Depth-based multivariate descriptive statistics with hydrological applications

Abstract

Hydrological events are often described through various characteristics which are generally correlated. To be realistic, these characteristics are required to be considered jointly. In multivariate hydrological frequency analysis, the focus has been made on modeling multivariate samples using copulas. However, prior to this step, data should be visualized and analyzed in a descriptive manner. This preliminary step is essential for all of the remaining analysis. It allows us to obtain information concerning the location, scale, skewness, and kurtosis of the sample as well as outlier detection. These features are useful to exclude some unusual data, to make different comparisons, and to guide the selection of the appropriate model. In the present paper we introduce methods measuring these features, which are mainly based on the notion of depth function. The application of these techniques is illustrated on two real‐world streamflow data sets from Canada. In the Ashuapmushuan case study, there are no outliers and the bivariate data are likely to be elliptically symmetric and heavy‐tailed. The Magpie case study contains a number of outliers, which are identified to be real observed data. These observations cannot be removed and should be accommodated by considering robust methods for further analysis. The presented depth‐based techniques can be adapted to a variety of hydrological variables.

Depth-based regional index-flood model

“…[28] A number of depth functions are available in the statistical literature such as Tukey, Mahalanobis, and Oja [Zuo and Serfling, 2000]. A simple illustration on the computation of Tukey depth is presented in Chebana and Ouarda [2011]. For high dimensions, most of the current depth functions are quite cumbersome to compute.…”

“…A simple illustration on the computation of Tukey depth is presented in Chebana and Ouarda [2011]. For high dimensions, most of the current depth functions are quite cumbersome to compute.…”

“…In principal, other depth functions can be used for this purpose. However, in the literature, usually one depth function is adopted for a given study and there are no comparisons or selection rules [e.g., B ardossy and Singh, 2008;Chebana and Ouarda, 2011].…”

Nonparametric catchment clustering using the data depth function

“…Application of the data depth function is relatively new in the field of water resources. It has been used in the field of regional flood frequency analysis (Chebana and Ouarda 2008, Wazneh et al 2013a, Wazneh et al 2013b (Chebana and Ouarda 2011b), regionalization of hydrological model parameters (Bardossy and Singh 2011) and robust estimation of hydrological model parameters (Bárdossy and Singh 2008), defining predictive uncertainty of a model , and in selection of critical events for model calibration (Singh and Bárdossy 2012). For more detailed information about the data depth function and its uses in field of water resources, please refer to Chebana and Ouarda (2011a), Chebana and Ouarda (2011c), Guerrero et al (2013), Krauße and Cullmann (2009) and Singh and Bárdossy (2012).…”

Optimal depth-based regional frequency analysis

“…For instance, Chebana and Ouarda (2011a) used these functions in an exploratory study of a multivariate sample including location, scale, skewness and kurtosis as well as outlier detection. In another study, Chebana and Ouarda (2011b) combined depth functions with the orientation of observations to identify the extremes in a multivariate sample. Bardossy and Singh (2008) used the statistical notion of depth to detect unusual events in order to calibrate hydrological models.…”