# Spatial Decision Support Layer

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. Decision support layers can range from simple Spatial Assignment Layer, that maps e.g. a real value to a geolocation (temperature at the geolocation) to more complex decision support layers that assign different mathematical functions to geolocations that could perform a specific calculation for the geolocation and the function itsself is different for different geolocation.

## 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 ${\displaystyle \mu }$.

• if the codomain ${\displaystyle M}$ is an arbitrary set (e.g. videos, websites, images or text documents) that are assigned to geolocation the we call that in general Spatial Assignment Layer (SAL).
• if the codomain ${\displaystyle M}$ is also a vector space, then we call the SAL a Spatial Decision Support Layer (SDSL).

As the basic example we could use scalar sets ${\displaystyle M=\mathbb {Q} ,\mathbb {R} ,\mathbb {C} }$ as basic codomain. With scalar codomains we can encode temperature, population density, contamination, radiation to geolocations and build decision support ontop of these layers. For a political map the codomain

${\displaystyle M:=\{USA,GB,SA,G,...\}}$

is a set of states and all geolocation are mapped to state to which the geolocation belongs. This set ${\displaystyle M}$ as codomain does not have a vector space structure and assignments allow not vector space operation in the codomain. Nevertheless this more general SAL can be helpful for decision support systems in general, to identify which legal framework has to be applied for decision support due to the national regulations that must be respected.

### 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

• Spatial Fuzzy Logic: 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
${\displaystyle \mu _{temp}:\mathbb {R} \longrightarrow [0,1]\qquad x\mapsto {\frac {1}{1+{\frac {(x-25)^{2}}{4}}}}}$
(Spatial Membership Functions) 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.
• Assignment of mathematical Function: In the examples about values (Spatial Assignment Layer SAL) or database records can be assigned to point in space and time. In a more general use case mathematical functions can be assigned to a geolocation that could perform a specific calculation for the geolocation or the area the mathematical function was assigned to. The mathematical functions change in time and space.

### Digital Signature

Explore the concept of digital signatures and explain why digital signatures are helpful for recipients of Spatial Decision Support Layers to be sure, that the digital product was not changed by someone else. Analyse different scenarios of your choice in which decision makers allocate resources based on decision support layers provided to them. How can digital signatures be used for exchanging decision support layers between organisations, companies, educational environments, ...? What are use-cases in which you would add also encryption to decision support layers (i.e. privacy of the data is relevant).