Bayes4EVT - Research Notebook

Datasets¶

We are primarily dealing with temperature measurements. In particular, we are looking at 2 meter temperature (t2m) above the Earths surface. Our region of interest is Spain and the period of interest spans 1960 until 2021. We are dealing directly with measurements from stations that are scattered throughout Spain, like Madrid, Bilbao, or Valencia. We want to take a step further and look for extreme temperatures. So this entails taking maximum daily values through our region and period of interest.

From a modeling standpoint, we hope to characterize the tails of distribution of these temperature measurements, i.e., the extreme values. This will be useful because we can make predictions about the occurrences of these extreme events. In addition, if we can correlate these extremes to external factors like Global Mean Surface Temperature (GMST), it makes a stronger statement about the trajectory of frequency of occurrence of these extreme values.

Models¶

We will take three approaches to modeling these extreme values: 1) block maximum (BM), 2) peak-over-thresholds (POT), and 3) temporal point process (TPP).

GEVD. For the BM method, we will use the generalized extreme value distribution (GEVD). This will be an annual maximum for each station. The annual BM is arguably the most reliable and stable method for classifying extremes. However, the downside is that we don’t take advantage of all of the values we would consider extreme.

GPD. For the POT method, we will use the generalized Pareto distribution (GPD). This will choose a threshold and model all values above the threshold. This method is more complex because one has to choose an appropriate threshold as well as decluster events. However, the upside is that we will have more occurrences of extremes which will be better for fitting models.

Point Process. We will choose the extreme distribution to be the PP which will be a combination of the GEVD and the Poisson-GPD but under the PP framework.

Covariates¶

Each of the above models assumes each observation is IID.

No Covariates¶

We will apply all models assuming no covariates. So we will find a set of parameters for each station independently.

p(y,\theta) = p(\theta) \prod_{n=1}^N p(y|\theta)

(1)

Spatial Covariates¶

We will utilize the neighbouring stations as covariates for the model.

p(y,\theta,f) = p(\theta) \prod_{n=1}^N p(y_n|z_n)p(z_n|f_n,\theta)

(2)

Global Mean Surface Temperature (GMST)¶

We will use the GMST as a proxy for the temporal dependency.

p(y,x,f,\theta) = p(\theta) \prod_{n=1}^N p(y_n|z_n)p(z_n|f_n, x_n,\theta)

(3)

Inference¶

Each of the above models will have various parameter estimators.

Maximum Likelihood Estimator (MLE)
Maximum A Posteriori (MAP)
Laplace Approximation (Laplace)
Markov Chain Monte Carlo (MCMC)

Experiments¶

Experiment I¶

We will apply different models using different inference methods on each station in Spain individually. We make the following assumptions:

I.I.D.
Stationary Process
No Spatial Dependencies
No Covariates

\begin{aligned} \mathcal{D} = \left\{ t_n, y_n \right\}_{n=1}^N && && t_n \in\mathcal{T}\subseteq\mathbb{R}^+ && && y_n\in\mathcal{Y}\subseteq\mathbb{R} \end{aligned}

(4)

Experiment Ia¶

For this process, we will exclusively use the GEVD for the data likelihood.

Results:

Experiment II¶

We will apply different models using different methods on each of the station assuming we can exploit the shared spatial information.

Experiment 2a - GEVD Model
Experiment 2b - GPD Model
Experiment 2c - TPP Model

Experiment III¶

We will apply different models using different methods on each of the station assuming we can exploit the shared spatial information. In addition, we will use the GMST as a proxy for the time component.

Experiment 3a - GEVD Model
Experiment 3b - GPD Model
Experiment 3c - TPP Model