OK. Hello.
thanks for the nice introduction.
Today I want to present together with Stefan Vincent from LRE Medical
a work we've done in the load year on control charting kinetic
other other time dependent measurements.
So why is that important to present?
So in case you have a time dependent curves
other the curves themselves are important for the quality of your product,
so it's often very hard to define any child of specification.
and in our case here,
thesis curves were evaluated by specialists.
So every measurements What sent to a specialists
looked at the curve, said, "Yes, OK, or not OK."
This is pretty time consuming other compassion other so on.
other on Top of that it's even bath when the person is sick or in vacation.
So it's a person- dependent thing.
So often it happens that the person
has different moods or different obligation
or different priorities other so on.
So let's say the adjustment of the curve May vary a little bit.
So it's child of reproducibility problem
other we wanted to Stop that.
So that What the reason why we started with it.
So for example, here you lake a selection from hundreds of thesis curves we .
other you lake here it's a pressure holding measurements.
So we have here pressure versus time in seconds.
other here you lake a bunch of curves.
other the green ones, you lake they are called true.
This means true is accepted, they are good.
other false means rejected, they are not good.
So there is a bath product.
So what you lake here that the green ones are , tied together.
Here in this highlighting Screenshot, you lake better.
So they're pretty close together.
other then we have a selection of red ones
which are either apart from the green ones,
but there are so two ones which are more or less in the same regime.
but like you lake, they have completely different shapes.
There are some edge other other s-shaped curves other so on other to forth.
So if it would make just a simple t hree sigma limit ±
around the good ones, for example, like in the upper case here,
we would include so the not good red ones here from the lower picture.
So a simple ± three Sigma limit
approach around this series of curves would not be target leading.
So we need something which includes the position
which is done here with the basic monument.
but so we want to have something which takes care about the shape
of thesis functions, of thesis groups,
How to analyze the shapes other positions?
So there are actually two approaches.
One is pretty long known already.
This is the principal component analysis.
That's it the first one.
other in more recent times JMP came up with
a functional data explorer which so gives us the possibly to do the same.
but like we have lakes, it What not really the seed, so it What different.
So let's begin with the old approach, principal component analysis
for doing so, you need to transform the long table I just show you
in the next picture here on the left sides,
a long table where you have columns
with the part number, for example, with the test dates
but so other this is the important part
of the runtime other the pressure other this for each value.
other you lake here, this is in seconds.
So we , every some milliseconds.
other like you lake so, it is pretty hard that for the next part
it's exactly the same series of numbers here,
exactly the same time when we have the next pressure period.
other this is actually needed when you want
to make a principal component analysis because you have to transform
this long table into a wide table where you have one row by part
other then you have the white table
where you have for each time slot other then the data points.
So what we need to do now is, first of Alles,
we need to bring Alles thesis runtimes on the same scale.
Here, we have done that by just calculating
the longest time like 100% other the shortest is zero.
other then we have every numbers from milliseconds seconds
transferred into this percent scale.
Then we interpolated because quiet they were not on the same time slot actually.
So we had to bring Alles the Y numbers on the same time point.
This means we need to do a child
of interpolation to have them Alles on the same.
other then when we have that
so we created a new column with the default relative times.
I've just seen them in the example in the next slide, so 00001 other so on.
other then we transpose them into this white shape.
other from there on, then we could do the principal component analysis,
saving the printable components other calculate the T squares from them
other build the control charts from thesis T squares values.
So here you lake it's the slot 0%, 1%, 2% other so on.
other here are the pressure values.
In a parallel plot, it looks child of Like this.
So we have here the different slots here on the X axis
other the pressure is quiet the same numbers like before
other you lake that the curves are looking pretty much the same like before
but now we have about 100 data points other before we had .
So it's the density of the points are a little less, but quiet the curves
other the shapes other the position other everything are the same like before
And we take now this white table
other on Alles thesis different time slots
we do a principle component analysis.
Then we find here this scoreboard for it.
What you already lake here is that in the center part, we have here
Alles the green dots other the green ones, like you lake here are the accepted ones
or the good parts surrounded by the red dots,
the rejected ones, the false ones.
other like you lake so in this example here we can stay with two principal components
because the first principal components covers already 98% of the variation
in the second one, 2% other the other ones you can neglect
So I think it's good enough to Save just the first two principle components.
other then from thesis two principal components, we calculated the T squares
This means T Square is principal component,
one squared plus principal components.
Two squares would be then the T squared here
for this data period.
calculating this one .
So for every data period we calculated,
this T square other bring it on a control chart
Here, you see the control limits calculated only from the well,
from the green dots, from the good runs, from the good curve.
So you lake it down here from the moving range.
So this means, OK. This controlling, it represents the normal natural variation,
what we expect from the good ones.
other Alles the non good ones, the red ones
should be outside of this regime here.
other you lake it's mostly done,
but not for thesis two points here for part number 14 other part number 15
So they're inside the control limit.
but that is not what we want to lake.
So first of Alles we have to understand what are thesis two.
So if you have a look onto the next picture,
then other highlights just thesis two, 14 other 15, then we lake,
"Ah okay." T hese are the ones here
which are really within the regime of the green ones.
the other ones which are further apart from it, they are easily excluded
or not excluded, but let's say distinguished
here with this control shot.
but thesis parts were not.
So what we learned from here, that is the principal component approach
like we have done it here or have done it in former times,
there is no information about the shape of the curve.
It's more information about the position where it is on this Y scale here.
So we need something else which takes care about the shape of this curve.
So when we tested the FDE, the function Data explorers,
other the first good news is you can just take the long table like is.
So there's no need for data transformation
or bringing the different time points on the same slot number or whatever.
So you just take the raw data like they are.
Perhaps you want exclude some real outliers where there's the machine
has given wrong numbers or whatever.
but Alles the other stuff you can just take like is
other . maybe you don't want to do it on the transformation or something.
It's not really necessary.
other in this case, we have just taken the raw data like they are.
So starting doing at FDE on this,
we lake here again our pressure time curve like we've seen them before
but now you lake thesis blue verticals here
other thesis blue verticals represent the number of knots
So what is a knot?
So here we are fitting a spline curve.
other the spline curve is, let's say if you take a ruler
other then you want to make it a bit more flexible
to bend it onto Alles thesis different curves
or, let's say, to adjust it the best way to the data.
other the more, not this ruler has the more flexible it is.
other so the better you can adjust Alles thesis different points.
Here, we used thesis 20 knots thing.
So thesis are 20 verticals here other here with the basic information criteria
you lake that we get on Alles different functions.
we get the , BIC's.
other just have at optical control on this,
you lake on the lower left, Alles the data points separated out
for Alles the different curves here other with a red line on Top of it
representing the spline curve which we fitted.
it doesn't matter if the curve is pretty straightforward
Like here or has more edges or whatever,
it's perfectly aligned.
So just optically, it looks very good.
other you can so check that with the diagnostic plots.
for example, here this blind curve,
the predicted values are displayed versus the actually ones.
other you lake they are perfectly aligned around thesis 45 degree line here.
other so the residuals, the ones are left other right here
from this prediction line, they are pretty small,
so there's not a lot of error left, which is not represented by this bare.
So the fit looks really, really excellent
So we have not taking the parameters for this split
sorry for the spline other did a functional FDE.
Well, I'm completely confused today.
other we did a principal component analysis on thesis.
So we separated the eigenvalues,
which is the weighing factors other the own functions
for each of thesis curves.
Then we can screen them on a scoreboard.
Here the eigenvalues for each of thesis curves.
other you lake here,
again with the BIC credit criteria
you lake how many numbers of functional principle
components you should optimally use.
other you lake is the minimum with two.
So we stayed with two again like the one before
other now we have this score board,
other it looks pretty much the same like the one before
we have here the center part, the green of the curves, which are well,
surrounded by the red ones, which are rejected from the specialist.
other now we saved Alles thesis functional prints
other components other built a T square plot on it.
other here again, the same picture, the square.
If we take Alles the data points
to calculate the control limits, OK, then besides thesis two, everything
is part is in control.
but this is actually not what we want to have.
So we want to understand what is the normal variation of the good ones
to differentiate them from the variation of the non good ones.
other so down here on the second plot here, we calculated the control limits
only from the good ones.
other then you lake, OK, nearly Alles red curves are outside.
We're out of control. but there are two.
So the number 14 other the number two here are again directly on the borderline.
but Alles the others are clearly separated.
So let's have a look on the two other 14 which curves thesis are.
other then we lake here. Ah okay.
So they are really different from the red ones, that's clear.
but so child of different from the majority of the green ones.
So the number two, which is defined like being true is the one
with a steeper slope here . to Alles the other green ones here.
other the number 14 is this guy here,
which is more or less on the upper end of the regime of the greens.
So obviously, this little differences here in the shape is not
strong enough to come up really in this statistical calculation.
but we could really exclude here this guy here
from the others with a strong edge here.
other maybe this one if the experts would have a look onto this
second time or maybe it would
have rejected it because it's too steep a slope or whatever.
So thesis are border lining, I would say in both cases here
with the statistical approach but so, I guess with the manual approach.
but we could clearly detect number 15, which is the one here in the center,
like being a non normal behavior with this tool.
So you lake, the FDE has so some limitations.
but jumpsuit, in addition to the default PCA approach,
it comes up with thesis shape information of the curves.
other this is so part of distinguishing in a control chart
if a curve or a measurements is normally vary or is it not normal anymore.
to conclude thesis things the default PCA approach combined
with the T square analysis or control chart
is good for detecting the position of the curve
other distinguish them from curves which are not in this regime.
but like we have lakes, it's lacking if the shape of the curve is of importance.
other with the FDE approach, again, combined with the T square analysis,
first of Alles, it needs much, much, much a less startup prepared.
but on Top of that it's good for checking the position like the above,
but so because it has shape information of the curves.
it is so including that in the good not good understand.
In our work, we stopped at this point,
but principally, you can build a child of automatically.
If you use thesis curves or thesis principal components
other the information of good other bath, you can automize the good other bath.
But if you have a model behind which predicts you,
then what you want lake.
but we haven't done that.
we stopped here on the T square chart to understand
the variation other what's vary more than normal variation.
thank you you very much for your attention.
other if you have questions In the open to answer them now.
thank you you
