Post date: Feb 23, 2014 9:5:40 PM
Hi Scott,
How I can compute the imprecision (i.e, the area representing the difference between the two envelopes) of a component A defined by A=kn(21,24) using "R" or "Risk calc"?
Many thanks
Mohamed:
If you mean the area between the left and right bounds of the c-box, you can just use the breadth function, in either Risk Calc or pbox.r.
kn <- function(k,n) return(env(beta(k, n-k+1), beta(k+1, n-k)))
A = kn(21,24)
breadth(A)
will give you 0.04483754 in R Entering
function kn() return env(beta($1, $2-$1+1), beta($1+1, $2-$1))
A = kn(21,24)
breadth(A)
in Risk Calc will get you 0.049666325. The small difference is because Risk Calc uses a somewhat coarser discretization than R does. If you need the two platforms to give the same answer, you can change the number of discretization levels in R by entering
Pbox$steps = 100
which will reduce them from the default of 200.
It is coincidental that you've asked this question. Just today I had some exchanges on how to measure uncertainty of p-boxes with Michael Beer, Vladik Kreinovich and Kari Sentz. I'll forward you our exchanges in case they might be of interest.
Cheers,
Scott
Thanks Scott,
But I have always a problem with the function "breadth(A)":
When I wrote :
A=beta(23,24) for example I obtain breadth(A)=0.005 instead of breadth(A)=0 because in my opinion there is no imprecision about A (We have just one distribution !!!!) ????
These results will hep me to :
Identify for which component (of our bridge system) information gain will cause the greatest information gain concerning the knowledge of the system behavior.
In both Risk Calc and pbox.r, precise probability distributions are modeled by left and right bounds that are coincident at the precise distribution. However, these bounds differ from each other according to the number of discretization levels. The software is trying to capture the representation error rigorously, in a way that Monte Carlo and similar approximations methods cannot do. This error is also propagated; otherwise we could not pretend to have a rigorous mathematical calculation system.
If you want the representation error to be small, you can increase the number of levels by changing Pbox$steps to a larger integer. However I would not recommend a number larger than about 2000, as the Cartesian product needed to handle such a structure is unwieldy. Of course, 2000 is not infinity, so there will still be some small area between the two curves enclosing the distribution from the left and from the right.
No matter how many discretization levels you use, the left and right bounds will be exactly at the distribution whenever they can be, as depicted below. I suggest that the breadth you're seeing for nominally precise distribution just defines the scale of differences in such breadths such that, any values you see at or below this value, you can presume are zeros. Values between these exact points are, by Williamson's convention, presumed to be conservatively consistent with monotonicity.
Make sense?
Scott
ok I understand.
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