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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
IID Model-17089.150320174.06643326
Spatial Model-16663.5828149462.91590443
Temporal Model-15571.707294264.73258625
Spatiotemporal Model-15198.3518349959.1095967
IID
Space
Time
Space-Time

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(t,s,θ)=μ0+μ1t+μ2(s)\boldsymbol{\mu}(t,\mathbf{s},\boldsymbol{\theta}) = \mu_0 + \mu_1t + \mu_2(\mathbf{s})
(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.

IID
Time
Space-Time

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.

IID
Time
Space-Time

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

Maps
IID
Time
Space-Time

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

Location Time-Weight

We will look at the location temporal-weight parameter, μ1\mu_1. This parameter dictates the positive (or negative) correlation between the time parameter and the location parameter.

Histogram
Time
Space-Time

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
Time
Space-Time

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

Maps
Time
Space-Time

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
(2)

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

Histogram

IID
Time
Space-Time

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

IID
Time
Space-Time

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

Map

IID
Time
Space-Time

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
(3)

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

Histogram

IID
Time
Space-Time

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

IID
Time
Space-Time

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

Maps

IID
Time
Space-Time

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)
(4)

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

Histogram
Map

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

Histogram

IID
Time
Space-Time

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

IID
Time
Space-Time

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

Maps

GMST - Scenario 0

IID
Time
Space-Time

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

GMST - Scenario 1

IID
Time
Space-Time

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

GMST - Scenario 2

IID
Time
Space-Time

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

Returns

Histogram

IID
Space
Time
Space-Time

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

IID
Space
Time
Space-Time

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

Maps

GMST - Scenario 0

IID
Space
Time
Space-Time

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

GMST - Scenario 1

IID
Space
Time
Space-Time

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

GMST - Scenario 2

IID
Space
Time
Space-Time

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

Differences

Case I - Scenario 0 and 1

The first case, we look at the absolute difference between the GMST scenarios 0 and 1. This corresponds to the difference in the pre-industrial climate and the actual climate.

Time
Space-Time

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

Case I - Scenario 1 and 2s

The first case, we look at the absolute difference between the GMST scenarios 1 and 2. This corresponds to the difference in the actual climate and the future climate.

Time
Space-Time

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

Bayes4EVT
Experiment 2a - Spain
Bayes4EVT
Experiment 1a - Spain (IID)