Very Short Introduction Plus
an Example of Weibull Engineering (WE) Basics
By Wes Fulton, CEO, Fulton
Findings (TM)
Copyright 2010-Present,
Fulton Findings. All Rights Reserved.
(originally
written 24 JUL 2010, last edited 1 OCT 2021)
The simple concepts in statistics can appear complicated to beginners
because many works on the subject use long and strange words. This brief
introduction uses shorter and more familiar words. All you need is a healthy
curiosity about the way things work.
. . . The pictures in this
overview represent some of the many uses for Weibull Engineering (WE) . . .
Bearings . . . Aeronautics .
. . Physics . . . Automotive . . . Dentistry . . . Welding . . . Gearing
Perhaps the only object
without variability is a good digital copy of a digital original. Practically
every other product and service has variability. For example, although very
similar . . . bearings with the same part number and manufactured one after the
other are not going to perform exactly the same. They will have differences in
how long they operate successfully. A tiny amount of variability in the type of
usage can also make a big difference in operating life capability. Along with
lifetime variability, there are other areas where variability effects are
important such as money markets, quality satisfaction levels, disease cure
rates, satellite reliability, maintenance scheduling, warranty analysis, safety
devices, and so on with an almost unending list of additional areas. The good
news is that variability can be modeled. Understanding variability and making
decisions about variability are straightforward with a proper variability
model.
Variability
models are called DISTRIBUTIONS.
From the correct distribution you can estimate the expected probability of
getting a particular result in test or in customer usage. Picking the
appropriate model for measurement variability is the entire focus of
statistics. In the following, you will notice that a distribution can be
presented either as a probability density function (PDF) or a cumulative
distribution function (CDF). Those acronyms stand for two different ways to
describe the same model . . . but more about that later. There is no math in
this introduction, but you can see the math if you want by looking at the
references listed at the end.
Exploration . . . Architecture
. . . Power Generation and Power Transmission . . . Military . . .
Communications
Back
around 1920, E.J. Gumbel began to
investigate in detail six different EXTREME
VALUE distributions for modeling the occurrence of rare events like
flooding and wind gusts and power surges. One of these six possible
extreme-value distributions is now called the Weibull distribution. It is one
of the most widely-used solutions for modeling how things vary, especially for
lifetime data (age-to-failure data) and reliability. The name Weibull is
usually pronounced in English-speaking countries as WAEE-BULL, but the name is no doubt pronounced differently in
Sweden where Waloddi Weibull was born. The subject
technique was promoted at first by the technically-gifted Waloddi
Weibull. He started investigating variability models (as well as many other
things) around 1930 eventually writing over 60 papers plus a book titled Fatigue Testing and Analysis of Results.
Weibull`s book was published in 1961 by Pergamon Press [1].
Weibull
distribution methods have been frequently updated with an explosion of use
since around 1950 and with new applications being added almost continually.
Now, there are easy-to-use computer programs available for Weibull modeling
along with Weibull classes to teach application. A main reference for this is
the handbook by Dr. Bob Abernethy
[2]. NOTE: He was `Dr. Bob` before there
was `Dr. Who`, `Dr. Dre`, `Dr. Phil`, `Dr. Oz`, etc.
Dr.
Bob`s book was the first book written specifically for Weibull Engineering . .
. or `WE` . . . and the book is now
titled The New Weibull Handbook(C).
The first version of it was published in the early 1980`s. It has since been
repeatedly updated by Dr. Bob adding the latest methods and describing the
latest software solutions to make it the de-facto world standard. Dr. Bob,
whose doctorate is in statistics, also invented
the engines for the SR-71 Blackbird spy plane! That plane was the eye in the sky for the United States
during the post-WWII `Cold War` (from about 1947 to around 1989) between the
Eastern Bloc countries and the Western Bloc countries. As of this writing, many
decades later and after having been retired, the SR-71 still holds the record
as the fastest self-powered manned aircraft with a top speed of 2,269 miles per
hour or very near Mach 3. The X-15
and X-43A and Gemini and Apollo
capsules and Space Shuttle are
technically faster, but those are either not 100% self-powered or not manned.
Compare the look of the venerable SR-71 to the futuristic silver-skinned spacecraft
in the much later movie Star Wars
Episode 1 (released 1999) . . . see the resemblance? Even science fiction
loves the all-too-real SR-71 design. The technical expertise of Dr. Bob plus his statistics expertise plus his simple writing style come together
to provide good reading, explaining things clearly for practical solutions to
real issues. So his book, Reference #2 below, is especially recommended for
further reading.
Compared
to other commonly used models, the Weibull distribution has a double advantage.
One, it is simpler, . . . and two, it is more versatile. It can exactly
duplicate distributions like exponential and Rayleigh, and by embracing
additional distributions like normal, lognormal, and Type I extreme-value (also
called Gumbel after E. J. Gumbel) we get into WE. It has a wider scope than
just the analysis of fatigue testing results. WE also includes
root-cause detection, event forecasting, spare parts projection, test planning,
optimum-replacement for lowest cost, accelerated testing, design comparison,
process reliability, manufacturing control, and cost control, as well as
others. It currently enjoys wide popularity with many people in the fields of
design, development, finance, fabrication, maintenance, operations, quality,
reliability, safety, and testing.
Locomotives . . .
Electronics . . . Transmissions . . . Machining . . . Food . . . Engines . . .
Construction
The Normal Distribution: Before some details of the
Weibull distribution get presented, let`s look at the well-worn normal distribution. The normal
distribution is the king of distributions, modeling many things well . . . but
not everything. Waloddi Weibull had an early paper on
the Weibull distribution rejected
because it was not the normal distribution! The normal distribution probability
density function (PDF) has only one basic shape (Figure 1 below) which may be
either wider-and-shorter or thinner-and-taller, but it always takes a bell-like
shape. It is sometimes called the bell
curve, and sometimes called the Gaussian
distribution in honor of Johann Carl
Friedrich Gauss. The normal distribution applies when modeling variability
in such cases as plus/minus error measurements, student test scores,
performance variability, X-bar quality control charts, and miss-distance for a
machining operation. There is also something called the central limit theorem which explains why the normal
distribution fits well when many additive effects are mixed in the data.
The
normal distribution model has 2 parameters, meaning that it only requires two
numbers for any application. Only the mean
value (referred to by the Greek letter Mu,
pronounced like mew) and the standard deviation value (Greek letter Sigma) are needed to completely
describe a normal distribution. The mean is a central tendency value, representing an expected middle value. Half
of the expected values from this distribution are below the mean and half are
above. For any symmetrical distribution like normal, the median value (50%) and
the mode value (highest PDF point) are the same as the mean value. That is not
necessarily true for nonsymmetrical models. The normal distribution standard
deviation is a measure of the amount of variability. Higher standard deviation
indicates higher variability being measured and also higher variability
expected in the future. However, product life and reliability and many
real-life measurements need something different.
Life-data
measurements exhibit variability not closely symmetrical around a central
value. Symmetry around a central value is required
for using the normal distribution effectively, and if used incorrectly for
reliability purposes the normal distribution can produce negative lifetime
estimates. So, the normal distribution is not generally the right choice for
modeling lifetime data or age-to-failure measurements for reliability purposes
or for that matter not the right choice either for most manufacturing
measurements like thickness and surface hardness.
The 2-parameter Weibull: The Weibull distribution
works well in modeling lifetime data and most manufacturing data. The Weibull
probability density function (PDF) can take many shapes (Figure 2 above) and
can fit to non-symmetrical measurements. Also, the simple 1-parameter and
2-parameter versions of the Weibull distribution will not produce a negative
value. That is a nice feature for life data analysis and for many actual
measurements that cannot be negative. The 2-parameter standard version of
Weibull is even simpler than the 2-parameter normal. The math is straightforward for the cumulative Weibull, but the
cumulative normal requires higher-math integral approximations . . . UGH!
The
standard Weibull is similar in complexity to the normal as it also has only two
parameters, characteristic value
(referred to by the Greek letter Eta
pronounced like ey-tah)
and slope value (Greek letter Beta pronounced like bey-tah). Shape parameter is another name for
Weibull slope, since the Weibull PDF shape changes with different Beta values.
The Weibull slope value moves in opposite direction to variability, such that
higher Beta indicates lower variability (desirable for higher quality). Eta is
the Weibull version of a central-tendency value. It is approximately
near the expected measurement. For some reason, the NIST explanation of
Weibull, and the Wikipedia explanation, and the referenced international
standard are all as of this writing out-of-step with each other when it comes
to Weibull naming convention. Other references may use different parameter
names than used here, however Eta
and Beta are used for the Weibull
2-parameter distribution in Dr. Abernethy`s handbook [2] and in the international
standard, IEC 61649, Edition 2, Weibull
Analysis [3].
NOTE:
Equations are omitted here for readability, but they are readily available in
the recommended references below and in many other references. Table 1 below
summarizes some of the reasons the Weibull distribution is gaining in usage.
TABLE 1: Distribution
Comparison Green Background =
Better
Distribution Model |
Normal |
Weibull (Standard) |
Quantity of Parameters
(Complexity . . . Lower is Better) |
2 |
2 |
Applications |
VERY MANY |
VERY MANY |
Good for Zero and Negative
as well as Positive Data (e.g. Residuals, Miss-Distance, etc.) |
YES |
NO |
Good for Data Symmetrical
Around a Central Tendency Value |
YES |
NO |
Good for Non-Symmetrical
Data |
NO . . . Models Well Only Perfectly Symmetrical Measurements |
YES |
Good for Positive-Only
Data Found with Most Manufacturing and Operational Measurements (e.g. Life
Data, Cycle Count, Wall Thickness, Case Depth, Performance, Thrust, Altitude,
Speed, Braking, Torque, Energy, Pulse Rate, Blood Pressure, etc.) |
NO . . . Always Predicts Some Probability of Negative Results |
YES |
Can Identify Type of
Failure Mechanism When Used for Reliability Analysis |
NO |
YES . . . Weibull Slope (Beta)
Can Often Identify Whether Measurements Indicate Infant Mortality or Wear Out |
1-Parameter Version
Available |
NO* |
YES |
3-Parameter Version
Available |
NO** |
YES |
Easy Monte Carlo
Simulation |
NO . . . Requires Many Lines of Code Executed for Each Repeated
Sample Value Generation |
YES . . . Requires Only 1 Line
of Code Executed for Each Repeated Sample Value Generation |
Size Factor Scalability* |
NO |
YES |
Probability Distribution
Function (PDF) |
Only Bell-Shaped |
Infinite Number of Unique Shapes
(Changes to Fit Data) |
Cumulative Distribution
Function (CDF) |
Requires Complex Integral Approximation with Numerical Methods |
Short Simple Equation (No
Approximation Required) |
User-Friendly Advanced
Mixture Solution Available |
NO |
YES |
*
Technically there can be a 1-Parameter Normal distribution with known Sigma, but it is hardly if ever
used due to limited scalability. Wallodi Weibull
realized that the normal distribution did not calculate correctly for the
situation where loads were distributed into different-sized parts. His rationale
for using a different distribution (later named for him) was that he needed
something that made sense for changing sizes, something that was scalable like
the Weibull distribution.
**
A 3-Parameter Normal distribution including an extra time-shift (t0) parameter
is not useful.
Figures
1 and 2 above represent probability density function (PDF) shapes. All PDF`s
(Weibull or normal or whatever) are identical in one respect, i.e. the area
underneath any PDF curve is exactly one (1) or 100%. So the PDF represents 100%
of where to expect a similar measurement. The amount of area under the PDF
curve, to the left of any point along the PDF plot horizontal measurement
scale, is the cumulative distribution function (CDF) form. The CDF is simply
another way to express the same model. The CDF representation is more useful
for answering questions about variability such as . . . How long can a product go with only a specific proportion of failures
expected? A Weibull CDF plot is displayed in Figure 3 below for the worked-out
example of WE analysis at the end of this introduction.
Weibull
often fits to the data better than the normal for some specific applications
due to Weibull`s multi-shape capability. The data itself selects the most
appropriate Weibull shape for best solution. Weibull also often fits better and
works better as a model for small samples. A small sample is taken here to be
twenty (20) or less measured occurrence values per Reference #2 below.
The 1-parameter Weibull: With very small sample
sizes the Weibull 1-parameter model, also called Weibayes, is the most accurate solution provided there is
sufficient and appropriate historical data to help. It works even down to zero
occurrences (a VERY small sample indeed!) as long as there are a few non-occurrences
(successes for reliability). With a good estimate of the Weibull slope (Beta)
already provided from prior experience, the Weibayes solution requires only
finding the Weibull characteristic value (Eta). This often produces a simpler
and more accurate model.
The 3-parameter Weibull: With a larger sample size,
more than 20 occurrences, the more complex 3-parameter version of Weibull
becomes very useful for modeling variability. The additional Weibull
distribution third parameter (t0 . . . pronounced like tee-zeeroh) represents a time shift for
occurrence age variability. The 3-parameter Weibull works well in cases where
either there is a delay in the onset of the occurrence mechanism (a failure
free period), or the aging process starts before the item officially begins
operation (prior deterioration).
Medicine . . . Chemistry . .
. Finance . . . Weather . . . anything with variability . . . and that would be
more than 99.9999% of everything!
Other Considerations: Knowing the root cause of an
occurrence mechanism provides tremendous help in determining corrective action.
Sometimes the Weibull solution can suggest the type of root cause given data is
generated only from a single root cause. With only one root cause being
analyzed the Weibull slope is usually above 1.0 (wear out) or below 1.0 (infant
mortality). Mixing different root causes together can complicate the
analysis even though there are reasonable solutions available for that (as long
as there are only two or three mechanisms and there is a larger quantity of
data). Carl Tarum wrote the first
easily-accessible software for advanced mixture analysis like this. Mixing many
different root cause mechanisms together in the same data set often produces a
Weibull solution with Beta slope-value near 1.0 (simplistic assumption of
constant occurrence rate, no matter what age) and with a reasonable goodness of
fit to the data. However, with many mechanisms mixed together there is
additional randomization and loss of resolution. This missing information could
otherwise be used to suggest appropriate corrective action. A major recommendation
is to focus on one root cause at a time if possible.
Once
the basic Weibull model is determined, it can provide the foundation for
forecasting like Abernethy Risk. An
additional piece of information needed to forecast is the expected usage rate such as the amount of aging
experienced per item each month. For example, distance travelled (miles or
kilometers) per vehicle per month would be the applicable usage rate for an
automotive application. Flight hours per engine per month might be applicable
for aircraft as well as patients per hospital room per month for hospitals.
Forecasting may be one of the most useful products of WE. Such calculated
expectations provide the basis for spare parts requirements and warranty
programs.
WE is
not limited to the analysis of life data. It is in demand for evaluating
process reliability, instrument calibration intervals, economic variability,
and quality control. Paul Barringer
pioneered the use of WE specifically for process reliability and instrumentation
calibration. These methods are extremely popular in the chemical, petro-chemical, and pharma industries. Dennis Keisic was influential in expanding the use of quality
methods beyond the normal distribution to include Weibull and lognormal. Weibull
or lognormal should be used for quality control monitoring instead of the
classically applied normal distribution where that different model is more
appropriate. Examples include `Six
Sigma` type quality control efforts in plating thickness variability,
rotating shaft wobble, progressive deterioration, and contamination level
monitoring. These last few applications are not modeled well with the normal
distribution as the measurements cannot be negative (normal will always predict
some probability of negative results). The standard Weibull and the lognormal
distributions are usually more accurate for these manufacturing applications,
since they only predict positive measurement values.
WORKED-OUT
EXAMPLE
The Issue - Your organization uses
hundreds of batteries all of a similar design. These batteries go into a data
recording module that cannot be monitored externally with sensors, as the
module is buried within a small medical device implanted into cancer patients.
The batteries are seldom required and only operate a few seconds at a time, but
the life requirement is in hours to minimize chance of failure. When a battery
dies, there is loss of data which requires an excessive cost in re-testing and
re-analysis to reproduce the desired data. This costly loss of data is
happening too often and your management does not like it. Testing on a new
sample of seven batteries provides the following operating hours from new to
failed state:
130 hours of battery life
before failure (Battery #1)
165 hours of battery life
before failure (Battery #2)
234 hours of battery life
before failure (Battery #3)
252 hours of battery life
before failure (Battery #4)
253 hours of battery life
before failure (Battery #5)
295 hours of battery life
before failure (Battery #6)
389 hours of battery life
before failure (Battery #7)
The Goal - Your management wants you
to find out how long any particular battery of the same design should be used
before replacement if the chance of failure is limited to only 2 percent (%).
Plus, you want to be able to defend your decision to management by being
conservative in your estimate of life capability.
The Analysis - You plot the battery test
data on Weibull CDF scaling. First you sort the data by their values (lowest to
highest). You then place the data points on the Weibull plot horizontally by
increasing data value, and vertically by increasing failure probability
estimated with order statistics. Then you fit a straight line (on this scaling)
to the data. NOTE: Good Weibull software
will do all of this for you, you just enter the data. The resulting
straight line from lower left to upper right is a standard Weibull solution
using graphical `rank regression` (rr). Another way to
get a solution repeatedly searches for highest data probability, giving a
`maximum likelihood estimate` (mle). The resulting
plot models the variability of battery life capability. The Weibull CDF plot of
the battery test results (Figure 3 below) shows graphical p-value estimate (pve%) of 79.51 for goodness-of-fit. A pve% value can go
anywhere from 0 to 100, and a value of 50 is nominal for data sampled from the
same distribution used for the plot. A pve% of 10 or higher is usually
acceptable. The pve% of 79.51 here is well above average (a good fit!). The
plot also indicates type of data (o/s) with 7 occurrences
and 0 suspensions. The fit line has Weibull slope Beta of 3.028
(above one). Any Weibull slope above one indicates wear out as the type of occurrence mechanism. Wear out means that
older items fail at a relatively faster rate than newer items. The wear out
indication for the batteries allows for a useful planned replacement interval.
A nominal life capability estimate of 75 hours comes by reading the horizontal
value of the fit line where it crosses 2% on the vertical scale (2% failure
probability).
This
nominal result is not a conservative estimate. A lower estimate line, called a confidence line, was added to the plot
for that. The curved confidence line on the plot is a 90% lower estimate of
life capability. There are several methods for estimating confidence bounds
like this. The code on the plot, W/rr/c%=pv-90, means that the solution for the straight
line is Weibull using the rank regression method with
lower(-) 90% confidence estimated by Pivotal (pv)
Monte Carlo. When read from the lower confidence line, the conservative
estimate of life capability goes down to 30 hours (horizontal scale). Note that
2% failure occurrence probability equates to 98% reliability. With additional
similar battery test data, the lower confidence line may get closer to the
nominal line. This might give an even higher estimate of life capability for
the same 98% reliability.
Adding
one cost factor for planned replacement (usually at lower cost) and a second
cost factor for emergency replacement due to failure (usually at higher cost)
would allow a more detailed optimum
replacement study to achieve minimum operational cost. It is possible here
because the failure mechanism appears to be wear out. Such a cost study is
mostly used for non-safety-related items.
The Result - You make a recommendation
for a pre-emptive planned replacement of each battery after 30 hours of
operation based upon your Weibull analysis. Later you have more testing
accomplished, and these results (consistent with before) reduce the uncertainty
by adding the latest data to original data thus bringing the conservative lower
bound higher and closer to nominal. That initial conservative estimate of 30
hours life is eventually raised to a very comfortable higher operational value.
There are very few problems with this battery after your corrective action is
implemented. Management likes you. You are promoted, you are happier, and the
Weibull estimate for your own life expectancy increases.
CONCLUSIONS
More
probability emphasis is coming to such fields as aerospace, energy production,
food production, financial markets, medicine, military, oil refining, physics,
public safety, and transportation. WE and similar probability-centered
analysis, like Reliability Centered Maintenance (RCM), lead the way in providing
useful answers to difficult questions. For more information visit http://www.WeibullNews.com on the web,
and view the references at the end.
If
you actually read through all of this and it clicked with you, you might be
smarter than you think. Quantum physics (not an easy topic) is practically 100%
based on probability stuff somewhat similar to this. Rocket scientists and
brain surgeons of the world . . . eat your heart out!
REFERENCES
1.
Weibull, Waloddi, Fatigue Testing and Analysis of Results, Pergamon Press, 1961 (the
only book by Weibull)
2.
Abernethy, Robert B. (Dr. Bob), The
New Weibull Handbook(c), self-published (first complete self-study reference
for Weibull Engineering) . . . here is the link for it on AMAZON:
https://www.amazon.com/dp/0965306232/
3.
IEC 61649, Edition 2, Weibull
Analysis (the official international standard) . .
. here is a link for it: https://webstore.iec.ch/publication/5698
4.
Select
Here for
the SAE Introduction to Weibull Solution
Methods
ABOUT
THE AUTHOR
Wes Fulton wrote the first widely-used
software for WE. He presents Dr. Bob`s Weibull Workshop for organizations and
companies and sometimes even for much-maligned governments around the world
including their military. He has two engineering degrees and a patent to his
name. He writes books for adults and for children.