Very Short Introduction Plus an Example of Weibull Engineering (WE) Basics
By Wes Fulton, CEO, Fulton Findings LLC
Copyright 2010-Present, Fulton Findings. All Rights Reserved.
(originally written on 24 JUL 2010, last edited on 2 MAR 2017)
The simple concepts in statistics can appear complicated to beginners
because many books 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 are going to 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 is straightforward with a proper variability model. Such 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 APPLIED 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 distribution . . . 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 in about 1920, six different EXTREME VALUE distributions were investigated in detail by E.J. Gumbel 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 Wallodi Weibull was born. The subject technique was promoted in the beginning 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. The 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 computer programs available for Weibull modeling along with Weibull classes to teach application. A main reference book for this is the handbook by Dr. Bob Abernethy [2]. This was the first book written specifically for Weibull Engineering . . . or WE . . . this handbook is now titled The New Weibull Handbook(C). The first version of the handbook was published in the 1982-83 time frame. It has since been repeatedly updated by Dr. Abernethy adding the latest methods and software solutions to make it the de-facto world standard. Dr. Abernethy, whose doctorate is in statistics, also invented the engines for the SR-71 Blackbird spy plane. That plane was the eye in the sky during the cold war between the U.S.A. and Russia. As of this writing, many decades later and after having been retired, that plane still holds the record as the fastest self-powered manned aircraft with a top speed of 2,269 miles per hour or about Mach 2.95. The X-15 and the X-43A and the Gemini and the Apollo and the Space Shuttle are technically faster, but those are not 100% self-powered. Compare the look of the old SR-71 to the silver-skinned spacecraft in the much later movie Star Wars Episode 1 . . . see the resemblance? Even science fiction loves the design of the real SR-71. 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 interval, accelerated testing analysis, design comparison, process reliability, manufacturing control, cost management, 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 . . . Transmission . . . Machining . . . Food . . . Engines . . . Construction
The Normal Distribution: Before some details of the Weibull distribution get presented, the well-worn normal distribution should be briefly described. 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 on 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 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 different additive effects are mixed in the data.
The normal distribution model has 2 parameters, meaning that it only requires two numbers for its description. To completely describe the normal distribution, only the mean value (referred to by the Greek letter Mu, pronounced like mew) and the standard deviation value (Greek letter Sigma) are required. 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 the mean. For any symmetrical PDF curve like the normal, the median value (50%) and the mode value (highest) are the same as the mean value. That is not necessarily true for any nonsymmetrical model. The 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 usually 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 the normal distribution can produce negative estimates for life. So the normal distribution is not generally the right choice for modeling lifetime data or age-to-failure measurements for reliability purposes.
The 2-parameter Weibull: The Weibull distribution works well in modeling lifetime data. The Weibull probability density function (PDF) can take many shapes (Figure 2 below) and can fit to non-symmetrical data. Also, the simple 1-parameter and 2-parameter versions of the Weibull distribution will not produce a negative value for life. That is a nice feature. 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 has characteristic value (referred to by the Greek letter Eta pronounced like ey-tah) and slope value (Greek letter Beta pronounced like bey-tah) for its two parameters. Shape parameter is another name for Weibull slope, since the Weibull PDF shape changes with different Beta values. The Weibull slope value moves in the opposite direction from the amount of variability, such that higher Beta indicates lower variability being measured and also lower variability expected in the future. Eta is the Weibull version of a central-tendency value, and 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 as of this writing all out-of-step with each other when it comes to Weibull parameter 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 handbook by Dr. Abernethy and also in the international standard, IEC 61649, Edition 2, Weibull Analysis [3]. Equations are omitted here for readability, however they are readily available in the recommended references below and many other references. Table 1 below summarizes some of the reasons that the Weibull distribution is gaining in usage.
TABLE 1:
Distribution Model |
Normal |
Weibull (Standard) |
Number of Parameters (Complexity) |
2 |
2 |
Applications |
THOUSANDS |
HUNDREDS |
Good for Zero and Negative as well as Positive Data (e.g. Residuals, Miss-Distance, etc.) |
YES |
NO |
Good for Positive-Only Data (e.g. Life Data, Wall Thickness, Case Depth, etc.) |
NO |
YES |
Good for Non-Symmetrical Data |
NO |
YES |
Can Identify Type of Failure Mechanism When Used for Reliability Analysis |
NO |
YES |
1-Parameter Version Available |
NO* |
YES |
3-Parameter Version Available |
NO** |
YES |
Easy Monte Carlo Simulation |
NO |
YES |
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 |
Relatively Short and Simple Equation (No
Approximation Needed) |
* Technically there can be a 1-Parameter Normal distribution with known Sigma, but with its limited scalability it is hardly if ever used. 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.
The figures above represent probability density function (PDF) shapes. Every PDF, 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 generally more useful when 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.
The Weibull fit to the data can be better than the normal distribution for some specific applications simply due to its multi-shape capability. The data itself selects the most appropriate Weibull shape for best solution. Weibull 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 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).
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). 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 can produce a simpler and more accurate model.
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 that the data is generated 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). Mixing many different root cause mechanisms together in the same data set often produces a Weibull solution with Beta slope near 1.0 and with a reasonable goodness of fit to the data. However, with many mechanisms mixed together there is additional randomization and loss of resolution needed to suggest appropriate corrective action. A major recommendation is to try and 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 doctor per month for hospitals. Forecasting may be one of the most useful aspects 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. Specifically, Weibull should be used for quality control monitoring instead of the classic normal distribution where Weibull is more appropriate. One additional appropriate area of application for WE is with regard to six sigma type quality control efforts like controlling 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 (but normal predicts some probability of negative results). The standard Weibull and the lognormal distributions are usually more accurate for these applications, since they only predict positive 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 amount of money 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 failure data with regard to operating hours since new:
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 data value, and vertically by failure probability estimated per order statistics. NOTE: Good Weibull software will do all of that for you, you just enter the data. Then a straight line (on this scaling) is fit to the data. The CDF plot scaling is done so that a straight line from lower left to upper right is a solution. This plot models the variability of battery life capability. The Weibull CDF plot of the battery test results (Figure 3 below) shows 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, such as Reliability Centered Maintenance (RCM), leads 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., The New Weibull Handbook(c), self-published (first complete self-study reference for Weibull Engineering)
3.
IEC 61649, Edition 2, Weibull Analysis (the official international standard)
4.
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