"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:

1. First we'll review the FORM/SORM algorithm.

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

3. Finally, we'll hope that all this will have caused the scales to fall from the eyes of at least a small fraction of True-Believers.

What is FORM/SORM?

The idea is based on the joint probability density of all the factors influencing failure or non-failure, including factors controlling demand and those effecting capacity.  This n-dimensional probability space is partitioned by some function, called the g-function into safe and non-safe regions.  The probability of an n-tupple of factors being in the non-safe 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 n-dimensional array of failure-causing 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 g-function.

Now, if this n-dimensional 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:

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

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

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

4. The point on the g-function closest to the origin is called the "Most Probable Point."(2)

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

6. 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 n-dimensional.  Still it is standard practice to illustrate the idea with only a one-dimensional demand and a one-dimensional capacity so that the concept can be plotted in a n=2 dimensional plot, like Figure 1 (next page).

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1. 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 lip-service 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 one-to-one mapping of any continuous univariate cdf to the unit square on (0,1) and then to the univariate normal density.
2. 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.

3. 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.

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