A Fuzzy C-Means approach for regionalization using a bivariate homogeneity and discordancy approach

In stochastic analysis for droughts, such as frequency or trend analysis, the absence of lengthy records typically limits the reliability of statistical estimates. To address this issue, “regional” or “pooled” analysis approach is often used. The main contribution of this study is to create regions based on bivariate criteria rather than univariate ones; the two variables are severity and duration. The methodology is applied to hydrological records of 36 unregulated flow monitoring sites in the Canadian “prairie” provinces of Alberta, Saskatchewan and Manitoba. Our criteria for a hydrological “region” to be suitable are that it should be homogeneous, that it should not be discordant, and that it should not be too small. Tests for homogeneity and non-discordancy are traditionally based on univariate L-moment statistics; for example there have been several applications of univariate L-moments to bivariate drought analysis by simply ignoring one of the variables. Instead, we use multivariate L-moments, also known as L-comoments. The approach uses site characteristics and a fuzzy clustering approach, called Fuzzy C-Means (FCM), to form the initial regions (clusters) and adjusts initial clusters based on partial or fuzzy membership of each site to other clusters to form final clusters that meet the criteria of homogeneity, lack of discordancy, and sufficient size. We also estimate return periods using a bivariate copula method.