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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.

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1  
FWIW, the above is called a probabalistic sum, and is a t-conorm: a valid fuzzy disjunction operator. –  badroit Dec 13 '11 at 4:18
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1 Answer

up vote 2 down vote accepted

I'm at least acquainted with the idea of fuzzy logic, as distinct from probabilities. Having said that: while you are not looking for a probabilistic sort of formula, you have stumbled upon one.

Taking max = 1, your formula can be finessed to the form

Fk+1
  = (1 − Fk) tk + Fk
  = Fk + tk − Fktk

which is the same thing as the "probability" of at least one of event A or event B happening, where A and B are independent random events, occurring with probability Fk and tk respectively. — This is why your construction is order independent (and in fact can be proven so using this observation).

To say the same thing in different terms: if you take the "fuzzy AND" (logical conjunction) of two qualities with fuzzy strengths x and y to be the product xy, then the formula x + y − xy is the corresponding "fuzzy OR" which will satisfy de Morgan's laws, i.e. NOT(x AND y) = (NOT x) OR (NOT y), if you take NOT x = (1 − x). Of course, you may not want this if you don't want a small positive fuzzy value to represent a large fuzzy value for its negation.

It seems to me that whatever you choose max to be, you can re-scale it to get identical behaviour to the case max = 1. So the utility of this interpretation is unlikely to hinge on that particular parameter.

For your other questions:

  • A collection in which number is important, but order is not, is a bag or multi-set (these are synonyms). Furthermore, that's all that it is; they're basically "set-like collections" which are equivalent to a histogram.

  • I would call the function "fuzzy OR" or "fuzzy disjunction".

  • If you want to make the fuzzy OR of a collection (a "bag") well-defined, it suffices to show commutativity (which makes it well-defined for bags of size at most two), and then show that it is associative — that is, that OR(x, OR(y,z)) = OR(OR(x,y), z) — which ensures that if it is well-defined for bags of size k, it is also well-defined for bags of size k + 1.

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Neil, many many thanks. This answer is of great help to me. –  badroit Oct 5 '10 at 21:12
    
Hmm... it also occurred to me that I was being a bit stupid. The max value can just be factored out and multiplied by the end result... gives the same value. Something like a "fuzzy AND" I guess. AND(OR(x, OR(y,z)),max). –  badroit Oct 5 '10 at 21:35
    
@badroit: I wouldn't sweat it, those forests can be quite stealthy some times. –  Niel de Beaudrap Oct 6 '10 at 7:14
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