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AI 4 Attribution

Feature Representation

CSIC
UCM
IGEO

Things to Consider:

  • Univariate --> Multivariate
  • Spatially Independent --> Spatially Dependent
  • Temporally Independent --> Temporally Dependent
  • Stationary --> Non-Stationary

Stationarity in Time Series

See example.

Stationarity in Spatial Data

See example.

Filtering

First, we perform some sort of filtering procedure to remove ... from the spatiotemporal cube.

y(x,t)=FFilter[y;θ](x,t),x∈ΩGlobe⊆RDst∈TGlobe⊆R+\begin{aligned} \boldsymbol{y}(\mathbf{x},t) = \boldsymbol{F}_\text{Filter}[\boldsymbol{y};\theta](\mathbf{x},t), && && \mathbf{x}\in\Omega_\text{Globe}\subseteq\mathbb{R}^{D_s} && && t\in\mathcal{T}_\text{Globe}\subseteq\mathbb{R}^+ \end{aligned}
(1)

In this case a kernel average of 3-5 days is applied.

Note:

Anomalies

First, we need to calculate the climatology of our dataset which is the global spatial average over some defined reference period. The equation for the climatology is given by:

Climatology Equation:yˉClimatology(t)=1Ns∑n=1Nsy(xn,t)Climatology Function:yˉClimatology:ΩGlobe×TReference→RDySpatial Domain:x∈ΩGlobe⊆RDsTemporal Domain:t∈TReference⊆R+\begin{aligned} \text{Climatology Equation}: && && \bar{y}_\text{Climatology}(t) &= \frac{1}{N_s}\sum_{n=1}^{Ns}\boldsymbol{y}(\mathbf{x}_n,t) \\ \text{Climatology Function}: && && \bar{y}_\text{Climatology}&: \Omega_\text{Globe}\times\mathcal{T}_\text{Reference} \rightarrow \mathbb{R}^{D_y} \\ \text{Spatial Domain}: && && \mathbf{x}&\in\Omega_\text{Globe}\subseteq\mathbb{R}^{D_s}\\ \text{Temporal Domain}: && && t&\in\mathcal{T}_\text{Reference}\subseteq\mathbb{R}^+ \end{aligned}
(2)

The reference period, TReference\mathcal{T}_\text{Reference}, is some defined period which captures majority of the trends we can expect to see. We also want this reference period to have minimal influence of anthropogenic activity. So, given the defined period of 1850-2023, we could take the reference period to be 1850-1880 (30 years).

To calculate the anomalies of the spatiotemporal cube, we subtract the climatology from the field.

Anomaly Equation:yˉAnomaly(x,t)=y(x,t)+yˉClimatology(t)Anomaly Function:yˉAnomaly:ΩGlobe×TGlobe→RDySpatial Domain:x∈ΩGlobe⊆RDsTemporal Domain:t∈TGlobe⊆R+\begin{aligned} \text{Anomaly Equation}: && && \boldsymbol{\bar{y}}_\text{Anomaly}(\mathbf{x},t) &= \boldsymbol{y}(\mathbf{x},t) + \boldsymbol{\bar{y}}_\text{Climatology}(t) \\ \text{Anomaly Function}: && && \boldsymbol{\bar{y}}_\text{Anomaly}&: \Omega_\text{Globe}\times\mathcal{T}_\text{Globe} \rightarrow \mathbb{R}^{D_y} \\ \text{Spatial Domain}: && && \mathbf{x}&\in\Omega_\text{Globe}\subseteq\mathbb{R}^{D_s}\\ \text{Temporal Domain}: && && t&\in\mathcal{T}_\text{Globe}\subseteq\mathbb{R}^+ \end{aligned}
(3)

What remains are the anomalies of the spatiotemporal field, y\boldsymbol{y}.

Data Reduction

Now, we perform a data reduction of the spatiotemporal field. In the simplest case, we can take the spatial average of the field at every time step. This will result in a single time series for the entire data cube.

Reduced Data Equation:y~anomaly(t)=1Ns∑n=1Nsyˉanom(x,t)Reduced Data Anomaly Function:y~anomaly:TGlobe→RDyTemporal Domain:t∈TGlobe⊆R+\begin{aligned} \text{Reduced Data Equation}: && && \tilde{y}_\text{anomaly}(t) &= \frac{1}{N_s}\sum_{n=1}^{N_s}\boldsymbol{\bar{y}}_{anom}(\mathbf{x},t) \\ \text{Reduced Data Anomaly Function}: && && \tilde{y}_\text{anomaly} &: \mathcal{T}_\text{Globe} \rightarrow \mathbb{R}^{D_y} \\ \text{Temporal Domain}: && && t&\in\mathcal{T}_\text{Globe}\subseteq\mathbb{R}^+ \end{aligned}
(4)
Modeling
Bayesian Inference
Modeling
Features (Learned)