"In theory there is no
difference between Theory and Practice.
In practice, there is."
How FORM/SORM is Supposed to Work
part
1 of 3
FORM
(First Order Reliability Method) has been used extensively by engineers for nearly two
decades. What has been
studiously ignored, however, is how well FORM/SORM assumptions hold up in
describing real data.
Here's the plan:

First we'll
review the FORM/SORM algorithm.

Next we'll use
the 68 HillberryVirkler fatigue crack growth specimens to see how well
FORM/SORM performs with real data.

Finally, we'll
hope that all this will have caused the scales to fall from the eyes of
at least a small fraction of TrueBelievers.
What is
FORM/SORM?
The idea is
based on the joint probability
density of all the factors influencing failure or nonfailure, including
factors controlling demand and those effecting capacity.
This ndimensional probability space is partitioned by some function,
called the gfunction
into safe and nonsafe regions. The probability of an ntupple
of factors being in the nonsafe region is the probability of failure.
This is a sound premise. The trouble arises in the dubious assumptions taken to
implement it.
In practice it is
not difficult to estimate the probability of failure of a given ndimensional
array of failurecausing factors, given their joint probability density.
However, the inverse problem  determining what collection of factors will
result in failure  is exceedingly difficult, often requiring an inversion
(solution) of the gfunction.
Now, if this ndimensional
problem could somehow be transformed into a single dimension, and made
Normal as well, then it would be easy to accomplish the inverse problem.
To do this FORM/SORM aficionados prescribe these steps:

"Transform" the real joint probability
density into an "equivalent"
multivariate normal density^{(1)},

Plot the
joint probability density of demand and capacity which is now multivariate
normal with zero means and identity covariance matrix,

Partition this probability
space into "safe" and "unsafe" regions with some
suitable
gfunction.
Since a
gfunction is often defined in terms of
probability of failure, this results in either circular reasoning or an
iterative solution.

The point on the
gfunction
closest to the origin is called the "Most Probable Point."^{(2)}

b
is defined as distance in standard deviation units from the center of the joint density to the "Most Probable
Point."

The
"transformed"
multivariate normal density is bisected by that line conveniently
producing a univariate normal density which can then be used to assign
probabilities.^{(3)}
Of course in all but
the very simplest situations there are many factors that influence both
demand and capacity so the resulting probability space is ndimensional.
Still it is standard practice to illustrate the idea with only a
onedimensional demand and a onedimensional capacity so that the concept
can be plotted in a n=2 dimensional plot, like Figure 1 (next
page).
___________________
Notes:

This is the mathematical equivalent of "Belling the Cat." While it is
sometimes possible to affect such transformations, in general they are
aren't feasible because they obscure interrelationships. In
practice any interrelationships are simply assumed to be zero,
but sometimes given lipservice in terms of their covariance.
Covariance is the simplest form for relating the variabilities among
parameters. Nonetheless, practitioners continue to insist
that they have accomplished this "transformation," using the onetoone
mapping of any continuous univariate
cdf to the unit square on (0,1) and then to the univariate normal density.

Even if there
were a "Most Probable Point" that doesn't mean that failure would
more probably occur there (unless the associated probability is greater than
50%) because the combined probabilities of the lesser probability
outcomes would make it more probable that failure would occur from one
of them. Consider this example: You purchase $10,000 worth of Lottery tickets, making you the
"Most Probable Winner" because you have far more opportunities than any other
individual.
But after the drawing you find to your chagrin that you did not win. Why?
Because the probability of anyone other than you is far greater than your probability,
even though yours was the largest individual winning probability. Focusing on the
"Most Probable Point" obscures the real issue of design safety. 
Each of these steps seems logical, yet the result
doesn't square with reality. For another example of "logical"
steps leading to an erroneous conclusion click
here.
___________________
