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ML4EO

Concept - Data Representation

CSIC
UCM
IGEO

Examples

  • Univariate Time Series
  • Multivariate Time Series
  • Univariate SpatioTemporal Series
  • Multivariate SpatioTemporal Series
  • Coupled Multivariate SpatioTemporal Series

Coordinates

Spatial

[Radius, Longitude, Latitude]\left[ \text{Radius, Longitude, Latitude}\right]
(1)
[Altitude/Depth, Longitude, Latitude]\left[ \text{Altitude/Depth, Longitude, Latitude}\right]
(2)
[Channel, X, Y]\left[ \text{Channel, X, Y}\right]
(3)

Temporal

Variable

  • Remote Sensing - Spectral Signature
  • Ocean - Temperature, Height, Salinity, Colour
  • Weather Station - Temperature, Precipitation, Wind Speed

Samples

  • Ensembles/Realizations
  • Patches

Dataset Structures

Univariate Time Series

D={tn,yn}n=1N,N=NTyn∈RDytn∈R+\begin{aligned} \mathcal{D} &= \left\{ t_n, y_n \right\}_{n=1}^N, && && N = N_T && && y_n \in\mathbb{R}^{D_y} && && t_n\in\mathbb{R}^+ \end{aligned}
(4)
Measurements:y∈RNTyt∈RTime Stamps:t∈RNTtn∈R+\begin{aligned} \text{Measurements}: && && \mathbf{y} &\in\mathbb{R}^{N_T} && && y_t \in\mathbb{R}\\ \text{Time Stamps}: && && \mathbf{t} &\in\mathbb{R}^{N_T} && && t_n \in\mathbb{R}^+ \end{aligned}
(5)

Shape

Irregular.

Example:

  • Extreme Events
  • Faulty Station
  • ARGO Floats

Regular.

Examples:

  • Single Weather Station - Max Temperature, Mean Temperature, Precipitation Accumulation
  • Global Mean Surface Temperature Anomaly

Multivariate Time Series

D={tn,yn}n=1N,N=NTyn∈RDytn∈R+\begin{aligned} \mathcal{D} &= \left\{ t_n, \mathbf{y}_n \right\}_{n=1}^N, && && N = N_T && && \mathbf{y}_n \in\mathbb{R}^{D_y} && && t_n\in\mathbb{R}^+ \end{aligned}
(6)
Measurements:Y∈RNT×Dyyn∈RTime Stamps:t∈RNTtn∈R+\begin{aligned} \text{Measurements}: && && \mathbf{Y} &\in\mathbb{R}^{N_T\times D_y} && && \mathbf{y}_n \in\mathbb{R}\\ \text{Time Stamps}: && && \mathbf{t} &\in\mathbb{R}^{N_T} && && t_n \in\mathbb{R}^+ \end{aligned}
(7)

Univariate SpatioTemporal Series

Coordinate-Based Representation

D={(tn,sn),yn}n=1Nyn∈RDytn∈R+\begin{aligned} \mathcal{D} &= \left\{ (t_n, \mathbf{s}_n), \mathbf{y}_n \right\}_{n=1}^N && && \mathbf{y}_n \in\mathbb{R}^{D_y} && && t_n\in\mathbb{R}^+ \end{aligned}
(8)
Measurements:Y∈RNT×Dyyn∈RDyTime Stamps:t∈RNTtn∈R+Spatial Coordinates:S∈RNT×Dssn∈RDs\begin{aligned} \text{Measurements}: && && \mathbf{Y} &\in\mathbb{R}^{N_T\times D_y} && && \mathbf{y}_n \in\mathbb{R}^{D_y}\\ \text{Time Stamps}: && && \mathbf{t} &\in\mathbb{R}^{N_T} && && t_n \in\mathbb{R}^+\\ \text{Spatial Coordinates}: && && \mathbf{S} &\in\mathbb{R}^{N_T \times D_s} && && \mathbf{s}_n \in\mathbb{R}^{D_s} \end{aligned}
(9)

Examples:

  • Weather Stations

Field-Based Representation

D={tn,yn}n=1N,N=NTD=DyDΩyn∈RD\begin{aligned} \mathcal{D} &= \left\{ t_n, \mathbf{y}_n \right\}_{n=1}^N, && && N = N_T && && D = D_y D_\Omega && && \mathbf{y}_n \in\mathbb{R}^{D} \end{aligned}
(10)
Measurements:Y∈RNT×Dyn∈RDTime Stamps:t∈RNTtn∈R+\begin{aligned} \text{Measurements}: && && \mathbf{Y} &\in\mathbb{R}^{N_T\times D} && && \mathbf{y}_n \in\mathbb{R}^D\\ \text{Time Stamps}: && && \mathbf{t} &\in\mathbb{R}^{N_T} && && t_n \in\mathbb{R}^+ \end{aligned}
(11)

Examples:

  • Gridded Weather Station Data Product
  • Sea Surface Height
  • Sea Surface Temperature
Concepts
PGM
Parameterizations
Parameterization - TOC