and I am a senior statistical writer at JMP,
but I also like to play around with functional data.
This project is on measurement systems analysis for curve data.
First, I'm just going to give a very brief background on MSA studies in general.
MSA studies determine how well a process
can be measured prior to studying the process itself.
It answers the question, how much measurement variation is
contributing to the overall process variation.
Specifically, the Gage R&R method, which is what I'll be using in my analysis,
determines how much variation is due to operation variation
versus measurement variation.
You can use a Gage R&R crossed MSA model when you have both a part and an operator.
The model you can see here for your measurement Y sub I J K,
that's going to be the Kth measurement made by the Jth operator on the Ith part.
In this model, you have a mean term,
a random effect that corresponds to the part,
a random effect that corresponds to the operator,
and a random effect that corresponds to the interaction or cross term.
You also have an error term.
This is simply a random effects model,
and all of these random effects are normally distributed random variables
with mean zero and some corresponding variance component.
When you fit this model,
you can use that to estimate the variance components
and then use those variance component estimates
to calculate the percentage gage R&R using the formula shown there.
In a standard MSA study,
all of your measurements are going to be single points.
But what happens if that's not the case?
What if instead you're measuring something like a curve?
That question was the motivation behind this project.
There was a client of JMP that was a supplier of automotive parts,
and they had a customer that specified
that a part needed to have a specific force by distance curve.
Obviously, the client wanted to design
their product to match the customer specified curve.
In order to do that,
they wanted to run a functional response DOE analysis
and JMP to design their product in order to do so.
However, before spending money on that experiment,
they wanted to perform an MSA on their ability to measure the parts force.
There are a lot more details about the actual data and this problem specifically
in an earlier 2020 white paper
titled Measurement Systems Analysis for Curved Data.
If you want any more details, look that up.
It should be on the community.
This in this graph, that's what the data looks like.
On the Y-axis, we have force, and on the X-axis, we have distance.
It looks like there are only 10 curves in this graph,
but there are actually 250 total curves.
There's just some clustering going on.
There are 10 different parts, five different operators,
and five replications per part operator combination.
A little bit about this data, obviously,
these measurements are curves and not points.
The data was collected evenly spaced in time, but not evenly spaced in distance.
There were some earlier projects
that tried a few different ways to perform some type of MSA study on this data.
They used some functional components, but stayed pretty true to a standard MSA.
When I looked at this data, I wanted to take a true functional approach
because I have a background in functional data.
Functional data analysis
is useful for data that are in the form of functions or curves.
There are many techniques to handle unequally spaced data,
a lot of which are available in the Functional Data Explorer platform in JMP.
My goal was to combine functional data methods with traditional MSA methods
to perform some type of functional measurement systems analysis.
My solution was to create a functional random effects model
by expanding the functional model using eigen function expansion,
rewriting that as a random effects or a mixed model
if you had any fixed effects also,
and then estimating the variance
components associated with the part and operator terms.
To go a little bit into the model notation.
For your functional model, you have Y sub I J K,
but this time at a particular distance, D,
to account for the functional nature of the data.
You're going to have a functional mean term,
a functional random effect that corresponds to the part,
a functional random effect that corresponds to the operator,
and a functional random effect that corresponds to the cross term
and also your error term.
Here, when you do the model expansion,
it's a little mathy, but essentially,
instead of having one variance component associated with the part
and one variance component associated with the operator,
you now have multiple variance components associated with each of those things.
That's going to account for the functional nature.
When you're fitting the model and estimating the variance components,
like I said, now you're going to have this
set of variance components that you can sum together
to estimate the functional variance component for part
and the same thing for operator and the cross term.
Once you have all those individual variance components,
you can use those to estimate the % gage R&R just like in a standard MSA.
How do I do this in JMP?
It's a multi step process that's outlined here,
and there are some more details in other slides.
But essentially, I estimate the mean curve in FDE and obtain the residual curves.
I then model the residual curves in FDE to obtain the eigen functions needed
for the eigen function expansion of the functional model
and save those eigen functions to the original data table.
I'm going to use those saved eigen functions in FitMix
to create a random effects model
or a mixed model if you also have fixed effects in your data.
I'm going to use nesting of the eigen function formula columns
and also the par and operator variables
to define the appropriate model specifications.
This is what your fit model window would look like.
Once I did all that for this data,
I was able to estimate the variance components and calculate the % gage R&R,
which in this case was 3.3030.
This indicated an acceptable measurement system
according to some ranges that were defined in this paper by Baren team.
That was it for the data analysis for my part.
This result was actually very similar to a worst-case scenario
that was obtained in a presentation in 2019.
It would be interesting to know if that was a coincidence
or if the results would be similar for different data as well.
Some thoughts that this project provoked.
Should we add a functional random effect for ID
to capture the within function correlation across distance?
This type of functional random effect
is actually really important in functional data
and is a big benefit of accounting for the functional nature of the data.
Unfortunately, in this data in particular.
Anytime I created a model with this term,
the corresponding variance components were zero,
so it didn't really capture anything extra, but it would be interesting
to see if it could be useful in different types of data.
I also think it would be interesting if we
could calculate a confidence interval for the % gage R&R.
There were also some minor, not issues,
but brought up questions of the residuals in the random effects model.
I observed a cyclical nature in those.
That's not always great.
I don't think it was a huge deal,
but I would like to have a good reason for why that was the case.
That's it. Thanks for listening.
If you want more details on this project,
it's very similar to a full 30-minute talk that I presented at Discovery Europe,
and so that video is on the community as well.
Thank you.
