Vector%20Generalized%20Additive%20Models%20and%20applications%20to%20extreme%20value%20analysis

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Vector Generalized Additive Modelsand
applications to extreme value analysis

• Olivier Mestre (1,2)
• Météo-France, Ecole Nationale de la Météorologie,
  Toulouse, France
• Université Paul Sabatier, LSP, Toulouse, France
• Based on previous studies realized in
  collaboration with
• Stéphane Hallegatte (CIRED, Météo-France)
• Sébastien Denvil (LMD)

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SMOOTHER

•  Smoothertool for summarizing the trend of a
  response measurement Y as a function of
  predictors  (Hastie Tibshirani)
• estimate of the trend that is less variable than
  Y itself
• Smoothing matrix S
• YS?Y
• The equivalent degrees of freedom (df) of the
  smoother S is the trace of S. Allows compare with
  parametric models.
• Pointwise standard error bands
• COV(Y)VS tS ?² given an estimation of ?²,
  this allows approximate confidence intervals
  (values 2?square root of the diagonal of V)

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SCATTERPLOT SMOOTHING EXAMPLE

• Data wind farm production vs numerical windspeed
  forecasts

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SMOOTHING

• Problems raised by smoothers
• How to average the response values in each
  neighborhood?
• How large to take the neighborhoods?
• ?
• Tradeoff between bias and variance of Y

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SMOOTHING POLYNOMIAL (parametric)

• Linear and cubic parametric least squares fits
  MODEL DRIVEN APPROACHES

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SMOOTHING BIN SMOOTHER

• In this example, optimum intervals are determined
  by means of a regression tree

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SMOOTHING RUNNING LINE

• Running line

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KERNEL SMOOTHER

• Watson-Nadaraya

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SMOOTHING LOESS

• The smooth at the target point is the fit of a
  locally-weighted linear fit (tricube weight)

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CUBIC SMOOTHING SPLINES

• This smoother is the solution of the following
  optimization problem
• among all functions f(x) with two continuous
  derivatives, choose the
• one that minimizes the penalized sum of squares
• Closeness to the data penalization of the
  curvature of f
• It can be shown that the unique solution to this
  problem is a natural cubic spline with knots at
  the unique values xi
• Parameter ? can be set by means of
  cross-validation

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CUBIC SMOOTHING SPLINES

• Cubic smoothing splines with equivalent df5 and
  10

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Additive models

• Gaussian Linear Model IEY?o?1X1?2X2
• Gaussian Additive model IEYS1(X1)S2(X2)
• S1, S2 smooth functions of predictors X1, X2,
  usually LOESS, SPLINE
• Estimation of S1, S2  Backfitting Algorithm 
• PRINCIPLE OF THE BACKFITTING ALGORITHM
• YS1(X1)e ? estimation S1
• Y-S1(X1)S2(X2)e ? estimation S2
• Y-S2(X2)S1(X1)e ? estimation S1
• Y-S1(X1)S2(X2)e ? estimation S2
• Y-S2(X2)S1(X1)e ? estimation S1
• Etc until convergence

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Additive models

• Additive models
• One efficient way to perform non-linear
  regression, but
• Crucial point
• ADAPTED WHEN ONLY FEW PREDICTORS
• 2, 3 predictors at most

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Additive models

• Philosophy
• DATA DRIVEN APPROACHES RATHER THAN MODEL DRIVEN
  APPROACH
• USEFUL AS EXPLORATORY TOOLS
• Approximate inference tests are possible, but
  full inferences are better assessed by means of
  parametric models

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Generalized Additive models (GAM)

• Extension to non-normal dependant variables
• Generalized additive models additive modelling
  of the natural parameter of exponential family
  laws (Poisson, Binomial, Gamma, Gauss).
• gµ?S1(X1)S2(X2)
• Vector Generalized Additive Models (VGAM) one
  step beyond

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Example 1

• Annual umber and maximum integrated intensity
  (PDI) of hurricane tracks over the North Atlantic

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Number of Hurricanes

• Number of Hurricanes in North Atlantic Poisson
  distribution

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Factors influencing the number of hurricanes

• GAM applied to number of hurricanes
  (YEAR,SST,SOI,NAO)

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GAM model

• Log(?) ?oS1(SST)S2(SOI)

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PARAMETRIC model

• broken stick model (with continuity constraint)
  in SOI, revealed by GAM analysis
• log(?) ?o?SOI(1)SOI?SSTSST SOIltK
• ?o?SOI(1)SOI?SOI(2)(SOI-K)?SSTSST
  SOI?K
• The best fit obtained for SOI value K1
• log-likelihood-316.16, to be compared with
  -318.71 (linearity)
• standard deviance test allows reject linearity
  (p value0.02)
• Expectation ? of the hurricane number is then
  straightforwardly computed as a function of SOI
  and SST

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EXPECTATION OF HURRICANE NUMBERS

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OBSERVED vs EXPECTED r0.6