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Statistical Engineering

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POD "floor" and POD "ceiling"
Some data do not support a POD curve that goes to zero on the left, or to one on the right.
2 + 2 = 5
Just because you can make a statistical statement doesn't make it true, no matter how much you wish that it were.
Parameter Estimates are NOT parameter values.
There is a profound difference between the mathematical behavior of a function whose parameter values are defined (e.g. the FORM/SORM paradigm) and the same function whose parameter values must be estimated from data.
How ?vs a POD Models Work
POD (Probability of Detection) is the probability that a signal, (? "ahat") will be larger than the decision threshold.
How hit/miss POD Models Work
POD (Probability of Detection) as a function of size is less straightforward for binary (yes/no) data when compared with data having a continuous response (?.
How WELL do POD Models Work?
In reality we only get to see ONE collection of data, and from that must estimate the most likely model for the unseen and unknown and unknowable "truth."
How the LogLikelihood Ratio Criterion Works
Constructing Confidence Bounds on Probability of Detection Curves based on how likely some alternatives to the maximum likelihood would be.
POD Short Course/Workshop
This two-day short course is based on the new (2007) MIL-HDBK-1823 and uses the mh1823 POD software.   The course provides the latest methods for measuring your NDE system's effectiveness and the workshop will use these state-of-the-art techniques to analyze your enterprise data.
MIL-HDBK-1823A, "Nondestructive Evaluation System Reliability Assessment"
2009 release of 2007 Update ? describes procedures for acquiring NDE data and statistical methods for analyzing it to produce POD(a) curves, 95% confidence bounds, noise analysis, and noise vs detection trade-off curves, and includes worked-out examples using real Hit/Miss and ?data.
Themes ...
... I'm not a philosopher - but I, like you, do occasionally ruminate on the human condition.
Hubris
"I don't need to understand your problem to solve it."
The Great Misunderstanding
Both statisticians and engineers recognize the mathematical competence of the other, and this is the cause of The Great Misunderstanding.
Quantitative Nondestructive Evaluation
It isn't the smallest crack you can find that's important ? it's the largest one you can miss.
Round Robin Testing ....
... testing can sometimes make you see something that isn't there.
False Positives and the ROC Curve ...
The relationship between POD and False Positives depends on more than the inspection itself.  It also depends on the frequency of defectives in the population being inspected.
Will ...
If you think that you dare not, you don't ...
Probability and Statistics ...
... are not one and the same. The differences are not nuanced. They are Apples and Oranges.
Reading List
I am often asked to recommend a "good statistics text."  Here are a few that I refer to often.
Two Secrets of Success
 
Monte Carlo Oversights
Most Engineering Monte Carlo simulations ignore the distinction between parameter values, and estimates of parameter values, resulting in a gross underestimation of the probability of "low-probability" events.
Repeated Inspections
Repeated inspections do not improve Probability of Detection (POD).
Central Limit Theorem Fine-Print
Readers have requested further explanation of when the CLT does not apply.
Pseudo-Proof that 2 equals 1
Seemingly logical steps can lead to a silly conclusion.  Unfortunately, not all silliness is as self-evident as this example.
The "Most Probable Point" is a fiction
First Order, and Second Order Reliability Methods (FORM/SORM) are based on a demonstrably false premise of a "Most Probable Point."
SEbullet_2.gif (863 bytes) Contrasting the Statistical with the Mathematical Properties of NESSUS/FORM
Engineers see reliability as an optimization problem on a known response surface. Statisticians view it differently.
SEbullet_2.gif (863 bytes) "Choosing" the Right Distribution
There is considerable folklore about choosing statistical distributions, as you might select the appropriate club from your golf bag.
SEbullet_2.gif (863 bytes) Frequentists and Bayesians
There is a continuing debate among statisticians over the proper definition of probability.
SEbullet_2.gif (863 bytes) "Probabilistics"
There is more to Monte Carlo simulation than replacing constants with probability densities.
SEbullet_2.gif (863 bytes) Bivariate Normal
Here is a simple algorithm for sampling from a bivariate normal distribution.
SEbullet_2.gif (863 bytes) Did you know ... ?
SEbullet_2.gif (863 bytes) Goodness-of-Fit
Goodness-of-Fit tests, like Anderson-Darling, tell you when you don't have a normal distribution.
SEbullet_2.gif (863 bytes) R-squared ...
... is an often misused goodness-of-fit metric, where bigger isn't always better.
SEbullet_2.gif (863 bytes) Other Measures
R-squared isn't the only way to judge how well the model works.
SEbullet_2.gif (863 bytes) Chronology of Crack Initiation
Tongue-in-cheek view contains insights.
SEbullet_2.gif (863 bytes) Curse of Dimensionality
Direct-sampling Monte Carlo requires the number of samples per variable to increase exponentially with the number of variables to maintain a given level of accuracy.
SEbullet_2.gif (863 bytes) Convergence in Distribution
We engineers are familiar with convergence to a point, but what of convergence to a distribution?
SEbullet_2.gif (863 bytes) Extreme Value Distributions
The largest, or smallest, observation in a sample has one of three possible distributions.  This is another example of "convergence in distribution."
SEbullet_2.gif (863 bytes) Joint, Marginal, and Conditional Probability
We engineers often play fast and loose with joint, marginal, and conditional probabilities - to our detriment.
SEbullet_2.gif (863 bytes) Correlation:
It's a lot more - and less - than you may think
SEbullet_2.gif (863 bytes) Outliers ...
Often infuriating, these can be very informative too.
SEbullet_2.gif (863 bytes) Wrong Grid?
Choosing the wrong grid can undermine your analysis, mislead your audience, and make you look foolish.
SEbullet_2.gif (863 bytes) Bayesian Thinking
... including an example from NDE
SEbullet_2.gif (863 bytes) Random Fatigue Limit on a P/C
Pascual and Meeker's RFL solves an old problem: how to have a runout model go through (rather than under) all the runout data.
SEbullet_2.gif (863 bytes) Free Translations
Not too Statistical, but still Fun!   Check it out!
SEbullet_2.gif (863 bytes) IntraOcular Trauma Test
Sometimes the best Goodness-of-Fit test is the easiest.
SEbullet_2.gif (863 bytes) Central Limit Theorem
Why is the Average of nearly anything always Normal ?
SEbullet_2.gif (863 bytes) Hiking the Grand Canyon, rim-to-rim!
Words and pictures are insufficient.
SEbullet_2.gif (863 bytes) Bayesian Updating
We use Bayesian Statistics every day without knowing it.
SEbullet_2.gif (863 bytes) Sums of Random Variables
Sometimes you need to know the distribution of some combination of things.  Here's an example.
SEbullet_2.gif (863 bytes) Distributional Inter-Relationships
There are myriad probability distributions.  But did you know that most are related to one another, and ultimately related to the Normal?

 

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Copyright ? 1998-2008 Charles Annis, P.E.
Last modified: June 08, 2014