Hmmm... this question might be a bit low-brow. I'm no mathematician.
Let's say that I have an unordered sequence of fuzzy values $T = (t_1 \ldots t_n)$, $T \in (0,1]^n$. Each value is a positive indication towards some observation (hence, the values are not probabilities in that a value of 0.1 does not indicate a value of 0.9 against, just that we have a weak positive affirmation of the observation).
To take an example, let's say that the observation is "whether or not there has been a hurricane in the US today": we have evidence from a weather forecast two days ago saying that there would be one today (0.1), there's been a high level of traffic on twitter with the keyword hurricane (0.2) and there's been breaking news reports about a hurricane in Florida (0.7).
I'm trying to formalise a means of aggregating these readings into a final score (the absolute value of the score makes no difference, but the relative value compared to other aggregations does). For what I'm doing, I want each fuzzy value to reduce the cumulative "uncertainty" about the observation. I also need to set an external $max\in(0,1]$ value on the aggregation result:
$F_0(T,max) = 0$;
$F_k(T,max) = (max - F_{k-1})t_k + F_{k-1}$;
$F(T,max) = F_n(T,max)$.
Taking the hurricane example, let $max = 1$:
$F_0 = 0$;
$F_1 = (1 - 0) * 0.1 + 0$;
$F_2 = (1 - 0.1) * 0.2 + 0.1 = 0.28$;
$F_3 = (1 - 0.28) * 0.7 + 0.28 = 0.784$.
This aggregation is nice (for my scenario) because it rewards high fuzzy values better than multiple medium or low values, it's order indepedent, and the answer is in $(0,max]$.
I'm having a lot of trouble formalising this fairly simple idea... don't have the necessary know-how or lingo (probably clear by now). So I have the following questions:
- Has anyone seen a similar such aggregation somewhere?
- How should I call this sequence of values, where the order is not important, and duplicates should be maintained? An unordered sequence? A set with duplicates? An unordered n-tuple?
- How should I formulate this function? Should I formulate it for an unordered pair first, and then for an unordered tuple? Is there a neat way of demonstrating commutativity?
Any help or nudges in the right direction would be greatly appreciated.