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Riesgo Hidrológico de EventosExtremos en Condiciones
No-Estacionarias
Jose D. Salas
Colorado State University, USA
J. Obeysekera
South Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
Lecture Outline• Introduction. Causes of change in extreme events• Review if basic concepts of probability and statistics• Models of extreme events for stationary and non-
stationary conditions• Parameter estimation and model selection for
stationary conditions• Other model alternatives for extreme events• Introduction to software R for modeling extreme
events
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Lecture Outline (cont.)• Introduction to software R for modeling extreme
events• Parameter estimation and model selection for non-
stationary conditions• Return period and risk for stationary and non-
stationary conditions• Examples of analysis of extreme events based on
stationary and non-stationary conditions• Hands on experience in using software R for
analyzing extreme events under stationary and non-stationary conditions
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Resources
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Riesgo Hidrológico de EventosExtremos en Condiciones
No-Estacionarias
Jose D. Salas
Colorado State University, USA
J. Obeysekera
South Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
Introduction: Causes of changes in extreme events
• Course objectives
• What is expected to be learned from the course
• Summary of the course content
• Format
• Materials and references
• Others
Introduction
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Outline
• Extreme hydrologic events
• Types of extreme value data
• Types of changes (non‐stationarity) in hydrologic data
• Causes of non‐stationarity
Introduction
Extreme Hydrologic Events
• Floods
• Low flows and droughts
• Max. and min. precipitation
• Max. and min. temperature (heat waves)
• Maximum wind
• Max. and min. storages (e.g. reservoir, groundwater levels, soil moisture, snow pack, glaciers)
• Max. sea levels
• Max. erosion rates, sediment transport, and sediment deposition
• Maximum water quality pollutant concentrations
• Many others
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Extreme Hydrologic Events
Drastic damages and changes produced by the extraordinary flood originated from Hurricane Mitch of October 1998. (a) a river and a bridge in Guatemala, Source: Time Magazine, Nov. 1998, and (b) the Choluteca River in Tegucigalpa, Honduras, Source: National Geographic Vol. 196(5), p. 111, Nov. 1999.
Extreme Floods
(a) (b)
Extreme Hydrologic Events
Tous Dam at Rio Jucar, Spain breached and caused a flood of about 15,200 m3/s along River Jucar in October of 1982. Source: Dr. J. Marco‐Segura, U. Valencia.
Extreme Floods
Fort Collins flood, July 1997
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Extreme Hydrologic Events
Extreme Floods
Extreme Hydrologic Events
Extreme Floods
Washed out road over Milford Dam near Milford, Kansas in July 1993(source: cover of ASCE Civil Engineering Magazine, Vol. 64(1), January.)
Mississippi flood of 1993, USA(source unknown)
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Extreme Hydrologic Events
Extreme Floods
Damage of extreme flood occurred in Venezuela, Dec. 15, 1999(source: Central University in Caracas, Prof. Marco P. Rivero)
Venezuela Venezuela
Extreme Hydrologic Events
Extreme Floods
Queensland, Australia, 2011
Pakistan, 2010 India, 2010 China
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Extreme Hydrologic Events
Extreme Floods
ʺThe frequency (of extreme weather situations) is way up,ʺ Andrew Cuomo, Governor of New York, 10/31/2012
Extreme Hydrologic Events
Extreme Droughts
Dust bowl, Midwest US 1930’s(source unknown)
Hayman fire near Cheesman Reservoir(source: Denver Water)
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Extreme Hydrologic Events
Extreme Droughts
A photo of a wildfire in the west in 2000 due to drought conditions, Aug.
6, 2000 in the Bitterroot Valley National Forest, MT.
Drought in the Amazon RiverSource: G. Chamorro, SENAMHI
Extreme Hydrologic Events
Extreme Droughts
Cheesman Reservoir, ColoradoDrought 2002
(source: Denver Water)
Dillon Reservoir, ColoradoDrought 2002
(source: Denver Water)
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Types of Extreme Value Data : Block Maxima
Blocks1 2 3 4 5
Block length could be:One YearSeason (wet and dry)
Example: Daily Precipitation
Types of Extreme Value Data : Peaks Over Threshold (POT)
• In hydrology, this is known as Partial Duration Series
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Threshold
Example: Daily Precipitation
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Definitions of Low Flows
Time series of daily flows, 1951‐2000
Definition of Low Flows (1)
Unit time period Tu
dd
d
v1 vjvm
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Definition of Low Flows (1)
The d‐day low flow for a given unit time period Tu is:
Q’ = min(v1, v2, …, vm)
where m = number of d‐day flows in a unit time period,
e.g. m=356 for d=10 and Tu=1 year
For N unit time periods (e.g. for N years of record) we
will get the sequence
Q’1 , Q’2 , . . . , Q’N
Then, frequency analysis must be performed to get the
T‐year d‐day low flow. Note that for low flows
T = 1/q (q=non‐exceedance probability)
Definition of Low Flows (2)
Low flow for a variable duration
In this case the duration of low flows is a random variable
A threshold flow Qo is selected and the following low flow
variables can be defined (refer to the following figure)
d* = max(d1, d2, . . . , dm) max. duration of low flows
v* = max(v1, v2, . . . , vm) max. deficit
I* = max(I1, I2, . . . , Im) max. intensity of low flows
Other quantities can be defined such as averages or
maximums, etc.
Series: d*1 , d*2 , . . . , d*N to do frequency analysis.
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Definition of Low Flows (2)
d1 d2 d3
v1 v2 v3
Example of low flow duration frequency analysis
The daily flows of Edisco River, SC for the period 1951‐2000
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Example of low flow duration frequency analysis
Extreme multiyear droughts
Poudre River annual flows at the Mouth of the Canyon (1884‐2002)
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Drought definition and properties
Critical droughts can be extracted from the time series
Types of Changes (non‐stationarity)in Hydrologic Data
• Trends (increasing and decreasing)
• Shifts (upward or downward)
• Mixed
• Seasonality (periodicity, cycles)
• Pseudo‐ cycles
• Clustering
• Others
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Trends (increasing floods)
(a) Abjerjona Basin (b) Little Sugar Creek Basin.
μt = μ0 ‐ ‐ ‐ μt = μ0 + a t
Trend in the mean and in the standard deviation
Trends (increasing and decreasing)in sea levels
(a) Key West, Florida (b) Adak, Alaska
Example: c = ‐ 1.34 mm/yr, b = 2.71 x 10‐5, e = 1079 mm,σ = 111.3 mm, and ε = ‐ 0.26 for the Adak gage.
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Shifts in Hydrologic Data
Niger River annual flows. Example of upwards and downwards shifts
St. Johns River (Florida) Annual FloodsShift related to the AMO
Shifts in Flood Data
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Annual Cycle in Hydrologic Data
Monthly streamflow data
Daily Cycle in Hydrologic Data
Hourly rainfall for the month of July for Denver Airport
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Causes of Non‐stationarity
• Human intervention in river basins (watersheds)Construction of hydraulic structuresDamsDiversionsTunnelsGroundwater pumpingOthers
Land use changesUrbanizationIrrigation and farming activitiesDeforestationBuilding transportation systemsOthers
Industrial development and operationsMining activitiesOthers
Causes of Non‐stationarity
• Human intervention in river basins (watersheds)
• Natural climate variabilityEffects of low frequency (large scale) phenomena such as ENSO (years) PDO (decades), and AMO (multi‐decades)
• Climate change due to increase of green‐house gasesincrease in air temperatureincrease in moisture of the airhydrologic effects uncertain and still debatable
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Causes of Non‐stationarity
Effect of construction of hydraulic structures along de Han River in Korea (source Dr. D.R. Lee)
Human intervention: Construction of hydraulic structures
Causes of Non‐stationarity
Effects of construction of hydraulic structures along de Han River in Korea (source Dr. D.R. Lee)
Human intervention: Construction of hydraulic structures
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Causes of Non‐stationarity
(a) Aberjona River, WinchesterMassachusetts
(b) Little Sugar Creek, Charlotte North Carolina
Human intervention: effects of urbanization
Causes of Non‐stationarity
Natural climate variability: Effects of low frequency
33 years
40 years
26 years 32 years
warm
coldcold
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Causes of Non‐stationarity
Effect of natural climate variability, PDO(source: Akintug and Rasmussen, 2005)
Effect of natural climate variability, AMOsource: Dr. J. Obeysekera, SFWMD
Natural climate variability: Effects of low frequency
Final Remarks
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Riesgo Hidrológico de EventosExtremos en Condiciones
No-Estacionarias
Jose D. SalasColorado State University, USA
J. ObeysekeraSouth Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
Review of Basic Concepts of Probability, Statistics, and Time Series
Outline
• Random events
• Probability of random events
• Random variable
• Probability laws
• Independent and dependent random variables
• Population moments
• Moments of linear functions of random variables
• Moments of non‐linear functions of random variables
• Central limit theorem
Review of Basic Concepts of Probability
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Review of Basic Concepts of Probability
Review of Basic Concepts of Probability
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Review of Basic Concepts of Probability
Review of Basic Concepts of Probability
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Review of Basic Concepts of Probability
Review of Basic Concepts of Probability
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Review of Basic Concepts of Probability
Review of Basic Concepts of Statistics
Outline
• Random sample
• Sample moments
• Moments of sample moments
• Estimation (estimates and estimators)
• Methods of estimation
• Properties of estimators
• Confidence limits on population parameters
• Confidence limits on population quantiles
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Review of Basic Concepts of Statistics
Review of Basic Concepts of Statistics
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Review of Basic Concepts of Statistics
Review of Basic Concepts of Statistics
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Review of Basic Concepts of Statistics
Review of Basic Concepts of Statistics
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Review of Basic Concepts of Statistics
Review of Basic Concepts of Statistics
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Review of Basic Concepts of Time Series
Review of Basic Concepts of Time Series
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Final Remarks
Riesgo Hidrólogico de EventosExtremos en Condiciones
No-Estacionarias
Jose D. SalasColorado State University, USA
J. ObeysekeraSouth Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
Models of Extreme Events for Stationary andNon‐Stationary Conditions
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Outline
• Typical models for extreme events (stationary conditions)
• General extreme value (GEV) model (stationary conditions)
• General extreme value (GEV) model (non‐stationary conditions)
Models of Extreme Events for Stationary and Non‐Stationary Conditions
• Typical models for extreme events (stationary conditions)
Stationary data
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Typical models for extreme events (stationary conditions)
Typical models for extreme events (stationary conditions)
Lognormal
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Typical models for extreme events (stationary conditions)
Gamma (Pearson)
Typical models for extreme events (stationary conditions)
Log‐Pearson Type III
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Typical models for extreme events (stationary conditions)
General extreme value
If
the model is called Logistic
Typical models for extreme events (stationary conditions)
Generalized logistic
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‐ Exponential if
‐ Uniform if
Typical models for extreme events (stationary conditions)
Generalized Pareto
Vilfredo Pareto1848‐1923
Typical models for extreme events (stationary conditions)
Kappa
Dr. Paul MielkeProfessor Emeritus, CSU
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Typical models for extreme events (stationary conditions)
Kappa
Typical models for extreme events (stationary conditions)
Mixtures and products
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Example: Mixture of three distrib. (Waylen & Caviedes, 1986)
Typical models for extreme events (stationary conditions)
Typical models for extreme events (stationary conditions)
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Typical models for extreme events (stationary conditions)
General Extreme Value (GEV) Model(stationary conditions)
• Assume block maxima coming from a series of independent and identically distributed (i.i.d.) observations:
• Distribution of Mn (assuming iids):
• Then for large n, and some an and bn:
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General Extreme Value (GEV) Model(stationary conditions)
G(z) belongs to one of these:
General Extreme Value (GEV) Model(stationary conditions)
Emil Gumbel1891‐1966
ʺIt seems that the rivers know the theory. It only remains to convince the engineers of the validity of this analysis.ʺ
Maurice Frechet1878‐1973
Waloddi Weibull1887‐1979
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General Extreme Value (GEV) Model(stationary conditions)
General Extreme Value (GEV) Model(stationary conditions)
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General Extreme Value (GEV) Model(stationary conditions)
Type III(ξ<0)Type I(ξ=0) Type II(ξ>0)
General Extreme Value (GEV) Model(stationary conditions)
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Non‐stationary data
Annual maximum floods
Built‐out?
Non‐stationary data
Mean and maximum sea levels for some sites in the USA
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General Extreme Value (GEV) Model(non‐stationary conditions)
General Extreme Value (GEV) Model(non‐stationary conditions)
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Final Remarks
Riesgo Hidrólogico de EventosExtremos en Condiciones
No-Estacionarias
Jose D. SalasColorado State University, USA
J. ObeysekeraSouth Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
Parameter Estimation and Model Selection forStationary GEV Models
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Outline• Parameter estimation and quantile estimation for GEV
modelsMOM, PWM, ML, BayesianProfile Likelihood
• Uncertainty of parameters and quantilesApproximate standard errors (delta method)Better approximation (profile likelihood)
• Model selectionLikelihood ratio (deviance statistic)AICDiagnostic plots
Parameter Estimation & Model Selectionfor Stationary GEV Models
Parameter estimation and quantileestimation for GEV models
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Parameter estimation and quantileestimation for GEV models
Parameter estimation and quantileestimation for GEV models
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Parameter estimation and quantileestimation for GEV models
Parameter estimation and quantileestimation for GEV models
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Parameter estimation and quantileestimation for GEV models
Parameter estimation and quantileestimation for GEV models
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Parameter estimation and quantileestimation for GEV models
Parameter estimation and quantileestimation for GEV models
Example: Umpqua flood data
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Parameter estimation and quantileestimation for GEV models
PWM estimates
Likelihood ratio test
ML estimates and standard errors
Calculations using R software
Parameter estimation and quantileestimation for GEV models
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Parameter estimation and quantileestimation for GEV models
Parameter estimation and quantileestimation for GEV models
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Parameter estimation and quantileestimation for GEV models
Parameter estimation and quantileestimation for GEV models
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Parameter estimation and quantileestimation for GEV models
Parameter estimation and quantileestimation for GEV models
Standard errors of parameters
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Parameter estimation and quantileestimation for GEV models
Variance‐covariance matrixEstimation of quantiles and
confidence limits
Parameter estimation and quantileestimation for GEV models
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General Extreme Value (GEV) Model(stationary conditions)
General Extreme Value (GEV) Model(stationary conditions)
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General Extreme Value (GEV) Model(stationary conditions)
General Extreme Value (GEV) Model(stationary conditions)
Estimation of quantiles for T=5, 10, 25, 50, 100, 250, 500 years and theirconfidence limits
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General Extreme Value (GEV) Model(stationary conditions)
General Extreme Value (GEV) Model(stationary conditions)
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General Extreme Value (GEV) Model(stationary conditions)
General Extreme Value (GEV) Model(stationary conditions)
*Keep θi constant and maximizeℓ(θ) with respect to all otherparameters, θ‐I
*Repeat for different θi
θi
ℓp(θi)
Profile likelihood for θi
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General Extreme Value (GEV) Model(stationary conditions)
Chi‐square distribution
θi
Confidence intervalfor θi
3.8415/2for k=1
Profile likelihood confidence limits for θi
General Extreme Value (GEV) Model(stationary conditions)
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General Extreme Value (GEV) Model(stationary conditions)
Confidence Interval
Confidence Interval
Shape Parameter 25‐yr quantile
Inference using profile likelihood
General Extreme Value (GEV) Model(stationary conditions)
218 390
267.6
‐0.11 0.24
0.0514
Delta method gives:(‐0.24,0.73)
Delta method givesZq:(193.7 341.57)
Results based on profile likelihood
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General Extreme Value (GEV) Model(stationary conditions)
General Extreme Value (GEV) Model(stationary conditions)
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General Extreme Value (GEV) Model(stationary conditions)
Model selection based on Deviance statistic and AIC
General Extreme Value (GEV) Model(stationary conditions)
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General Extreme Value (GEV) Model(stationary conditions)
General Extreme Value (GEV) Model(stationary conditions)
Results for the UmpquaFlood data based on GEV
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Final Remarks
Riesgo Hidrológico de EventosExtremos en Condiciones
No-EstacionariasOther model alternatives for extreme events
Credit:Jana Sillmann
Jose D. SalasColorado State University, USA
J. Obeysekera (‘Obey’)South Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
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Types of Extreme Value Data : Block Maxima (review)
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Blocks1 2 3 4 5
Block length could be:One YearSeason (wet and dry)
Example: Daily Precipitation
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Models based on r-largest statistics
Daily Rainfall (Fort Lauderdale, Florida)
5 largest values of Rainfall every year
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Models based on r-th largest statistics
, … . ,
!
1/
, … ,
138
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Modeling the r-th Largest Order values
, … , 1,2, . . ,
Likelihood
, , 1
/
x 1
Example: 10 largest sea-levels in Venice
Year r1 r2 r3 r4 r5 r6 r7 r8 r9 r10
1931 103 99 98 96 94 89 86 85 84 79
1932 78 78 74 73 73 72 71 70 70 69
1933 121 113 106 105 102 89 89 88 86 85139
Example
r l
1 -222.7 111.1(2.6) 17.2 (1.8) -0.077 (0.074)
5 -732.0 118.6(1.6) 13.7(0.8) -0.088 (0.033)
7 916.5 119.1(1.47) 13.25(0.7) -0.09(0.029)
10 1139.1 120.5 (1.36) 12.8 (0.55) -0.113 (0.019)
r-largest order statistics for Venice sea levels
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(*ignore trend for the moment)
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Global Warming Protesters
Peaks Over Threshold (POT) Models
Motivation: Can we
use all “extreme”
values above a given
threshold?
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Threshold Example: Daily Precipitation
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Peaks Over Threshold - Theory
Stochastic model:
It can be shown that (Coles,2001):
then
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y>0
Generalized Pareto Distribution (GPD)
For large enough u and Y=X-u,
Case: 0 Threshold Selection is important
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0
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Can we get the equivalent GEV parameters from GPD?
Need , , from , is the same for both GEV and GPD
Need to estimate , Need some new formulation to estimate the
third parameter
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Poisson-GPD Model
The number of N exceedances of the threshold level, u, has a Poisson distribution with mean
Assuming N ≥ 1, the excess values Y1 ,..,Yn
are IIDs from GPD
For z > u, the probability that the annual maximum of Y is,
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Poisson-GPD model (cont.)
147
This will be identical to the form of the GEV if,
Knowing (=# above u/total #) we can now compute the equivalent GEV parameters
A more elegant approach: Point Process Model (PP)
Interpretation of extreme value behavior that unifies all models (GEV, GPD, Poisson-GPD)
Leads to a “likelihood” (to be discussed) that enables a more natural formulation of non-stationarity in threshold excesses than from GPD alone
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Poisson Process (PP) Model (preview)
Domain, D=[0,T] x u,∞ X is viewed as two-
dimensional, non-homogenous Poisson process (t, x u
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0 T
u
A
t1 t2
x
The rate is,
The number of points in A, N(A) is a Poisson Process with mean:
, also
Poisson Process (PP) Model (Cont.)
Advantages: Direct estimation of GEV parameters
is not a function of u (unlike in GPD)
Easier to implement in the “non-stationary” setting (to be discussed later)
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Summary of Models
Block Maxima GEV
Gumbel
Others
Peaks Over Threshold (POT) GPD
Poisson-GPD
PP
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Generalize Pareto Distribution (GPD) –Maximum Likelihood Estimation
Recall:
Two parameters to estimate:
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MLE for GPD - Spreadsheet
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=-$K$5*LN($Q$5)-(1+1/$Q$6)*SUM(M7:M364)
File:Fortgpd.xlsx
Poisson Process (PP) Model (review)
Domain, D=[0,T] x u,∞ X is viewed as two-
dimensional, non-homogenous Poisson process (t, x u
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0 T
u
A
t1 t2
x
The rate is,
The number of points in A, N(A) is a Poisson Process with mean:
, also
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Poisson Process (PP) Model –Inference
Advantages: Direct estimation of GEV parameters
is not a function of u (unlike in GPD)
Easier to implement in the “non-stationary” setting (to be discussed later)
Fort Collins Precipitation data again
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Point Process (PP) Model - MLE
Likelihood function (Coles 2001, chapter 7)
Ii is a small interval around xi , I is the full range
GEV parameters directly:
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MLE for PP - Spreadsheet
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=-$O$6*(1+$O$10*($O$7-$O$8)/$O$9)^(-1/$O$10)-$K$8*LN($O$9)-(1+1/$O$10)*SUM(L10:L367)
File:Fortpp.xlsx
Dependent Sequences – Declustering
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Average Daily Temperature
U
U
Clusters for different r
r=1,2r=3
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Declustering Procedure
Use an empirical rule to define clusters of exceedences
Identify maximum excess within each cluster
Assume cluster maxima are independent, and that the conditional excess is distributed as a generalized Pareto
Fit the GPD to the cluster maxima
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Change to Return Level Formula
Rate at which clusters must be taken into account. m observation return level:
is the extremal index
Let nu = number of exceedances above u
nc = number of clusters above u
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Declustering of Fort Collins Rainfall> load("Fort.RData")
> attach(a)
> dcRain1<-dclust(Rain,u=0.4,r=1)
> names(dcRain1)
[1] "xdat.dc" "ncluster" "clust"
>gpd<-gpd.fit(Rain,0.4,npy=214)
>gdc=gpd.fit(dcRain1$xdat.dc,0.4,npy=214)
gpd$nexc=358 dcRain1$ncluster=288
gpd$mle: 0.394(0.031) 0.164(0.058)
gdc$mle: 0.446(0.037) 0.130(0.060)
Quantile(100-yr)
gpd: 5.05 (3.84,7.89)
gdc: 4.79 (3.71,7.16)
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Non-stationary GEV models Parameters are function of time (or any
other variable-”covariates”)
Examples:
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Non-stationary GEV: Examples (cont.)
163
Change Point Models
Math Classic..
164
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Final Remarks
165
Riesgo Hidrológico de EventosExtremos en Condiciones
No-EstacionariasIntroduction to modules of the software R for
modeling extreme events
Credit:Jana Sillmann
Jose D. SalasColorado State University, USA
J. ObeysekeraSouth Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
84
What is R
Most cost effective statistical package (why?)
R is a system for statistical computation and graphics.
Also a programming language, high level graphics, interfaces to other languages
The R language is a dialect of S which was designed in the 1980s
167
What is R (Cont.)
an effective data handling and storage facility,
a suite of operators for calculations on arrays, in particular matrices,
a large, coherent, integrated collection of intermediate tools for data analysis,
graphical facilities for data analysis and display either on-screen or on hardcopy, and
a well-developed, simple and effective programming language which includes conditionals, loops, user-defined recursive functions and input and output facilities
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Installation of R
http://www.r-project.org/
Select Comprehensive R Archive Network (CRAN) mirror site
Download R for Windows (or for the particular operating system)
Install base (install R for the first time)
Provides basic functions which let R function as a language
Base package and contributed packages
For a complete list of functions, use library(help = "base")
169
Help for R
http://www.r-project.org/
http://www.rseek.org/
http://archive.org/details/TheRBook
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Downloading R
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Downloading R
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Downloading R
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Downloading R
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Doing more..
Need to get “contributed packages”
http://cran.r-project.org/web/packages/available_packages_by_name.html
> help("install.packages")
Eg. Installing package “extRemes” > install.packages("extRemes")
Will ask for CRAN mirror site
>getwd() to find current working directory
>setwd(“path”) to set the working directory - or use R.gui()
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Data structures 1/6 Vector A list of numbers, such as (1,2,3,4,5) R: a<-c(1,2,3,4,5) or a=c(1,2,3,4,5) Command c creates a vector that is assigned to object a
Factor a = c(1,2,2,4,4,5)
A list of levels, either numeric or string R: b<-as.factor(a) Vector a is converted into a factor b:1 2 2 4 4 5
with Levels: 1 2 4 5
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Data structures 2/6 Data frame A table where columns can contain numeric and
string values R: d<-data.frame(a, b)
Matrix All columns must contain either numeric or string
values, but these can not be combined R: e<-as.matrix(d) Data frame d is converted into a matrix e
R: f<-as.data.frame(e) Matrix e is converted into a dataframe f
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Data structures 3/6
List Contains a list of objects of possibly different types. R: g<-as.list(d) Converts a data frame d into a list g
>g=gev.fit(x) g is now a list object >names(g) check names of the objects in g "trans" "model" "link" "conv" "nllh"
"data" "mle" "cov" "se" "vals"
g$mle gives the values of mle object:301.8081211 169.9874853 0.3544377
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Data structures 4/6 Some command need to get, for example, a matrix, and do
not accept a data frame. Data frame would give an error message.
To check the object type: R: class(d)
To check what fields there are in the object: R: d R: str(d)
To check the size of the table/matrix: R: dim(d)
To check the length of a factor of vector: R: length(a)
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Data structures 5/6
Some data frame related commands: R: names(d) Reports column names
R: row.names(d) Reports row names
These can also be used for giving the names for the data frame. For example: R: row.names(d)<-c("a","b","c","d","e") Letters from a to e are used as the row names for
data frame d Note the quotes around the string values!
R: row.names(d)
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Data structures 5/6 Naming objects:
Never use command names as object names!
If your unsure whether something is a command name, type to the comman line first. If it gives an error message, you’re safe to use it.
Object names can’t start with a number
Never use special characters, such as å, ä, or ö in object names.
Underscore (_) is not usable, use dot (.) instead: Not acceptable: good_data
Better way: good.data
Object names are case sensitive, just like commands
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Reading data 1/2
Command for reading in text files is:read.table(”suomi.txt”, header=T, sep=”\t”)
This examples has one command with three arguments: file name (in quotes), header that tells whether columns have titles, and sep that tells that the file is tab-delimited.
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Reading data 2/2 It is customary to save the data in an object in
R. dat<-read.table(”suomi.txt”, header=T, sep=”\t”)
Here, the data read from file suomi.txt is saved in an object dat in R memory.
The name of the object is on the left and what is assigned to the object is on the right.
Command read.table( ) creates a data frame.
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Using data frames Individual columns in the data frame can be accessed using
one of the following ways:
Use its name: dat$year
dat is the data frame, and year is the header of one of its columns. Dollar sign ($) is an opertaor that accesses that column.
Split the data frame into variables, and use the names directly: attach(dat)
Use subscripts
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Subscripts 1/2 Subscripts are given inside square brackets
after the object’s name: dat[,1] Gets the first column from the object dat
dat[,1] Gets the first row from the object dat
dat[1,1] Gets the first row and it’s first column from the
object dat
Note that dat is now an object, not a command!
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Subscripts 2/2 Subscripts can be used for, e.g., extracting a subset of the data:
dat[which(dat$year>1900),]
Now, this takes a bit of pondering to work out…
First we have the object dat, and we are accessing a part of it, because it’s name is followed by the square brackets
Then we have one command (which) that makes an evaluation whether the column year in the object dat has a value higher than 1900.
Last the subscript ends with a comma, that tells us that we are accessing rows.
So this command takes all the rows that have a year higher 1900 from the object dat that is a data frame.
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Writing tables To write a table: write.table(dat, ”dat.txt”, sep=”\t”)
Here an object dat is written to a file called dat.txt. This file should be tab-delimited (argument sep).
Write.csv(dat, ”dat.csv”, row.names=FALSE)
This file can be opened in Excel
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Commonly used R commands in this course
Command What it does
>R start R>?plot Get help on 'plot' command>ls() Check the object space>class(a) Check the data type of object a
>head(a) What the data object looks like (few rows printed)>names(a) names of object a>a=c(1,2,3,4,5) Create a vector object a with values 1:5>c=cbind(a,b) Combine vector objects a and b into the object c
>plot(x,y) plot y versus x
>is.na(a) A logical vector showing which values in a are missing
>sum(is.na(a)) Number of missing values in object a>b=a[‐20,] Remove 20th row of a and place it in b
>a=read.csv("file.csv",header=T)Read data in "file.csv" which has a header and place in to object a
>write.csv(a,"file.csv") Write the object a into csv file called "file.csv"
>a=seq(5,100,5) Create a sequence of numbers, 5 to 100 with increments of 5
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Creating a fancy plot
fout= “myplot.png”
#set the device for plotting
png(fout,pointsize=12,units="in",width=8,height=6,res=350)
par(mar=c(5.1,6.1,3.1,3.1))
par(font=2,font.lab=2,font.axis=2,cex=1.5,cex.axis=1.1,cex.lab=1.1)
#now plot
plot(yrs,x,pch=19,xlab="Year",ylab=“Discharge”, col="red")
title(“myplot”)
#set the device off
dev.off()
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Reading and Manipulating Data>dat=read.csv(“sample.csv”,header=TRUE)
#notice missing values denoted as “NA”
>class(dat)
>head(dat)
>attach(dat)
>amean=tapply(Rain,Year,mean,na.rm=T)
>momean = tapply(Rain,Month,sum,na.rm=T)
>sum(is.na(dat))
#setting missing values to zero
>dat[is.na(dat[,3]),3]=0
>plot(Rain)
>boxplot(Rain~Month)
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Compiling & Running a functionmyplot <- function(Year,Rain,outflag=0) {
if(outflag != 0 ) png("rainplt.png")
amean = tapply(Rain,Year,sum,na.rm=T)
yrs = as.numeric(names(amean))
plot(yrs,amean,pch=19,ylab="Rain",col="red",cex=1.5)if(outflag !=0 ) dev.off()return(list(yrs=yrs,amean=amean))}
>v = myplot(Year,Rain)
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> v$yrs[1] 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905> v$amean1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 52.35 62.47 73.28 46.77 40.95 55.96 34.54 37.02 25.51 42.34 52.11
Quitting R
Use command q() or menu choise File->Exit. R asks whether to save workspace image. If you
do, all the object currently in R memory are written to a file .Rdata, and all command will be written a file .Rhistory.
These can be loaded later, and you can continue your work from where you left it.
Loading can be done after starting R using the manu choises File->Load Workspace and File-> Load History.
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A powerful IDE for R (www.rstudio.com)
Extremes Package Demo
library(extRemes)
extremes.gui()
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Task ismev/extRemes evd evir fExtremes POT VGAM
Data generation
gev gen.gev rgev rgev gevsim, gumbelsim rgev
gpd gen.gpd rgpd rgpd gpdSim rgpd rgpd
GEV fitting
MLE gev.fit fgev gev, gumbel gevFit, gumbelFit gev
Profile Likelihoodgev.prof,gev.profxi,gev.parameterCI profile.evd
Diagnostics gpd.giag
GPD fitting
Threshold mrl.plot mrlplotmeplot,
findthreshmrplot, findThreshold,
mxfPlot mrlplot,lmomplot
gpd.fitrange tcplot tcplot
MLE gpd.fit fpot gpd gpdFit fitgpd gpd
Profile Likelihoodgpd.prof, gpd.profxi, gpd.parameterCI profile.evd gevrlevelPlot
gpd.pfrl,gpd.pfscale, gpd.pfshape
Fisherbasedgpd.firl,gpd.fiscale,
gpd.fishape
bootstrappingboot.matrix, boot.sequence
PP fittingMLE pp.fit fpot pot pointProcess fitpp
decluster declust, decluster deCluster
CovariatesGEV gev.fit fgev
GPD gpd.fit ?
PP pp.fit 195
Final Remarks
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Riesgo Hidrológico de EventosExtremos en Condiciones
No-EstacionariasParameter estimation and model selection for
non‐stationary conditions
Credit:Jana Sillmann
Jose D. SalasColorado State University, USA
J. ObeysekeraSouth Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
Climate change
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Extreme Data Types
Stationary Stochastic properties do not change with time
Non-Stationary Stochastic properties vary with time Trends (mean, frequency and intensity)
Cycles (diurnal, annual)
Because of co-variation with other variables (e.g. Flood = f(El Nino phenomenon)
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Non-stationary GEV models Parameters are function of time (or any
other variable-”covariates”)
Examples:
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Nonstationary GEV Models (cont.)
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Change Point Model “Seasons” model
Parameter Estimation (Non-stationary case)
Non-stationary GEV model: Zt = GEV( (t), (t), (t))
Shape parameter is difficult to estimate and it is unrealistic to model it as a smooth a function of time
Log-likelihood:
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t , t , t T
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Parameter Estimation (Cont.)
Non-stationary Gumbel
Two parameters, but both functions of time
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Parameter Estimation (Cont.)
Example 1
Parameters, = ( 0 , 1 , , )
Example 2
Parameters, = ( 0 , 1 , 2 , , )
Approximate Standard errors derived using the same techniques that we discussed for stationary case
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Math Classic..
Example: Aberjona River, Winchester, Massachusettes
Typical of basins where land use changes have caused increasing floods
Nonstationarymean, variability?
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R and Excel Solutions a=read.csv("Aberjona_1102
500.csv",header=T)
cov=yrs-mean(yrs)
cov=as.matrix(cov,ncol=1)
gev.fit(x,ydat=cov,mul=1)
Output:
$nllh 410.931
$mle 319.3587323 2.8814896 163.3669736 0.3040809
$se 25.3638274 1.0190117 21.1793573 0.1337115
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Confidence Limits for parameters: Approximate Normality of MLEs
For large n:
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ExpectedInformationMatrix
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Response variable: Discharge
[1] "cov.selected = Cov"
Convergence successfull![1] "Convergence successfull!"
[1] "Maximum Likelihood Estimates:“ MLE Stand. Err.
MU: (identity) 319.35873 25.36373
Cov: (identity) 2.88149 1.01901
SIGMA: (identity) 163.36697 21.17922
Xi: (identity) 0.30408 0.13371
[1] "Negative log-likelihood: 410.931027606394"
Parameter covariance:
[,1] [,2] [,3] [,4]
[1,] 643.318930 7.19927261 358.945815 -1.33843641
[2,] 7.199273 1.03838421 4.686030 -0.05198826
[3,] 358.945815 4.68603046 448.559343 -0.37964699
[4,] -1.338436 -0.05198826 -0.379647 0.01787876
extRemes package
Confidence Limits using Bootstrapping
Recall that Z ~ GEV( t , t ,
Then e = [Z- t / tis GEV~(0,1,
The basic approach is to bootstrap the residuals, e and fit multiple models
Fit Z ~ GEV( (t), (t), )
Compute residuals, e
For k = 1, Nsamples Resample residuals with
replacement = enew
Recreate z = Znew using ( (t),(t),
Fit Znew ~ GEV( (t), (t), )
Compute ZT for the desired T
Compute 5th and 95th interval using Nsamples of ZT
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Model Selection (Likelihood Ratio Test) Let a model, M1=f(θ(1), θ(2)) and M0=f(θ(1)=0, θ(2)). M0 is a subset of M1. The
question is, Is model M1 any better than M0? Deviance Statistic:
Reject M0 in favor of M1 if
where cα is the (1-α) quantile of the χ2k
distribution dimension of θ(1)
Alternatively:
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α
cα
Model Selection based on AIC & BIC
Akaike’s Information Criteria (AIC) AIC(k) = -2llh (k)+2 k, k=number of parameters
Select the model which has the minimum AIC
Bayesian Information Criterion (BIC) BIC(k) = -2llh(k) + k ln T k = number of parameter
T = sample size
Select the model which has the minimum BIC
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Model Diagnostics (Non-stationary) case
~ , , Standardized variable
11
Note: follows a Gumbel distribution
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Model Diagnostics (Cont.)
Propbability plot:
, exp exp 1,2, … ,
Quantile plot:
, log log ; 1,2, … ,
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Modeling Non-stationarity -Summary
Fit various modeling using MLE
Select an appropriate model using AIC, BIC and Likelihood Ratio Test as criteria
Compute Return Level, and Risk for a given Return Period
Compute confidence intervals for parameters and return level
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Example: Aberjona River, Winchester, Massachusettes
Typical of basins where land use changes have caused increasing floods
Nonstationarymean, variability?
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Aberjona River – First Stationary Case
>dat=read.csv("Aberjona_1102500.csv",header=TRUE)
>head(dat) (yrs x)>attach(dat)>gum<- gum.fit(x) #fit Gumbel$gum$mle: {337.4(28.1) 209.3(23.0)}$gum$nllh= 419.7192> gum.diag(gum)>AICgum=2*(gum$nllh+2) =843.4384
>gev<- gev.fit(x) #fit GEVgev$mle {301.8(25.5) 170.0(22.3)
0.354(0.13)} gev$nllh=414.9693>gev.diag(gev)
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Modeling Non-staionarity using covariates
g = gev.fit(x, ydat = cov, mul=c(1,..), sigl=c(1,..), shl=c(1,..), siglink = exp, muinit=c(…),…)
Covariate matrix, cov
Year SOI AMO
1945 …. ….
1946 …. ….
R-Demo
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Aberjona River - Model Checking & Non-stationary Modeling>D= -2*(gev$nllh-gum$nllh)
> chi=qchisq(0.95,1)
> p=pchisq(D,1,lower.tail=FALSE)
AICgev=2*(gev$nllh+3)
D = 9.499 ; chi =3.84; p=0.002;
AICgev= 835.9386
D > Chi, and AICgev < AICgum
GEV is chosen over Gumbel
#Probability Weighted Moments
> lmr<-lmom.ub(x)
> pwmgev<-pargev(lmr)
#nonstationarity in location parameter
>cov <- as.matrix(yrs - mean(yrs),ncol=1)
>gevmu <- gev.fit(x,ydat=cov,mul=1)
gevmu$nllh=410.931
gevmu$mle {319.4(25.4) 2.88(1.02) 163.4(21.2) 0.304 (0.134)}
> D= -2*(gevmu$nllh-gev$nllh)
> chi=qchisq(0.95,1)
> p=pchisq(D,1,lower.tail=FALSE)
> AICgevmu=2*(gevmu$nllh+4)
D=8.076 chi=3.841459 p=0.004 AICgevmu=829.8621
> Select GEV-MU(t) over GEV!
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Aberjona River – Model Comparison
IndexModel Name Parameters nllh Comparison D chi p AIC
1 gum , 419.72 843.4
2 gummu (t), 414.01 2 vs. 1 11.42 3.84 0.001 834.0
3 gumsc , (t) 412.21 3 vs. 2 15.01 3.84 0.000 830.4
4 gummusc (t), (t) 406.23 4 vs. 2 15.56 3.84 0.000 820.5
5 gev , , 414.97 5 vs. 1 9.50 3.84 0.002 835.9
6 gevmu (t), , 410.93 6 vs. 5 8.08 3.84 0.004 829.9
7 gevsc , (t), 411.99 7 vs. 6 5.96 3.84 0.015 832.0
8 gevmusc (t), (t), 405.79 8 vs. 7 10.29 3.84 0.001 821.6
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Non-stationary Model –GEV{ t t ,
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Confidence Limits using Bootstrapping
Recall that Z ~ GEV( t , t ,
Then e = [Z- t / tis GEV~(0,1,
The basic approach is to bootstrap the residuals, e and fit multiple models
Fit Z ~ GEV( (t), (t), )
Compute residuals, e
For k = 1, Nsamples Resample residuals with
replacement = enew
Recreate z = Znew using ( (t),(t),
Fit Znew ~ GEV( (t), (t), )
Compute ZT for the desired T
Compute 5th and 95th interval using Nsamples of ZT
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Snippets of code#initial guess R0library(BB)p0 = 1/T0yp0 = -log(1-p0)R0 = mu0 - (sig/xi)*(1-yp0^(-xi))#compute Return level for the desired value of T – finding root using dfsaned = dfsane(par=R0,fn=myfun,T=T,Nmax=Nmax,mut=mut,sig=sig,xi=xi,quiet=TRUE,control=list(trace=FALSE))Rm = d$parconv <- d$convergence
myfun <- function(x,T,Nmax,mut,sig,xi) {pt = 1- exp(-((1+(xi/sig)*(x-mut))^(-1/xi)))pt[is.nan(pt)]<-1ptc = cumprod(1-pt)px =matrix(1,Nmax,1)for(i in 1:Nmax) {if(i == 1) px[i] = pt[1] else px[i] = ptc[i-1]*pt[i]}t=1:Nmaxex = sum(t*px)y <- T - ex#print(ex)y}
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Rest of the code (for boostrapping)for(k in 1:Nsample) {enew <- sample(e,replace=TRUE)znew <- enew * g$vals[,2]+g$vals[,1]gnew<-gev.fit(znew,ydat=cov,mul=1)muslope = gnew$mle[2]mu0=gnew$vals[yrs==lastyr,1]+(consyr-lastyr)*muslopesig = gnew$mle[3]xi = gnew$mle[4]mut = mu0 + (t-1)*muslopeR0new = mu0 - (sig/xi)*(1-yp0^(-xi))
.. Continued to the right
d = dfsane(par=R0new,fn=myfun,T=T0,Nmax=Nmax,mut=mut,sig=sig,xi=xi,quiet=TRUE,control=list(trace=FALSE))Rmnew = d$parcvals[k] <- d$convergenceRms[k]<-Rmnew….}
#5th and 95th confidence limits?r <- quantile(Rms,probs=c(0.05,0.95))
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Results - Parameters
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Results – Return Level (Design Quantile)
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Trend with a change point Little Sugar Creek at Archdale
Drive in Charlotte, North Carolina (Gumbel Distribtion)
>cov = yrs
>cov[yrs <= 1945] <- 1945
cov:1945 1945 1945 1945 1945 1945 1945 1945 1945
1945 1945 1945 1945 1945 1945 1945 1945 1945 1945 1945 1945 1945 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975………………
>cov=matrix(cov-mean(cov),ncol=1)
>gum=gum.fit(x)
>gumu=gum.fit(x,ydat=cov,mul=1)
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Two Change points Mercer Creek,
Washington State
>yr1 = 1970
>yr2 = 1986
>cov=yrs
>cov[yrs <= yr1] <- yr1
>cov[yrs >= yr2] <- yr2cov: 1970 1970 1970 1970 1970 1970
1970 1970 1970 1970 1970 1971 1972 1973 1975 1977 1978 1980 1981 1982 1983 1985 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986 1986
>gev = gev.fit(x)
>gevmu=gev.fit(x,ydat,mul=1)
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Sea Level Rise – Modeling extremesKey West tide gauge>a=read.csv("KeyWest.csv",header=TRUE)> head(a)Year Mean Max1913 1495.4 20421914 1481.2 2070>attach(a)>gevmu=gev.fit(Max,mul=1,ydat=as.matrix(Year-mean(Year)))>g=lm(Mean~Year) #regression
Non-linear change possible in the future Relation shop mean and max! Can we use that property? – later
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Offset=552 mm
Sea Level Rise Projections (in US)
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Modeling with Covariatesdata(fremantle)
Year SeaLevel SOI
1897 1.58 -0.67
1898 1.71 0.57
…attach(fremantle)ydat=cbind(Year-mean(Year),SOI-mean(SOI))gev=gev.fit(SeaLevel)gevmu=gev.fit(SeaLevel,ydat=ydat,mul=1)gevmusoi =gev.fit(SeaLevel,ydat=ydat, mul=c(1,2))
So which model is better?
D<- -2*(gevmu$nllh-gev$nllh)
chi<-qchisq(0.95,1)
p<-pchisq(D,1,lower.tail=FALSE)
AIC=2*(gevmu$nllh+4)
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Fremental Extreme Sea Levels –Model Comparison
IndexModel Name Parameters nllh Compare D chi p AIC
1 gev , , -43.567 NA NA NA NA -81.13
2 gevmu (t), , -49.914 2 vs. 1 12.69 3.84 0.0004 -91.83
3 gevmusoi (t,soi), , -53.898 3 vs. 2 7.97 3.84 0.0047 -97.79
3 gevmusoi (t,soi), , -53.898 3 vs. 1 20.66 5.99 ~0 -97.79
gevmu is better than gev
Adding SOI appears to improve the model further (compare 3 vs. 2)
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Atlantic Multi-Decadal Oscillation (AMO) index as a covariate St. Johns River, Florida. Floods
influenced by the phase of AMO
>dat=read.csv(“Stjohn.csv”,header=T)
>attach(dat)
Year StjQ
1944 3300
1945 9230
>ydat=rep(1,1,length(Year))
>ydat[yrs <= 1969] = -1
>ydat<-matrix(ydat,ncol=1)
>gev=gev.fit(StjQ)
>gevmu=gev.fit(StjQ,ydat=ydat,mul=1)
gev$nllh=400.7574
gevmu$nllh=396.4719
D=8.571; Chi=3.841; p=0.003
AICgev=403.8; AICgevmu=400.5
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Math Classic..
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118
Final Remarks
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Riesgo Hidrólogico de EventosExtremos en Condiciones
No-Estacionarias
Jose D. SalasColorado State University, USA
J. ObeysekeraSouth Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
Return Period and Risk for Stationary andNon‐stationary Extreme Conditions
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Return Period and Risk for Stationary andNon‐stationary Extreme Conditions
Return Period and Risk for Stationary andNon‐stationary Extreme Conditions
120
Return Period and Risk Under Stationary Conditions
Return Period Under StationaryConditions
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Return Period Under StationaryConditions
Design flood and constant values of exceeding (p) and non‐exceeding (q = 1‐p) probabilities throughout years 1 to t,
i.e. stationary condition (Salas and Obeysekera, 2013)
Distribution of the “Waiting Time”for Stationary Conditions
122
Distribution of the Waiting Time(also known as “First Arrival Time”)
Flood occurrenceFlood occurrence
Return Period Under StationaryConditions
123
Return Period Under StationaryConditions
Return Period Under StationaryConditions
124
Hydrologic Risk Under StationaryConditions
Hydrologic Risk Under StationaryConditions
125
Return Period and Risk Under Non‐Stationary Conditions
Fig.5 Example of non‐stationary annual flood data
Developments to deal with Non‐Stationarity
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Return Period and Risk Under Non‐Stationary Conditions
Return Period and Risk Under Non‐Stationary Conditions
127
Distribution of the “Waiting Time”for Non‐Stationary Conditions
Distribution of the “Waiting Time”for Non‐Stationary Conditions
128
Return Period for Non‐Stationary Conditions
Return Period for Non‐Stationary Conditions
129
Hydrologic Risk for Non‐Stationary Conditions
Hydrologic Risk for Non‐Stationary Conditions
130
Hypothetical Example for DeterminingT and R for Non‐stationary Conditions
Final Remarks
131
Riesgo Hidrólogico de EventosExtremos en Condiciones
No-Estacionarias
Jose D. SalasColorado State University, USA
J. ObeysekeraSouth Florida Water Management District, USA
Laboratorio Nacional de Hidráulica, Universidad Nacional de Ingeniería, Lima Perú
Examples of Analysis of Extreme Events forNon‐stationary Conditions
Examples of Analysis of Extreme Events for Non‐stationary Conditions
Outline
• Examples based on the exponential distribution
• Examples of increasing floods
• Examples of increasing and decreasing sea levels
• Examples of shifting flood regimes
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Example Using an Exponential Distribution
Example Using an Exponential Distribution
133
Examples of Analysis of Extreme Events for Non‐stationary Conditions
Examples of Analysis of Extreme Events for Non‐stationary Conditions
134
Examples Using the GEV forIncreasing Flood Events
Examples Using the GEV forIncreasing Flood Events
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Examples Using the GEV forIncreasing Flood Events
Examples Using the GEV forIncreasing Flood Events
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Examples Using the GEV forIncreasing Flood Events
Examples Using the GEV forIncreasing Flood Events
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Examples Using the GEV forIncreasing Flood Events
Examples Using the GEV forIncreasing Flood Events
Aberjona River Basin, Winchester, Massachusetts (Vogel et al, 2011)
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Examples Using the GEV forIncreasing Flood Events
Aberjona River Basin, Winchester, Massachusetts
Examples Using the GEV forIncreasing Flood Events
Aberjona River Basin, Winchester, Massachusetts
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Examples Using the GEV forIncreasing Flood Events
Examples Using the GEV forIncreasing Flood Events
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Examples Using the GEV forIncreasing Flood Events
Examples Using the GEV forIncreasing Flood Events
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Examples Using the GEV forIncreasing Flood Events
Examples Using the GEV forIncreasing Flood Events
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Examples Using the GEV forIncreasing Flood Events
Examples Using the GEV forIncreasing Flood Events
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Examples Using the GEV forIncreasing and Decreasing Sea Levels
(a) (b)
Examples Using the GEV forIncreasing and Decreasing Sea Levels
(a) (b)
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Examples Using the GEV forIncreasing and Decreasing Sea Levels
(a) (b)
Examples Using the GEV forIncreasing and Decreasing Sea Levels
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Further Remarks
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