamowski et al. 2013). In this context, water scarcity has become more severe in many regions experiencing rapid population growth and economic expansion (Feng and Luo 2011; Qian et al. 2014; Jia et al. 2015; Jiang et al. 2018). According to Haimes (2009), risk is a combination of probability and losses. Therefore, the accurate prediction of water shortage probability is critical for the implementation of effective measures to minimize losses, especially in situations of drought. Many methods can be used to predict risk. For example, one way is by building a probability distributi
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