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Transform data to a common distribution and check the

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Unformatted text preview: that the extRemes function, return.level, will know whether the objects refer to a GEV or GPD fit. # Without de-clustering. class( phx.fit0) < − "gpd.fit" return.level( phx.fit0, make.plot=FALSE) # With de-clustering. class( phxdc.fit0) < − "gpd.fit" return.level( phxdc.fit0, make.plot=FALSE) Non-Stationarity Sources • Trends: climate change: trends in frequency and intensity of extreme weather events. • Cycles: Annual and/or diurnal cycles often present in meteorological variables. • Other. Non-Stationarity Theory No general theory for non-stationary case. Only limited results under restrictive conditions. Can introduce covariates in the distribution parameters. Non-Stationarity Phoenix minimum temperature 75 70 65 Minimum temperature (deg. F) Phoenix summer minimum temperature 1950 1960 1970 Year 1980 1990 Non-Stationarity Phoenix minimum temperature Recall: min{X1, . . . , Xn} = − max{−X1, . . . , −Xn}. Assume summer minimum temperature in year t = 1, 2, . . . has GEV distribution with: µ(t) = µ0 + µ1 · t log σ (t) = σ0 + σ1 · t ξ (t) = ξ Non-Stationarity Phoenix minimum temperature Note: To convert back to min{X1, . . . , Xn}, change sign of location parameters. But note that model is Pr{−X ≤ x} = Pr{X ≥ −x} = 1 − F (−x). ˆ µ(t) ≈ 66.170 + 0.196t ˆ log σ (t) ≈ 1.338 − 0.009t ˆ ξ ≈ −0.21 Likelihood ratio test for µ1 = 0 (p-value < 10−5), for σ1 = 0 (p-value ≈ 0.366). Non-Stationarity Phoenix minimum temperature Model Checking. Found the best model from a range of models, but is it a good representation of the data? Transform data to a common distribution, and check the qq-plot. 1. Non-stationary GEV to exponential εt = ˆ ξ (t) ˆ 1+ [Xt − µ(t)] ˆ σ (t) ˆ −1/ξ (t) 2. Non-stationary GEV to Gumbel (used by ismev/extRemes εt = ˆ 1 Xt − µ(t) ˆ log 1 + ξ (t) ˆ ˆ σ (t) ξ (t) Non-Stationarity Phoenix minimum temperature Model Checking. Found the best model from a range of models, but is it a good representation of the data? Transform data to a common distribution, and check the qq-plot. Q−Q Plot (Gumbel Scale): Phoenix Min Temp 4 q 3 q q q 2 q qq 1 q qq q q qq qq 0 q qqqq qqq qqq q qqq q qqq qq −1 Empirical q q q qq q −1 0 1 Model 2 3 Non-Stationarity Physically based covariates Winter maximum daily temperature at Port Jervis, New York Let X1, . . . , Xn be the winter maximum temperatures, and Z1, . . . , Zn the associated Arctic Oscillation (AO) winter index. Given Z = z , assume conditional distribution of winter maximum temperature is GEV with parameters µ( z ) = µ0 + µ 1 · z log σ (z ) = σ0 + σ1 · z ξ (z ) = ξ Non-Stationarity Physically based covariates Winter maximum daily temperature at Port Jervis, New York ˆ µ(z ) ≈ 15.26 + 1.175 · z ˆ log σ (z ) = 0.984 − 0.044 · z ξ (z ) = −0.186 Likelihood ratio test for µ1 = 0 (p-value < 0.001) Likelihood ratio test for σ1 = 0 (p-value ≈ 0.635) Non-Stationarity Cyclic variation 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...
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This note was uploaded on 01/26/2014 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.

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