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DUANE BONING: OK, so today we're going to talk about some
of the probability models associated with variation
in manufacturing processes.
The last couple of classes, we've
tried to go and do two things.
One is give you a little bit of perspective
on a couple of different kinds of--
or actually, a fairly wide family
of manufacturing processes, give you
a little bit of exposure to semiconductor processes
as well as a couple of different mechanical--
mechanically-oriented processes, and give you
some ideas or ways to think about those processes,
including just the physical action that's going on,
as well as some of the sources of variation.
So we're going to dive in a little bit
today on dealing with data taken from some manufacturing
processes, and start to look at random components of that
and systematic components of that.
But back on the theme of trying to make sure
that you have at least some exposure, given everybody's
different backgrounds, to these processes,
we would like to make sure you have the opportunity
to at least see a couple of the kinds of tools
and equipment in the facilities--
especially those associated with semiconductor manufacturing,
if you haven't previously seen those.
So let me turn it over to Hayden,
who can check in with you well the problem
is that as well, but also about arranging opportunities
for you to see a little bit more, if you would like.
HAYDEN TAYLOR: So yes, on the problem set, well,
if you still have questions about it,
then feel free to email or call today.
See me afterwards.
And problem set 2 is now up on the website.
That's due on Tuesday, the 26th of this month.
So there's a bit longer for that,
and I'm willing to start helping you with that too.
So thank you.
DUANE BONING: Any questions for Hayden?
Any big issues with the problem set?
AUDIENCE: Actually, I have a--
DUANE BONING: Go ahead.
HAYDEN TAYLOR: OK.
AUDIENCE: So I have a quick question.
So if you can refer to the Question 3 of the Problem Set
1, and you say it's proportional to the water and the time--
the KOH.
And when you talk about these [INAUDIBLE],,
you're talking about the concentration
or you're talking about the molarity?
HAYDEN TAYLOR: What am I saying is proportional to KOH.
DUANE BONING: The square brackets around the KOH.
HAYDEN TAYLOR: Oh, I see.
AUDIENCE: The very last figure [INAUDIBLE]..
HAYDEN TAYLOR: Yeah, yeah.
AUDIENCE: --proportional.
The [? edge 3 ?] is proportional.
So it's proportional to the concentration,
or it's proportional to the molarity?
HAYDEN TAYLOR: Well, for a fixed volume, it's the same.
There are two lines on that plot,
and the models that are put forward by [? Seidel ?]
to explain the data.
The data are the open circles on that plot.
And there are two models that have plotted.
And what I want you to use are primarily
the experimental data.
So look at the curved line that is plotted on that graph.
AUDIENCE: OK and for the curved line that you
want us to look at, the x-axis should be the concentration,
right?
HAYDEN TAYLOR: That's right.
Yes, the x-axis--
AUDIENCE: [INAUDIBLE]
HAYDEN TAYLOR: --is the weight of KOH.
In other words, that's the number
of grams of KOH per gram of water.
AUDIENCE: Yeah, I just want to clarify.
That's it.
HAYDEN TAYLOR: Yeah.
Thank you.
AUDIENCE: Thanks.
Thank you.
HAYDEN TAYLOR: Anybody else?
OK.
DUANE BONING: I was just also going to say,
it's sort of a manufacturing process
that we use for making up these problem sets.
And they are not expected to be completely defect-free,
so we expect part of the process is
detecting errors or questions.
And then we try to correct them.
Also, as a follow-up to that the schedule for the next problem
set, just looking ahead for the next couple of weeks,
the reason that the problem set is due not next Thursday,
but rather, the Tuesday after that--
a little bit longer than a week--
is that here at MIT, next Monday, this coming Monday
is President's Day, a holiday.
And then Tuesday is one of these strange days.
You folks out in Singapore-- you may recall this.
Tuesday is a Monday.
So it's on a Monday class schedule,
so we will not be meeting.
OK?
Go to whatever classes you would normally have on Monday.
Go to those on Tuesday out here to MIT.
And so then our next lecture will be a week from today,
on Thursday.
OK, so also going with the material
that we're starting to talk about now
and the second problem set that's just been released
is another reading assignment.
And in particular, what we're trying
to do this year is focus--
do sort of the concentrated dive of May and Spanos Chapter 4.
In the past, sometimes we've assigned
instead Chapters 2 and 3 of Montgomery,
which are much longer.
Chapter 4 in Spanos is a shorter overview
of basic statistical distributions
and some very basic manipulation and use of statistics
that we'll talk about today.
So we're not formally assigning chapters 2 and 3 in Montgomery,
but the intent here--
what I mean by this reading assignment is,
if you don't have very much background on statistics,
you may want to read Chapters 2 and 3.
Or you can skim them to know what's
in there, and in particular, that's probably the right place
to go for a longer explanation, if you need more material.
If May and Spanos is too condensed
for you or you'd like additional examples,
there's a very nice articulation in Montgomery.
Now, I have not made a formal assignment of May and Spanos
Chapter 3, but that is where you should
go if you want to read more or need
to read more about semiconductor fabrication process physics.
That chapter is a very nice, again,
relatively condensed description,
articulation of the key process steps used in microfabrication.
We'll, throughout the term, be popping in and out
of a few of those unit processes,
and I just want you to be aware that's what's in that chapter.
And you will quite likely want to go in and skim that or read
that, and dive in on some of those
processes if you need more background
on some of those things.
And I think, especially before this tour
that Hayden is organizing, it'd be at least really good
to skim chapter 3.
Again, it's mostly on the process physics,
but every now and then, it does at least
show you a little bit about the equipment that's used.
And it's very nice to be able--
for Hayden to be able to point to some of the equipment
and components of that, and have you visualize what's going on
in the process physics, or be able to ask questions
about that.
OK, so what I'm going to do here at the start
is just run through some example processes.
And in particular, these are little not quite
manufacturing processes.
They are more like research scale implementations
of some basic manufacturing processes that we have actually
generated some data from, or students have generated data
from over the years.
So I'm going to show you some actual measurements
from a few simple cases.
And we're going to look at that data,
and what I think we're going to see is variation.
And the puzzle for you and for us--
and to some extent, for me, because
in many of these processes, I was not
involved in the generation of the data--
our puzzle here is to look at the data
and try to think about what might be going on in the data.
So here's some-- a basic turning process.
So we've got a work piece.
We've got under computerized numerical controls, CNC.
We've got the ability to cut into that work piece, which
is rotating, to try to get to diameter work pieces
of a specified dimension.
So here's the diameter measured on a particular location
on that work piece as a function of 47--
I think it's 47 or 48--
different repetitions.
Now, I believe this may have involved
more than one set of students, or more than one class.
We can think of that as a shift change
in a manufacturing process.
We're simply gathering here a measurement of a diameter
on that work piece.
So talk to me.
Tell me what you see.
Is it uniform?
Not especially-- although you always
have to be really careful looking,
because on any kind of plot, you want
to really look at the spread.
Here it's kind of magnified, because here's 0.7 inches,
and we're down to about 0.697 up to 0.702.
That's not full scale.
But if we zoom in around whatever nominal is--
and in fact, it's not even clear what the target dimension was--
I might guess and think that it was 0.7 inches,
because that's a nice round number, but who knows?
So it's definitely not uniform.
What else do you see?
Yeah?
AUDIENCE: The data is sort of drifting up over time.
DUANE BONING: OK.
So it looks like there might be a long-term trend of a drift
over the number of runs.
What might lead you to think that that would
be a physically reasonable--
AUDIENCE: [INAUDIBLE]
DUANE BONING: Right.
So if the tool wears-- and in fact, if the tool wears
or some long-term change in the state of the equipment--
that's often the source of these long-term trends.
Other things people see--
Yeah?
AUDIENCE: It's fairly systematic.
It constantly goes up and down, and up and down.
There's only two or three points before it goes back up.
DUANE BONING: Yeah, that's very interesting.
Even the variation here--
it seems to be cyclical, or periodic, or something
systematic going on in there.
Again, I wasn't there for the generation of this data,
but what might some reasons for that be?
AUDIENCE: The worker might be trying
to correct with the subsequent one,
and he overshoots constantly.
DUANE BONING: Interesting-- yes.
That's a good idea.
I guess, in fact, one can imagine, especially
if you're taking a measurement after every tool,
and the target was 0.7, say, then maybe, near the beginning,
we're starting to see a deviation.
Maybe this one was a little low, and they're
trying to get back up, making an overcorrection-- because
near the end here, it looks like there's a little bit more
centered around 0.7.
AUDIENCE: I think that might be because
of backlash [INAUDIBLE].
DUANE BONING: OK.
AUDIENCE: [INAUDIBLE]
DUANE BONING: So backlash being an operator adjustment
that is made as perhaps part of this compensation strategy?
AUDIENCE: [INAUDIBLE]
DUANE BONING: Interesting--
AUDIENCE: [INAUDIBLE]
DUANE BONING: Anybody in Singapore--
any other observations or ideas?
AUDIENCE: Since measurement is measured
at a different place of the things,
so maybe you can see every three point
will be a larger diameter.
I think maybe this is measured at the other side
of the thing [INAUDIBLE].
And maybe for the lower-- for the smaller diameter,
it's measured in the inner side.
DUANE BONING: Yes, it's quite possible--
at least on this plot.
We might learn more if we look at more data.
I read this as run number, rather than measurement number.
So somehow, perhaps, this is intended
to be representative of the whole part,
but it's possible those are multiple measurements
on different part--
portions of an overall turning.
You could start to look and hypothesize on that.
Is the 1, 2, 3-- is that the same pattern you see each time?
And it's not clear whether we see exactly that same pattern
or not.
Other ideas?
AUDIENCE: [INAUDIBLE]
DUANE BONING: Right.
So run number is a stand-in for time, and these are occurring.
So your point here is--
AUDIENCE: [INAUDIBLE]
DUANE BONING: Sure.
AUDIENCE: [INAUDIBLE]
DUANE BONING: Right.
So it may not be equipment state,
or indirectly, it may be also the rest of the facility
environment or the line.
And over time, who knows?
Maybe this thing starts heating up,
or the work pieces that are coming in
have a different temperature--
quite possible that there could be some other long-term state
change.
We do notice there's this shift change.
And referring back to this long-term trend,
if we actually look at this early part,
it may or may not be drifting that much.
It may actually be fairly steady.
And what we're actually seeing in here
is an overall mean shift, but maybe not even a long-term
trend.
So one might actually pose, maybe even apply
some statistical methods to look and say,
is there a statistically significant mean difference
between this set and this set of data?
So it may look like a long-term trend,
or it may simply be a reflection that there
is a setup change on the equipment when
a new shift comes in, or there may
be just inherent differences in how the operator interacts
with the equipment.
How they load the part might be slightly different.
Any other ideas?
Let's go back to the--
because I think you had mentioned
something interesting about possible oscillation in here.
What might other sources or causes of oscillatory behavior,
or what looks like two distinct sets of data here--
in fact, I'm not completely sure that it's purely oscillatory,
because you can come down here, and it
seems like sometimes you've got two things that
are down low, and then one up, or three things that are low,
and one up.
AUDIENCE: Well, this is a little bit farfetched,
but maybe you could have [? stock ?] material
of maybe 1 meter, and they always
cut it into three pieces.
So they always have the first piece of stock material,
and they had some material properties that
led to have [INAUDIBLE] maybe.
And the middle piece led to middle dimensions,
and the last piece led to smaller dimensions.
DUANE BONING: I like that.
That might be very interesting.
So if I were generalizing that, it may well be--
very often, there may be more than one set
of starting material.
So it may have been all one piece and cut into three,
and so they're randomly picking one of the three,
and therefore, you're getting a different mix of those three
points in here.
But in general, it may well be in many cases
that you've got two different incoming streams.
If I were looking at this, instead of thinking of three,
it kind of almost looks to me like this set of data here--
it's kind of randomly around some value,
and this is a different set of data around some other value.
So it may be that there's just two boxes apart sitting there,
and they're just pulling, randomly, starting material out
of one of the two.
Yeah?
AUDIENCE: I also don't know if there's
any control on how far from the [INAUDIBLE] the materials
[INAUDIBLE] you could put it in, be a little bit farther,
and then it would have a larger cantilever
arm than the next user.
DUANE BONING: Right, right.
So there could be lots of sources
of both random and deterministic variations.
So we talked in here about things like a mean shift.
Maybe there's a deviation between these two shift
changes.
And that would be a either deterministic or--
systematic maybe is a better word of a deviation.
And then, within different sets, we
may have no detailed explanation of why
we have these small deviations, and a random model
for some of those components might be appropriate.
So that's part of the idea that we're after.
So over here on the side, we're just referring back again to
of our variation equation.
Maybe I'll put that up so that we can also
refer to that, because I think we've touched on some
of these different components.
We said that our deviations in our output
y you can often characterize as some component that
has a sensitivity to disturbances in our process
parameters, and those disturbances,
and some component that relates to our controllability
when we make small changes [INAUDIBLE]
and then control inputs.
Among other things, we said, well,
maybe some of our input feedstock, maybe delta a's--
changes in the parameters.
We also-- and I thought it was very interesting, talking
about overcompensation in the control scenario.
It might be, in fact, that there are
deviations or errors in active control, changes
to the settings.
There may also be-- and we also talked about it here--
is recognition that the function y of our alpha may, in fact,
be time-dependent, or that--
put it another way--
some of the delta a's have time components and time trends
to them.
We'll come back, and I'm sure come up with more examples
as we look a little bit further.
And I think this is some of the same process.
It has different run numbers on it, but this is explicitly--
it looks like there's, again, a dimension and run
number along the horizontal axis, but in this case,
there's the yellow, our inner; the purple, our middle;
and the blue, our outer dimensions.
So you can imagine here that perhaps measurements
are being made at different locations on the work piece.
And I believe that this line right here
is, again, a shift change.
So talk to me a little bit about this.
What else do we notice in this case?
There's one important difference in this data compared
to this data that I'm looking for in particular.
What additional characteristics-- or what else
could you say about this data?
AUDIENCE: [INAUDIBLE] I guess the spread on the [INAUDIBLE]..
DUANE BONING: OK.
So one observation is, if we were just looking--
well, right here, this is a good one,
where inner, middle, and outer are all close together.
But your point is, generally, we're
seeing a very huge spread in the measurement
within that particular part than we do over here.
AUDIENCE: [INAUDIBLE]
DUANE BONING: Much tighter band--
so now you're talking like a band here?
AUDIENCE: Yeah, that [INAUDIBLE]..
DUANE BONING: Good, good--
yep?
AUDIENCE: [INAUDIBLE]
DUANE BONING: Interesting-- yeah,
so it's a situation here where we don't know
the specifications, so it's a little hard to actually know
which is better--
because your point you were making is, in this case,
the blue's always above--
or almost always above the purple,
always above the yellow.
And it may well be that it's meant
to have a slight taper to it.
We'd need to more.
And actually, that's a really important point
in manufacturing processes.
You can't just look at the data and always know
what's going on.
You also have to have information from design,
know what the intended results are.
So certainly, just to highlight a couple
of the things [INAUDIBLE] mentioned,
one here is, very often, we are also
interested in within part variation.
So assuming that maybe they all really
were supposed to have the same dimension,
then this spread here is very different than this
spread here.
So characterization of within part variation
may be very important in understanding that.
And then the other thing here is, unlike this--
we still have all that gook on here,
but in here we were basically having a similar band,
and it was all just kind of moving or shifting.
But the part-to-part variation did not seem to change.
The variance of the process did not seem to change.
The mean shifted.
So that's a very common characteristic
that we're worried about.
And your point here is that, in this case,
this looks like a much tighter band, or the standard deviation
of the invariance of the process in a run-to-run sense
is very different than in this process.
And so it may be somebody with a slightly steadier hands,
or a more consistent process, or a more consistent operator,
or the equipment-- maybe some change was made--
improvement to the equipment saying,
hm, that little screw is a little loose.
Let me tighten that down.
Who knows?
But something clearly changed in order
to apparently improve the process.
So those are a couple more characteristics.
AUDIENCE: [INAUDIBLE]
DUANE BONING: Yes-- question?
AUDIENCE: Professor--
DUANE BONING: Yes?
AUDIENCE: Since you mentioned that maybe
with the shift change there's some improvement
in the [INAUDIBLE] of data, and that's
why it gives us a smaller band right now.
But I noticed that the outer, middle, and inner data--
they are pretty close to each other,
as in the difference between these three sets of data
are much wider in the previous shift than in this right now.
So can we still say that maybe this--
the object we are measuring is tapered?
Because it doesn't look like it's tapered right now.
DUANE BONING: Absolutely-- well, but the point
was made that we don't actually know the design intent,
so this was a hypothesis in here.
I think it's probably unlikely that there
was an intended taper, and I'll agree with you.
But in here, it looks like there is
somewhat closer within part correspondence
in these dimensions.
OK, so let me go to one more.
Here's another one.
This is brake bending of a metal sheet.
The basic idea in this process is
we're trying to change the geometry under the application
of force in a bending process.
So we might have some tooling.
We have some piece of metal, say, and we press down,
and the intent is to create some permanent deformation
in the part.
An important component here is that, when
we remove that force, remove that pressure,
there can be spring back in that part--
so little bit of the inherent characteristics of this.
And so here's some data looking at using the same tooling,
and I believe the same set of downforce, with--
I believe this is a measurement of the resulting angle.
Alpha is not a good variable to use here, is it?
We call that just angle of the resulting component.
And what else is changing here is that the source material
is changing.
So in some cases, we're using aluminum, some cases steel--
and also with different thicknesses.
So I won't even ask you to share--
the most obvious observation is, depending on the material type,
we certainly get different angles, different degrees
of spring back.
That's the most obvious thing.
What else do you see in this?
Yeah?
AUDIENCE: It depends on the thickness of the material.
DUANE BONING: OK.
AUDIENCE: So for the same material [INAUDIBLE]
the springback, it will be more.
DUANE BONING: Right.
OK, and one could certainly imagine good physical reasons
for that, that thicker material is
going to perhaps spring back more readily.
And one could even go off and imagine
trying to build a first principle physical model that
would tell you what the resulting angle
or degree of spring back would be as a function of that.
And in fact, it's probably worth it
to go and consult some literature
and see if there is physical insight.
But if I were to ask you to actually tell me what you think
the result might be if we had a 0.45-inch steel part,
what would your first inclination be?
Would it be to go to the technical literature
and look up a physical model for this?
AUDIENCE: [INAUDIBLE] averaging between the [INAUDIBLE]..
DUANE BONING: Yeah.
You would be basically using some averaging
to deal with noise and manufacturing variation,
but you would be building a very simple empirical model based
on the data that you have, and then interpolating in some way.
And here we've only got two data points,
so you might pick the middle.
We don't know.
You might want to actually go and design some experiment
where you add a third point so you
know if there's some non-linear dependence, and so on.
So that's certainly an important thing.
There is a very clear input-output,
a deterministic effect that, again, you
would like to be able to understand enough to be
able to deal with that.
And the same may also be true of other discrete design choices--
discrete choice being the material,
whether it's aluminum or steel.
You would like to know, is it a similar trend?
One thing that's very interesting
here is it appears that the delta
as a function of those thicknesses
is about the same-- very nearly the same
in the two material cases.
And then there's a material delta.
So I'm already starting to think, OK,
I can have a very simple additive empirical model that
has a delta effect due to the material type.
It's a binary choice--
or a binary coefficient, depending on that.
And then an additional component is a function of thickness,
and maybe that function of the thickness is independent.
That delta from amine is independent [INAUDIBLE]
material type.
OK, anything else going on?
Yes?
AUDIENCE: [INAUDIBLE] there is kind
of a drop in the middle of the set.
[INAUDIBLE]
DUANE BONING: Yes-- very interesting,
and maybe a hint almost of something, but then it--
yeah.
So there might be something systematic going on there.
The human eye is wonderful for looking for patterns, isn't it?
Actually, one of the challenges often
is using both engineering judgment and when
the data is simple enough to detect trends, but also
have statistical methods when the volume of data is huge
or there are many things changing at the same time
to detect possible shifts and trends.
Anything else?
Yeah, you had more.
AUDIENCE: OK.
[INAUDIBLE]
DUANE BONING: Yes.
AUDIENCE: [INAUDIBLE]
DUANE BONING: So you're worried about this, and maybe this
also.
AUDIENCE: Yeah.
DUANE BONING: In fact, we don't even
know if that's a still component or an aluminum.
Yeah.
So in some sense here, if I were looking at this data,
I almost start to think these are relatively--
well, you'd have to ask as a function in comparison
to specifications to actually say
how well-controlled that is, but there is relatively tight bands
with a couple of these shifts, and then almost outlier points.
Now, would you think that these points
are coming from the same source of deviation that
might be at work in this band?
Might be-- but it's such a large deviation that it
starts to feel unlikely.
And another use of statistical methods
will actually be to quantify, how likely is it
that we would observe by chance alone
a deviation within the natural variation band of such a point?
And the key reason is, if you see something big like that,
there might be a [? point cause. ?]
And it's quite possible in here that the operator is
making a measurement on each part and said, whoops--
something just happened here.
Something got misadjusted on the tool.
I need to readjust it and get it back in.
Maybe there's even a little bit of something going on there.
Or it might have been a single event that
wasn't-- didn't require adjustment.
Maybe you go in, you investigate,
and you go, that one piece of source material coming in,
that piece of metal coming in was much thicker
than it was intended to be.
AUDIENCE: [INAUDIBLE]
DUANE BONING: Yeah, maybe that's a piece of aluminum right
in there, or this was-- exactly--
this was a steel [? 0.3 ?] in the aluminum set,
or steel [? 0.3 ?] in the first one of those.
Absolutely-- that's a cool observation.
Yeah, that point seems to fit down in that distribution,
doesn't it?
Very interesting-- so a couple of the points
here-- the kinds of things we might be looking for here
are deterministic and systematic kinds
of effects that we might want to use modeling for.
And then we also are starting to get
close to the statistical process control thinking, which
is to say we would like to be able to know what
the inherent or natural variation of the process
is having to do with these sorts of bands,
and then be able to detect with high degree of confidence
that something strange has happened,
and I better take action.
And that SPC, or statistical process control.
Maybe we're going to see many of the same characteristics here.
This is just a little observation
from injection molding.
So this is a plastic molded part.
We've got a wit, the part along this dimension,
and down here we've got a number of the run.
And also notice--
I'll just draw your attention here--
there are clear intentional bands here,
where this first segment here says holding time
was five seconds, injection press 40%.
And we're changing here-- this is
a holding time of 10 seconds.
Here's another holding [INAUDIBLE] of five seconds
and 60%--
and whatever is lurking under there.
I think it's a holding time of 10 seconds.
We could find out, but--
talk to me about this one.
Oh, I guess also here, notice, we're
plotting two pieces of data.
We've got a width and then some average.
So it's quite possible that what we're doing here
is this is a point width at a well-known location,
and than the average is we take that measurement
on many, many locations around the part.
Or it's possible we also actually run multiple parts,
little mini batches of-- who knows-- maybe five parts,
and I'm plotting the average across that little batch--
that five-part batch, which is another very common thing
to do in SPC. So tell me a little bit about this.
What do you see?
Yeah?
AUDIENCE: [INAUDIBLE]
DUANE BONING: Yeah.
So you're talking about this band here to here.
Right.
In fact, this might be the result
of a designed experiment, where somebody
is trying to explore the effective different process
conditions, these four different cases.
And part of the goal of that may be
to identify not only things that are closer to a mean target,
but are also more robust--
inherently robust, meaning they have a smaller delta a,
or are the process itself has a smaller
sensitivity among at that operating point to whatever
inherent disturbance, whether it be temperature deviations
or what have you.
So that's a good observation.
It's also kind of amazing to me here that the average looks
rock steady in these cases, but these individual width
measurements have a substantial deviation.
AUDIENCE: Is that the average of all [INAUDIBLE]??
DUANE BONING: We don't really know.
I don't know.
Yes, it's quite possible that this is just a plot.
So that's your point.
This may be [INAUDIBLE] suspiciously so, right?
This may be the average across that ensemble,
and all that is doing it's not saying
it's an average of multiple measurements
or multiple samples, but that's just highlighting
that there's this delta.
And that's got to be it, because--
exactly.
There's zero deviation in those average points.
OK, but these are some of the same kinds of themes.
We're, again, looking at offsets from deterministic process
conditions.
We're looking at inherent variation, ranges.
We may be doing design experiments.
You could start to think of empirical models that
would help to guide us to an optimal point, the point being
there that maybe these four conditions give you
a little bit of an exploration across two different process
parameters.
One approach would be maybe your target is 40.85 or 40.87.
You might pick the one that was closest to it.
But if you actually built a model, including
a model with continuous parameters,
you could then interpolate and drive even closer to a target.
Yeah-- another question or comment?
AUDIENCE: [INAUDIBLE]
DUANE BONING: Right, one would guess.
AUDIENCE: [INAUDIBLE]
DUANE BONING: Right.
AUDIENCE: [INAUDIBLE]
DUANE BONING: Yes.
So that's actually a very interesting point.
When we start talking a little about more about
design of experiments, about the synergistic effects,
it may not be a simple additive model.
You're saying, as we go here, from 5 to 10,
we got an increase.
Here, when we went to 5 to 10, we got a decrease.
So the effect of holding time may be completely
in the opposite direction at two different pressures--
very good point.
So that means there is some strong interaction
between those two-- or mutual dependence between those two
parameters.
Man, we're getting very abstract in this data.
We've got measurements on the output and we've got--
some number of run number on the side.
This is actually looking very similar to some other things.
We see definite shift effects.
We see mean effects.
Again, we see perhaps an outlier.
The point here was that-- and this didn't come up earlier--
we keep looking and trying to put blame back
on the manufacturing equipment or the source material.
You also have to look carefully at your measurement apparatus.
There's inherent measurement error.
Even if the parts were almost perfect,
very often, you're going to have an inherent measurement
limitation and deviations due to measurement error.
And you may have loading problems
into the measurement apparatus.
So it's not always an indication that there's part deviations.
In some sense, the measurement equipment
is manufacturing process, tool, or equipment
itself that is also subject to deviations.
And in fact, part of the applications of some
of the statistical methods we'll talk about seek
to characterize or do gauge studies of the accuracy
and capability of your measurement equipment
and the components of variation that affect that.
So for example, there's a pure reproducibility
and there's a repeatability component
to typical gauge studies.
One component, just real quickly,
would be I've loaded the part on the measurement apparatus,
and then I just fire whatever the measurement sequence is
multiple times, and think of that as a pure repeatability--
somehow the physical-- maybe there's--
it's an optical measurement.
There's scatter from extraneous light sources nearby off
of that.
And then there might be a separate component,
if I pulled that same part out, took it out, and put it back
on, and did that sequence of operations,
including the operator operations.
Then you have a reproducibility component that might be,
in fact, much larger.
So you'd need to understand, and maybe decompose and break down
to those different components.
And then, if you started to see that one
was a source of deviation, then you
might look at strategies to control or compensate for that.
Yeah?
AUDIENCE: [INAUDIBLE]
DUANE BONING: Yeah.
AUDIENCE: At one of my internships,
the company was getting a lot of [INAUDIBLE]..
They were really trying to figure out
[INAUDIBLE],, and when it went bad,
at the supplier [INAUDIBLE].
DUANE BONING: Absolutely, absolutely--
in some sense, it's often the analogy of it's not working--
oh, check the plug on the wall.
Always check your measurement equipment and apparatus
before you start diving and tearing your hair out, going
too deep into debugging of the manufacturing process.
Check each step along the way--
glad they discovered that pin problem back in the--
at the case.
Oh, here's more.
What do we see here?
Again, some output as a function of run number [INAUDIBLE]
again, maybe this is another sheet
bending kind of component.
We see effects of shift change multiple times here.
These are different shifts, maybe different shifts
or different parts.
What's very interesting here is that, under some conditions,
we have a nice stable, not drifting in time--
fairly nice, stable, or stationary mean,
with some deviation--
every now and then, maybe something
that looks outside of the distribution.
But in some cases, holy cow--
that's a very different effect than what's going on in here.
And it's back to this is a very clear drift that is definitely
not just a one-time shift.
Yeah?
AUDIENCE: When I looked at the [INAUDIBLE],, and the first
[INAUDIBLE].
I'm wondering if maybe the [INAUDIBLE]..
DUANE BONING: Perhaps-- or maybe this worker
finds a way that corrects for it in a comparatively effective
fashion, whereas the strategy used in this case over here
is not nearly as effective or quick at trying to get back,
if that was the target.
And so in fact, one of the challenges often in SPC
or other control algorithms-- or not even algorithms,
but control practices, especially involving people--
is one shift or one set of operators or engineers
in charge of a process may learn things
that don't necessarily get captured
and conveyed to other shifts.
OK, so what is going on in lots of these examples is perhaps
the most important point here--
is that, very often, there are two highly different
characteristics that keep coming up.
There's systematic or deterministic effects
having to do with the type of the material, settings
on the equipment, choice of process conditions.
And then there's also random components.
There's the inherent spread.
And a key point here is inherent spread you can almost never
completely get rid of.
I guess, philosophically, you could ultimately maybe appeal
to quantum mechanics, that there would always be some deviation.
I think very rarely--
perhaps in some semiconductor processes,
we do approach truly inherent physical randomness.
But generally, it's that the understanding is limited
in what is contributing to that small degree of spread,
or it's impossible to completely control those--
that inherent randomness.
But then we also have big disturbances.
They may still be random events, but they are not
this little band, but they are big events with a clear cause.
We may not always be able to discern the cause,
or may not always be able to discover it, but in many cases,
we've been hypothesizing about many
of these big shift changes or a big drift.
And a bit of investigation and more knowledge
about the process would readily show us
what's going on in those cases.
And in those, when there's something
systematic or deterministic going on,
the whole idea is you make some changes to the equipment
to stamp those out.
And what we're often going to be dealing with
is designing experiments and whatnot
to try to understand systematic components and eliminate them,
maybe as part of the initial design
implementation of a process.
Or in an ongoing process, when we see a deviation,
we've got to go and debug it, find out what caused that,
and eliminate it.
And the natural occurrence then is
that, over time, as you have learning about your process
and manufacturing system, you're generally squeezing out
these deterministic sources.
It's something-- was repeatable, systematic,
and therefore, you can come up with strategies
either to eliminate it or to compensate for it.
Maybe you have to have a control strategy that
recognizes that source of variation
and keep squeezing it out.
And what you generally are left with
are these random components, hopefully
in a very nice, narrow band.
But we still need to understand and model those,
and so that's actually where--
another place where the statistical analysis tools are
really crucial, because then we want to be able to understand,
what is the natural or reducible--
irreducible component that we don't know how to stamp out one
at a time--
and use that information to be able to detect
in that small scatter when something unusual happens.
And that's the idea of statistical process control.
So what we want to do is characterize the process--
particularly just the random component,
assuming that we've been able to squeeze out
many of those other systematic components.
Why do we want to characterize that process?
There's some of the things that I've mentioned,
and that you've mentioned.
If I see one of these points that
looks a little bit different than the distribution,
we said that [? by i, ?] but very often, the question is,
maybe there's a point that's a little bit above some
of the other points.
Then you would often want to ask the question,
do I really believe the output changed, or is it
part of that natural variation?
And if we start to think about some natural variation
and models for that, including distributions--
which we'll get to in just a moment--
then we can start to ask that question
in a very systematic way, in terms of actual probabilities.
Other reasons is we may go back to,
did the input actually cause the change?
Did I make some output change?
A very important point here is, how confident
are we have these answers?
Very often here, we were appealing to it
looks like that's a bigger change, or this is--
it was a mean offset.
Well, that's OK to try to convince your management
or convince your operators that, look, something changed,
but in many cases, you'd actually
like to be able to quantify how confident you are that that was
either a deviation point or that some specific decision that you
made had a true effect.
And so the probability and statistical tools
are really, really important for that.
So what are some ways that we can actually
characterize and model the random components?
Well, that's where we're getting into random processes
and random variables.
And I hope you're seeing this is the basis
for statistical process control I've referred to.
And we'll dive into some of the tools and techniques, as well.
Design of experiments-- so if we were
making that pressure change or that time
change, how big of an effect is it?
How certain are we within the random natural variation
that the deviation-- or the controlled decision I
made actually had a big effect?
And then finally, even with compensation strategies--
like this-- one of the very first things that we said--
in that oscillating case, there might be inherent noise.
And it's quite possible that somebody
could be making process control feedback
changes to try to compensate.
But if they're trying to compensate for something that
doesn't have, in fact, a connection between the control
decision and the true source of noise,
they may, in fact, be injecting more noise in the process.
So understanding actually how feedback control works
when we've got random, and not just the systematic component,
is very important.
What are some ways of describing randomness?
Actually, philosophically, there's
a couple of very different ways of thinking--
if you open up a statistic book, in the first few pages,
we'll talk about this.
And from a manufacturing perspective,
well, we're really dealing with data.
In many ways, the data tells us a lot.
And one approach here is to look at actual large ensembles
or collections of data, and look at histograms,
frequency distributions.
And we'll look at some examples of that.
A very different approach is to start with a pure probability
model that assumes there is a universe of
and an infinite number of samples,
and there are characteristics of that infinite number
of samples.
And that gives us the tools to put empirical observations,
like frequency histograms, on a mathematical basis that we
can reason with.
So for example, a way of--
let me jump forward here-- this is some thermoforming data.
Again, this is some dimension, and then,
over this collection of parts, we have a--
this bar chart, this frequency histogram.
And what that's really saying is, for each of the bends
in that histogram, I'm simply plotting the frequency--
or relative frequency that some output was
between the bounds of that bin.
And so what we are doing is converting from historical data
into bins.
This is still discrete bins, but then we're
associating that and interpolating forward
to say, well, the probability in most applications--
if I were to do some more part, we
think that this histogram, in some sense,
represents a larger set of data, and I
can use it to talk about the probabilities of occurrences
of new data in here.
So what do you see in here?
Whoops-- there's a little insert here with the raw data.
Now, if I hid that from you--
it's hard for me--
I'm trying to put my hand over this.
If I hid that from you here, and I hid that from you here,
and you just looked at this data, typical things you would
say about it--
it looks normal, or Gaussian distributed.
Maybe you would think almost it's completely random.
It's got to mean and the deviation.
It's got a normal distribution.
Now look back at the data.
It's kind of drifting around.
Just a peek forward to other things to be cautious about--
don't always believe or just look at your statistics.
Always go back to the raw data, because if this
were a characteristic of my process,
I would say there is a random component,
I think, here, but then I'm also seeing maybe
some systematic drift that--
I'd like to go in and squeeze that out.
And then, if I were able to get that out of my process,
then I might have a random component
that doesn't have that deterministic or systematic
component.
And it might have a distribution,
but it might be much, much tighter.
So an important point here is purely
empirical data doesn't always--
it may look random, but it may have embedded in it--
if you were to deconvolve all of the sources of variation,
this big random--
or this big normal distribution may have mixed into it
lots of different systematic components,
as well as underlying natural variation.
Imagine, for example, that I actually went in
and I got rid of this long-term drift.
Would you believe that the remaining variation is all
random?
I bet I could go and I could plot it, and I bet it'd look--
the remaining deviations would look mostly normal as well.
You think so?
Well, who knows?
You might go in and do some of the other kinds of hypotheses
about systematic sources that we were talking
about in the other cases.
For example, it looks like there may be a bit of an up,
down-- there might be something cyclical or periodic going on.
That might be another systematic source.
It could masquerade as a random normal distribution.
So there's always that mix in.
However, in many cases, maybe you also want to live with--
it's not worth it to you, or too expensive to go
in and eliminate, or you tried and we're not
able to eliminate also some of these wandering drifts.
And in that case, maybe the best you can do
is at least characterize that remaining variation, which
has a systematic component.
But you would still like to have a probability model for it
so that you could detect deviations that were outside
of that variation source.
So if I saw a big shift up, I would
say I've got a point way out here
in the tail of the distribution that's highly unlikely.
And you'd still want to use that.
So now, we might want more compact ways of describing
this distribution.
Actually, in the age of computers,
I think, with lots of big disks and lots of great data
collection, one of the most underutilized opportunities
is keeping around all the raw data,
and actually using empirical distribution.
So in some sense, it's kind of a holdover from the days
when it was really tough to keep track and use
vast amounts of data.
And we almost always would start to say,
OK, I'd like a compact way of describing this,
and I want to describe it as something that only has
maybe a couple of parameters.
It has a mean and some variance component.
But that's a different philosophical point.
There's, actually, I think, a huge amount
of statistical methods that are underutilized
that deal with sampling or resampling from raw data
that actually can capture subtle things going on in the data.
We're, in this course, not going to be using those things.
We're pretty much going to seek compact descriptions
of randomness in ways that we can reason about.
There are data-intensive ways of doing
the same kinds of reasoning.
So one important point here is we're often
going to take this discrete binned kind of information--
we want to characterize it with perhaps a continuous parameter
variable and a compact distribution,
such as a normal or Gaussian distribution.
And this is especially good when the outputs are themselves
truly continuous parameters, like a width,
a dimension of the part.
And one point here, just to remind you-- especially
when we go to continuous parameters,
you've got to be a little bit careful
about talking about the probability of your part
or your variable taking on any specific value
in that continuous distribution.
Back here when we had things binned,
I could talk about the probability
of measuring something in that bin,
within that range, 1.485 to 1.487.
But now, if I were to ask, what is the likelihood
that I measure exactly 1.486, be careful.
That is a very, very discrete value that--
when we talk about continuous probabilities,
the probability of getting any one value--
and it's truly continuous--
is 0.
You're never going to see exactly that.
Now, with real measurement tools,
we never have perfect continuous measurements either.
There is always a precision and a discreetness
capable from the tool.
So this is an abstraction dealing
with the mathematical niceties.
I'd just point out it's not exactly true
and measured reality and data.
So very often, when we talk about probabilities,
we got to be careful.
The probability of any one value is
0 what we talk about really is the probability
of our measurement or whatever being in some range that's
inherent in the distribution.
But we can also talk about, for example, the probability
that the value be in the range from minus
infinity up to some value y star,
and that is a reasonable question
to ask in terms of probabilities.
And that is embedded in this notion
of a cumulative probability function, or cumulative density
function, or CDF.
Just to illustrate that a little bit more,
we can do the same thing, have a cumulative frequency
with discrete data.
So if I had that underlying histogram here,
I could also build a cumulative frequency histogram
that simply was giving me the probability
that a part or measured value was up to, but less
than some particular value.
And that's simply the integration
across the PDF up to that cumulative density function.
And we can do the same thing that we
did with the discrete probability frequency
relationship.
So we can approximate the frequency with a PDF.
We can also look at the cumulative frequency
and approximate that with a cumulative density
function, or a CDF.
So we can have that kind of an equivalent.
And those equivalents also, again, help us interpolate
and ask probability questions also
on values that may not have been exactly lined up with our bins.
So very often, we're going to be, I guess,
talking either about the density function--
the PDF of some variable--
or the cumulative probability.
And there is this inherent relationship between the two.
I describe the cumulative probability as an integration
up.
The probability that some value is less than or equal to
would be an integral up to some x star.
And so that value right there corresponds to that.
And then the inverse is--
relationship is true as well.
If I had a cumulative density function,
I can differentiate that to get down to the probability density
function.
So this is some basic statistical concepts
that you probably have seen before.
Many of you have seen it in much more detail in 2.853.
OK, so very often, the histogram does
suggest an underlying approximate or convenient
probability density function.
Very often, we've talked about normal distributions,
but a key thing to look at is always look at your raw data,
because there may be other perfectly reasonable continuous
or discrete probability density or probability mass
functions that apply in the discrete case,
such as a uniform distribution.
And so one of the things that we'll see
is the family of some typically emerging or typically observed
kinds of distribution models.
So very often, what we're doing is
we're gathering data, looking at these histograms,
and looking to see, is there a consistent pattern, such
as we saw with the underlying normal distribution?
And we're often then trying to intuit
what that reasonable underlying distribution may be.
There are ways-- and we may touch on them a little bit--
to be able to test whether or how
good a particular distribution model may fit with that data.
And on the problem set, in fact, you'll
explore things like a qqnorm plot.
And I'm not going to go into detail here,
but essentially, that--
assume a model-- for example, what the probability density
function probabilities would be associated with certain kinds
of observations, if it were a normal distribution--
and then you plot your data against that
and start to see how well the data fits
that particular model.
Let's go back to that original set
of data, the [? c and c turning. ?]
If I were to plot that as a distribution,
I might get something like this.
I might argue that my bins may be too coarse here,
but what do you see?
This is a little tricky, right?
If I were to plot that data as a distribution,
I might be tempted to say, OK, that's one big Gaussian
distribution.
You could try that.
But again, if we look back at the data,
another interpretation here is, really--
you can start to see it--
it's bimodal.
It's a mix of two distributions.
So I guess the simple point here is don't always
assume it's going to be just a single distribution that's
underlying it.
There may be a mixture.
Similarly, here's another set.
And very often, again, plotting your empirical data
in time order tells you a lot more, but also
be sensitive to those kinds of questions.
I could have started the whole class off and never
shown you the raw data, and just started plotting distributions.
And in some cases, things might be more observable there.
Usually it's easier to see perhaps lurking up here,
when there's such a very clear time shift.
But here, this is the histogram associated with these,
and it sure looks like one normal distribution here
and another normal distribution here.
Now, by the way, if these things were ping
ponging back and forth, or they were randomly picking
and it was one of those source material deviations,
it might actually be hard to detect that,
whereas the bimodality-- the fact
that I've actually got what looks like two mixtures-- may
actually appear more easily in a histogram kind of form.
There's another example.
Here we start to see, again, these two shifts,
and now you can start to convert these perhaps
to two statistical distributions, two
Gaussians with different means.
And now you can also estimate parameters,
like what the mean shift is.
So once you've got distributions,
now you've got mathematical approaches
for being able to estimate those parameters.
So this is just going back to some of that other data.
OK, so some of the key points here out of this portion
is, even if there's no strong input effect,
there may still be disturbances.
And we'll often see a consistent histogram pattern emerge.
And what we often want to do is use
that either to detect deterministic deviations
and squeeze them out, or once we've done all those process
improvements and gotten those all squeezed out,
we may still have an underlying inherent distribution.
Maybe it's random variation associated
with a Gaussian distribution.
And the key idea is we can even use,
and importantly, we want to use knowledge of that underlying
pattern.
It's not a systematic pattern.
It's a random pattern associated with that distribution--
so that we can predict behavior, and in particular, set
limits on normal behavior.
This is a nice case.
Maybe this is what we'll consider normal behavior.
And again, I want to be able to statistically
say, what's the likelihood that some
of these points or a new point belong to the natural variation
or not?
So we're going to typically use analytic probability density
functions to do that.
I think we've already talked about most of these things.
I've been emphasizing, by the way, continuous values,
these width or geometry parameters,
but there's also a set of different statistical
distribution than the normal distribution associated
especially with the occurrence-- random occurrence of point
defects.
And a little later in the term, we'll
actually do some yield modeling dealing
with things like Poisson distributions
and in these sorts of discrete distributions
that are more appropriate to discrete events
or discrete occurrences.
We're going to be, in the first part here,
dealing pretty much with continuous parameters.
So this is just laying out, again, a little bit.
We've talked about some of these.
I want to build up just very briefly,
get you going on some of the basic definitions.
And then Chapter 4 in Spanos really articulates
these much more.
So we have our probability density function
and cumulative density function, and there is that relationship
between the two.
And then we can ask about certain moments
or characteristics of these continuous probability density
functions.
These should be familiar to.
We can ask, what is the mean or the expected value of one
of these distributions?
And here I'm using t as a stand-in for time or run
number.
And if I have a probability density function associated
with the random variable, and even time, then
there we have a definition of the expected value
of x as a function of time.
Now, I'm kind of being pedantic here and giving
this definition, because we also saw trends--
long-term trends in our data.
And we could ask what the mean is as a function of time.
Again, we're in a situation where I'm trying
to squeeze out those drifts.
So usually we're in a situation where we want to avoid
non-stationary drifting processes or analyses where
some of these functions--
the probability density function and its parameters,
like the mean--
drift over time.
We typically want to be having stationary processes where
the mean and other moments, like variance, are
independent of time.
So in the stationary case, which is what we're mostly
going to be dealing with, now I've
removed the time dependence on the underlying
statistical model.
Let's say, from time to time, it should be stable.
And then we have a mean, this mu of x-- the true,
or theoretical true mean--
that we would calculate with the same formula.
But again, the idea of the stationary process--
important characteristic there is that the mean is a constant,
and not wandering or drifting.
Similarly, in the case when it's a stationary process,
the variance, defined here as the second moment--
the expectation of the distance squared of your data
from the mean--
[INAUDIBLE] expectation of that--
is the variance.
That's just the definition.
And in the stationary case, that is also
a constant of the process--
has a nice alternative little description,
if you work through the mathematics,
where it can also be calculated as the expectation
of the square of your data minus the mean squared.
By the way, you can ask the same questions-- variance, mean--
for any continuous distribution.
They're perfectly good moments.
In some cases, they completely characterize the distribution.
In other cases, there may need to be additional moments.
And an example here is we could ask,
what is the mean of a uniform distribution-- uniform, where
the probability density is equal across an entire range?
And so one could calculate those as well.
Everybody is seeing the normal distribution.
Again, we can actually have an explicit description
for what the PDF is.
We'll often be talking about the unit standard normal or unit
normal distribution.
The full distribution here actually has the mean in it,
and it has the variance in it.
So for your particular process, those
may be very different for whatever collection of data.
In order to be able to talk generically about properties
of the normal distribution, and to use tabulated values and so
on, this normalization, where we subtract off the mean
to get a 0 mean description, and then we
divide by the standard deviation,
gives us the unit normal distribution.
And that's exactly always this distribution
in terms of z units.
Now we've made it independent to the mean, independent
of the variance.
And the exact values, the probability densities
for any values of z are well-known and tabulated.
So we've made it 0 mean and made it unit norm--
unit variance.
There is also a cumulative distribution here.
I'm describing this as p of z, so now we
can ask the cumulative distribution
function for that unit normal, and plot that.
For the normal PDF, what's nice is
this is the whole distribution, and all
we need are those two parameters, mean and variance,
and it completely characterizes the process.
And then there's some statistical operations
that we can do for when we-- that are really convenient when
we have normal distributions.
For example, if we have multiple effects, we can look,
and the sum of multiple normal random variables
still has a normal distribution.
So I'm just giving you a little bit of a peek here.
You can do some reading, which is probably
more effective than me just flashing by these things.
There are important operations, for example,
where, if we have an output y that
is a sum of multiple random variables,
there are things like the mean of the overall y
is simply the sum of the mean of each
of these individual random variables.
And if there's a scaling in the mean, that simply scales.
There are also operations when these--
if these are independent random variables, the variances add.
And you've got to be careful any constant.
If you run that through the definition of variance,
those are also squared.
So you'll see the basic description
of some of these mathematical operations
and these distributions in May and Spanos.
And you'll also start to get a feel
for using these distributions to ask
some very standard questions, like what
is the probability that you would
observe a value in some range?
So I think, with that, we'll close.
If you want to get started on problem set 2,
there's some problems that--
this is just basically giving you some familiarity
with manipulating things with Gaussian distributions
and other statistics.
And then we'll start building up more detail next Thursday.