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Overview

In this application, we are interested in modeling temperatures, and in particular, temperature extremes.

Quick Walk-Through

Datasets.

The first set of datasets will feature a range of different

Global Mean Surface Temperature Anomaly. This is our first univariate time series that we will be using to explore data-driven modeling techniques. It is a basic dataset which captures the change in mean surface temperature over the Earths surface. It has already been pre-processed to remove all artifacts and

Equations
Univariate Time Series:D={tn,yt}n=1NTyn∈RMultivate Time Series:D={tn,yt}n=1NTyn∈RDyMultivate Spatiotemporal Series:D={(tn,sn),yt}n=1NTyn∈RDyCoupled Multivate Spatiotemporal Series:D={(tn,sn),xn,yt}n=1NTyn∈RDy\begin{aligned} \text{Univariate Time Series}: && && \mathcal{D} &= \left\{t_n, y_t \right\}_{n=1}^{N_T} && && y_n\in\mathbb{R} \\ \text{Multivate Time Series}: && && \mathcal{D} &= \left\{t_n, \mathbf{y}_t \right\}_{n=1}^{N_T} && && \mathbf{y}_n\in\mathbb{R}^{D_y} \\ \text{Multivate Spatiotemporal Series}: && && \mathcal{D} &= \left\{(t_n, \mathbf{s}_n), \mathbf{y}_t \right\}_{n=1}^{N_T} && && \mathbf{y}_n\in\mathbb{R}^{D_y} \\ \text{Coupled Multivate Spatiotemporal Series}: && && \mathcal{D} &= \left\{(t_n, \mathbf{s}_n), \mathbf{x}_n, \mathbf{y}_t \right\}_{n=1}^{N_T} && && \mathbf{y}_n\in\mathbb{R}^{D_y} \\ \end{aligned}
(1)

Sub-Topics

Basics. We will feature some basics of the Bayesian modeling workflow. There will be some basics like defining a prior, likelihood, and posterior. We will also have. All code will be done with a high-level PPL, i.e., numpyro.

Bayesian Hierarchical Modeling. For many small datasets, we have a lot of hierarchical dependencies. By introducing the notion of a fully-pooled, non-pooled, and partially-pooled model for a simple model, we will introduce the notion of a latent-variable model. This will be an underlying approach for all remaining models throughout these applications.

Game of Dependencies. Many people do not have a solid grasp about dependencies within spatiotemporal data. We will walk-through this step-by-step by slowly introducing more complex representations, e.g., IID, time conditioning, dynamical models, and state-space models.

Extreme Values. Extreme values are a worlds apart from the standard mean and standard deviation. In this section, we will introduce this from fir We will introduce this from the perspective of Temporal Point Processes (TPPs). Some staple distributions will include the Poisson Process, the GEVD, and the GPD.

Marked Temporal Point Processes. The end-goal for extreme values (and many schemes in general) is to make predictions, i.e., what is the probability that there will be an extreme event, λ, with a certain intensity, zz, at a certain location, s\mathbf{s}, or time, tt, given historical observations, H\mathcal{H}, and some covariate parameter, xx.

Useful Skills

Unstructured Measurements. We will immediately start dealing with unstructured datasets which are ubiquitous in the geosciences. We will learn how to model data with an unstructured representation. We will also learn how we can transform them to a more structured representation which is more useful for other other applications.

Real-Time Applications. We will be dealing with time series directly. This gives us the advantage that we will be able to train our models, use them for predictions, and then update them as we obtain more observations.

Immediate Applicability. We will have some immediate applicability because extreme events are things that we have to deal with on a daily basis.

Topics

Datasets

We will use some standard datasets that are found within the AEMET database.

D={(tn,sn),xn,yn}n=1N,yn∈RDyxn∈RDysn∈RDstn∈R+\begin{aligned} \mathcal{D} &= \left\{(t_n, \mathbf{s}_n), \mathbf{x}_n, \mathbf{y}_n \right\}_{n=1}^{N}, && && \mathbf{y}_n\in\mathbb{R}^{D_y} && \mathbf{x}_n\in\mathbb{R}^{D_y} && \mathbf{s}_n\in\mathbb{R}^{D_s} && t_n\in\mathbb{R}^{+} \end{aligned}
(2)
  • Covariante, xnx_n
    • Global Mean Surface Temperature Anomaly (GMSTA)
  • Measurement, yny_n:
    • Daily Mean Temperature (TMax)
    • Daily Maximum Temperature (TMean)

Global Mean Surface Temperature

This is an anomaly dataset.

y=y(t),y:R+→Rt∈T⊆R+\begin{aligned} y &= y(t), && && y:\mathbb{R}^+\rightarrow\mathbb{R} && && t\in\mathcal{T}\subseteq\mathbb{R}^+ \end{aligned}
(3)

Time

Δt=1 YearT=[1930,2023]NT=\begin{aligned} \Delta t &= \text{1 Year} \\ \mathcal{T} &= [1930,2023]\\ N_T &= \end{aligned}
(4)

Weather Station Data Spain

yn=y(sn,tn),y:RDs×R+→RDy\begin{aligned} \mathbf{y}_n &= \boldsymbol{y}(\mathbf{s}_n,t_n), && && \boldsymbol{y}:\mathbb{R}^{D_s}\times\mathbb{R}^+ \rightarrow \mathbb{R}^{D_y} \\ \end{aligned}
(5)

Our coordinates are as follows.

sn∈Ω⊆RDstn∈T⊆R+yn∈Y⊆RDy\begin{aligned} \mathbf{s}_n \in\Omega\subseteq\mathbb{R}^{D_s} && && t_n\in\mathcal{T}\subseteq\mathbb{R}^+ && && \mathbf{y}_n \in \mathcal{Y}\subseteq\mathbb{R}^{D_y} \end{aligned}
(6)

We can also showcase the discretized version

Time

Δt=1 DayT=[1960,2020]NT=\begin{aligned} \Delta t &= \text{1 Day} \\ \mathcal{T} &= [1960, 2020] \\ N_T &= \end{aligned}
(7)

Spatial Coordinates

Ω=Point Cloud[670 Stations]Ds=[LongitudeLatitudeAltitude]\begin{aligned} \Omega &= \text{Point Cloud} && [\text{670 Stations}]\\ D_s &= \begin{bmatrix} \text{Longitude} \\ \text{Latitude} \\ \text{Altitude} \end{bmatrix} \end{aligned}
(8)

Variables

Dy=[Max TemperatureMean TemperatureMinimum TemperaturePrecipicationWind Speed]D_y = \begin{bmatrix} \text{Max Temperature} \\ \text{Mean Temperature} \\ \text{Minimum Temperature} \\ \text{Precipication} \\ \text{Wind Speed} \end{bmatrix}
(9)

Extreme Events

D={yn}n=1N\mathcal{D} = \left\{y_n\right\}_{n=1}^N
(10)

We will use some standard techniques to extract the extreme events from the time series.

  • Block Maximum (BM)
  • Peak-Over-Threshold (POT)
  • Temporal Point Process (TPP)

Data Likelihood

y∼p(y∣z)y \sim p(y|z)
(11)

We need to decide which likelihood we want to use. For the mean data, we can assume that it follows a Gaussian distribution or a long-tailed T-Student distribution. However, for the maximum values, we need to assume an EVD like the GEVD, GPD or TPP.

  • Mean -> Gaussian, T-Student
  • Maximum -> GEVD, GPD, TPP

Parameterizations

p(y,z,θ)=p(y∣z)p(z∣θ)p(θ)p(\mathbf{y},\mathbf{z},\boldsymbol{\theta}) = p(y|z)p(z|\theta)p(\theta)
(12)

We have to decide the parameterization we wish to use.

  • IID
  • Hierarchical
  • Strong-Constrained Dynamical, i.e., Neural ODE/PDE
  • Weak-Constrained Dynamical, i.e., SSM, EnsKF

State Estimation

z∗(θ)=arg min⁡J(z;θ)z^*(\theta) = \argmin \hspace{2mm}\boldsymbol{J}(z;\theta)
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  • Weak-Constrained
    • Gradient-Based - 4DVar
    • Derivative-Free - Ensemble Kalman Filter
    • Minimization-Based - DEQ

Predictions

u∼p(u∣z∗)p(z∗∣D)u \sim p(u|z^*)p(z^*|\mathcal{D})
(14)

Our final task is to use a our new model to make predictions.

  • Feature Representation
  • Interpolation
  • HindCasting
  • Forecasting

Questions

Parameter Estimation

  • Naive Density Estimator, p(y,θ)=p(θ∗∣D)p(y,\theta)=p(\theta^*|\mathcal{D})
  • Can we maximize the data likelihood?, θ∗=arg min⁡L(θ)\theta^* = \argmin \boldsymbol{L}(\theta)
  • What are the distribution of the parameters? θ∼p(θ∗∣D)\theta\sim p(\theta^*|\mathcal{D})
  • What parametric form fits the best? f,θ,AIC,BIC,NLLf,\theta,AIC,BIC,NLL
  • How sensitive is it to the initial condition? x0∼p(x0∣θ)x_0\sim p(x_0|\theta)
  • What is the predictive uncertainty?, y∼p(y∣z)p(z∣x∗,θ∗)p(θ∗∣D)y\sim p(y|z)p(z|x^*,\theta^*)p(\theta^*|\mathcal{D})
  • Can we introduce structure to reduce the complexity? LocalLinear, AutoRegressive, Cycle

Representation

  • What temporal resolution for GEVD?, e.g. Decade, Year, Season, Month
  • What temporal resolution for GPD?, e.g., Day, 3 Days, 5 Days, 7 Days
  • What spatial resolution for GEVD?, e.g., Individual Stations, Spatial Clusters, Coarse Grid, Fine Grid

Predictions

  • Can we forecast? Does it make sense? zT+τz_{T+\tau}
  • Can we hindcast? Does it makes sense? zT−τz_{T-\tau}

Appendix

There are various things we need to go over which will help us deal with extreme events.

Applications
Application Track
Extremes
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