Complex network analysis and robustness evaluation of spatial variation of monthly rainfall

Abstract

Recently, complex network-based approaches are shown to be efficient for spatial analysis of rainfall variation. One of the most critical limitations of correlation-based networks is using some assumed threshold levels to identify the existence of links that lead to different topological and community structures. A hypothesis test is formulated for the robustness analysis of recovered community structures for monthly rainfall data of Tasmania, Australia. To this aim, variation of information (VI) curves are constructed for the original and random networks. Then Gaussian process regression method is applied for these curves at different correlation threshold values (i.e., 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, and 0.90) to obtain maximum likelihoods as forms of Bayes factors. The networks are analyzed on a local and global scale by using node strength, node efficiency, edge density, and global efficiency measures to reveal the features of the rainfall network in the basin. Mainly, local strengths and efficiencies show that the networks are more efficient for threshold values higher than of 0.70. Global measures (i.e., edge density and global efficiency) decrease as the threshold increases except for the threshold of 0.80. In a rainfall forecasting exercise, using the robust network with the threshold value of 0.80 increases the coefficient of determination, Nash–Sutcliffe, and Kling-Gupta efficiencies from 0.29, 0.27, and 0.43 to 0.41, 0.36, and 0.61, respectively, leading to about 41%, 33%, and 41% improvement. Therefore, the results could be useful for determining robust network structures for various hydrological purposes such as filling missing values, regional flood analysis, and forecasting.

Appendix 1

The formulas of the performance indices for assessing the forecasting performance can be found below:

i. Coefficient of determination ($R^{2}$)

$$R^{2} = \frac{{\left( {\sum\nolimits_{i = 1}^{N} {\left( {P_{i} - \overline{P}} \right)\left( {\hat{P}_{i} - \tilde{P}} \right)} } \right)^{2} }}{{\sum\nolimits_{i = 1}^{N} {\left( {P_{i} - \overline{P}} \right)^{2} \sum\nolimits_{i = 1}^{N} {\left( {\hat{P}_{i} - \tilde{P}} \right)^{2} } } }}$$

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where $N$ is the length of data set, $\overline{P}$ and $\tilde{P}$ are the mean values of observed and forecasted values, $P_{i}$ and $\hat{P}_{i}$ are observed and forecasted values, respectively. It varies between 0 (no relation) and 1 (perfect fit), and describes the amount of observed variance explained by the model.

ii. Nash–Sutcliffe efficiency (NSE)

$$NSE = 1 - \frac{{\sum\nolimits_{i = 1}^{N} {\left( {P_{i} - \hat{P}_{i} } \right)^{2} } }}{{\sum\nolimits_{i = 1}^{N} {\left( {P_{i} - \overline{P}} \right)^{2} } }}$$

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NSE is the normalization of the forecasting error by the variance of the observed series. It varies between 1.0 (perfect fit) and $- \infty$. A NSE value below zero indicates that the mean value of the observed series can be considered as a predictor model.

iii. Kling-Gupta efficiency (KGE)

$$KGE = 1 - \sqrt {\left( {r - 1} \right)^{2} + \left( {\alpha - 1} \right)^{2} + \left( {\beta - 1} \right)^{2} }$$

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where $r$ is the correlation coefficient, $\alpha$ represents variability and is the ratio of the standard deviation of forecasted and observed rainfall time series, and $\beta$ represents bias and is the ratio of the mean values of forecasted and observed time series. KGE overcomes systematic underestimation of peaks and variability in the NSE and it varies between 1.0 (perfect fit) and $- \infty$.