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Dynamical Systems Formulation

Firstly, we have some weather stations

Spatial Coordinates:s∈Ω⊆RDs\begin{aligned} \text{Spatial Coordinates}: && && \mathbf{s}&\in\Omega\subseteq\mathbb{R}^{D_s} \end{aligned}
(1)

In our case, the vector, s\mathbf{s}, gives us the longitude, latitude, and altitude of the weather station. In addition, we have some measurements across time.

Temporal Coordinates:t∈T⊆R+\begin{aligned} \text{Temporal Coordinates}: && && t &\in \mathcal{T} \subseteq \mathbb{R}^+ \end{aligned}
(2)

There are two time domains that we have access to. For the global mean surface temperature (GMST), we have a time series that is available from 1800 till present day (2020). For the daily maximum temperature measurements, we have a time series that is available from 1960 till present day (2020).

First, we have our measurements which are maximum temperature values.

Measurements:y=y(t,s)y:R+→Rt∈Ty⊆R+s∈Ω⊆RDs\begin{aligned} \text{Measurements}: && && y &= y(t,\mathbb{s}) && && y:\mathbb{R}^+\rightarrow\mathbb{R} && && t \in \mathcal{T}_y\subseteq \mathbb{R}^+ && \mathbf{s}\in\Omega\subseteq\mathbb{R}^{D_s} \end{aligned}
(3)

Next, we have our GMST measurements which is the temperature anomaly (in degrees Celsius) over our time domain. This is a single time series so there is no dependency on spatial coordinates.

Covariate:x=x(t)x:R+→Rx∈Tx⊆R+\begin{aligned} \text{Covariate}: && && x &= x(t) && && x:\mathbb{R}^+\rightarrow\mathbb{R} && && x \in \mathcal{T}_x\subseteq \mathbb{R}^+ \end{aligned}
(4)

ODE Formulation

The DMT is formulated as an ordinary differential equation (ODE). First, we will define it as a system of ODEs whereby we have a state variable

State:z=[xy],z∈R2\begin{aligned} \text{State}: && && \mathbf{z} &= \begin{bmatrix} x \\ y \end{bmatrix}, && && \mathbf{z}\in\mathbb{R}^2 \end{aligned}
(5)

Now, we can define an equation of motion which describes the temporal dynamics of the system.

Equation of Motion:dzdt=f(z,t,θ),f:R2×R+×Θ→R\begin{aligned} \text{Equation of Motion}: && && \frac{d\mathbf{z}}{dt} &= \boldsymbol{f}(\mathbf{z},t,\theta), && && \boldsymbol{f}:\mathbb{R}^2 \times \mathbb{R}^+ \times \Theta \rightarrow \mathbb{R} \end{aligned}
(6)

We also have initial measurements of the system

Initial Values:z(0)=[x(0)y(0)]:=z0\begin{aligned} \text{Initial Values}: && && \mathbf{z}(0) &= \begin{bmatrix} x(0) \\ y(0) \end{bmatrix} := \mathbf{z}_0 \end{aligned}
(7)

From the fundamental theory of calculus, we know that the solution of said ODE is a temporal integration wrt time

zt=z0+∫0tf(z0,t,θ)dt\begin{aligned} \mathbf{z}_t = \mathbf{z}_0 + \int_0^t \boldsymbol{f}(\mathbf{z}_0, t, \theta)dt \end{aligned}
(8)

Conventionally, we use ODE solvers like Euler, Heun, or Runge-Kutta.

zt=ODESolve(f,z0,t,θ)\mathbf{z}_t = \text{ODESolve}(\boldsymbol{f}, \mathbf{z}_0, t, \theta)
(9)

Non-Dimensionalization

We will reparameterize this ODE to remove some dependencies on time. The above equation is divided by

dydt×dtdx=dydx=f(y,t,θ)g(x,t,θ):=h(y,x,θ)\frac{dy}{dt}\times \frac{dt}{dx} = \frac{dy}{dx} = \frac{f(y,t,\theta)}{g(x,t,\theta)} := h(y,x,\theta)
(10)

Parameterization

There are many special forms of ODEs which are known from the literature.

1st Order ODE:f(y,x,θ)=f1(x)−f2(x)⋅y\begin{aligned} \text{1st Order ODE}: && && \boldsymbol{f}(y,x,\theta) &= \boldsymbol{f}_1(x) - \boldsymbol{f}_2(x)\cdot y \end{aligned}
(11)

An example form would the following:

f(y,x,θ)=a0+a1x+a2y\boldsymbol{f}(y,x,\theta) = a_0 + a_1 x + a_2 y
(12)

Constant Form. The first form assumes that we have a constant change in DMT wrt the GMST

Constant EOM:f(y,x,θ)=a0Linear Solution:y(x)=y0+a0t\begin{aligned} \text{Constant EOM}: && && \boldsymbol{f}(y,x,\theta) &= a_0 \\ \text{Linear Solution}: && && y(x) &= y_0 + a_0 t \end{aligned}
(13)

Linear Form. The first form assumes that we have a constant change in DMT wrt the GMST

Linear EOM:f(y,x,θ)=a0+a1tQuadratic Solution:y(x)=y0+a0t+12a1t2\begin{aligned} \text{Linear EOM}: && && \boldsymbol{f}(y,x,\theta) &= a_0 + a_1 t\\ \text{Quadratic Solution}: && && y(x) &= y_0 + a_0 t + \frac{1}{2}a_1t^2 \end{aligned}
(14)

Multiplicative Form. The first form assumes that we have a constant change in DMT wrt the GMST

Linear EOM:f(y,x,θ)=a2yExponential Solution:y(x)=y0exp⁡(a2t)\begin{aligned} \text{Linear EOM}: && && \boldsymbol{f}(y,x,\theta) &= a_2 y\\ \text{Exponential Solution}: && && y(x) &= y_0 \exp \left( a_2t \right) \end{aligned}
(15)
Bayes4EVT
Experiment 1 - Formulation
Bayes4EVT
Experiment 1b - Spain (Time)