Skip to article frontmatterSkip to article content

Below is my preferred notation for modeling.

Spatial Coordinates

s∈Ω⊆RDs\mathbf{s} \in \Omega \subseteq \mathbb{R}^{D_s}
(1)

For example, we can have:

Cartesian:s=[x,y,z]Spherical:s=[Longitude,Latitude,Depth/Altitude]\begin{aligned} \text{Cartesian}: && && \mathbf{s} &= [x,y,z]\\ \text{Spherical}: && && \mathbf{s} &= [\text{Longitude},\text{Latitude},\text{Depth/Altitude}] \end{aligned}
(2)

Note: in many cases, we have coordinate transformations whereby we can move between different coordinate systems. For example, we can simply move between Cartesian coordinate system and the spherical coordinate system. There are also many examples of the Coordinate Reference System (CRS) which allows users to use their own custom system depending upon their field of view.

Temporal Coordinates

t∈T⊆R+t \in \mathcal{T} \subseteq \mathbb{R}^+
(3)

For example, we can have:

t=[seconds]t=[hours]\begin{aligned} t &= [\text{seconds}] \\ t &= [\text{hours}] \end{aligned}
(4)

Observations

Measurements we can actually observe.

y=y(s,t),y:RDs×R+→RDys∈Ωy⊆RDst∈Ty⊆R+\begin{aligned} \boldsymbol{y} &= \boldsymbol{y}(\mathbf{s},t), && && \boldsymbol{y}: \mathbb{R}^{D_s}\times\mathbb{R}^+\rightarrow\mathbb{R}^{D_y} && && \mathbf{s}\in\Omega_y\subseteq\mathbb{R}^{D_s} && && t \in \mathcal{T}_y \subseteq \mathbb{R}^+ \end{aligned}
(5)

Covariate

Data which we believe is conditionally important for our model.

x=x(s,t),x:RDs×R+→RDxs∈Ωx⊆RDst∈Tx⊆R+\begin{aligned} \boldsymbol{x} &= \boldsymbol{x}(\mathbf{s},t), && && \boldsymbol{x}: \mathbb{R}^{D_s}\times\mathbb{R}^+\rightarrow\mathbb{R}^{D_x} && && \mathbf{s}\in\Omega_x\subseteq\mathbb{R}^{D_s} && && t \in \mathcal{T}_x \subseteq \mathbb{R}^+ \end{aligned}
(6)

Quantity of Interest (QoI)

The true quantity we are interested in estimating.

u=u(s,t),u:RDs×R+→RDxs∈Ωu⊆RDst∈Tu⊆R+\begin{aligned} \boldsymbol{u} &= \boldsymbol{u}(\mathbf{s},t), && && \boldsymbol{u}: \mathbb{R}^{D_s}\times\mathbb{R}^+\rightarrow\mathbb{R}^{D_x} && && \mathbf{s}\in\Omega_u\subseteq\mathbb{R}^{D_s} && && t \in \mathcal{T}_u \subseteq \mathbb{R}^+ \end{aligned}
(7)

Latent Variables

Unknown, unobserved variables

z=z(s,t),z:RDs×R+→RDxs∈Ωz⊆RDst∈Tz⊆R+\begin{aligned} \boldsymbol{z} &= \boldsymbol{z}(\mathbf{s},t), && && \boldsymbol{z}: \mathbb{R}^{D_s}\times\mathbb{R}^+\rightarrow\mathbb{R}^{D_x} && && \mathbf{s}\in\Omega_z\subseteq\mathbb{R}^{D_s} && && t \in \mathcal{T}_z \subseteq \mathbb{R}^+ \end{aligned}
(8)
Software
PPL
Appendix
Glossary