In this section, we show that the EGP model based on the truncated Beta distribution can be used to model non-zero precipitation, which corresponds to exceedances above the very low threshold of 0, while maintaining the tail behavior.
The daily summer precipitation (May–October) recorded at the Montréal-Trudeau International Airport meteorological station (Québec, Canada) from 2000 to 2020 are investigated. This dataset can be loaded using the ExtendedExtremes.dataset
data = ExtendedExtremes.dataset("pcp")
filter!(row -> row.Date>= Date(2000,1,1), data)
filter!(row -> month(row.Date) in 5:10, data)
dropmissing!(data)
first(data,5)1 -73.75 45.47 MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A 7025250 2000-05-01 1.0 2 -73.75 45.47 MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A 7025250 2000-05-02 0.0 3 -73.75 45.47 MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A 7025250 2000-05-03 0.0 4 -73.75 45.47 MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A 7025250 2000-05-04 0.5 5 -73.75 45.47 MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A 7025250 2000-05-05 0.0
The EGP parameter estimation with maximum likelihood is performed with the fit_mle
u = 0.0
y = data.pcp[data.pcp .> u] .- u;
fd = fit_mle(ExtendedGeneralizedPareto{TBeta}, y)ExtendedGeneralizedPareto{TBeta{Float64}}(
V: TBeta{Float64}(α=0.03797646874817642)
G: GeneralizedPareto{Float64}(μ=0.0, σ=11.75360977328577, ξ=-0.01981634739270602)
)
Several diagnostic plots for assessing the accuracy of the EGP model fitted to the Montréal data are can be shown with the diagnosticplots
set_default_plot_size(16cm, 16cm)
ExtendedExtremes.diagnosticplots(y, fd)
Return Period
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
105
106
107
108
109
10-4.0
10-3.5
10-3.0
10-2.5
10-2.0
10-1.5
10-1.0
10-0.5
100.0
100.5
101.0
101.5
102.0
102.5
103.0
103.5
104.0
104.5
105.0
105.5
106.0
106.5
107.0
107.5
108.0
10-5
100
105
1010
10-4.0
10-3.8
10-3.6
10-3.4
10-3.2
10-3.0
10-2.8
10-2.6
10-2.4
10-2.2
10-2.0
10-1.8
10-1.6
10-1.4
10-1.2
10-1.0
10-0.8
10-0.6
10-0.4
10-0.2
100.0
100.2
100.4
100.6
100.8
101.0
101.2
101.4
101.6
101.8
102.0
102.2
102.4
102.6
102.8
103.0
103.2
103.4
103.6
103.8
104.0
104.2
104.4
104.6
104.8
105.0
105.2
105.4
105.6
105.8
106.0
106.2
106.4
106.6
106.8
107.0
107.2
107.4
107.6
107.8
108.0
h,j,k,l,arrows,drag to pan
i,o,+,-,scroll,shift-drag to zoom
r,dbl-click to reset
c for coordinates
? for help
?
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Return Level
Return Level Plot
Data
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-50
0
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-200
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-95
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h,j,k,l,arrows,drag to pan
i,o,+,-,scroll,shift-drag to zoom
r,dbl-click to reset
c for coordinates
? for help
?
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
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0.4
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0.0
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-0.5
0.0
0.5
1.0
-0.40
-0.38
-0.36
-0.34
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0.82
Density
Density plot
Model
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h,j,k,l,arrows,drag to pan
i,o,+,-,scroll,shift-drag to zoom
r,dbl-click to reset
c for coordinates
? for help
?
-100
-80
-60
-40
-20
0
20
40
60
80
100
120
140
160
180
-80
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Empirical
Quantile Plot
Model
-1.5
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0.0
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2.5
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-0.9
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2.0
-1
0
1
2
-1.00
-0.95
-0.90
-0.85
-0.80
-0.75
-0.70
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
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-0.05
0.00
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1.00
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1.45
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1.95
2.00
h,j,k,l,arrows,drag to pan
i,o,+,-,scroll,shift-drag to zoom
r,dbl-click to reset
c for coordinates
? for help
?
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
-1.0
-0.9
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1.0
1.1
1.2
1.3
1.4
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1.6
1.7
1.8
1.9
2.0
-1
0
1
2
-1.00
-0.95
-0.90
-0.85
-0.80
-0.75
-0.70
-0.65
-0.60
-0.55
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-0.45
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0.00
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1.00
1.05
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1.45
1.50
1.55
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1.95
2.00
Empirical
Probability Plot
The diagnostic plots consist in the probability plot (upper left panel), the quantile plot (upper right panel), the density plot (lower left panel) and the return level plot (lower right panel). These plots can be displayed separately using respectively the probplotqqplothistplotreturnlevelplot