# Spatial Decision Support Layers

GIS Layers

A Spatial Decision Support System[1] operates on Layers as input data. Rule based system processes the layers and creates a decision support output (e.g. spatial multi-criteria evaluation and expert knowledge[2]). The output of spatial decision support system can be (among others) a spatial decision support layer again.

## Mathematics: Spatial Decision Support Layer as Mapping

In terms of mathematical definition a spatial decision support layer is a mapping

${\displaystyle \mu :\Omega \longrightarrow M}$

where ${\displaystyle M}$ is the codomain (target set) of the mapping ${\displaystyle \mu }$.

### Domain of Spatial Decision Support Layer

${\displaystyle \Omega \subset \mathbb {R} ^{4}}$ includes the geographic coordinate system ${\displaystyle \subset \mathbb {R} ^{2}}$, the height ${\displaystyle \subset \mathbb {R} }$ and the time index ${\displaystyle \subset \mathbb {R} }$. So ${\displaystyle \Omega \subset \mathbb {R} ^{4}}$ is a subset of a threedimensional vector space ${\displaystyle \mathbb {R} ^{4}}$. The geographic coordinate system is a coordinate system used in geography that enables every location on Earth to be specified by a set of numbers, letters or symbols. In specialized works, "geographic coordinates" are distinguished from other similar coordinate systems, such as geocentric coordinates and geodetic coordinates. See, for example, Sean E. Urban and P. Kenneth Seidelmann[3] The coordinates are often chosen such that one of the numbers represents a vertical position, and two or three of the numbers represent a horizontal position. A common choice of coordinates is latitude, longitude and elevation[4].

### Output of Decision Support Layer

Heatmap and temporal change as GIF animation - Frames of animation represent different time stamps ${\displaystyle t\in T}$

Think of the output set ${\displaystyle M}$ in the following categories, that are explain by examples:

• (number) e.g. temperature is ${\displaystyle 30^{o}C}$ at time ${\displaystyle t}$, at geolocation ${\displaystyle (x,y)}$ and at altitude ${\displaystyle z}$
${\displaystyle \mu (x,y,z,t)=30}$ with ${\displaystyle \omega :=(x,y,z,t)\in \Omega }$
• (set of objects nearby) the spatial decision support layer provides object that nearby, e.g. an geotagged ambulance ${\displaystyle (A,\omega _{A})}$ and a health care facility ${\displaystyle (H,\omega _{H})}$:
${\displaystyle \mu (x,y,z,t)=\{(A,\omega _{A}),\,(H,\omega _{H})\}}$ with the geolocation ${\displaystyle \omega _{A},\omega _{H}\in \Omega }$ of ${\displaystyle A}$ and ${\displaystyle H}$.
The decision support layer answers the question,
 What are nearby health care services that I can get access to?


Ambulance might be nearer and mobile, while the health care facility might be far away but it might be far away but it could provide better health services. Decision makers will decide which resource will be used dependent on the disease or injury of a patient and the decision support layers provides the information, which resources are in reach of the patient's location. The example provides ${\displaystyle A}$ and ${\displaystyle H}$ as spatial objects, that contain the specification of the ambulance ${\displaystyle A}$ and the health care facility at ${\displaystyle H}$.

## Computer Science: Spatial Decision Support Layer as UML-Class

• Learn about Spatial Fuzzy Logic and create a spatial decision support layer for temperature at specific time ${\displaystyle t\in T\subset \mathbb {R} }$.
• We use a membership function ${\displaystyle \mu _{temp}}$ that maps a temperature ${\displaystyle \subset \mathbb {R} }$ into the real number between 0 and 1 (i.e. the interval ${\displaystyle [0,1]\subset \mathbb {R} }$) by the following definition
${\displaystyle \mu _{temp}:\mathbb {R} \longrightarrow [0,1]\qquad x\mapsto {\frac {1}{1+{\frac {(x-25)^{2}}{4}}}}}$
With definition above the temperature is optimal for mosquitoes (${\displaystyle =1}$) at a temperature ${\displaystyle 25^{o}C}$. If the temperature is higher than ${\displaystyle 25^{o}C}$ (e.g. ${\displaystyle 35^{o}C}$) or lower than ${\displaystyle 25^{o}C}$ the fuzzy value is decreasing for lower and higher temperatures example the domain can be defined as the set of real numbers ${\displaystyle \Omega :=\mathbb {R} }$, so that the membership function ${\displaystyle \mu _{temp}:\Omega \rightarrow [0,1]}$ could take all temperatures in degrees Celsius as input variable.
${\displaystyle Temp:\Omega \longrightarrow \mathbb {R} ,\qquad (x,y,z,t)\mapsto Temp(x,y,z,t)}$
• Missing height altitude/time of argument: When the height/altitude is not provided, the altitude of the surface is use. Explain why this concept is helpful for decision makers.
• How much time does it take to access a health care facility for get a specific health care service. This could vary in space and time due to environmental conditions.
${\displaystyle Time:\Omega \longrightarrow \mathbb {R} ,\qquad (x,y,z,t)\mapsto Time(x,y,z,t)}$
${\displaystyle Time(x,y,z,t_{1})=3.5h}$ means that it takes 3.5 hours to reach a health care facility at time ${\displaystyle t}$. In a rainy season ${\displaystyle t_{2}}$ the value might change to ${\displaystyle Time(x,y,z,t_{2})=10h}$ (i.e. 10h travel time).
• The function ${\displaystyle Temp}$ maps the the tupel ${\displaystyle (x,y,z,t)\in \Omega }$ with the longitude ${\displaystyle x}$, the latitude ${\displaystyle y}$, the elevation above sealevel ${\displaystyle z}$ and the time index ${\displaystyle t\in T\subset \mathbb {R} }$ to the temperature ${\displaystyle Temp(x,y,z,t)}$ at the geolocation ${\displaystyle (x,y)}$ at altitude ${\displaystyle z}$ and time index ${\displaystyle t}$.
• Explain, why the altitude ${\displaystyle z}$ and time index ${\displaystyle t\in T\subset \mathbb {R} }$ are important input parameters for the temperature layer!
• Composition ${\displaystyle \mu :=\mu _{temp}\circ F}$ defines a spatial decision support layers. Explain the purpose for vector control units working for a Public Health Agency.