Re: Summary, was Re: Every time ..., was Re: General formula

[David Chia <rsedc@xxxxxxxxxxxxxxxxxxxxxxx>]

> Unless A and B reveal their structure to us, so that we may identify
> and eliminate duplicated components, we cannot apply such a formula.
> And if the duplicated components [ai,bi] for some of the i, all exhibit
> different expected lifetimes, there is no opportunity for the "grouping"
> you mentioned, which says nothing more than SUM[C] to n terms = n*C for
> a constant C.  They will be "groups of 1", and we are reduced to the
> equation I gave:
> 
>   1/TC  =  1/TA + 1/TB - ( SUM[1/TCi] for ci in INTERSECT(A,B) )
> 
> The problem is in identifying this intersection, and not simply treating
> attributes A and B as atoms.

In practice the relationship between A and B might not be simple nor
deterministic. The first order life equation for totally independent
attributes looks simple and straight forward. However, if there are
observable correlations among the attributes, second (or higher) order
life equations might be required and there is also the question of
which type of higher order life equation to be used. Furthermore, since
the relationship (if any) between the attributes most probably will not
be known deterministic, the coeffs of the first order term in the
higher order life equations most probably will not be ones.

If historical data (from the environment and in the context where the
cert is to be used) is available and the distributions of the attribute
lifetimes are roughly statistically normal, the expected cert life
equation might be able to be determined through stepwise regression
which will only include significant first or higher order terms.
Example of a simple second order life equation

   1/T = SUM a(i)/T(i) + SUM SUM b(i,j)/T(i)/T(j)
          i               i   j

where a(i) and b(i,j) are coeffs determined from the statistical regression.
b(i,j) is a measure of the 'intersection' you mentioned.

If historical data is not available, life equation determined from other
'similar' situation could be used for rough approximation but due care
should be taken for possible errors.


David Chia