Risk Management/Modelling of Risks and Response Maps

This learning resources has the objective to learn about modelling Risk and Response Maps. A map represents: Risk management allocated the resources according to the calculated risk. As a first simple approach we try to identify a "good" response strategy and allocate the available resources in way, that the resources will have the highest impact on risk mitigation.
 * spatial aspects of risk and
 * spatial aspects of the response according to risk.

We have to deal with uncertainty where a disaster like earth quakes happens or where certain communicable diseases occur in epidemiology.

The one-dimensional normal distribution is well-know in most scientific disciplines. First of all we extend this content of probability density to the two dimensional case.

Probability Density function


The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix $$\boldsymbol\Sigma$$ is positive definite. In this case the distribution has density



\begin{align} f_{\mathbf x}(x_1,\ldots,x_k) & = \frac{\exp\left(-\frac 1 2 ({\mathbf x}-{\boldsymbol\mu})^\mathrm{T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right)}{\sqrt{(2\pi)^k|\boldsymbol\Sigma|}} \\[5pt] & = \frac{\exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^\mathrm{T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right)}{\sqrt{|2\pi\boldsymbol\Sigma|}}, \end{align} $$

where $${\mathbf x}$$ is a real k-dimensional column vector and $$|\boldsymbol\Sigma|\equiv \operatorname{det}\boldsymbol\Sigma$$ is the determinant of $$\boldsymbol\Sigma$$. Note how the equation above reduces to that of the univariate normal distribution if $$\boldsymbol\Sigma$$ is a $$1 \times 1$$ matrix (i.e. a single real number).

Probability density function on Maps
For maps the domain of the probability density is an area on earth characterized e.g. by latitude und longitude. Probability distribution in general will not look like multivariate normal distribution. A decision-maker will be able to identify areas on the map where it is more likely that a disaster occurs than in other areas of the world. One example of a visual representation of an probability density functions is the OpenLayers HeatMap for Earthquakes.

Impact density function
Considering Risk as


 * $$ Risk = Probability \times Impact \qquad (\ast)$$

the probability density describes one constituent of risk. Furthermore we need an impact density function. An area with a high probability of earthquakes might have a low risk because no humans are living in that area. The term of "population density" used in this case is an example of an impact density functions. Valleys in the population density function indicate that only a few people live in that area and peaks in the population density function show that many people per square kilometer are living in that area (e.g. Vesuvius close to the city Naples has erupted many times since and is the only volcano on the European mainland to have erupted within the last hundred years. Today, it is regarded as one of the most dangerous volcanoes in the world because of the population of 3,000,000 people living nearby and its tendency towards violent, explosive eruptions of the Plinian type, making it the most densely populated volcanic region in the world..

If $$A\subset \Omega$$ is an area in domain $$\Omega$$ of the population densitiy $$f$$

f: \Omega \rightarrow \R, \qquad  \mu_f(A) = \int_{A} f(x) \, dx. $$ then $$\mu(A)$$ is the number of people living in a area $$A$$.

Risk density function
Let $$p:\Omega \rightarrow [0,1]$$ is a probabilty function, that defines the probability $$p(x)$$, that someone gets affected by an event at geolocation $$x \in \Omega$$ (e.g. event=vulcanic erruption). The spatial risk density is defined by:

r: \Omega \rightarrow \R, \qquad  \mu_r(A) = \int_{A} p(x) \cdot f(x) \, dx. $$ The risk value $$\mu_r(A)$$ is the expected number of people affect by an event in the area $$A\subset \Omega$$. In a first step $$\Omega$$ can be regarded as surface of the earth and $$A$$ is the area of interest for a decision maker or risk manager.

Remark: Keep in mind that a probability function is different from a probability density functions, because if everyone in the area at all geolocations of $$\Omega$$ will be defninitely affected by an event, than the probability functions $$p(x)=1$$ for all geolocations $$x \in \Omega$$. In general this violates the property of a probability density function:

p: \Omega \rightarrow \R, \qquad  \mu_p(\Omega) = \int_{\Omega} p(x) \, dx = 1. $$

Spatial Fuzzy Logic and membership functions




Spatial properties
In the previous chapter we mapped probability density or an impact density to a geolocation with a map. Keep in mind that the integral of probability density over the domain $$\Omega \subset \R^2$$ is 1.



f: \Omega \rightarrow \R, \qquad  \mu(\Omega) = \int_{\Omega} f(x) \, dx = 1. $$

That does not mean that the probability density function fullfils the property $$f(x) \leq 1$$ for all $$x \in \Omega$$ (see normal distribution).

A spatial property in general maps a property to a geolocation. Examples for spatial properties are:
 * boolean properties $$ f(x_1,x_2)= true$$ represents e.g that the geolocation with latitude $$x_1$$ and longitude  $$x_2$$ belongs to Italy and $$ f(x_1,x_2)= false$$ if the geolocation $$(x_1,x_2)$$ does not belong to Italy. In disaster management  $$ f(x_1,x_2)= true$$ could represent that electricity is available at geolocation $$(x_1,x_2)$$.
 * temperature: $$ f(x_1,x_2)= 20^o C$$ means that currently a temperature $$20^o C$$ of the geolocation $$(x_1,x_2)$$.
 * population density: $$ f(x_1,x_2)= 5500$$ means e.g. that "5500 people per square kilometer" is the population density at geolocation $$(x_1,x_2)$$. If $$A\subset \Omega$$ is an area in the domain $$\Omega$$ the total population living in the area is the following integral of the population density $$f$$:



f: \Omega \rightarrow \R, \qquad  \mu(A) = \int_{A} f(x) \, dx. $$

If $$A := [a_1, b_1] \times [a_2 ,b_2 ]\subset \Omega := \R^2$$ is a rectangle, then is integral is a double integral (two dimensions):



\mu(A) = \int_{a_2}^{b_2} \int_{a_1}^{b_1} f(x_1,x_2) \, dx_1 \, dx_2. $$

2D/3D/4D Domain of Maps
The D in the title stands for Dimension of vector space. The domain for Risk and response map, for assigning digital values or records to a geolocation could have the following dimensions:
 * 2D: Longitude, Latitude
 * 3D Longitude, Latitude, Elevation/Altitude
 * 4D Longitude, Latitude, Elevation/Altitude, Time

Spatial Decision Support Layers are basic elements for Spatial Decision Support Systems, which has 4D domain as default.

Remark: In a spherical geometry mathematical laws are not true in general (sum of angles in a triangle on the sphere is not equal to $$180^o$$)

Learning Task

 * Learn about Spatial Decision Support Layers
 * Search the web for a health risk map based on the contamination soil. Decribe a workflow for decision makers that want to perform risk mitigation strategies based on risk maps.
 * What are resources that could support risk mitigation for contaminated area? Think in two directions:
 * Remove the contamination of soil (clean up)
 * Improve Risk Literacy of population, so that they are not exposed to contamination! How could you create a spatial representation of risk knowledge and could that be helpful to determine the response according to the identified risk?
 * Try to collect types of resources that could used for risk mitigation in general. Create a workflow how to create reponse map for decision makers, that supports risk management.