In this talk, we're going to describe
repeated measures degradation and its implementation in JMP.
I'm going to present the background, motivation,
some technical ideas and examples.
Then I'll turn it over to Peng, who will do a demonstration showing
just how easy it is to apply these methods in JMP 17.
Here's an overview of my talk.
I'm going to start out with some motivating examples,
and then explain the relationship between degradation and failure and the advantages
of using degradation modeling in certain applications.
Then I'll describe the motivation for our use of Bayesian methods
to do the estimation.
To use Bayes' methods for estimation, you need to have prior distributions.
I'll talk about the commonly used, noninformative and weekly informative
prior distributions.
Also in some applications,
we will have informative prior distributions and how those can be used.
Then I'll go through two examples,
and at the end I'll have some concluding remarks.
Our first example is crack growth.
We have 21 notched specimens.
The notches were .9 " deep, and that's like a starter crack.
Then the specimens are subjected to cyclic loading,
and in each cycle, the crack grows a little bit.
When the crack gets to be 1.6 " long, that's the definition of a failure.
We can see that quite a few
of the cracks have already exceeded that level, but many of them have not.
Traditionally, you could treat those as right censored observations.
But if you have the degradation information,
you can use that to provide
additional information to give you a better analysis of your data.
The basic idea is to fit a model to describe the degradation paths,
and then to use that model to induce a failure time distribution.
Our second example is what we call Device B,
a radio frequency power amplifier.
Over time the power output will decrease
because of an internal degradation mechanism.
This was an accelerated test where units were subjected
to higher levels of temperature,
2 37, 195 and 150 degrees C.
The engineers needed information about the reliability of this device
so that they could determine how much redundancy to build into the satellite.
The failure definition in this case was
when the power output dropped to -.5 decibels.
We can see that all of the units had already failed
at the higher levels of temperature,
but at 150 degrees C, there were no failures yet.
But there is lots of information about how
close these units were to failure by looking at the degradation paths.
Again, we want to build a model for the degradation paths,
and then we use that to induce a failure time distribution.
In this case, the use condition is 80 degrees C.
We want to know the time at which units
operating at 80 degrees C, would reach this failure definition.
Once again, we build a model for the degradation paths.
We fit that model, and then from that we can get a failure time distribution.
Many failures result from an underlying degradation process.
In some applications, degradation is the natural response.
In those situations, it makes sense to fit a model
to the degradation and then use the induced failure time distribution.
In such applications,
once we have a definition for failure, which we call a soft failure
because the unit doesn't actually stop operating when it reaches that level
of degradation, but it's close enough to failures that engineers say we would
like to replace that unit at that point, just to be safe.
Now, in general, there's two different
kinds of degradation data, repeated measures degradation,
like the two examples that I've shown you, and destructive degradation,
where you has to destroy the unit in order to make the degradation measurement.
For many years, JMP has had
very good tools for handling degradation data.
I'm focused in this talk
on the repeated measures degradation methods
that are being implemented in JMP 17.
There are many other applications
of repeated measures degradation, for example, LED or laser output,
the loss of gloss in an automobile coating and degradation of a chemical compound,
which can be measured with techniques such as FTIR
or any other measured quality characteristic
that's going to degrade over time.
There are many applications for repeated measures degradation.
There are many advantages of analyzing degradation data if you can get them.
In particular,
there is much more information in the degradation data
relative to turning those degradation data
into failure time data.
This is especially true if this heavy censoring.
Indeed, it's possible to make inferences about reliability
from degradation data in situations where there aren't any failures at all.
Also, direct observation
of the degradation process allows us to build better models
for the failure time distribution because we're closer
to the physics of failure.
Now, several years ago, when we were planning the second edition
of our reliability book, we made a decision to use
more Bayesian methods in many different areas of application.
One of those is repeated measures degradation.
Why is that?
What was the motivation for using Bayes' methods
in these applications?
I used to think that the main motivation
for Bayesian methods was to bring prior information into the analysis.
Sometimes that's true,
but over the years, I've learned that there are many
other reasons why we want to use Bayesian methods.
For example, Bayesian methods do not rely
on large sample theory to get confidence intervals.
It relies on probability theory.
Also, it turns out that when you use Bayes' methods with carefully chosen,
noninformative or weekly informative prior distributions,
you have credible interval procedures that have very good coverage properties.
That is, if you ask for 95 % interval,
they tend to cover what they're supposed to cover with 95% probability.
In many applications, there are many non-Bayesian approximations
that can be used to set confidence intervals.
When you do Bayesian inference, it's very straightforward.
There's really only one way to do it.
Also, Bayesian methods can handle with relative ease,
complicated model data combinations
for which there's no maximum likelihood software available.
For example, complicated combinations
of nonlinear relationships, random parameters and sensor data,
the Bayes' methods are relatively
straightforward to apply in these complicated situations.
Finally, last but certainly not least, Bayesian methods do allow an analyst
to incorporate prior information into the data analysis.
But I want to point out that the revolution we've had
in the world of data analysis to use more Bayesian methods,
most analysts are not bringing
informative prior information into their analysis.
Instead, they use weekly informative or noninformative priors
so that they don't have to defend the prior distribution.
But in many applications,
we really do have solid prior information that will help us get better answers.
I will illustrate that in one of the examples.
Bayesian methods require
the specification of a prior distribution.
As I said, in many application, analysts do not want to bring
informative prior information into the modeling and analysis.
What that requires is some default prior
that's noninformative or weekly informative.
There's been a large amount of theoretical research on this subject
over the past 40 years, leading to such tools as reference priors,
Jeffrey's priors, and independent Jeffrey's priors
that have been shown to have good frequentist coverage properties.
One of my recent and current research areas
is to try and make these ideas operational
in practical problems, particularly in the area of reliability.
A simple example of this is if you want to estimate a location parameter
in the log of a scale parameter, a flat prior distribution leads
to credible intervals that have exact coverage properties.
That's very powerful.
Also, flat prior distributions can be well approximated by a normal distribution
with a very large variance, and that leads to weekly informative priors.
Again, it's somewhat informative,
but because the variance is very large, we call it weekly informative.
The approach that I've been taking to specify prior distributions is to find
an unconstrained parameterization, like the location parameter
and the log of the scale parameter that I mentioned above,
and then use a noninformative
or weekly informative flat or normal distribution
with very large variances as the default prior.
Then it's always good idea to use some
sensitivity analysis to make sure that the prior are approximately noninformative.
That is, as you perturb the prior parameters,
it doesn't affect the bottom line results.
JMP uses very sensible, well-performing methods to specify
default prior distributions that are roughly in line
with what I've described here.
Having those default prior distributions
makes the software user-friendly,
because then the user only has to specify prior distributions
where they have informative information that they want to bring in.
Here's just an illustration to show that as the standard deviation
of a normal distribution gets larger, you approach a flat prior distribution.
Now, as I said, in some applications, we really have prior information
that we want to bring in that is informative prior information.
When we have such information,
we will typically describe it with a symmetric distribution,
like a normal distribution, although some people prefer to use
what we call a location- scale t distribution
because they have longer tails.
In most applications where we have this
informative prior information, it's only on one of the parameters.
Then we're going to use noninformative,
or weekly informative prior distributions
for all of the other parameters.
Let's go back to alloy a.
What we're going to do is we're going to fit a model to the degradation paths,
and then use that model to induce a failure time distribution.
Now if you look in an engineering textbook
on fatigue or materials behavior,
you'll learn about the Paris crack-g rowth model.
It's always nice to have a model that agrees with engineering knowledge.
JMP has implemented this Paris crack-growth model.
Here's the way it would appear in a textbook.
Then on the right here we have the JMP implementation of that.
It's one of the many models that you can choose to fit to your degradation data.
Now, c and m here, which in JMP is c and b2,
are materials parameters, and they are random from unit to unit.
The K function here is known as a stress intensity function.
For the crack we're studying here,
the stress intensity function has this representation.
Now this is a differential equation,
because we've got a of t here, and also a here,
you can solve that differential equation and get this nice closed form.
This is the model that's being fit within JMP.
Again the parameters b1 and b2 will be random from unit to unit.
Now here's the specification of the prior distribution.
I've illustrated here two different
prior distributions, the default prior in JMP,
and the prior distribution that we used in the second edition
of our reliability book, which we call SMRD2,
statistical methods for reliability addition two.
Now the way we specify prior distributions
in SMRD2 and JMP is doing this as well, is with what I call a 99 % range.
For example, we say that we're going
to describe the mean of the b1 parameter
by a normal distribution that has 99 % of the probability
between -15 and 22.
That's a huge range.
That is weekly informative.
Then we have similar, very wide ranges for the other mean parameter here.
Then for the Sigma parameter JMP,
following usual procedures for these uses a half Cauchy distribution,
which has a long upper tail,
and therefore, again, is weekly informative.
Now in our reliability book, we used much tighter ranges.
But interestingly, the two different prior distributions here
give just about the same answer, because both are weekly informative.
That is, the ranges are large relative to,
let us say, the confidence interval
that you would get using non-Bayes' methods.
Now, in addition to specifying the prior distributions, which again,
JMP makes very easy because it has these nice default priors,
you also have the ability to control the Markov chain Monte Carlo algorithm.
The only default that I would change here
is typically I would run more than one chain.
I changed the one here to four.
The reason for doing that is twofold.
First of all, in most setups, including JMP,
you can run those simultaneously, so it doesn't take any more computer time.
The other thing is we want to compare those four different Markov chains
to make sure that they're giving about the same answers.
We call that mixing well.
If you see a situation where one of those chains is different
from the others, that's an indication of a problem.
If you have such a problem,
then the usual remedy is to increase the number of warmup laps,
which is set to be 10 by default, but you can increase that.
What that does it allows JMP to tune
the MCMC algorithm to the particular problem so that it will sample correctly
to get draws from the joint posterior distribution.
In all of my experiences using JMP, and Peng has suggested that he's had
similar experiences that by increasing that high enough,
with any examples that we've tried, JMP will work well.
But with 10 for most applications, that is a sufficiently large number.
Here's the results.
Here's a table of the parameter estimates.
Well, typically in reliability applications,
we're not so much interested in the estimates themselves.
We're going to be interested in things
like failure distributions, which we look at in a moment.
Then in this plot,
we have estimates of the sample paths
for each of the cracks.
Again, you can see the failure definition here.
As Peng will show you,
JMP makes it easy to look at MCMC diagnostics.
It's always a good idea to look
at diagnostics to make sure everything turned out okay.
What you do is you export posterior draws from JMP,
and then JMP has set up there some scripts to create
these various different diagnostics.
For example, there's a script to make a trace plot or a time series plot.
I always like to compare those for the different chains.
Then there's another one to make what we call a pairs plot,
or scatterplot matrix of the draws.
That's what we see here.
Then as I said, we can use those draws
to generate estimates of the failure time distribution.
JMP implements that by using the distribution profiler here.
We can estimate fraction failing is a function of time
for any given number of cycles.
Now let's go to the Device B RF Power A mplifier,
again, an accelerated repeated measures degradation application.
We're going to need a model that describes the shape of the paths
and the relationship the temperature has on the rates of degradation.
Again, the use condition in this application is 80 degrees C,
and we're going to want to estimate
the failure time distribution at 80 degrees C.
In SMRD2, this is the way we would describe the path model
that fits device b.
We call this an asymptotic model,
because as time gets large, we eventually reach an asymptote.
In this equation, X is the transformed temperature.
We call it an Uranus transformation of temperature.
X 0 is an arbitrary centering value.
Beta one is a rate constant for the underlying degradation process.
Beta three is the random asymptote.
Those two parameters, the rate constant and the asymptote,
are random from unit to unit, and we're going to describe
that randomness with a joint or bivariate lognormal distribution.
Beta two, on the other hand, is a fixed, unknown parameter that is the effect
of activation energy that controls how temperature affects the degradation rate.
This is where the X0 comes in.
Typically we choose X 0 to be somewhere
in the range of the data
or at a particular temperature of interest,
because beta one would be the rate constant at X 0.
Again, there's a large number of different models that are available.
Here is how you would choose this particular asymptotic model.
This corresponds to the same equation we have in SMRD 2.
The only difference is that JMP uses a slightly different numbering convention
for the parameters.
That was done to be consistent
with other things that are already in JMP elsewhere.
Again, we have to specify prior distributions,
but JMP makes that easy
because they provide these defaults, these weekly, informative defaults.
Here I have the default that JMP would have that we're going to use
if we did not have any prior information to bring in.
I'm going to do that analysis,
but I'm also going to bring in the information that engineers have.
In particular, we only have information for b3
and so that's being specified here.
But all the other entries in the table are exactly the same as the JMP default,
again, making it really easy to implement these kinds of analyses.
Here's the results.
Once again, here we have a table giving the parameter estimates,
credible intervals and so forth.
In this plot, again,
we have estimates of the sample paths for all of the individual units.
Again, we have the failure definition here,
but what we really want are estimates of the failure time distribution
at 80 degrees C.
Again, we're going to do that by using a profiler.
On the left here we have the estimate
of fraction failing is a function of time at 80 degrees C for the default priors.
On the right we have the same thing,
except that we've used the informative prior distribution for B 3.
Immediately you can see that prior information has allowed us
to get much better estimation precision.
The confidence interval is much more narrow.
Interestingly, the point estimate of fraction failing
actually increased when we brought in that prior information.
In this case, the prior information would allow the engineers to get
a much better estimate
of fraction failing as a function of time.
Then to make that important decision
about how much redundancy to build into the satellite.
Let me end with some concluding remarks.
Repeated measures degradation analysis
is important in many reliability applications.
It is also important in many other areas of application,
like determining expiration dates
for products like pharmaceuticals and foodstuffs.
When will the quality be
at such a low level that the customers no longer happy?
Also, in certain circumstances,
we can bring prior information into the analysis, potentially allowing us
to lower cost of our degradation experiments.
JMP 17 has powerful, easy- to- use methods for making
lifetime inferences from repeated measures degradation data.
Now I'm going to turn it over to Peng,
and he's going to illustrate these methods to you using a demonstration in JMP.
Thank you, Professor.
Now, the demo time.
The purpose of a demo is to help you to begin exploring the stable art approach
to analyze repeated measures degradation data.
I will show you how to locate the sample data tables using JMP,
how the information is organizing in the report
and highlight some important information that you need to know.
First, there are three repeated measures degradation data samples
among JMP sample data tables.
Alloy a, d evice b, are two examples with embedded scripts.
The s laser does not have an embedded script.
Alloy a is an example without an x variable.
Device b is example with an x variable.
To find them go to the help and click sample index.
Try to find the outline node called reliability/ survival.
Unfold it and should see a lloy a is here, and device b is here.
To find GAS laser,
you need to go to the sample data folder
on your computer by clicking this button, open the sample data folder.
Now I'm going to open alloy a.
Then I'm going to analyze menu, reliability and survival,
choose the repeatedly measure degradation.
Now we see the launch dialog.
Our assign length, crack lengths goes to y,
specimen goes to label system ID and million cycles go to times.
Then I'm going to click okay.
This is the initial report.
It's a fifth linear models for individual batches.
I'm going to select the third model here by clicking this video button.
This fifth initial model of a Paris model of this alloy a data.
I'm going to click this, go to Bayesian estimation button
and generate a configuration interface.
Here we see the model formula.
Here are the default settings for the priors.
Then we are not going to change anything right now,
and we are just going to use the default prior to fit our model.
Now I'm going to click
this fit model button here and let it run.
Then I'm going to explain what are in the report.
In the end, how to get the failure distribution profiler.
Now I'll click the button, the algorithm start run
and the progress dialogue said that the first step is tuning.
The underlying algorithm will go through some
round of warm up laps, is procedure.
The algorithm is trying to learn the shape of the posterior distribution,
for example, where the peak, how wide is the span, et cetera.
In the end, they will try to figure out
what is a good thinning value to draw samples from the posterior distribution,
such that the samples are as little auto correlated as possible.
Then the algorithms enter the second step.
The dialogue says, this step is collecting posterior samples.
In this step, an automatic thinning of 80 is applied.
The dialogue shows how much time in total the algorithm had been running.
In the second stage, the dialogue also shows expected completion time.
By such, I hope it can help users
to adjust to their expectation accordingly.
Sometimes excessive, long expected completion time is a sign of a problem.
Then we wait a little bit, and the algorithm should finish soon.
Okay, now the algorithm has finished.
Let's see what's in the report.
First is the completion time.
If we left your computer run over time,
you may want to know in the morning on next day.
The second is a copy of your settings, including priors, number of iterations,
random seed and other thing shows.
Third part is posterior estimates.
Be it a summary or the posterior samples,
there are two links on the site to allow you to export posterior samples.
I'm going to emphasize the first link.
One purpose of using these first link
to export posterior examples is to inspect potential problems.
Two main concerns are: convergence and effective sample size.
Let's look at it.
The table have parameters in individual columns.
Each row a posterior sample.
There are several embedded scripts.
The most important one is the first one.
I'm going to click this green triangle to run the script.
The script simply run time series on individual columns
and show their time series plot.
In the context of MCMC, this plot is known as the trace plot.
What do we see here?
What I call good results.
The series are stationary and no significant auto correlation.
Loosely speaking, when I say stationary in this context, I specifically mean plots
looks like these.
They are straight equal with band of random dots.
Okay, let me close the report and the table.
We are seeing good results here.
Also the data and a fitted model also shows the results is good.
Now we are ready to ask for a fitted time distribution profiler.
To do that, go to the report outline node menu
and select show like distribution profilers.
Most entries in this dialogue has sensible default values
and then we only need to supply one of the failure definition.
I'm going to enter 1.6 to this upper failure definition.
Before I'm going to click okay, I'm going to reduce this number
of SMRD2 realizations to 5,000 to save me some time.
Then I'm click on okay.
This is also a computational intensive procedure,
but not as expensive as MCMC in general.
It should finish quickly.
You can use the profilers to get
the failure probability and the quantile estimates.
I'm not going to elaborate further,
because profiler is a very common and important feature in JMP.
Okay, this is the end of the demonstration,
and you are ready to explore by yourself.
But here are a couple of tips that might be useful.
First, before you save the script to the table,
go to check this save posterior to script option
before you save the script to the table.
By this, next time you run the save script,
software will bring back the fitted model instead of going
through the lengthy MCMC procedure once again.
The second thing that I want to bring to your attention is
we have seen good examples, good results, but there are bad ones.
Here are some bad examples.
This bad example means either fail to converge,
or there are high auto correlations.
To address them my first suggestion is to increase
the number of warm up laps.
Second suggestion where we turn off auto-thinning
and apply a large thinning value, a thinning number manually.
If those suggestions don't work, it's likely that the model
or its configurations are not appropriate for the data.
You may need help.
Okay, this are all we would like to illustrate this time,
and I hope you can start to use this information to explore
the state of art approach to analyze repeated measures degradation data.
Thank you.
