Aggregating numerous blurry values for an offered monitoring

Hmmm ... this inquiry could be a little bit reduced - eyebrow. I'm no mathematician.

Allow is claim that I have an unordered series of blurry values $T = (t_1 \ldots t_n)$, $T \in (0,1]^n$. Each value is a favorable sign in the direction of some monitoring (therefore, the values are not chances because a value of 0.1 does not show a value of 0.9 versus, simply that we have a weak favorable affirmation of the monitoring).

To take an instance, allow is claim that the monitoring is "whether there has actually been a storm in the United States today": we have proof from a weather prediction 2 days ago claiming that there would certainly be one today (0.1 ), there is been a high degree of website traffic on twitter with the keyword storm (0.2) and also there is been damaging report concerning a storm in Florida (0.7 ).

I'm attempting to formalise a method of accumulating these analyses right into a last rating (the outright value of ball game makes no distinction, yet the loved one value contrasted to various other gatherings does). Wherefore I'm doing, I desire each blurry value to lower the collective "unpredictability" concerning the monitoring. I additionally require to set an exterior $max\in(0,1]$ value on the gathering outcome:

$F_0(T,max) = 0$ ;

$F_k(T,max) = (max - F_{k-1})t_k + F_{k-1}$ ;

$F(T,max) = F_n(T,max)$.

Taking the storm instance, allow $max = 1$:

$F_0 = 0$ ;

$F_1 = (1 - 0) * 0.1 + 0$ ;

$F_2 = (1 - 0.1) * 0.2 + 0.1 = 0.28$ ;

$F_3 = (1 - 0.28) * 0.7 + 0.28 = 0.784$.

This gathering behaves (for my circumstance) due to the fact that it compensates high blurry values much better than numerous tool or reduced values, it is order indepedent, and also the solution remains in $(0,max]$.

I'm having a great deal of problem formalising this rather straightforward suggestion ... do not have the essential recognize - just how or language (possibly clear now). So I have the adhering to inquiries:

  • Has any person seen a comparable such gathering someplace?
  • Just how should I call this series of values, where the order is trivial, and also replicates should be kept? An unordered series? A set with matches? An unordered n - tuple?
  • Just how should I create this function? Should I create it for an unordered set first, and afterwards for an unordered tuple? Exists a cool means of showing commutativity?

Any kind of aid or pushes in the appropriate instructions would certainly be substantially valued.

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2019-05-18 23:06:52
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Answers: 1

I'm at the very least accustomed with the suggestion of blurry reasoning, as distinctive from chances. Having claimed that: while you are not seeking a probabilistic type of formula, you have actually come across one.

Taking max = 1, your formula can be operated to the kind

F k+1
= (1 − F k ) t k +F k
= F k +t k − F k t k

which coincides point as the "chance" of at the very least among occasion A or occasion B taking place, where An and also B are independent arbitrary occasions, accompanying chance F k and also t k specifically.-- This is why your building and construction is order independent (and also actually can be confirmed so utilizing this monitoring).

To claim the very same point in various terms: if you take the "blurry AND" (sensible combination) of 2 top qualities with blurry toughness x and also y to be the item xy , after that the formula x +y xy is the equivalent "blurry OR" which will certainly please de Morgan is regulations, i.e. NOT ( x AND y ) = (NOT x ) OR (NOT y ), if you take NOT x = (1 − x ). Certainly, you might not desire this if you do not desire a tiny favorable blurry value to stand for a huge blurry value for its negation.

It appears to me that whatever you pick max to be, you can re - range it to get the same practices to the instance max = 1. So the energy of this analysis is not likely to rest on that certain parameter.

For your various other inquiries:

  • A collection in which number is necessary, yet order is not, is a bag or multi - set (these are basic synonyms). In addition, that is all that it is ; they are primarily "set - like collections" which amount a pie chart.

  • I would certainly call the function "blurry OR" or "blurry disjunction".

  • If you intend to make the blurry OR of a collection (a "bag") well - specified, it is adequate to show commutativity (that makes it well - specified for bags of dimension at the majority of 2), and afterwards show that it is associative-- that is, that OR ( x , OR ( y , z )) = OR (OR ( x , y ), z )-- which makes certain that if it is well - specified for bags of dimension k , it is additionally well - specified for bags of dimension k +1.

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2019-05-21 07:44:33
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