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Convenient to estimate dicult to interpret in

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Unformatted text preview: end as cities grow). Model lower tail as upper tail after negation. Peaks Over Thresholds (POT) Approach Dependence above threshold # Fit without de-clustering. phx.fit0 < − gpd.fit( -Tphap$MinT, -73) # With runs de-clustering (r=1). phx.dc < − dclust( -Tphap$MinT, u=-73, r=1, cluster.by=Tphap$Year) phxdc.fit0 < − gpd.fit( phx.dc$xdat.dc, -73) Peaks Over Thresholds (POT) Approach Point Process: frequency and intensity of threshold excesses Event is a threshold excess (i.e., X > u). Frequency of occurrence of an event (rate parameter), λ > 0. Pr{no events in [0, T ]} = e−λT Mean number of events in [0, T ] = λT . GPD for excess over threshold (intensity). Peaks Over Thresholds (POT) Approach Point Process: frequency and intensity of threshold excesses Relation of parameters of GEV(µ,σ ,ξ ) to parameters of point process (λ,σ ∗,ξ ). • Shape parameter, ξ , identical. • log λ = − 1 log 1 + ξ u−µ ξ σ • σ ∗ = σ + ξ ( u − µ) More detail: Time scaling constant, h. For example, for annual maximum of daily data, h ≈ 1/365.25. Change of time scale, h, for GEV(µ,σ ,ξ ) to h h σ =σ h ξ 1 and µ = µ + ξ σ 1− h h −ξ Peaks Over Thresholds (POT) Approach Point Process: frequency and intensity of threshold excesses Two ways to estimate PP parameters • Orthogonal approach (estimate frequency and intensity separately). Convenient to estimate. Difficult to interpret in presence of covariates. • GEV re-parameterization (estimate both simultaneously). More difficult to estimate. Interpretable even with covariates. Peaks Over Thresholds (POT) Approach Point Process: frequency and intensity of threshold excesses Fort Collins, Colorado daily precipitation Analyze daily data instead of just annual maxima (ignoring annual cycle for now). Orthogonal Approach ˆ = 365.25 · No. Xi > 0.395 ≈ 10.6 per year λ No. Xi ˆ ˆ σ ∗ ≈ 0.323, ξ ≈ 0.212 Peaks Over Thresholds (POT) Approach Point Process: frequency and intensity of threshold excesses Fort Collins, Colorado daily precipitation Analyze daily data instead of just annual maxima (ignoring annual cycle for now). Point Process ˆ µ ≈ 1.384 ˆ σ = 0.533 ˆ ξ ≈ 0.213 ˆ ˆ = 1 + ξ ( u − µ) ˆ λ σ ˆ ˆ −1/ξ ≈ 10.6 per year Risk Communication Under Stationarity Unchanging climate Return level, zp, is the value associated with the return period, 1/p. That is, zp is the level expected to be exceeded on average once every 1/p years. That is, Return level, zp, with 1/p-year return period is zp = F −1(1 − p). For example, p = 0.01 corresponds to the 100-year return period. Easy to obtain from GEV and GP distributions (stationary case). Risk Communication Under Stationarity Unchanging climate For example, GEV return level is given by σ zp = µ − [1 − (− log(1 − p))]−ξ ξ 0.2 Similar for GPD, but must take λ into account. 0.1 0.0 GEV pdf 0.3 Return level with (1/p)−year return period p 1−p z_p Risk Communication Under Stationarity Unchanging climate Compare previous GPD fits (with and without de-clustering). Must first assign the class name, “gpd.fit" to each so...
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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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