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Application of Extreme Value Theory (EVT) in River Morphology
Power and Water University of Technology Application of Extreme Value Theory (EVT) in River Morphology By: Mahkameh Zare Supervised by: Dr. S. Mousavi and Dr. M.R.M. Tabatabai Department of Water and environmental Engineering Power and Water University of Technology February 2nd 2013
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Introduction Main Question When is a river discharge called “Flood”? High discharge?! High water elevation?! Annual maximum?! France, 1992 Flood event definition A flood event can be defined as a high river flow, exceeding a pre-defined threshold.
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Introduction Main Question When is a river discharge called “Flood”? High discharge?! High water elevation?! Annual maximum?! France, 1992 Flood event definition A flood event can be defined as a high river flow, exceeding a pre-defined threshold.
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Power and Water University of Technology
EVT models Introduction AMF POT Comparison Application Conclusion Sketch of the EVT Case study EVT models EVT Models AMF The maximum peak of each year is selected to be modeled POT Peaks above threshold are extremes Markovian All data above threshold are extremes being modeled
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Power and Water University of Technology
EVT models Introduction AMF POT Comparison Application Conclusion Sketch of the EVT Case study EVT models EVT Models AMF The maximum peak of each year is selected to be modeled POT Peaks above threshold are extremes Markovian All data above threshold are extremes being modeled
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Power and Water University of Technology
EVT models Introduction AMF POT Comparison Application Conclusion Sketch of the EVT Case study EVT models EVT Models AMF The maximum peak of each year is selected to be modeled POT Peaks above threshold are extremes Markovian All data above threshold are extremes being modeled
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Power and Water University of Technology
EVT models Introduction AMF POT Comparison Application Conclusion Sketch of the EVT Case study EVT models EVT Models AMF The maximum peak of each year is selected to be modeled POT Peaks above threshold are extremes Markovian All data above threshold are extremes being modeled
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Power and Water University of Technology
EVT models Introduction AMF POT Comparison Application Conclusion Sketch of the EVT Case study EVT models EVT Models AMF The maximum peak of each year is selected to be modeled POT Peaks above threshold are extremes Markovian All data above threshold are extremes being modeled
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Power and Water University of Technology
EVT models Introduction AMF POT Comparison Application Conclusion Sketch of the EVT Case study EVT models EVT Models AMF The maximum peak of each year is selected to be modeled POT Peaks above threshold are extremes Markovian All data above threshold are extremes being modeled
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Power and Water University of Technology
EVT models Introduction AMF POT Comparison Application Conclusion Sketch of the EVT Case study EVT models EVT Models AMF The maximum peak of each year is selected to be modeled POT Peaks above threshold are extremes Markovian All data above threshold are extremes being modeled
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Probability distribution of maxima Generalized extreme value distribution GEV parameters Gumbel VS GEV Conventional approaches Using standard statistical techniques to estimate F from observed data. Accepting that F is unknown and looking for approximate families of models for It is usual to adopt a distribution and then to estimate the relevant parameters of that distribution But... there are weaknesses 1- A technique is required to choose the distribution 2- Subsequent inferences presume this choice to be correct and don’t allow the uncertainty such a selection involves Central limit theorem Regardless of the distribution F for the population Drawback Very small discrepancies in the estimate of F can lead to substantial discrepancies in
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Probability distribution of maxima Generalized extreme value distribution GEV parameters Gumbel VS GEV Conventional approaches Using standard statistical techniques to estimate F from observed data. Accepting that F is unknown and looking for approximate families of models for It is usual to adopt a distribution and then to estimate the relevant parameters of that distribution But... there are weaknesses 1- A technique is required to choose the distribution 2- Subsequent inferences presume this choice to be correct and don’t allow the uncertainty such a selection involves Central limit theorem Regardless of the distribution F for the population Drawback Very small discrepancies in the estimate of F can lead to substantial discrepancies in
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Probability distribution of maxima Generalized extreme value distribution GEV parameters Gumbel VS GEV Conventional approaches Using standard statistical techniques to estimate F from observed data. Accepting that F is unknown and looking for approximate families of models for It is usual to adopt a distribution and then to estimate the relevant parameters of that distribution But... there are weaknesses 1- A technique is required to choose the distribution 2- Subsequent inferences presume this choice to be correct and don’t allow the uncertainty such a selection involves Central limit theorem Regardless of the distribution F for the population Drawback Very small discrepancies in the estimate of F can lead to substantial discrepancies in
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Probability distribution of maxima Generalized extreme value distribution GEV parameters Gumbel VS GEV Conventional approaches Using standard statistical techniques to estimate F from observed data. Accepting that F is unknown and looking for approximate families of models for It is usual to adopt a distribution and then to estimate the relevant parameters of that distribution But... there are weaknesses 1- A technique is required to choose the distribution 2- Subsequent inferences presume this choice to be correct and don’t allow the uncertainty such a selection involves Central limit theorem Regardless of the distribution F for the population Drawback Very small discrepancies in the estimate of F can lead to substantial discrepancies in
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion GEV parameters Generalized extreme value distribution GEV parameters Gumbel VS GEV Location ( ) which defines where the mass is Scale ( ) which controls how “smallest extremes” increase Shape ( ) which controls how tail evolves
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion GEV parameters Generalized extreme value distribution GEV parameters Gumbel VS GEV Location ( ) which defines where the mass is Scale ( ) which controls how “smallest extremes” increase Shape ( ) which controls how tail evolves
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion GEV parameters Generalized extreme value distribution GEV parameters Gumbel VS GEV Location ( ) which defines where the mass is Scale ( ) which controls how “smallest extremes” increase Shape ( ) which controls how tail evolves
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Power and Water University of Technology
Gumbel VS GEV Introduction AMF POT Comparison Application Conclusion Power and Water University of Technology Generalized extreme value distribution GEV parameters Gumbel VS GEV Fitting Gumbel to heavy-tailed data Zoom to the tail... Gumbel can not model the tail properly So Gumbel is not reliable
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Gumbel VS GEV Power and Water University of Technology Generalized extreme value distribution GEV parameters Gumbel VS GEV Fitting Gumbel to heavy-tailed data Zoom to the tail... Gumbel can not model the tail properly So Gumbel is not reliable
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Power and Water University of Technology
Believe in annual maximum flood as extreme? Introduction AMF POT Comparison Application Conclusion Power and Water University of Technology GEV parameters Gumbel VS GEV Believe in annual maximum flood?! Question Is it really convenient to take annual maximums as extremes?
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Power and Water University of Technology Declustering Generalized Pareto Distribution (GPD) Model parameters Threshold selection Parameter and confidence estimation Imagine a time series of X1,X2,... The excesses over the threshold u of a time series X1,X2, ... are the observations (called exceed- ances) Xi − u such that Xi > u. A cluster is defined as a group of consecutive exceedances. A peak excess is defined as the largest excess in a cluster of ex- ceedances.
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Power and Water University of Technology Declustering Generalized Pareto Distribution (GPD) Model parameters Threshold selection Parameter and confidence estimation Imagine a time series of X1,X2,... The excesses over the threshold u of a time series X1,X2, ... are the observations (called exceed- ances) Xi − u such that Xi > u. A cluster is defined as a group of consecutive exceedances. A peak excess is defined as the largest excess in a cluster of ex- ceedances.
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Declustering Power and Water University of Technology Generalized Pareto Distribution (GPD) Model parameters Threshold selection Parameter and confidence estimation Imagine a time series of X1,X2,... The excesses over the threshold u of a time series X1,X2, ... are the observations (called exceed- ances) Xi − u such that Xi > u. A cluster is defined as a group of consecutive exceedances. A peak excess is defined as the largest excess in a cluster of ex- ceedances.
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Stochastic behavior of extreme events Generalized Pareto Distribution (GPD) Model parameters Threshold selection Parameter and confidence estimation Let be a sequence of independent and identically distributed random variables, having distribution function F. This parallels the use of GEV as an approximation to the distribution of maxima of long sequence when the parent population is unknown If F were Known But this is not the case . So approximations are sought
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Stochastic behavior of extreme events Generalized Pareto Distribution (GPD) Model parameters Threshold selection Parameter and confidence estimation Let be a sequence of independent and identically distributed random variables, having distribution function F. This parallels the use of GEV as an approximation to the distribution of maxima of long sequence when the parent population is unknown If F were Known But this is not the case . So approximations are sought
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Power and Water University of Technology
Introduction AMF POT Comparison Application Conclusion Asymptotic model Power and Water University of Technology Generalized Pareto Distribution (GPD) Model parameters Threshold selection Parameter and confidence estimation Let be a sequence of independent random variables with common distribution function and Suppose Defined on where If Which is called generalized pareto distribution
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