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Syllabus and description
Univariate frequency analysis is a widely used tool to study hydrological phenomena and perform quantitative analyses about the risk assessment. With the advent of powerful statistical packages (such as S-plus and R) and the development of copula theory, a multivariate analysis is increasingly feasible in hydrological studies. This one-week short course will focus on the statistical theory and applications of multivariate frequency analysis from the point of view of copulas approach. The course will provide the theoretical and applied knowledge for conducting hydrological simulations using copula methods with a particular emphasis on extreme events across climate regimes.

Monday, July 20th

Opening and Introduction Lectures. (Columbia University, instructors: Upmanu Lall, Salvatore Grimaldi, Nicola Chiara).
Potential application of Copula function in Hydrology and Civil engineering.
Univariate extreme value theory: order statistics, extreme value theory, hazard, return period, and risk (Instructors: Carlo De Michele)
Afternoon Lab: (Polytechnic Institute of New York University). Introduction to statistical softwares: S-plus and R. Importing and exporting data. Graphical representations of a time series. Exporting a graph. Writing, compiling and executing a script. Some examples. Using the help-on-line. Developing a simple routine for preliminary univariate frequency analysis. (Instructors: Francesco Serinaldi)

Tuesday, July 21st

Morning Lectures (Polytechnic Institute of New York University): Dependence: bivariate concepts of dependence, measures of association, tail dependence. Bivariate analysis via copulas: 2-copulas, Archimedean copulas, return periods via copulas. Multivariate analysis via copulas: multivariate copulas, Archimedean copulas. (Instructors: Gianfausto Salvadori, Fabrizio Durante, Francesco Serinaldi)

Afternoon Lab (Polytechnic Institute of New York University):   Exploratory data analysis: Probability integral transformation, graphical tools to detect bivariate dependence: χ plot, Genest’s τ plot.  Writing routines to compute Kendall’ tau and application of Genest’s codes for detecting dependence on real world data (Instructors: Fabrizio Durante, Francesco Serinaldi, Ivan Kojadinovic)

Wednesday, July 22nd

Morning Lectures (Polytechnic Institute of New York University): Multivariate extreme value theory: multivariate extreme value distributions, multivariate dependence, multivariate return periods. Extreme values copulas, dependence function, tail dependence. (Instructors: Gianfausto Salvadori, Fabrizio Durante, Francesco Serinaldi)
Afternoon Lab (Polytechnic Institute of New York University): Inference procedures: likelihood based methods: EML, CML, IFM. Non parametric estimation: empirical copula, estimation by sample measures of dependence. Writing routines for CML, Kendall’s tau based estimation and empirical copula assessment (Instructors: Fabrizio Durante, Francesco Serinaldi, , Ivan Kojadinovic).

Thrusday, July 23rd

Morning Lectures (Polytechnic Institute of New York University): Hydrological applications of copulas: AMC evaluation, spillway safety control,  combination of flows from upstream sites definition of the distribution of the results of a set of statistical test performed on spatially dependent series, definition of a regional return period, intersite effects in regional frequency analysis, studies the drought duration and severity, description of the spatial variability in geostatistics. (Instructors: Carlo De Michele, Salvatore Grimaldi)
Afternoon Lab (Polytechnic Institute of New York University): Evaluation criteria for copulas: diagnostic graphs and formal tests. Simulation of copulas samples. Writing routines for KC plot. (Instructors: Fabrizio Durante, Francesco Serinaldi).

Friday, July 24th

Morning (Polytechnic Institute of New York University)  Students projects.
Afternoon (Columbia University) 3’00 p.m.  Final student presentation of the projects.


Ministero dell'Ambiente e della Tutela del Territorio e del Mare

Institute for Environmental Protection and Research

 Honors Center of Italian Universities - H2CU 
Sapienza University of Rome

University of Tuscia,
Viterbo, Italy


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