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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 declustering.
phx.fit0 < − gpd.fit( Tphap$MinT, 73) # With runs declustering (r=1).
phx.dc < − dclust( Tphap$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 reparameterization (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/pyear return period is
zp = F −1(1 − p).
For example, p = 0.01 corresponds to the 100year 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 declustering). 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.
 Spring '08
 Staff

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