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1 0 gumbel type limit as 0 2 0 frchet type 3

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Unformatted text preview: gs to one of the following three types. Background on Extreme Value Analysis (EVA) Extremal Types Theorem I. Gumbel G(z ) = exp − exp − z−b a II. Fréchet z ≤ b, 0, G (z ) = exp − , −∞<z <∞ z −b −α a , z > b; III. Weibull G (z ) = exp − − 1, with parameters a, b and α > 0. z −b α a , z < b, z≥b Background on Extreme Value Analysis (EVA) Extremal Types Theorem The three types can be written as a single family of distributions, known as the generalized extreme value (GEV) distribution. G(z ) = exp − 1 + ξ z−µ σ −1/ξ , + where y+ = max{y, 0}, −∞ < µ, ξ < ∞ and σ > 0. Background on Extreme Value Analysis (EVA) GEV distribution Three parameters: location (µ), scale (σ ) and shape (ξ ). 1. ξ = 0 (Gumbel type, limit as ξ −→ 0) 2. ξ > 0 (Fréchet type) 3. ξ < 0 (Weibull type) Background on Extreme Value Analysis (EVA) Gumbel type • Light tail • Domain of attraction for many common distributions (e.g., normal, lognormal, exponential, gamma) 0.0 0.1 pdf 0.2 0.3 0.4 0.5 Gumbel 0 1 2 3 4 5 Background on Extreme Value Analysis (EVA) Fréchet type • Heavy tail • E [X r ] = ∞ for r ≥ 1/ξ (i.e., inﬁnite variance if ξ ≥ 1/2) • Of interest for precipitation, streamﬂow, economic impacts pdf 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Frechet 0 1 2 3 4 5 Background on Extreme Value Analysis (EVA) Weibull type • Bounded upper tail at µ − σ ξ • Of interest for temperature, wind speed, sea level 0.0 0.1 0.2 pdf 0.3 0.4 0.5 Weibull 0 1 2 3 4 5 6 Background on Extreme Value Analysis (EVA) Normal vs. GEV ## The probability of exceeding increasingly high values ## (as they double). # Normal pnorm( c(1,2,4,8,16,32), lower.tail=FALSE) # Gumbel pgev( c(1,2,4,8,16,32), lower.tail=FALSE) # Fréchet pgev( c(1,2,4,8,16,32), shape=0.5, lower.tail=FALSE) # Weibull (note bounded upper tail!) pgev( c(1,2,4,8,16,32), shape=-0.5, lower.tail=FALSE) Background on Extreme Value Analysis (EVA) Normal vs. GEV # Find Pr{X < x} for X = 0, . . . , 20. cdfNorm < − pnorm( 0:20) cdfGum < − pgev( 0:20) cdfFrech < − pgev( 0:20, shape=0.5) cdfWeib < − pgev( 0:20, shape=-0.5) # Now find Pr{X = x} for X = 0, . . . , 20. pdfNorm < − dnorm( 0:20) pdfGum < − dgev( 0:20) pdfFrech < − dgev( 0:20, shape=0.5) pdfWeib < − dgev( 0:20, shape=-0.5) Background on Extreme Value Analysis (EVA) Normal vs. GEV par( mfrow=c(2,1), mar=c(5,4,0.5,0.5)) plot( 0:20, cdfNorm, ylim=c(0,1), type="l", xaxt="n", col="blue", lwd=2, xlab="", ylab="F(x)") lines( 0:20, cdfGum, col="green", lty=2, lwd=2) lines( 0:20, cdfFrech, col="red", lwd=2) lines( 0:20, cdfWeib, col="orange", lwd=2) legend( 10, 0.05, legend=c("Normal", "Gumbel", "Frechet", "Weibull"), col=c("blue", "green", "red", "orange"), lty=c(1,2,1,1), bty="n", lwd=2) Background on Extreme Value Analysis (EVA) Normal vs. GEV plot( 0:20, pdfNorm, ylim=c(0,1), typ="l"...
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