{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ShortCourseMalta

# Transform data to a common distribution and check the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online