GEV Flood Quantile Estimators with Bayesian Shape-Parameter GLS Regression

1
GEV Flood Quantile Estimators with Bayesian
Shape-Parameter GLS Regression
Dirceu Silveira Reis Jr., Jery R. Stedinger and
Eduardo Savio Martins Fundação Cearense de
Meteorologia e Recursos Hídricos FUNCEME,
Fortaleza, Brazil, and Cornell University,
Ithaca, NY, USA.
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The Challenge

• Wish to estimate extreme flood quantiles
• (such as 99 percentile) and flood risk
  distribution
• by fitting 3-parameter GEV distribution using
  annual maximum flood records of length 25 to 100
  years.
• Some records have censored values - only know
  floods were below or above a threshold, due to
  zero values and recording limitations for
  historical records.

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The Problem

• Would like to use maximum likelihood estimators
  because of asymptotic properties, and ability to
  incorporate censored data.
• But absurd MLEs of ? can result in moderate
  samples
• Large uncertainty in extreme flood quantile
  estimates
• BEWARE Statisticians often call ? ? - ?.

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To reduce uncertainty

• Use regression to derive regional shape parameter
  ? and its variance using basin characteristics.
• At-site ? estimates obtained with L-moments.
• Small-sample variance and cross-correlations of
  ?-estimators derived by Monte Carlo analysis.
• Combine at-site data regional information as a
  prior probability distribution on ? to define
• Generalized Maximum Likelihood Estimators (GMLE)

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Outline

• GLS Regression Model
• Classical model error estimator
• Bayesian analysis
• GLS Regional ? for Illinois River Basin
• GEV for Illinois Sites with Regional ?
• Conclusions

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GLS Regression

• GOAL
• Obtain efficient estimators of a hydrologic
  statistic as a function of physiographic basin
  characteristics.
• MODEL
• ? ab1log(Area)b2(Slope) . . . Error

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GLS Model

• Why use Generalized Least Squares
• different record lengths mi gt different
  precisions
• PLUS cross-correlation among estimators
• Basic Model for k

L(???) Cov ?i, ?j ??? I ???? where ????
computed using average ? with record lengths mi
and cross-correlations ?i,j between concurrent
flows.

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GLS Analysis Solution

• GLS regression model (Stedinger Tasker, 1985,
  1989)
• ? X b ?
• with parameter estimator b for b
• XT L(???)-1 X b XT L(???)-1 ?
• Estimate model-error ??? using moments
• (? - X b)T L(???)-1 (? - X b) n - p
• with L(???) ??? I ????
• n dimension of ??vector p dimension of b

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Measures of Precision

• Model error variance
• Var? E ?i - xiT ?2 ??2
• Parameter (sampling) error variance
• Varb (XT?-1X)-1
• Variance of Prediction for new site
• VPnew E ?0 - x0T b2 ??2 x0T (XT?-1X)-1
  x0

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Pseudo R2 for GLS
Consider the GLS model

• Not interested in total error ?
• that includes sampling error ??
• which cannot hope to explain
• how much of critical model
• error ? can we explain,
• where Var? ????

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Key Lesson

• For range of problems, GLS provides a flexible
  procedure for regionalization when only imperfect
  estimates of hydrologic parameters are available.
• Generally provides unbiased estimators
• of model error variance ??? and Var(b),
• and efficient estimators b of ??

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Moment Estimators Drawbacks

• Estimated ??2 often equals zero when sampling
  variance is dominant error.
• Moments procedure does not provide natural
  measure of precision of the model error variance
  ??2
• Asymptotic MLE approximations have trouble with
  bound at zero.

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Likelihood function - model error ??2 (Tibagi
River, Brazil, n17)
Maximum of likelihood may be at zero, but
larger values are very probable. Zero clearly
not in middle of likely range of values.
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Advantages of Bayesian Analysis

• Provides posterior distribution of
• parameters ?
• model error variance ??2, and
• predictive distribution for dependent variable

Bayesian Approach is a natural solution to the
problem
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Bayesian GLS Model

• Prior distribution x(?, ??2)
• Parameter b are multivariate normal (Q)
• Model error variance ??2
• Exponential dist. (?) E??2 ? 24
• Likelihood function
• Assume data is multivariate N X?, ?

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Quasi-Analytic Bayesian GLS

• Joint posterior distribution

• Marginal posterior of sd2

where integrate analytically normal likelihood
prior to determine f in closed-form.
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Example of a posterior of ??2 (Tibagi, Brazil, n
17)
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Quasi-Analytic Result
From joint posterior distribution
can compute marginal posterior of b
and moments by 1- dimensional num. integrations
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Example Illinois River Basin

• Stations 62 stations in midwest USA
• Record length 14 to 90 years
• Covariates
• Drainage area
• Main channel slope
• Length
• Area of lakes
• Forest cover
• Soil permeability index
• Dummy variables Z1 and Z2 for regions

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Best Regional Models of k?Illinois River basin
(USA)
ASV average sampling variance Avg xT Var?
x AVP average variance of prediction ??2
Avg xT Var? x
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What did we learn?

• OLS results in too large a model error variance
  estimate because includes sampling error.
• Weighted Generalized Least Squares produce
  different results cross-correlation matters.
• Moment versus Bayesian estimator of model error
  variance was important different model error
  variances result different models selected.
• Obtained informative regional ? distribution.
• Pseudo-R2 55.

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GEV ?-estimates using prior information
(variances in parentheses)
Sites are in Illinois Basin. Geophysical prior
has mean -0.1, var (0.1222).
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Flood Frequency for Site 2 (90 years)
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Conclusions

• GMLE - regional prior for site with n 90,
  reduced by 25 uncertainty variance in 100-yr
  flood for site with n 27, reduced 100-year
  flood by 50 and variance 60 times!
• GLS reflects precision due to different record
  lengths plus cross-correlations among hydrologic
  statistics thus provides more realistic
  description of sampling errors.
• Bayesian GLS
• quasi-analytic procedure
• full realistic posterior of ? and ??? plus
  moments
• provides regional estimates of ? and their
  precision

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References

• Coles, S.G. and M.J. Dixon, Likelihood-Based
  Inference for Extreme Value Models, Extremes,
  2(1), 5-23, 1999.
• Martins, E.S. and J.R. Stedinger, Generalized
  Maximum Likelihood GEV Quantile Estimators for
  Hydrologic Data, Water Resour. Res., 28(11),
  3001-3010, 2000.
• Martins, E.S., and J.R. Stedinger,
  Cross-correlation among estimators of shape,
  Water Resour. Res., 38(11), doi10.1029/2002WR0015
  89, 2002.
• Reis, D.S., Jr., Flood Frequency Analysis
  Employing Bayesian Regional Regression and
  Imperfect Historical Information, Ph.D. Thesis,
  Cornell University, Ithaca, NY, USA, 2005.
• Reis, D. S., Jr., J.R. Stedinger, and E.S.
  Martins, Bayesian GLS Regression with application
  to LP3 Regional Skew Estimation, Proceedings
  World Water Environmental Resources Congress
  2003, Editors P. Bizier and P. DeBarry,
  Philadelphia, PA, American Society of Civil
  Engineers, June 23-26, 2003.
• Reis, D. S., Jr., J.R. Stedinger, and E.S.
  Martins, Bayesian GLS Regression with application
  to LP3 Regional Skew Estimation, accepted Water
  Resour. Res., , May 2005.
• Stedinger, J.R., and G.D. Tasker, Regional
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