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Unformatted text preview: , col="blue", lwd=2, xlab="x", ylab="f(x)") lines( 0:20, pdfGum, col="green", lty=2, lwd=2) lines( 0:20, pdfFrech, col="red", lwd=2) lines( 0:20, pdfWeib, col="orange", lwd=2) Example Fort Collins, Colorado daily precipitation amount http://ccc.atmos.colostate.edu/~odie/rain.html • Time series of daily precipitation amount (in), 1900–1999. • Semi-arid region. • Marked annual cycle in precipitation (wettest in late spring/early summer, driest in winter). • No obvious long-term trend. • Recent flood, 28 July 1997. (substantial damage to Colorado State University) Example Fort Collins, Colorado precipitation Fort Collins daily precipitation q q 0.04 q q 0.03 q q q q q q q 0.02 0.01 q q q 0.00 Precipitation (in) q q q q q q q q q q q q q q q q q q q Jan q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Mar q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q May q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Jul q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Sep q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Nov Example Fort Collins, Colorado precipitation Annual Maxima Fort Collins annual maximum daily precipitation 3 2 1 Precipitation (in) 4 1997 1900 1920 1940 Year 1960 1980 2000 Example Fort Collins, Colorado precipitation How often is such an extreme expected? • Assume no long-term trend emerges. • Using annual maxima removes effects of annual trend in analysis. • Annual Maxima fit to GEV. # Fit Fort Collins annual maximum precipitation to GEV. fit < − gev.fit( ftcanmax$Prec/100) # Check the quality of the fit. gev.diag( fit) Example Fort Collins, Colorado precipitation Fit looks good (from diagnostic plots). Parameter Estimate (Std. Error) Location (µ) 1.347 (0.617) Scale (σ ) 0.533 (0.488) Shape (ξ ) 0.174 (0.092) Example Fort Collins, Colorado precipitation # Is the shape parameter really not zero? # Perform likelihood ratio test against Gumbel type. fit0 < − gum.fit( ftcanmax$Prec/100) Dev < − 2*(fit0$nllh - fit$nllh) pchisq( Dev, 1, lower.tail=FALSE) Likelihood ratio test for ξ = 0 rejects hypothesis of Gumbel type (p-value ≈ 0.038). Example Fort Collins, Colorado precipitation −107 −108 −109 −110 Use gev.profxi and locator(2) to find CI’s. −111 Profile Log−likelihood −106 −105 95% Confidence intervals for ξ , using profile likelihood, are: (0.009, 0.369). −0.1 0.0 0.1 0.2 Shape Parameter 0.3 0.4 Example Fort Collins, Colorado precipitation Return Levels # Currently must assign the class "gev.fit" to t...
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