In a VC/RC/US model, the probability of each possible outcome is explicitly calculated (RC), the range of variability is incorporated by making separate calculations (VC) for each value of the variability parameter(s), and the uncertainty is simulated (US). We start with the simplest model, where there is no variability to worry about.
Randomness model only (RC/US)
We'll illustrate this approach with an example. Imagine you wish to calculate the number of outbreaks next year in some region. The explicit calculation model is provided here: VC RC US - 1.
In an explicitly calculated model like this, it is a simple matter to include uncertainty about any parameters of the model. For example, if we were not confident that the coin was truly fair but instead wish to describe our estimate of the probability of heads as a Gamma(0, 1/10, 22) (for example if our estimate comes from observing 22 outbreaks in 10 years) distribution, we can simply enter the Gamma distribution, which necessitates a small change to the model to ensure the same value is used everywhere.
The links to the VC RC US - 1 software specific models are provided here:
The explicit calculation model would take the form of the spreadsheet:
In the spreadsheet, we have used the Excel function POISSON(x,2.2,0) which returns the probability of x outbreaks where the Poisson intensity is 2.2 outbreaks/year. Statistical results, like the mean and standard deviation shown in the spreadsheet model, can also be determined explicitly as needed. The formula calculating the mean and standard deviation are array formulae, entered into Excel with Ctrl+Shift+Enter.
The explicit calculation model would take the form of the spreadsheet:
In the spreadsheet, we have used the Excel function POISSON(x,2.2,0) which returns the probability of x outbreaks where the Poisson intensity is 2.2 outbreaks/year. Statistical results, like the mean and standard deviation shown in the spreadsheet model, can also be determined explicitly as needed. The formula calculating the mean and standard deviation are array formulae, entered into Excel with Ctrl+Shift+Enter.
The separation of uncertainty and randomness is simple and clear when using a model that explicitly calculates the randomness, since we use probability calculation formulas for the randomness and MC simulation for the uncertainty. You can view VC RC US - 2 and investigate its behavior. Taking say 30 iterations of the model will give you an effective appreciation of the contribution of uncertainty and allow you to produce a second order plot of the output:
The links to the VC RC US - 2 software specific models are provided here:
Variability and randomness model (VCRCUS)
We can now extend the model to consider more than one area, each with a different level of outbreak risk (variability). VC RC US - 3 model uses the identity:
Poisson(a) + Poisson(b) = Poisson(a+b)
to determine the number of outbreaks in total over the four areas. Although the model is very simple, it already illustrates the limitations of combining several probability distributions in a calculation model. The complexity of a VC/RC risk model increases exponentially with additional elements. We have used the above Poisson identity which makes the "Total' calculation very simple, but in doing so the probabilities are calculated separately for each distribution. In other words, the probability of having (say) 1 outbreak in total is not calculated as a function of the other probabilities, because it gets too complicated:
P("Total' = 1) = P(Area1 = 1)* P(area 2 =0)* P(Area3 = 0)*P(Area4=0)
+ P(Area1 = 0)* P(area 2 =1)* P(Area3 = 0)*P(Area4=0)
+ P(Area1 = 0)* P(area 2 =0)* P(Area3 = 1)*P(Area4=0)
+ P(Area1 = 0)* P(area 2 =0)* P(Area3 = 0)*P(Area4=1)
[Try calculating the probability of 2 total outbreaks… it gets very laborious]
Therefore, you can't easily see the effect of outbreaks in each Area on the total outbreaks, and therefore, nearly all real-world risk problems cannot be evaluated with VC/RC calculation models.
The links to the VC RC US - 3 software specific models are provided here:
