# Spatial Processes

Tags :: Spatial Statistics

## Realizations

Can be thought of as 2 dimensional with points denoted as \[ s_z(s_1, s_2)’ = s = (u,v)’ = s = (x, y)' \]

A process is denoted as \[ \{Y(s) : s \in \mathcal{R}\} \] where \(\mathcal{R}\) is the spatial region with area \[|\mathcal{R}| = \int_{u \in \mathcal{R}} du \] The spatial region has different properties for the 3 different frameworks:

- Geostastical models have a fixed and continuous \(\mathcal{R}\),
- Lattice/Areal models have fixed and finite \(\mathcal{R}\),
- And the \(\mathcal{R}\) for Spatial Point Process models are random

While point processes have a hypothetical bounds this is ** not** \(\mathcal{R}\).

Typically we are interested in the process \[ \{Y(s_1), \dots, Y(s_n)\}, \; \{Y_1, \dots, Y_n\} \] where a realization from the process is \[ \{y_1, y_2, \dots, y_n\} \] in plain words, all this means is \(Y(s_i)\) is a random process, and \(y(s_i)\) is observed from that process. The same random process can create realizations that look different but have the same spatial dependence.

Start thinking more of spatial dependence? What are some processes with really small/large spatial dependence

## Modeling

Typically we have one realization of a spatial proccess, we have to make **assumptions** to proceed. We have to **model** the process.

We can often develop sufficient models charatarized by only two components

- First order effects (1st moment): mean
- Often deterministic or charactarizing large-scale spatial structure
- Function of covariates

- Second order effects (2nd moment): spatial dependence
- Covariance/correlation
- Deviations around mean

The second order effects are what is key in spatial statistics. The Spatial process can be thought of as \[ \text{Spatial Process} = \mu + \text{correlated process} \] or rather as a regression model

\begin{aligned} Y_i &= \beta_0 + \beta_1 x_i + \epsilon_i \\ \epsilon & \sim N(\mu, \sigma^2) \end{aligned}

## Spatial data

Three components of spatial data

- Features
- Points, lines, areas, etc..

- Support
- Size, shape, orientation of feature

- Attributes
- Values associated with features

The components can apply to all spatial surfaces but are mostly commonly associated with regions on the earth (both terrestrial and non).