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  • Duration?
  • Mean of Globe -> Correlated with Mean Location of Extremes

Metrics

Log-Likelihood

Table 1:Table with results for each model

ModelNLLError
GEVD-124.73832354.06643326
GPD (Q95, 3D)-1358.7062424420.17945954
GPD (Q98, 3D)-567.9748068114.90358918
GPD (Q99, 3D)-291.2815345411.48212622
GEVD
GPD (Q95, 3D)
GPD (Q95, 3D)
GPD (Q98, 3D)
GPD (Q99, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Parameters

Location Parameter

For the location parameter, recall the formulation

μ(s,θ)=μ0\boldsymbol{\mu}(\mathbf{s},\boldsymbol{\theta}) = \mu_0
(1)

So for this experiment, each model has a location bias parameter, μ0\mu_0. However, the temporal model includes the location-bias parameter and the location-temporal weight parameter, μ1\mu_1. In addition, the spatial model includes all parameters in the above equation.

Location-Bias

Histogram

This is the histogram of all samples of the location-bias parameter, μ0\mu_0, for each station in Spain.

GEVD
GPD (Q90, 3D)
GPD (Q95, 3D)
GPD (Q98, 3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Mean Histogram

This is the histogram of the mean of the location-bias parameter, μ0\mu_0, for each station in Spain.

GEVD
GPD (Q90, 3D)
GPD (Q95, 3D)
GPD (Q98, 3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Maps
GEVD
GPD (Q95, 3D)
GPD (Q95, 3D)
GPD (Q98, 3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Sigma

σ∗=σ+κ(y0−μ)\sigma^* = \sigma + \kappa (y_0 - \mu)
(2)

This parameter is only present for the GPD distribution.

Histogram

GPD (Q95, 3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Mean Histogram

GPD (Q98,3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Map

GPD (Q95, 3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Scale

Recall, the parameterization for the scale parameter is given by

σ(t;θ)=σ0\sigma(t;\boldsymbol{\theta}) = \sigma_0
(3)

where σ0\sigma_0 is the scale parameter per station. This means that each model will have the same scale parameterization.

Histogram

GEVD
GPD (Q95, 3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Mean Histogram

GEVD
GPD (Q95, 3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Map

GEVD
GPD (Q95, 3D)
GPD (Q95, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Concentration

Recall, the parameterization for the shape parameter is given by

κ(t;θ)=κ0\kappa(t;\boldsymbol{\theta}) = \kappa_0
(4)

where κ0\kappa_0 is the shape parameter per station. This means that each model will have the same shape parameterization.

Histogram

GEVD
GPD (Q95, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Mean Histogram

GEVD
GPD (Q95, 3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Maps

GEVD
GPD (Q95, 3D)
GPD (Q98, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Rate

We can relate the GEVD parameters to the GPD. This gives us a rate parameter, λ, which is the expected number of events that exceed some threshold, y0y_0, per year.

λ=σ+κ(y0−μ)\lambda = \sigma + \kappa (y_0 - \mu)
(5)

However, we need to define an exceedence threshold, y0y_0. We will do a simple 95% quantile for each independent station with a declustering of 3 days. Then we can calculate the rate, λ.

Threshold

GEVD

Figure 38:Madrid Daily Maximum Temperature Time Series A histogram of the threshold parameter, y0y_0, for all stations.

Histogram

GEVD

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Mean Histogram

GEVD

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Maps

GEVD

Figure 2:Madrid Daily Maximum Temperature Time Series The return period for the iid model.

Returns

Histogram

GEVD
GPD (Q90, 3D)
GPD (Q95, 3D)
GPD (Q98, 3D)
GPD (Q99, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Mean Histogram

GEVD
GPD (Q90, 3D)
GPD (Q95, 3D)
GPD (Q98, 3D)
GPD (Q99, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The negative log-likelihood loss (equation (5)) for each time step within the time series.

Maps

GEVD
GPD (Q90, 3D)
GPD (Q95, 3D)
GPD (Q95, 3D)
GPD (Q99, 3D)

Figure 2:Madrid Daily Maximum Temperature Time Series The return period for the iid model.

Bayes4EVT
Experiment 1b - Spain (Time)
ML4EO
Overview