Using an effective method to measure the brain functional connectivity is an important step to study the brain functional network. The main methods for constructing an undirected brain functional network include correlation coefficient (CF), partial correlation coefficient (PCF), mutual information (MI), wavelet correlation coefficient (WCF), and coherence (CH). In this paper we demonstrate that the maximal information coefficient (MIC) proposed by Reshef et al. is relevant to constructing a brain functional network because it performs best in the comprehensive comparisons in consistency and robustness. Our work can be used to validate the possible new functional connection measures.

We show that there is a bijection between the renormalizations and proper completely invariant closed sets of expanding Lorenz map, which enables us to distinguish periodic and non-periodic renormalizations. Based on the properties of the periodic orbit with minimal period, the minimal completely invariant closed set is constructed. Topological characterizations of the renormalizations and $\alpha$-limit sets are obtained via consecutive renormalizations. Some properties of periodic renormalizations are collected in Appendix.

We study the multifractal properties of water level with a high-frequency and massive time series using wavelet methods (estimation of Hurst exponents, multiscale diagram, and wavelet leaders for multifractal analysis (WLMF)) and multifractal detrended fluctuation analysis (MF-DFA). The dataset contains more than two million records from 10 observation sites at a northern China river. The multiscale behaviour is observed in this time series, which indicates the multifractality. This multifractality is detected via multiscale diagram. Then we focus on the multifractal analysis using MF-DFA and WLMF. The two methods give the same conclusion that at most sites the records satisfy the generalized binomial multifractal model, which is robust for different times (morning, afternoon, and evening). The variation in the detailed characteristic parameters of the multifractal model indicates that both human activities and tributaries influence the multifractality. Our work is useful for building simulation models of the water level of local rivers with many observation sites.