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DUANE BONING: OK.

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As you'll see with
today's lecture,

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I made a small change
in the lecture sequence.

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I basically combined the last
part of Tuesday's lecture

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00:00:32,060 --> 00:00:35,540
with what I was going to talk
about next week because they

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really go together better
anyway, which is multivariate

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SPC together with a couple of
the advanced control charting.

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And so next week we'll do
the yield modeling lecture

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00:00:47,210 --> 00:00:51,840
rather than try to
squish that in today.

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00:00:51,840 --> 00:00:55,050
So what I want to
do today is first

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have a very brief warm up.

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Just get us back into
the swing of things

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with conventional control charts
and then talk a little bit

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about these alternative
control charts

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00:01:03,080 --> 00:01:07,520
with a little bit of increased
sensitivity for rapid detection

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of shifts, as well as drifts.

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But primarily
targeted at shifts.

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And these are the moving
average-- exponentially

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00:01:16,790 --> 00:01:19,700
weighted moving average and
the cumulative sum chart.

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00:01:19,700 --> 00:01:24,260
And then we'll move into
multivariate SPC, which

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00:01:24,260 --> 00:01:28,940
is a little bit of a tutorial
on multivariate statistics

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in general.

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00:01:30,900 --> 00:01:31,910
So that's the plan.

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So to get us warmed up we'll
do a rapid design of a control

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chart.

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00:01:38,750 --> 00:01:40,880
How to do this
relatively quickly.

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00:01:40,880 --> 00:01:43,400
I will tell you that
we have a model based

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on lots of historical
data of this process

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that when it is in
control it's operating

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as a normally distributed
process with a mean of 1

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and a variance of 1.

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00:01:54,380 --> 00:01:58,280
We're going to be taking
samples every nine parts.

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00:01:58,280 --> 00:02:00,720
And then the next 9 and so on.

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So what should the--

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00:02:02,290 --> 00:02:03,200
here's the easy one.

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Wake you up in the morning.

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What should the center line be?

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00:02:07,682 --> 00:02:08,273
AUDIENCE: 5.

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DUANE BONING: 5.

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00:02:08,940 --> 00:02:09,710
OK.

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00:02:09,710 --> 00:02:14,700
Everybody-- how about
the plus minus 3 sigma?

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00:02:14,700 --> 00:02:20,910
We're quite happy
with the 1 in 370 ARL

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with normal typical plus
minus 3 sigma control limits.

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Where should the plus minus
3 sigma control limits be?

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00:02:31,840 --> 00:02:33,190
I hear a 6 and 4.

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Why?

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AUDIENCE: [INAUDIBLE].

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DUANE BONING: I'm sorry?

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AUDIENCE: [INAUDIBLE]

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DUANE BONING: OK.

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Which is?

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00:02:42,396 --> 00:02:48,252
AUDIENCE: Mu plus
sigma [INAUDIBLE]..

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DUANE BONING: Mu plus
sigma plus minus 3 sigma.

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3 sigma over square
root of sample size.

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And so the 3 sigma and the
3, which comes from the root

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[? end ?] make it a
nice plus minus 1.

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And there are some data.

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OK.

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So I plotted 99.

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We ran the process 99 runs.

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We have 11 samples.

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Here's the question.

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Is the process in control?

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We have a yes.

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We have another yes.

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I contend you don't know.

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It's a little bit tricky here.

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AUDIENCE: Not the 1 [INAUDIBLE].

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DUANE BONING: Yeah, I guess.

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That's actually not
what I have in mind.

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You could go in and start down
the path of looking and saying,

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00:03:50,650 --> 00:03:54,450
OK, what's the probability
of having, for example three

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rising points in a row?

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And all the other [INAUDIBLE].

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But I'll say none
of that's happening.

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So I will say that the mean--

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there's nothing too strange
happening with the mean

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that you can tell
with this chart.

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Is the process in control?

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AUDIENCE: [INAUDIBLE]
variance [INAUDIBLE]----

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DUANE BONING: Yeah.

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AUDIENCE: [INAUDIBLE]
variance is getting huge

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00:04:18,686 --> 00:04:20,172
compared to earlier points.

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DUANE BONING: Oh.

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00:04:20,880 --> 00:04:22,995
So you're saying the
variance is getting larger.

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AUDIENCE: [INAUDIBLE]
variance-- yeah.

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00:04:24,210 --> 00:04:25,043
DUANE BONING: Maybe.

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00:04:25,043 --> 00:04:26,250
It's hard to tell with this.

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00:04:26,250 --> 00:04:28,290
This Is a mean chart.

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00:04:28,290 --> 00:04:31,500
All it's telling you is
the tracking of the mean.

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00:04:31,500 --> 00:04:34,410
We said that the process
model is that we have

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00:04:34,410 --> 00:04:35,940
this mean and this variance.

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00:04:35,940 --> 00:04:38,550
And so I guess the
main point here

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is you don't really know if
you're just tracking the mean.

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00:04:41,670 --> 00:04:46,300
What's happening also with the
variance within your sample?

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00:04:46,300 --> 00:04:48,730
And in fact, on
the next page this

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is the raw data-- all 99 points
where we grouped 9 by 9 by 9

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00:04:53,830 --> 00:04:56,120
to get the sample.

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00:04:56,120 --> 00:05:00,970
And if you saw this, you might
be a little less comfortable.

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00:05:00,970 --> 00:05:03,250
Now you could always go
back and look at raw data.

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00:05:03,250 --> 00:05:05,720
And raw data is always
really good to do.

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00:05:05,720 --> 00:05:07,390
But what you would
also want to do

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00:05:07,390 --> 00:05:11,930
is a typical S chart or a range
chart or something like that.

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00:05:11,930 --> 00:05:13,570
And in this case,
if you design the S

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00:05:13,570 --> 00:05:19,030
chart with those
apriori pieces of data--

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00:05:19,030 --> 00:05:25,150
sample size is 9 and the
process variance is 1--

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00:05:25,150 --> 00:05:29,080
with the C4 correction factor
and so on-- all of that just

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00:05:29,080 --> 00:05:32,030
out of the formula--

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00:05:32,030 --> 00:05:34,210
you get these control limits.

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00:05:34,210 --> 00:05:37,630
And sure enough,
at about sample 10

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00:05:37,630 --> 00:05:39,070
we're above the control limit.

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Sample 11 is even more
above the control limit.

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And that's indicative
of essentially

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the increased variance
of those two groups.

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00:05:49,400 --> 00:05:52,000
And I will tell you, I
generated this data with JMP.

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00:05:52,000 --> 00:05:56,470
And sure enough, right
at the whatever it was--

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00:05:56,470 --> 00:05:59,020
the 77th point--

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00:05:59,020 --> 00:06:03,220
I re-sampled from a normal
distribution with something

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00:06:03,220 --> 00:06:06,350
like a variance of 1.9 or
2 or something like that.

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00:06:06,350 --> 00:06:12,550
So the main point here was you
need to monitor both aspects--

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00:06:12,550 --> 00:06:16,720
both the mean and the variance
to know whether your process

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00:06:16,720 --> 00:06:23,080
model is still operative.

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00:06:23,080 --> 00:06:25,060
We're saying that for
a normal distribution

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00:06:25,060 --> 00:06:28,600
the mean and variance are
really telling you everything.

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00:06:28,600 --> 00:06:30,520
Now, the other sub
point in here is

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that if you were actually
looking at this run data,

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there is a lot of noise.

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00:06:34,900 --> 00:06:36,970
There is inherent process noise.

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And part of the
goal here is you're

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00:06:38,860 --> 00:06:43,630
trying to detect a mean
in the face of that noise.

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00:06:43,630 --> 00:06:47,770
And so one way of thinking about
sampling and calculating x bar

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00:06:47,770 --> 00:06:50,510
is that, indeed,
you are filtering.

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00:06:50,510 --> 00:06:51,370
OK.

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00:06:51,370 --> 00:06:55,030
So you want to
filter to suppress

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00:06:55,030 --> 00:06:58,120
the additional variance
effect when you're

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looking for the mean phrase.

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00:07:00,880 --> 00:07:04,930
But again, you've got to also
monitor for the variance.

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00:07:04,930 --> 00:07:08,170
But this noise makes
it a little bit tricky

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00:07:08,170 --> 00:07:11,400
in terms of being able
to detect the means.

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00:07:11,400 --> 00:07:12,790
So you want to filter.

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00:07:12,790 --> 00:07:17,020
And one of the ideas of these
alternative control charts

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00:07:17,020 --> 00:07:20,860
is explicitly think about
the filtering action

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00:07:20,860 --> 00:07:22,030
that you're doing--

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00:07:22,030 --> 00:07:25,390
a little bit almost from
a more signal processing

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00:07:25,390 --> 00:07:27,970
point of view, but
not necessarily

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00:07:27,970 --> 00:07:33,670
have to wait for an
additional nine runs

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before you call that a
sample, aggregate it,

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00:07:36,670 --> 00:07:40,630
and then plot it, and then
wait for another nine runs.

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One alternative approach here
is to take a running average.

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So I take a window, say, of
9 points and measurements,

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00:07:51,010 --> 00:07:54,400
calculate my statistics,
and then for each new run

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move my window over and
recalculate my statistics

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00:07:58,000 --> 00:08:00,070
so that I'm in
some sense updating

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what my best guess of the
current state of the process

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is after every run.

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00:08:08,380 --> 00:08:10,780
And the advantages of
this are that you can get

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more averages per unit of data.

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OK.

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00:08:16,000 --> 00:08:23,750
You can go ahead and use the
run data alone if you wanted to.

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00:08:23,750 --> 00:08:27,140
You could actually
plot for example,

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00:08:27,140 --> 00:08:30,070
the individual measurements and
even do a hybrid sort of thing

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where you did a
running average just

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for the standard deviation.

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But the real advantage of these
is they essentially improve

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your ability to detect.

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00:08:40,610 --> 00:08:43,159
They improve the
filtering action

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potentially because you can
play with that window size

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to filter more but also be able
to react potentially faster.

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So I'm going to
describe a couple

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of these alternative charts.

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00:08:57,527 --> 00:08:59,360
And I'm going to focus
really on the running

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average detection of the mean.

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00:09:01,770 --> 00:09:06,860
But the same idea also applies
to filtering of the variance

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00:09:06,860 --> 00:09:08,360
across multiple runs.

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00:09:08,360 --> 00:09:17,810
So there's also running S
charts or moving range charts

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00:09:17,810 --> 00:09:20,340
and those sorts of things.

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00:09:20,340 --> 00:09:23,330
So don't forget the lesson
we just learned a minute ago.

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00:09:23,330 --> 00:09:25,310
You got to monitor
both mean and variance.

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00:09:25,310 --> 00:09:29,030
But for the next 10,
15 slides I'm only

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00:09:29,030 --> 00:09:33,300
going to talk about running
averages on the average.

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00:09:33,300 --> 00:09:35,730
So one idea of the
running average

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00:09:35,730 --> 00:09:38,450
is a simple running
average or moving average

190
00:09:38,450 --> 00:09:43,880
where we simply take
some n values of the past

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00:09:43,880 --> 00:09:46,130
and calculate say
a simple average.

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00:09:46,130 --> 00:09:48,830
And similarly, we can do
that for the variance.

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00:09:48,830 --> 00:09:52,340
So in that case, this
simplest picture,

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00:09:52,340 --> 00:09:54,500
this simple moving
average is just

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00:09:54,500 --> 00:09:57,080
an equally weighted window.

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00:09:57,080 --> 00:09:58,640
You pick your window size.

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00:09:58,640 --> 00:10:02,300
Maybe it's a window
size of 9 and you

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00:10:02,300 --> 00:10:04,760
calculate a new moving
average statistic, which

199
00:10:04,760 --> 00:10:06,800
I've labeled m sub i here.

200
00:10:06,800 --> 00:10:10,520
And so that's pretty
trivial formula.

201
00:10:10,520 --> 00:10:12,140
Now how do you set
the control limits

202
00:10:12,140 --> 00:10:15,890
on a moving average chart?

203
00:10:15,890 --> 00:10:19,970
Well, you want plus
minus 3 sigma control

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00:10:19,970 --> 00:10:23,990
limits based on the
variance that you

205
00:10:23,990 --> 00:10:27,080
expect in your statistic.

206
00:10:27,080 --> 00:10:28,220
Right?

207
00:10:28,220 --> 00:10:32,750
So previously we'd
have sample size of 9.

208
00:10:32,750 --> 00:10:35,180
And we'd say, well,
what's the variance?

209
00:10:35,180 --> 00:10:39,920
It's 1 over-- it's
sigma over root n

210
00:10:39,920 --> 00:10:42,980
is the standard
deviation associated

211
00:10:42,980 --> 00:10:45,110
with our normal sample size.

212
00:10:45,110 --> 00:10:47,090
And that same formula
ends up working here

213
00:10:47,090 --> 00:10:49,280
because we've got
equal weighting.

214
00:10:49,280 --> 00:10:50,360
Right?

215
00:10:50,360 --> 00:10:57,590
So if we simply apply, again,
assuming each of the points

216
00:10:57,590 --> 00:11:00,950
are truly in process control
and I do not have correlation

217
00:11:00,950 --> 00:11:04,130
in time between them--

218
00:11:04,130 --> 00:11:07,010
it's very easy to see
what the variance is.

219
00:11:07,010 --> 00:11:10,640
Now you'll notice
something a little strange

220
00:11:10,640 --> 00:11:13,380
right here at the beginning.

221
00:11:13,380 --> 00:11:17,420
This is kind of an artifact
of the startup of the chart.

222
00:11:17,420 --> 00:11:20,660
Let's say I have just started
the use of the control chart

223
00:11:20,660 --> 00:11:23,370
and I've only got
five samples so far,

224
00:11:23,370 --> 00:11:26,960
but I've got a moving
average window of 9.

225
00:11:26,960 --> 00:11:28,790
Well, I don't have
nine data points yet.

226
00:11:28,790 --> 00:11:32,690
I have to form my moving average
really with a window size

227
00:11:32,690 --> 00:11:34,590
only of 5 so far.

228
00:11:34,590 --> 00:11:37,670
And because I'm taking
the variance divided

229
00:11:37,670 --> 00:11:41,930
by 5 instead of divided
by 9, the variance

230
00:11:41,930 --> 00:11:45,050
of my average of a five
point sample is larger.

231
00:11:45,050 --> 00:11:48,500
And you can see
essentially each sample

232
00:11:48,500 --> 00:11:50,840
until I get to my
full window size of 9,

233
00:11:50,840 --> 00:11:53,610
has slightly larger
control chart limits.

234
00:11:53,610 --> 00:11:54,110
OK.

235
00:11:54,110 --> 00:11:57,630
So that's what's going on there.

236
00:11:57,630 --> 00:11:58,130
OK.

237
00:11:58,130 --> 00:12:00,320
And you can also
sort of see here now

238
00:12:00,320 --> 00:12:02,700
the filtering action at work.

239
00:12:02,700 --> 00:12:10,140
Notice that each
of these samples

240
00:12:10,140 --> 00:12:13,770
don't vary dramatically
from the previous one.

241
00:12:13,770 --> 00:12:18,540
They are no longer uncorrelated
from one sample to the next.

242
00:12:18,540 --> 00:12:21,270
In fact, with this
equal weighting,

243
00:12:21,270 --> 00:12:27,720
every new time point
shares 88% of the data

244
00:12:27,720 --> 00:12:31,340
with the previous data point.

245
00:12:31,340 --> 00:12:34,410
So there's a huge amount of
correlation from one point

246
00:12:34,410 --> 00:12:35,740
in time in the next.

247
00:12:35,740 --> 00:12:40,470
So you should not
try to use things

248
00:12:40,470 --> 00:12:44,070
like the WECO rules
associated with three data

249
00:12:44,070 --> 00:12:47,760
points in a row or four or five
data points in a row rising.

250
00:12:47,760 --> 00:12:51,690
Those things are
right out because this

251
00:12:51,690 --> 00:12:54,990
is a running control chart or
moving average control chart.

252
00:12:54,990 --> 00:12:57,240
And you've now got correlation.

253
00:12:57,240 --> 00:13:01,200
It's benefit though is that
it is doing that filtering.

254
00:13:01,200 --> 00:13:05,850
And you can take
whatever current point

255
00:13:05,850 --> 00:13:08,670
that you get as a
sort of your best

256
00:13:08,670 --> 00:13:13,440
guess of what the current
state of the overall mean is.

257
00:13:13,440 --> 00:13:16,560
And so you would
trigger on alarms

258
00:13:16,560 --> 00:13:18,140
above or below the
control limits.

259
00:13:21,920 --> 00:13:25,520
So simple moving average--
this is relatively simple.

260
00:13:25,520 --> 00:13:30,620
Any questions on this
before we do a unequally

261
00:13:30,620 --> 00:13:33,790
weighted moving average?

262
00:13:33,790 --> 00:13:35,135
Pretty clear.

263
00:13:35,135 --> 00:13:36,760
AUDIENCE: I have a
question, Professor.

264
00:13:36,760 --> 00:13:37,510
DUANE BONING: Yes.

265
00:13:37,510 --> 00:13:38,260
Good.

266
00:13:38,260 --> 00:13:41,470
These questions have been
wonderful by the way.

267
00:13:41,470 --> 00:13:45,220
The last couple of classes, I
think the questions have really

268
00:13:45,220 --> 00:13:46,310
been right on target.

269
00:13:46,310 --> 00:13:47,780
It's been a lot of fun.

270
00:13:47,780 --> 00:13:50,520
So we have high expectations
for your question.

271
00:13:50,520 --> 00:13:52,658
[LAUGHTER]

272
00:13:53,838 --> 00:13:54,380
AUDIENCE: OK.

273
00:13:54,380 --> 00:13:57,110
So that's a lot of
pressure, but my question

274
00:13:57,110 --> 00:13:59,300
is that for the
window size on 9,

275
00:13:59,300 --> 00:14:02,840
so do the two windows
next to each other-- they

276
00:14:02,840 --> 00:14:05,630
overlap by 8 data points?

277
00:14:05,630 --> 00:14:07,490
DUANE BONING: Right.

278
00:14:07,490 --> 00:14:08,040
AUDIENCE: OK.

279
00:14:08,040 --> 00:14:08,540
That's it.

280
00:14:08,540 --> 00:14:11,710
DUANE BONING: That's
exactly right.

281
00:14:11,710 --> 00:14:12,710
AUDIENCE: OK, thank you.

282
00:14:15,320 --> 00:14:18,500
DUANE BONING: Now, one problem
with that overlap of eight data

283
00:14:18,500 --> 00:14:24,290
points is, by the time you
get to 8 points in the past,

284
00:14:24,290 --> 00:14:26,762
it may be--

285
00:14:26,762 --> 00:14:30,260
that's a long time ago,
depending on how rapidly you're

286
00:14:30,260 --> 00:14:31,670
sampling or whatever.

287
00:14:31,670 --> 00:14:35,480
And compared to the data
point that you just took,

288
00:14:35,480 --> 00:14:37,340
you might think that
the data point you just

289
00:14:37,340 --> 00:14:39,950
took represents the current
state of the process

290
00:14:39,950 --> 00:14:45,240
a lot better than the
0.789 time points ago.

291
00:14:45,240 --> 00:14:47,720
And so there is a more
general approach here,

292
00:14:47,720 --> 00:14:51,140
where you can pick some
weighting for how much you

293
00:14:51,140 --> 00:14:57,910
want to fold in data from
different points in time.

294
00:14:57,910 --> 00:15:01,690
And a classic way
to do the weighting

295
00:15:01,690 --> 00:15:05,230
is to use an exponentially
weighted moving

296
00:15:05,230 --> 00:15:08,260
average, where you might
define the weighting function--

297
00:15:08,260 --> 00:15:10,900
these a sub i's here--

298
00:15:10,900 --> 00:15:15,070
I guess I've switched
notation to w here--

299
00:15:15,070 --> 00:15:18,010
when I'm at the current sample
point or the current time point

300
00:15:18,010 --> 00:15:23,860
t, I might weight points
backwards in time by index i,

301
00:15:23,860 --> 00:15:28,210
i going from 1 to 9
to 1,000-- whatever--

302
00:15:28,210 --> 00:15:30,820
by a function looking
something like this,

303
00:15:30,820 --> 00:15:37,880
with some r factor, where
r is weighting factor.

304
00:15:37,880 --> 00:15:40,570
So what happens here,
as I might think

305
00:15:40,570 --> 00:15:46,600
of r as being how much I
believe my current data point--

306
00:15:46,600 --> 00:15:50,410
might be I believe my
current data point by 90%,

307
00:15:50,410 --> 00:15:54,640
and then the next data point,
1 minus r, is only 10%.

308
00:15:54,640 --> 00:15:57,200
So I'm only mixing
in 10% of that.

309
00:15:57,200 --> 00:15:59,290
And then the next data
point beyond that--

310
00:15:59,290 --> 00:16:02,290
it's maybe 0.1 times 0.1.

311
00:16:02,290 --> 00:16:05,200
It's only 0.01,
so I'm only mixing

312
00:16:05,200 --> 00:16:07,660
in 1% of the
previous data point.

313
00:16:07,660 --> 00:16:15,190
So very rapidly, I have this
drop-off as a function of data

314
00:16:15,190 --> 00:16:16,375
points in the past--

315
00:16:18,970 --> 00:16:20,340
this is my i index--

316
00:16:24,120 --> 00:16:28,710
so that I much more heavily
weight recent history,

317
00:16:28,710 --> 00:16:32,910
and then exponentially
decay into the past.

318
00:16:35,513 --> 00:16:37,180
But I still have this
degree of freedom.

319
00:16:40,305 --> 00:16:41,430
I can pick r.

320
00:16:41,430 --> 00:16:45,960
I can pick the relative
weighting of more recent data

321
00:16:45,960 --> 00:16:49,910
compared to old data.

322
00:16:49,910 --> 00:16:52,710
Now, one way of
looking at this is,

323
00:16:52,710 --> 00:16:58,790
actually, I have access
to all past data,

324
00:16:58,790 --> 00:17:02,750
and I would apply
this out to 10 data

325
00:17:02,750 --> 00:17:08,420
points, or 50, or 100 with the
appropriate weighting function.

326
00:17:08,420 --> 00:17:10,849
And I guess, with computer
systems these days,

327
00:17:10,849 --> 00:17:12,980
it would be no problem
to actually apply it

328
00:17:12,980 --> 00:17:14,930
in that fashion.

329
00:17:14,930 --> 00:17:18,410
There's a beautiful thing about
the exponentially weighted

330
00:17:18,410 --> 00:17:21,470
moving average,
which is I actually

331
00:17:21,470 --> 00:17:25,300
don't have to keep track
of all of the raw data.

332
00:17:25,300 --> 00:17:27,050
There's another way
of thinking about what

333
00:17:27,050 --> 00:17:30,320
the EWMA, this exponentially
weighted moving average

334
00:17:30,320 --> 00:17:31,830
is doing.

335
00:17:31,830 --> 00:17:35,540
And that's that I have
my previous best estimate

336
00:17:35,540 --> 00:17:39,590
of the state of my variable--
my previous best estimate

337
00:17:39,590 --> 00:17:40,880
of the mean--

338
00:17:40,880 --> 00:17:42,860
and now I have one
new data point,

339
00:17:42,860 --> 00:17:44,660
and I use that
one new data point

340
00:17:44,660 --> 00:17:47,690
to update my best estimate.

341
00:17:47,690 --> 00:17:52,170
My best estimate had embedded
in it all of the past history,

342
00:17:52,170 --> 00:17:54,020
so now, with the
additional new weight,

343
00:17:54,020 --> 00:17:58,820
I can just do a simple update
of my best state estimate.

344
00:17:58,820 --> 00:18:03,050
And there is essentially
a recursive formula

345
00:18:03,050 --> 00:18:07,280
that makes this explicit, and
it's exactly the same, thing

346
00:18:07,280 --> 00:18:10,730
if you now apply it
iteratively out to all history,

347
00:18:10,730 --> 00:18:14,690
as actually performing that
computation of applying

348
00:18:14,690 --> 00:18:17,210
the weights on all past data.

349
00:18:17,210 --> 00:18:20,630
So that is to say,
let's say, a sub i

350
00:18:20,630 --> 00:18:25,250
is my best estimate of whatever
my statistic is-- my mean.

351
00:18:25,250 --> 00:18:28,640
Again, I might weight
my current data point,

352
00:18:28,640 --> 00:18:31,950
my most recent data
point, say, 90%,

353
00:18:31,950 --> 00:18:34,760
which would be
really rapid update.

354
00:18:34,760 --> 00:18:37,430
I really believe my
most recent data.

355
00:18:37,430 --> 00:18:42,200
And then my 1 minus r
applies to my previous--

356
00:18:42,200 --> 00:18:47,430
oops-- to my previous estimate.

357
00:18:54,350 --> 00:18:57,230
So I'm believing only
10% my previous estimate,

358
00:18:57,230 --> 00:19:00,223
and 90% my new estimate.

359
00:19:00,223 --> 00:19:01,640
So it makes it a
little bit easier

360
00:19:01,640 --> 00:19:03,980
in terms of the
computation, but I

361
00:19:03,980 --> 00:19:06,140
think it's also
conceptually nice in terms

362
00:19:06,140 --> 00:19:08,870
of what the action here is.

363
00:19:08,870 --> 00:19:14,310
Now, one can go through and do
a little bit of the mathematics.

364
00:19:14,310 --> 00:19:18,980
It's actually a little easier
to do it on the full summation

365
00:19:18,980 --> 00:19:23,810
version, but you just do
basic expectation math

366
00:19:23,810 --> 00:19:27,230
on the variance of an EWMA--

367
00:19:27,230 --> 00:19:30,350
again, drawing from a
normal distribution that

368
00:19:30,350 --> 00:19:32,120
is in control.

369
00:19:32,120 --> 00:19:35,750
And one can estimate
the standard deviation

370
00:19:35,750 --> 00:19:38,030
of this a statistic.

371
00:19:38,030 --> 00:19:40,470
And it's kind of a
little bit messy.

372
00:19:40,470 --> 00:19:45,020
It's got the underlying process
variation in it, the fact

373
00:19:45,020 --> 00:19:47,300
that you are still--

374
00:19:47,300 --> 00:19:51,080
you can still be doing sampling.

375
00:19:51,080 --> 00:19:56,300
For each x sub i, that might
be a sample of n data points.

376
00:19:56,300 --> 00:19:59,530
Very often, you might simply
be doing it one data point,

377
00:19:59,530 --> 00:20:02,240
so n, very often, is just 1--

378
00:20:02,240 --> 00:20:07,220
and then this formula with r
in it, including this 1 minus 1

379
00:20:07,220 --> 00:20:11,690
minus r to the 2t, where t
is how far along you are.

380
00:20:11,690 --> 00:20:14,150
So it's kind of nasty,
but one thing that's nice

381
00:20:14,150 --> 00:20:16,400
is, once you've start
started up your control chart

382
00:20:16,400 --> 00:20:18,590
and you're away from
that little artifact

383
00:20:18,590 --> 00:20:20,330
that we saw with
the moving average,

384
00:20:20,330 --> 00:20:24,350
where the variance
limits change--

385
00:20:24,350 --> 00:20:28,100
once you get far enough
away from that start up,

386
00:20:28,100 --> 00:20:33,530
your t is large, and that this
whole term just drops out--

387
00:20:33,530 --> 00:20:35,420
drops out more fast when--

388
00:20:35,420 --> 00:20:39,830
more rapidly for
large values of r,

389
00:20:39,830 --> 00:20:42,900
because I'm really weighting
more towards recent history.

390
00:20:42,900 --> 00:20:44,990
But even for small
or modest values

391
00:20:44,990 --> 00:20:48,230
of r, after you get
out 10, 20 data points,

392
00:20:48,230 --> 00:20:50,220
it typically drops away.

393
00:20:50,220 --> 00:20:52,340
So now you've got an
estimate for large T

394
00:20:52,340 --> 00:20:55,610
for the variance, sigma
a of your statistic,

395
00:20:55,610 --> 00:20:58,670
and you can simply
plot plus/minus

396
00:20:58,670 --> 00:21:01,760
3 sigma, or whatever
control limit you want,

397
00:21:01,760 --> 00:21:05,380
depending on your
false alarm rate alpha.

398
00:21:05,380 --> 00:21:08,070
So this is pretty cool.

399
00:21:08,070 --> 00:21:10,320
One quick observation
here is, again, we still

400
00:21:10,320 --> 00:21:12,720
have a lot of degrees
of flexibility

401
00:21:12,720 --> 00:21:15,000
around the choice of r.

402
00:21:15,000 --> 00:21:21,105
And if we look at this formula
here, and the effect of r--

403
00:21:21,105 --> 00:21:25,920
I sort of plotted this
out in this next chart.

404
00:21:25,920 --> 00:21:29,790
This effect of r on the sigma
multiplier-- oops-- sorry--

405
00:21:29,790 --> 00:21:33,820
is this factor right here.

406
00:21:33,820 --> 00:21:38,430
So in other words, as r changes,
how do my control limits move?

407
00:21:38,430 --> 00:21:41,580
What is this multiplier
of my sigma over root

408
00:21:41,580 --> 00:21:44,880
n from my normal sampling?

409
00:21:44,880 --> 00:21:46,410
And you can look at it here.

410
00:21:46,410 --> 00:21:51,330
And as r gets larger,
well, that gets larger--

411
00:21:51,330 --> 00:21:53,010
as r approaches 1.

412
00:21:53,010 --> 00:21:55,750
Typically, you don't
ever go above 1.

413
00:21:55,750 --> 00:21:57,630
You can't believe
your most recent data

414
00:21:57,630 --> 00:21:59,700
point more than 100%.

415
00:21:59,700 --> 00:22:04,400
It's a ratio, or
fraction of the percent.

416
00:22:04,400 --> 00:22:09,450
You can see that,
as r gets larger,

417
00:22:09,450 --> 00:22:15,040
this multiplier starts
to approach some limit.

418
00:22:15,040 --> 00:22:21,180
And finally, when r is equal to
1, this multiplier is simply 1.

419
00:22:21,180 --> 00:22:27,140
In the limit, when r is equal
to 1, what are we doing?

420
00:22:27,140 --> 00:22:30,700
We're doing 100% update based
on my most recent data point.

421
00:22:30,700 --> 00:22:33,480
In other words, I'm only looking
at my most recent data point.

422
00:22:33,480 --> 00:22:35,790
My variance is simply
the process variance

423
00:22:35,790 --> 00:22:37,230
coming from sampling.

424
00:22:37,230 --> 00:22:40,600
This all reduces to
[? simply ?] sigma

425
00:22:40,600 --> 00:22:44,190
a being by sampling
variance, whatever

426
00:22:44,190 --> 00:22:46,770
size sample I was doing
within each formulation.

427
00:22:46,770 --> 00:22:55,080
So in that case, I have
no factor or no multiplier

428
00:22:55,080 --> 00:23:00,630
on that sigma A. But for other
values of r, what's happening?

429
00:23:00,630 --> 00:23:06,430
For smaller values
of r, what do we see?

430
00:23:09,030 --> 00:23:15,690
The sigma A is smaller than
the sigma x over root n.

431
00:23:15,690 --> 00:23:20,310
We're getting reduction in the
variance because of filtering,

432
00:23:20,310 --> 00:23:21,870
the variance of our estimate.

433
00:23:21,870 --> 00:23:25,110
So that was the whole idea
is I can tighten my control

434
00:23:25,110 --> 00:23:30,450
limits, essentially by
the filtering action.

435
00:23:30,450 --> 00:23:33,960
So as I get smaller
and smaller r's, I

436
00:23:33,960 --> 00:23:36,120
do more and more
aggressive filtering

437
00:23:36,120 --> 00:23:38,190
with lots of past data.

438
00:23:38,190 --> 00:23:40,980
I get tighter and
tighter limits,

439
00:23:40,980 --> 00:23:45,720
but of course, I'm also
suppressing recent data

440
00:23:45,720 --> 00:23:50,070
little bit more heavily,
so there is this trade-off.

441
00:23:50,070 --> 00:23:53,040
Let's look at some examples.

442
00:23:53,040 --> 00:23:56,130
Oh, well, I guess I made
these points already here.

443
00:23:56,130 --> 00:24:01,320
Again, the variance of this
estimate for the mean--

444
00:24:01,320 --> 00:24:03,240
your best guess of the mean--

445
00:24:03,240 --> 00:24:07,650
is smaller than you would
have had with an x bar chart,

446
00:24:07,650 --> 00:24:11,370
because the x bar chart,
remember, for a sample

447
00:24:11,370 --> 00:24:14,170
size of n, is simply that.

448
00:24:14,170 --> 00:24:16,080
So we do get this suppression.

449
00:24:16,080 --> 00:24:20,460
And it is very nice that
it also, in the limit,

450
00:24:20,460 --> 00:24:24,930
works out for both the
sample size of 1 case--

451
00:24:24,930 --> 00:24:26,610
these formulas still hold--

452
00:24:26,610 --> 00:24:29,302
and also the full
update case, or the r

453
00:24:29,302 --> 00:24:32,280
equals 1, where we have
completely unfiltered data.

454
00:24:32,280 --> 00:24:37,360
You just plot in your pure
run data on your chart.

455
00:24:37,360 --> 00:24:43,530
So what happens if I use
much smaller r's, like 0.1?

456
00:24:43,530 --> 00:24:45,750
I got much tighter
control limits,

457
00:24:45,750 --> 00:24:51,270
but I've got long delays
potentially then in detecting--

458
00:24:51,270 --> 00:24:52,800
I've got a lot of
noise suppression,

459
00:24:52,800 --> 00:24:54,690
but again, I get
back to long delays

460
00:24:54,690 --> 00:24:58,780
in detecting or responding
to a true shift in the mean.

461
00:24:58,780 --> 00:25:01,890
So I want to look at
a few examples here

462
00:25:01,890 --> 00:25:05,010
to give you a feel of the
trade-off between different r's

463
00:25:05,010 --> 00:25:12,000
for some data, and compare it as
well to a typical x bar chart.

464
00:25:12,000 --> 00:25:19,440
OK, so here's a set of data, and
we're seeing a comparison here

465
00:25:19,440 --> 00:25:21,645
between the x bar chart--

466
00:25:24,540 --> 00:25:32,720
so you can see here are the
x bar upper and lower control

467
00:25:32,720 --> 00:25:34,030
limits.

468
00:25:34,030 --> 00:25:40,970
And sure enough, after
150 runs, where at--

469
00:25:40,970 --> 00:25:45,530
I don't know-- about run
90, we make this shift

470
00:25:45,530 --> 00:25:46,410
in the process--

471
00:25:46,410 --> 00:25:50,000
so we shift the whole
process by a half

472
00:25:50,000 --> 00:25:53,090
a standard deviation of
the underlying process.

473
00:25:53,090 --> 00:25:56,460
And sure enough, by about--

474
00:25:56,460 --> 00:26:00,470
I don't know-- run 140, we
finally got a data point

475
00:26:00,470 --> 00:26:02,840
in the tail of the
mean distribution--

476
00:26:02,840 --> 00:26:04,370
the regular x bar--

477
00:26:04,370 --> 00:26:06,200
that fell out.

478
00:26:06,200 --> 00:26:09,110
And we could actually calculate
the average run length

479
00:26:09,110 --> 00:26:13,790
to detect a mean shift of 0.5
using some of the techniques we

480
00:26:13,790 --> 00:26:16,270
talked about last time.

481
00:26:16,270 --> 00:26:19,720
And we might not be happy
with that very long ARL,

482
00:26:19,720 --> 00:26:24,280
and the question is, would an
EWMA chart help us with this?

483
00:26:24,280 --> 00:26:26,170
And this is an
example where we're

484
00:26:26,170 --> 00:26:32,020
using about a 30% update
based on each new data point,

485
00:26:32,020 --> 00:26:35,230
and plotted in
purple with the blue

486
00:26:35,230 --> 00:26:38,170
and the yellow are the
upper and lower control

487
00:26:38,170 --> 00:26:40,930
limits for the EWMA.

488
00:26:40,930 --> 00:26:47,640
And what you see is the
effect of averaging.

489
00:26:47,640 --> 00:26:53,560
Now, I always find it's
easiest for me to conceptualize

490
00:26:53,560 --> 00:26:57,070
the action of the EWMA chart
is just kind of like a moving

491
00:26:57,070 --> 00:27:00,640
average-- the equally weighted
moving average, but weighted

492
00:27:00,640 --> 00:27:03,160
a little bit more
towards recent history.

493
00:27:03,160 --> 00:27:07,690
But the whole idea is
I get that aggregation

494
00:27:07,690 --> 00:27:10,540
from any kind of a
moving average so that,

495
00:27:10,540 --> 00:27:13,180
right after this
shift right here,

496
00:27:13,180 --> 00:27:15,760
now, when I take into account--

497
00:27:15,760 --> 00:27:21,400
with an r of 0.3, I'm taking
into account three, four, five

498
00:27:21,400 --> 00:27:22,540
data points--

499
00:27:22,540 --> 00:27:25,930
I've got kind of an
effective window weighted

500
00:27:25,930 --> 00:27:29,080
about 30% towards my most
recent, but including

501
00:27:29,080 --> 00:27:30,730
some past history--

502
00:27:30,730 --> 00:27:34,450
that all of the
plus delta nu data

503
00:27:34,450 --> 00:27:39,760
points all average, and then
my normal process variation

504
00:27:39,760 --> 00:27:42,440
just cancels out each other.

505
00:27:42,440 --> 00:27:46,510
And so it's very
intuitive, I think,

506
00:27:46,510 --> 00:27:51,820
that this EWMA is going
to aggregate or respond

507
00:27:51,820 --> 00:27:56,230
to or sense the delta
shift in the mean more

508
00:27:56,230 --> 00:28:03,580
than an individual x bar
chart would have been doing--

509
00:28:03,580 --> 00:28:05,770
so that it responds
in this case,

510
00:28:05,770 --> 00:28:10,330
I don't know, after about
15 or 20 data points.

511
00:28:10,330 --> 00:28:15,690
You can start to see excursions
beyond the control limit.

512
00:28:15,690 --> 00:28:16,723
Question--

513
00:28:16,723 --> 00:28:18,390
AUDIENCE: At the
beginning of the chart,

514
00:28:18,390 --> 00:28:20,803
why are the control
limits [INAUDIBLE]

515
00:28:20,803 --> 00:28:21,720
spreading [INAUDIBLE]?

516
00:28:21,720 --> 00:28:27,100
The last time, they
were [INAUDIBLE]

517
00:28:27,100 --> 00:28:28,790
DUANE BONING: Yes,
yes-- good point.

518
00:28:31,650 --> 00:28:35,400
I think you can go back to this
formula and see if that works.

519
00:28:42,720 --> 00:28:45,150
I bet, if you look at
the effective the t

520
00:28:45,150 --> 00:28:53,510
action, formulaically
or analytically,

521
00:28:53,510 --> 00:28:57,130
I think that, as t starts
out being very small--

522
00:29:04,980 --> 00:29:08,130
so if t is 0, this
is essentially 1.

523
00:29:12,000 --> 00:29:18,240
Well, let's say t was 1 and
r was 0.3, I've got 0.7.

524
00:29:18,240 --> 00:29:19,750
I've got a reduction.

525
00:29:19,750 --> 00:29:23,950
So I think, for
small t, I've only

526
00:29:23,950 --> 00:29:26,890
got a limited amount of
data, and so I'm not--

527
00:29:26,890 --> 00:29:29,770
I don't have the long
tail in the EWMA.

528
00:29:29,770 --> 00:29:32,500
So I think I've only got
a small amount of data

529
00:29:32,500 --> 00:29:34,750
with a larger weight
in my control limits

530
00:29:34,750 --> 00:29:36,110
I think are smaller.

531
00:29:36,110 --> 00:29:41,200
But then, as t gets larger,
I've got a little bit more

532
00:29:41,200 --> 00:29:43,150
and more data until t--

533
00:29:43,150 --> 00:29:45,310
so I think if I were
to plot this 1 minus 1

534
00:29:45,310 --> 00:29:47,200
minus r to the 2t--

535
00:29:47,200 --> 00:29:54,220
kind of has a funky
shape that dies out.

536
00:29:54,220 --> 00:29:56,980
But I wouldn't worry too
much about the startup times.

537
00:29:56,980 --> 00:29:59,620
There are artifacts when you
start up a control chart,

538
00:29:59,620 --> 00:30:01,810
but once you just start
a control chart anyway,

539
00:30:01,810 --> 00:30:04,720
you're still feeling
out the process,

540
00:30:04,720 --> 00:30:07,930
so I wouldn't worry too
much about the startup

541
00:30:07,930 --> 00:30:08,900
times on these.

542
00:30:08,900 --> 00:30:09,400
Yeah?

543
00:30:09,400 --> 00:30:18,768
AUDIENCE: [INAUDIBLE] How do
you get this side [INAUDIBLE]

544
00:30:18,768 --> 00:30:19,560
DUANE BONING: Yeah.

545
00:30:19,560 --> 00:30:22,920
So the question is-- if you
didn't hear that in Singapore--

546
00:30:22,920 --> 00:30:25,110
was, when would you
use an EWMA or what

547
00:30:25,110 --> 00:30:29,170
are the factors in using an
EWMA from an x bar chart?

548
00:30:29,170 --> 00:30:34,920
And I think, if you've
got a lot of noise

549
00:30:34,920 --> 00:30:39,600
and you're really worried
about detecting rapidly a shift

550
00:30:39,600 --> 00:30:42,510
in the process, an EWMA--

551
00:30:42,510 --> 00:30:44,790
and we'll see the
CUSUM in a minute--

552
00:30:44,790 --> 00:30:50,160
has improved sensitivity to
rapid detection of the shift.

553
00:30:50,160 --> 00:30:53,410
But it comes at one
really important cost,

554
00:30:53,410 --> 00:30:59,280
which is a loss in transparency
in what's going on.

555
00:30:59,280 --> 00:31:01,320
The x bar, I think,
conceptually,

556
00:31:01,320 --> 00:31:03,930
is a lot easier to
understand, have

557
00:31:03,930 --> 00:31:07,290
everybody on the line
understand what it is.

558
00:31:07,290 --> 00:31:11,280
The data, if you also explode
out and show the full data--

559
00:31:11,280 --> 00:31:14,610
the x bar and the s--

560
00:31:14,610 --> 00:31:17,610
gives you a good feel
directly to the statistics

561
00:31:17,610 --> 00:31:19,020
of the process.

562
00:31:19,020 --> 00:31:20,940
Here you look at this
and you lose touch,

563
00:31:20,940 --> 00:31:27,240
I think, a little bit
with the statistics of--

564
00:31:27,240 --> 00:31:29,550
and the relationship
to your raw data.

565
00:31:29,550 --> 00:31:31,440
It's a little harder to
get your mind around.

566
00:31:36,660 --> 00:31:40,530
I don't know-- 10, 15 years ago,
I fell in love with the EWMA

567
00:31:40,530 --> 00:31:42,870
chart and approach--

568
00:31:42,870 --> 00:31:47,080
not only for charting,
but also for doing

569
00:31:47,080 --> 00:31:49,950
simple run-by-run control.

570
00:31:49,950 --> 00:31:52,410
And so I kind of
like it, but I can

571
00:31:52,410 --> 00:31:55,980
empathize with it being a
little bit harder to explain

572
00:31:55,980 --> 00:31:59,250
and maintain on the line.

573
00:31:59,250 --> 00:32:02,580
It's also got a little bit
more calculation, so back

574
00:32:02,580 --> 00:32:05,010
in the days of--

575
00:32:05,010 --> 00:32:07,380
and it's still true in some
of the manufacturing lines.

576
00:32:07,380 --> 00:32:10,050
I know, when we went on the
tour and some of the plants--

577
00:32:10,050 --> 00:32:14,702
very, very up-to-date
plants in Singapore

578
00:32:14,702 --> 00:32:16,410
and other places around
the world, people

579
00:32:16,410 --> 00:32:18,510
are still often taking
measurements by hand,

580
00:32:18,510 --> 00:32:22,890
and doing the hand calculation,
and writing it on a chart.

581
00:32:22,890 --> 00:32:24,270
This has an extra step.

582
00:32:24,270 --> 00:32:27,390
You've got to do
an extra average.

583
00:32:27,390 --> 00:32:34,300
So in modern day, when
almost all the calculations

584
00:32:34,300 --> 00:32:35,388
are automated.

585
00:32:35,388 --> 00:32:36,930
And the data collection
is automated.

586
00:32:36,930 --> 00:32:40,350
I think there's even more
reason to go with an EWMA.

587
00:32:40,350 --> 00:32:45,510
So the thing I like about
the best about the EWMA is I

588
00:32:45,510 --> 00:32:50,040
think of it as giving me my
best current estimate based

589
00:32:50,040 --> 00:32:53,670
not only on just my last
data point, but a little bit

590
00:32:53,670 --> 00:32:55,650
of filtering,
including past history.

591
00:32:55,650 --> 00:33:01,770
It's giving my best educated
state of the current process.

592
00:33:01,770 --> 00:33:05,790
And why I like that for
then run-by-run control is,

593
00:33:05,790 --> 00:33:08,880
based on my current
model, my updated--

594
00:33:08,880 --> 00:33:11,100
on a recursive
basis, updated model

595
00:33:11,100 --> 00:33:14,740
of the current state of the
process I can make a decision.

596
00:33:14,740 --> 00:33:19,620
So here the decision in SPC is
the process in control or not

597
00:33:19,620 --> 00:33:24,720
'But if you think of this
being not just the EWMA not--

598
00:33:24,720 --> 00:33:26,140
giving you more information.

599
00:33:26,140 --> 00:33:30,810
It's not just giving you,
am I in control or not,

600
00:33:30,810 --> 00:33:32,620
given the random spread?

601
00:33:32,620 --> 00:33:35,190
But if you interpret
this little purple data

602
00:33:35,190 --> 00:33:39,030
point here as my best
estimate of where

603
00:33:39,030 --> 00:33:42,420
the mean in the
recent past has been,

604
00:33:42,420 --> 00:33:45,930
if there is a little bit
of drift or movement maybe.

605
00:33:45,930 --> 00:33:51,120
I do active control and
not pure SPC control.

606
00:33:51,120 --> 00:33:53,850
I'm not making any
longer just the decision.

607
00:33:53,850 --> 00:33:55,080
Nothing has changed.

608
00:33:55,080 --> 00:33:57,030
I'm saying there
are small changes,

609
00:33:57,030 --> 00:34:00,180
and maybe I want to go and
make a small modification

610
00:34:00,180 --> 00:34:02,860
or an adjustment
on the equipment.

611
00:34:02,860 --> 00:34:03,360
Now.

612
00:34:03,360 --> 00:34:07,050
If, you are in true
statistical process control,

613
00:34:07,050 --> 00:34:08,739
that would be the
wrong thing to do.

614
00:34:08,739 --> 00:34:12,040
You'd be over-adjusting, and
we'll talk about that later.

615
00:34:12,040 --> 00:34:16,230
But that's another thing I like
about the EWMA is very often,

616
00:34:16,230 --> 00:34:20,310
there is a little bit of time
correlation in the process.

617
00:34:20,310 --> 00:34:26,679
And the EWMA is tracking
those small changes

618
00:34:26,679 --> 00:34:29,260
and filtering out
some of the noise.

619
00:34:29,260 --> 00:34:30,639
It's a good question.

620
00:34:30,639 --> 00:34:31,540
I'll take a question.

621
00:34:31,540 --> 00:34:35,207
There was somebody had
a question in Singapore.

622
00:34:35,207 --> 00:34:35,790
AUDIENCE: Yes.

623
00:34:35,790 --> 00:34:40,230
So under smaller r will lead
to the narrow control limit,

624
00:34:40,230 --> 00:34:41,715
right?

625
00:34:41,715 --> 00:34:42,840
DUANE BONING: That's right.

626
00:34:42,840 --> 00:34:45,380
AUDIENCE: That's so the
narrow control limit--

627
00:34:45,380 --> 00:34:48,480
you will have a smaller beta?

628
00:34:53,389 --> 00:34:56,449
DUANE BONING: So a smaller
beta would be less chance--

629
00:34:56,449 --> 00:34:57,580
AUDIENCE: Smaller beta?

630
00:34:57,580 --> 00:34:58,330
DUANE BONING: Yes.

631
00:34:58,330 --> 00:35:00,630
AUDIENCE: Yeah,
[INAUDIBLE] type II error.

632
00:35:00,630 --> 00:35:03,960
DUANE BONING: Right-- but
its power is then larger,

633
00:35:03,960 --> 00:35:08,790
so the chance of detecting
a true out of control point

634
00:35:08,790 --> 00:35:11,430
increases.

635
00:35:11,430 --> 00:35:14,850
But there's a very important
additional factor, which

636
00:35:14,850 --> 00:35:17,540
is there's additional time lag.

637
00:35:17,540 --> 00:35:19,290
So I'm going to show
a couple of examples,

638
00:35:19,290 --> 00:35:22,410
because you can see
in other examples

639
00:35:22,410 --> 00:35:25,890
the effective the filtering and
the time response that it takes

640
00:35:25,890 --> 00:35:28,710
to detect can go up.

641
00:35:28,710 --> 00:35:30,773
AUDIENCE: OK.

642
00:35:30,773 --> 00:35:32,440
DUANE BONING: So
another question here--

643
00:35:32,440 --> 00:35:33,120
AUDIENCE: I'm was just curious.

644
00:35:33,120 --> 00:35:35,640
What sample size were you
using for the x bar chart?

645
00:35:35,640 --> 00:35:36,950
DUANE BONING: I don't know.

646
00:35:36,950 --> 00:35:40,710
I'm not sure what
was used in the--

647
00:35:40,710 --> 00:35:41,790
as a sample size.

648
00:35:41,790 --> 00:35:44,350
I think you're just seeing
the plot of the sample,

649
00:35:44,350 --> 00:35:47,025
so the sample size is actually
the same in both cases.

650
00:35:52,370 --> 00:35:56,390
But I don't what was being used.

651
00:35:56,390 --> 00:36:00,590
In some sense, it doesn't
actually matter too much.

652
00:36:00,590 --> 00:36:02,780
The EWMA is looking--

653
00:36:02,780 --> 00:36:05,500
is an EWMA over sample.

654
00:36:09,020 --> 00:36:10,970
I might have 1,000
parts an hour,

655
00:36:10,970 --> 00:36:16,400
and every 15 minutes, I
might do a sample size of 10.

656
00:36:16,400 --> 00:36:18,820
So I'm only getting
sample size 10, lots more.

657
00:36:18,820 --> 00:36:23,960
And what's being plotted here,
in some sense, is the sample.

658
00:36:23,960 --> 00:36:27,070
So I have an x bar where I
might plot those samples.

659
00:36:27,070 --> 00:36:28,900
Here what I'm doing--

660
00:36:28,900 --> 00:36:34,240
the EWMA has enough-- it has
the flexibility to extend to n

661
00:36:34,240 --> 00:36:36,700
equals 1 sampling, if
that's what I'm doing.

662
00:36:36,700 --> 00:36:39,340
But in some sense,
it could also be

663
00:36:39,340 --> 00:36:45,160
used where, instead of each of
those n equals 10 samples was

664
00:36:45,160 --> 00:36:46,570
the only data I looked at.

665
00:36:46,570 --> 00:36:48,370
Each one of those is my t--

666
00:36:48,370 --> 00:36:53,220
point in t, and my previous
sample of size 10 is t minus 1.

667
00:36:56,260 --> 00:37:00,520
So in some sense, it's doing
the whole sample-by-sample

668
00:37:00,520 --> 00:37:02,270
comparison.

669
00:37:02,270 --> 00:37:06,550
So my point here
was both the EWMA

670
00:37:06,550 --> 00:37:12,650
and the x bar are using, in
this derivation, the same sample

671
00:37:12,650 --> 00:37:13,150
size.

672
00:37:13,150 --> 00:37:15,100
I'm not differentiating in that.

673
00:37:15,100 --> 00:37:18,640
All I'm doing in this
case really, with the EWMA

674
00:37:18,640 --> 00:37:21,880
is using multiple
samples back in time,

675
00:37:21,880 --> 00:37:25,770
and the x bar only uses one.

676
00:37:25,770 --> 00:37:32,070
So in fact, if I go to r
equals 1, the two converge.

677
00:37:32,070 --> 00:37:33,240
They are the same.

678
00:37:33,240 --> 00:37:38,190
The r equals EWMA is exactly
the same as the x bar.

679
00:37:38,190 --> 00:37:39,860
AUDIENCE: But the
larger my sample size

680
00:37:39,860 --> 00:37:44,160
is the less of a spread
I get between my two--

681
00:37:44,160 --> 00:37:45,486
[INAUDIBLE]

682
00:37:45,486 --> 00:37:47,640
DUANE BONING: They would
both go together, though,

683
00:37:47,640 --> 00:37:49,015
because I had to
have the factor.

684
00:37:49,015 --> 00:37:52,350
So back here on slide 12,
if I change my sample size,

685
00:37:52,350 --> 00:37:56,520
both sigma A and my
upper control limits

686
00:37:56,520 --> 00:38:00,550
and lower federal limits for the
x bar shrink exactly the same.

687
00:38:00,550 --> 00:38:04,840
So I would scale both
of them, if I increase.

688
00:38:04,840 --> 00:38:10,600
So what I was talking about here
is with different values of r.

689
00:38:10,600 --> 00:38:14,840
So this was an r equals 0.3.

690
00:38:14,840 --> 00:38:17,740
Here's a smaller
r, r equals 0.1.

691
00:38:17,740 --> 00:38:19,240
So what's happening here?

692
00:38:19,240 --> 00:38:22,600
I'm only accepting my most
recent data point 10%,

693
00:38:22,600 --> 00:38:25,870
and mixing past history and 90%.

694
00:38:25,870 --> 00:38:28,900
That's more averaging
of the past.

695
00:38:28,900 --> 00:38:31,570
I'm using more past
history, and so

696
00:38:31,570 --> 00:38:38,030
in this case, what you see is
kind of a more slowly drifting,

697
00:38:38,030 --> 00:38:39,750
more heavily--

698
00:38:39,750 --> 00:38:43,190
excuse me-- more heavily
filtered version.

699
00:38:43,190 --> 00:38:47,830
And here, with the mean
shift occurring about here,

700
00:38:47,830 --> 00:38:50,890
you can start to
see in this case

701
00:38:50,890 --> 00:38:59,017
a big mean shift at 0.5 with,
I guess, a sample size of 5.

702
00:38:59,017 --> 00:39:00,850
You do need to know the
sample size in order

703
00:39:00,850 --> 00:39:05,260
to know the effect of the
mean shift and ability

704
00:39:05,260 --> 00:39:08,660
to detect with either chart.

705
00:39:08,660 --> 00:39:10,000
So somewhere in here--

706
00:39:10,000 --> 00:39:11,260
I guess around there--

707
00:39:11,260 --> 00:39:15,730
I got I got lucky, and
I'm starting to sample now

708
00:39:15,730 --> 00:39:19,960
with the x bar picking
up that detection.

709
00:39:19,960 --> 00:39:24,760
But with the EWMA, in some
sense, I've still got 90%.

710
00:39:24,760 --> 00:39:29,770
It takes 9 or 10 data points
for the more recent data

711
00:39:29,770 --> 00:39:33,310
to outweigh the past
data, and so I've actually

712
00:39:33,310 --> 00:39:37,450
got, in this case, a
time lag in detecting

713
00:39:37,450 --> 00:39:43,880
the mean shift with the
EWMA compared to the x bar.

714
00:39:43,880 --> 00:39:46,620
So a question here first--

715
00:39:46,620 --> 00:39:48,584
AUDIENCE: I don't see
how we can compare them

716
00:39:48,584 --> 00:39:52,680
because they actually are
different time scales.

717
00:39:52,680 --> 00:39:54,810
All the x bar will
effectively-- let

718
00:39:54,810 --> 00:40:04,390
us say the sample
size is [INAUDIBLE]

719
00:40:04,390 --> 00:40:06,970
DUANE BONING: So the question
was, how do you compare

720
00:40:06,970 --> 00:40:08,740
the time scales on these?

721
00:40:08,740 --> 00:40:12,080
So here's what I
think is going on.

722
00:40:12,080 --> 00:40:14,650
And this is sort of what I
was talking about a second m

723
00:40:14,650 --> 00:40:17,320
but it's probably still
a little confusing.

724
00:40:17,320 --> 00:40:22,240
I believe that this n
equals 5 sample size means

725
00:40:22,240 --> 00:40:27,100
that, inside of this data point,
if I were to explode that,

726
00:40:27,100 --> 00:40:32,090
it actually has five
data points inside of it.

727
00:40:32,090 --> 00:40:37,420
And I'm only plotting the
average of those five on here.

728
00:40:37,420 --> 00:40:42,160
This data point also has
those same five data points,

729
00:40:42,160 --> 00:40:45,310
but then it also
grabbed the data

730
00:40:45,310 --> 00:40:48,100
that was lurking in
the previous one.

731
00:40:48,100 --> 00:40:53,800
And it has its five data
points and its five--

732
00:40:53,800 --> 00:40:55,040
whoops.

733
00:40:55,040 --> 00:40:55,540
Yah!

734
00:41:01,900 --> 00:41:02,830
Don't want to--

735
00:41:08,340 --> 00:41:11,400
I think it was trying
to exit me out here.

736
00:41:14,200 --> 00:41:15,960
So while it's doing
that, the point

737
00:41:15,960 --> 00:41:21,840
here was that the n equals
5 is not looking back

738
00:41:21,840 --> 00:41:23,910
five data points in the x bar.

739
00:41:23,910 --> 00:41:26,800
It's buried in each of
the individual points--

740
00:41:26,800 --> 00:41:38,330
so that both EWMA
and x bar are not

741
00:41:38,330 --> 00:41:42,440
affected by this n equals
5 sampling differently.

742
00:41:42,440 --> 00:41:47,510
But then the time scale
is correct in terms of--

743
00:41:47,510 --> 00:41:50,300
x bar only looks at
my most recent sample.

744
00:41:50,300 --> 00:41:54,050
The EWMA does look
back in history,

745
00:41:54,050 --> 00:41:57,170
and that's why you've got
this additional time lag.

746
00:41:57,170 --> 00:42:00,170
Was that getting at your
question or your observation,

747
00:42:00,170 --> 00:42:02,370
or is your question
a little different?

748
00:42:02,370 --> 00:42:03,850
AUDIENCE: [INAUDIBLE]

749
00:42:03,850 --> 00:42:05,060
DUANE BONING: OK.

750
00:42:05,060 --> 00:42:05,960
OK.

751
00:42:05,960 --> 00:42:08,617
And there was a
question in Singapore.

752
00:42:08,617 --> 00:42:09,200
AUDIENCE: Yes.

753
00:42:09,200 --> 00:42:11,630
So my question is
sort of built on what

754
00:42:11,630 --> 00:42:13,110
[? Stephen ?] was asking.

755
00:42:13,110 --> 00:42:17,510
So for a smaller r, you
have a higher probability

756
00:42:17,510 --> 00:42:19,152
of detecting a mean shift.

757
00:42:19,152 --> 00:42:21,697
But at same time,
for a smaller r,

758
00:42:21,697 --> 00:42:23,780
the time it takes for you
to detect the mean shift

759
00:42:23,780 --> 00:42:25,070
is longer.

760
00:42:25,070 --> 00:42:27,800
How does these two coexist?

761
00:42:27,800 --> 00:42:30,530
Well, you have a higher
chance of detecting it,

762
00:42:30,530 --> 00:42:32,840
but it takes longer as well.

763
00:42:32,840 --> 00:42:35,210
DUANE BONING: Yes.

764
00:42:35,210 --> 00:42:36,980
They're a trade-off.

765
00:42:36,980 --> 00:42:41,660
And so what you are basically
given is a degree of freedom

766
00:42:41,660 --> 00:42:44,030
that you get to play
with, or experiment with,

767
00:42:44,030 --> 00:42:46,650
or adapt to your process.

768
00:42:46,650 --> 00:42:49,940
So what are the factors in
adapting to your process?

769
00:42:49,940 --> 00:42:53,000
If your process is
really noisy and you

770
00:42:53,000 --> 00:42:54,980
think it's going to
be hard to detect

771
00:42:54,980 --> 00:43:00,020
a mean shift in the
phase of that noise,

772
00:43:00,020 --> 00:43:01,970
then you do more averaging.

773
00:43:01,970 --> 00:43:03,620
You'd have a smaller r.

774
00:43:03,620 --> 00:43:05,990
If you're trying to detect--

775
00:43:05,990 --> 00:43:08,600
maybe with a higher
risk of a false alarm,

776
00:43:08,600 --> 00:43:12,230
but you're trying to
detect more quickly,

777
00:43:12,230 --> 00:43:16,670
then I'd have a larger
r, so it responds faster,

778
00:43:16,670 --> 00:43:18,720
but doesn't filter as much.

779
00:43:18,720 --> 00:43:22,610
So you can see here, with
the r equals 0.1 versus the r

780
00:43:22,610 --> 00:43:23,990
equals 0.3--

781
00:43:23,990 --> 00:43:25,730
this is the same
data, I believe.

782
00:43:25,730 --> 00:43:27,110
Is this the same data?

783
00:43:27,110 --> 00:43:29,250
I guess not-- it's
just another example.

784
00:43:29,250 --> 00:43:38,240
Here's an r equals 0.3 kind
of case, where, for this data,

785
00:43:38,240 --> 00:43:40,400
this looks like about
the right trade-off.

786
00:43:40,400 --> 00:43:43,160
I'm getting
reasonable filtering.

787
00:43:43,160 --> 00:43:45,980
The black dots are my EWMA.

788
00:43:45,980 --> 00:43:50,480
And for a mean shift here,
I've got very rapid--

789
00:43:50,480 --> 00:43:54,980
I've got rapid detection,
much more rapid than,

790
00:43:54,980 --> 00:43:58,730
in fact, the x bar,
which doesn't really

791
00:43:58,730 --> 00:44:00,890
trigger finally until way--

792
00:44:00,890 --> 00:44:05,790
another 20, 25 runs later.

793
00:44:05,790 --> 00:44:10,070
So I think what you'd have to
do is, again make the trade-offs

794
00:44:10,070 --> 00:44:13,340
between responsivity,
the size of the shift

795
00:44:13,340 --> 00:44:15,440
that you're trying
to detect, and how

796
00:44:15,440 --> 00:44:20,960
much cost you to have different
delays in detecting the shift.

797
00:44:20,960 --> 00:44:23,540
So we've got two questions
here-- one here first.

798
00:44:23,540 --> 00:44:26,490
AUDIENCE: [INAUDIBLE]
It's almost

799
00:44:26,490 --> 00:44:28,808
like you are taking
a larger sample size.

800
00:44:28,808 --> 00:44:34,290
[INAUDIBLE] You're taking
the 5 samples for--

801
00:44:34,290 --> 00:44:39,180
5 measurements per sample, plus
5 previously or slightly lower

802
00:44:39,180 --> 00:44:42,510
[INAUDIBLE] plus 5 [INAUDIBLE]
plus 5 [INAUDIBLE] So

803
00:44:42,510 --> 00:44:44,850
you're just taking
more samples per--

804
00:44:44,850 --> 00:44:46,890
more measurements per
data point, right?

805
00:44:46,890 --> 00:44:47,640
DUANE BONING: Yes.

806
00:44:47,640 --> 00:44:49,342
So that's a fair way
of looking at it.

807
00:44:49,342 --> 00:44:50,800
That's a good way
of looking at it.

808
00:44:50,800 --> 00:44:53,820
So just so everybody could
hear, with the x bar,

809
00:44:53,820 --> 00:44:58,590
I take my n equals whatever
sample, but with the EWMA,

810
00:44:58,590 --> 00:45:00,780
I still got that from
my most recent update,

811
00:45:00,780 --> 00:45:02,550
but I'm doing
additional filtering

812
00:45:02,550 --> 00:45:04,770
or averaging of past history in.

813
00:45:04,770 --> 00:45:07,590
And the net effective that
is I've got a net aggregate--

814
00:45:07,590 --> 00:45:10,800
more number of
samples, if you will--

815
00:45:10,800 --> 00:45:13,020
unequally weighted, but a more--

816
00:45:13,020 --> 00:45:15,360
and therefore, that
makes perfect sense

817
00:45:15,360 --> 00:45:19,440
that your control
limits come down.

818
00:45:19,440 --> 00:45:21,720
That's a very good
way of looking at it.

819
00:45:21,720 --> 00:45:23,740
AUDIENCE: So in the
previous example,

820
00:45:23,740 --> 00:45:26,868
does EWMA also
trigger before x bar?

821
00:45:26,868 --> 00:45:27,910
Because x bar [INAUDIBLE]

822
00:45:27,910 --> 00:45:30,118
DUANE BONING: Oh, yes-- good
point, good point-- yes.

823
00:45:32,740 --> 00:45:35,445
AUDIENCE: Is there ever a
case when x bar will trigger

824
00:45:35,445 --> 00:45:37,537
[INAUDIBLE]

825
00:45:37,537 --> 00:45:39,370
DUANE BONING: Good
question-- are there ever

826
00:45:39,370 --> 00:45:42,100
situations where x bar
might trigger still

827
00:45:42,100 --> 00:45:47,470
before an EWMA for
an x bar chart?

828
00:45:47,470 --> 00:45:49,420
I don't quite know
the trade-offs

829
00:45:49,420 --> 00:45:53,140
and where you get in the
limits of the trade-off.

830
00:45:56,200 --> 00:45:58,510
You've always got
chance at work,

831
00:45:58,510 --> 00:46:02,050
and I suspect that,
with a large enough--

832
00:46:02,050 --> 00:46:06,970
so the following scenario
probably comes closest.

833
00:46:06,970 --> 00:46:13,030
Let's say you had a very large
mean shift, but a very small r.

834
00:46:13,030 --> 00:46:17,920
With an x bar, I react
entirely to a sample drawn

835
00:46:17,920 --> 00:46:19,810
from the very large mean shift.

836
00:46:19,810 --> 00:46:23,590
Let's say it was a 2 sigma
or a 3 sigma means shift.

837
00:46:23,590 --> 00:46:25,690
But I have a very small r.

838
00:46:25,690 --> 00:46:29,020
Maybe I'm only
calling that 0.05.

839
00:46:29,020 --> 00:46:32,550
That 5% mix in of that
most recent data point

840
00:46:32,550 --> 00:46:35,620
is probably swamped out
by my past data still,

841
00:46:35,620 --> 00:46:38,920
and so in that case, I think
my EWMA would likely not

842
00:46:38,920 --> 00:46:42,140
detect the mean shift.

843
00:46:42,140 --> 00:46:44,380
So I think of the EWMA
as especially good

844
00:46:44,380 --> 00:46:46,420
for improving
responsivity and detection

845
00:46:46,420 --> 00:46:49,900
ability for small effects,
because it's aggregating,

846
00:46:49,900 --> 00:46:52,370
just like the CUSUM that
we'll see in a second.

847
00:46:52,370 --> 00:46:56,020
But I think, if there are
really big-- bigger effects,

848
00:46:56,020 --> 00:46:59,710
you lose by the filtering
in the time response.

849
00:46:59,710 --> 00:47:02,005
Was that an example
you're going to suggest?

850
00:47:02,005 --> 00:47:07,430
AUDIENCE: [INAUDIBLE]

851
00:47:07,430 --> 00:47:08,680
DUANE BONING: Oh, good point--

852
00:47:08,680 --> 00:47:12,128
AUDIENCE: [INAUDIBLE]

853
00:47:12,128 --> 00:47:12,920
DUANE BONING: Yeah.

854
00:47:12,920 --> 00:47:19,118
AUDIENCE: [INAUDIBLE]

855
00:47:19,118 --> 00:47:19,910
DUANE BONING: Yeah.

856
00:47:19,910 --> 00:47:20,600
Yeah.

857
00:47:20,600 --> 00:47:25,150
So other rules here would
help-- yeah, absolutely.

858
00:47:25,150 --> 00:47:32,370
OK, so I guess this is
sort of saying a little bit

859
00:47:32,370 --> 00:47:33,060
the same thing.

860
00:47:33,060 --> 00:47:37,610
This, I think, is really
good for small mean shifts--

861
00:47:37,610 --> 00:47:38,630
that are persistent.

862
00:47:38,630 --> 00:47:42,950
It's not a one-time shot, that
I've got a delta mu mean shift,

863
00:47:42,950 --> 00:47:45,060
and this allows us to aggregate.

864
00:47:45,060 --> 00:47:48,300
There are additional
ideas we can use.

865
00:47:48,300 --> 00:47:49,800
Yeah-- question [INAUDIBLE]

866
00:47:49,800 --> 00:47:53,630
AUDIENCE: [INAUDIBLE]

867
00:47:53,630 --> 00:47:55,940
DUANE BONING: No, you
would not want to--

868
00:47:55,940 --> 00:47:58,130
so I think Hayden's
point here was

869
00:47:58,130 --> 00:48:01,430
on the purple beta
points you might apply,

870
00:48:01,430 --> 00:48:03,230
which was just an x bar.

871
00:48:03,230 --> 00:48:06,860
You might apply the WECO
but you should not--

872
00:48:06,860 --> 00:48:09,560
there might be variants
you could derive

873
00:48:09,560 --> 00:48:14,100
some of the other WECO
rules, but I don't

874
00:48:14,100 --> 00:48:15,600
know how you would do that.

875
00:48:15,600 --> 00:48:16,500
It's not intuitive.

876
00:48:16,500 --> 00:48:18,930
I think, for the
EWMA, you pretty much

877
00:48:18,930 --> 00:48:23,220
are just trying
to track the mean,

878
00:48:23,220 --> 00:48:28,350
and not additional trends or--

879
00:48:28,350 --> 00:48:31,320
I would not use any of
the additional WECO rules

880
00:48:31,320 --> 00:48:32,440
with an EWMA chart.

881
00:48:35,060 --> 00:48:37,370
So there's another chart
that is very often used.

882
00:48:37,370 --> 00:48:39,500
It's called a CUSUM I
just want to give you

883
00:48:39,500 --> 00:48:41,150
a quick feel for
that, and then move

884
00:48:41,150 --> 00:48:47,840
to some multivariate issues,
which is also targeted

885
00:48:47,840 --> 00:48:50,780
at this idea that, when
I have a small mean,

886
00:48:50,780 --> 00:48:54,650
shift what I'd like to do
is use the fact that it is

887
00:48:54,650 --> 00:48:58,010
a persistent mean shift that
I'm then sampling around

888
00:48:58,010 --> 00:48:59,820
within the process noise.

889
00:48:59,820 --> 00:49:02,000
And if there were a
simple way to aggregate

890
00:49:02,000 --> 00:49:06,500
the mean shift algorithm
many data points in the past,

891
00:49:06,500 --> 00:49:11,480
I'd build up a signal that's
larger and might improve

892
00:49:11,480 --> 00:49:17,780
my ability to detect that effect
and reduce the average run

893
00:49:17,780 --> 00:49:20,780
length for detecting
that effect.

894
00:49:20,780 --> 00:49:24,200
And this CUSUM,
or cumulative sum,

895
00:49:24,200 --> 00:49:28,200
is another way of thinking
of it as a filter--

896
00:49:28,200 --> 00:49:30,140
and in this case,
just a discrete time

897
00:49:30,140 --> 00:49:34,730
integrator-- which is looking
at the sum of deviations

898
00:49:34,730 --> 00:49:39,300
of each of my sample point
from the assumed grand mean.

899
00:49:39,300 --> 00:49:42,350
So I'm not just filtering
and plotting the average,

900
00:49:42,350 --> 00:49:43,880
but I'm saying--

901
00:49:43,880 --> 00:49:48,837
asking the question, am I still
in control at my assumed x bar

902
00:49:48,837 --> 00:49:51,800
bar, my mu 0?

903
00:49:51,800 --> 00:49:54,560
And if I'm not,
then this statistic,

904
00:49:54,560 --> 00:49:59,780
this sum of being to one
side of it, the c sub j

905
00:49:59,780 --> 00:50:03,140
here will grow if I've
got a plus mean shift,

906
00:50:03,140 --> 00:50:05,900
or become more and more
negative and grow if I've

907
00:50:05,900 --> 00:50:08,780
got a negative mean shift--

908
00:50:08,780 --> 00:50:12,740
so that I'll convert a mean
shift to an integration

909
00:50:12,740 --> 00:50:14,570
of a mean shift.

910
00:50:14,570 --> 00:50:16,430
The mean shift is a step.

911
00:50:16,430 --> 00:50:21,350
The integration of a
step response is a ramp.

912
00:50:21,350 --> 00:50:25,940
So what we would hope
to do with a CUSUM chart

913
00:50:25,940 --> 00:50:28,970
is improve our mean
shift sensitivity,

914
00:50:28,970 --> 00:50:32,060
and basically have charts
that look like this.

915
00:50:32,060 --> 00:50:39,310
When I'm in control, I might
wander around the mean.

916
00:50:39,310 --> 00:50:41,510
Sometimes I'll be
above, sometimes below,

917
00:50:41,510 --> 00:50:45,460
so this c sub j, this
cumulative statistic,

918
00:50:45,460 --> 00:50:47,110
does wander around a bit.

919
00:50:47,110 --> 00:50:50,050
But if I do have a
mean shift, on average,

920
00:50:50,050 --> 00:50:53,560
I keep adding to that C sub J.

921
00:50:53,560 --> 00:50:55,330
Sometimes I might go
down a little bit,

922
00:50:55,330 --> 00:50:58,580
because I still have something
a little bit below the mean.

923
00:50:58,580 --> 00:51:02,530
And so what you
create is this ramp.

924
00:51:02,530 --> 00:51:07,880
You create a slope caused
by that mean shift.

925
00:51:07,880 --> 00:51:11,260
Now, one can also now
apply some statistics

926
00:51:11,260 --> 00:51:19,330
and design, control limits
for that C sub J statistic,

927
00:51:19,330 --> 00:51:22,360
or use a couple of
other approaches.

928
00:51:22,360 --> 00:51:28,430
And one approach is this
so-called V-Mask that says it--

929
00:51:28,430 --> 00:51:34,060
what I would like to do is, if,
in fact, I've got a mean shift,

930
00:51:34,060 --> 00:51:35,980
this slope--

931
00:51:35,980 --> 00:51:37,690
let me erase a little bit here--

932
00:51:40,840 --> 00:51:48,410
whoops-- so there are two
different approaches--

933
00:51:48,410 --> 00:51:50,260
and I just want
to briefly mention

934
00:51:50,260 --> 00:51:54,460
both of those-- for detecting
things on these charts.

935
00:51:54,460 --> 00:51:59,026
One is draw a control limit
for the aggregate of the C sub

936
00:51:59,026 --> 00:52:07,120
J. The other is to look and say,
once I've got a mean shift, I

937
00:52:07,120 --> 00:52:11,980
start building data points with,
on average, some slope that

938
00:52:11,980 --> 00:52:18,700
is different than a slope that
I would expect by random chance

939
00:52:18,700 --> 00:52:24,460
alone, just forming the sum
statistic from a normally

940
00:52:24,460 --> 00:52:28,440
sampled process.

941
00:52:28,440 --> 00:52:33,150
The tabular CUSUM-- this
idea of actually looking

942
00:52:33,150 --> 00:52:35,890
at the statistics of--

943
00:52:35,890 --> 00:52:38,490
let me skip this for a second--

944
00:52:38,490 --> 00:52:41,490
this normalized
statistic z and s,

945
00:52:41,490 --> 00:52:46,157
and looking and deciding on what
control limits are, and so on--

946
00:52:46,157 --> 00:52:47,490
I'm actually going to skip that.

947
00:52:47,490 --> 00:52:49,080
There's about five
slides in here,

948
00:52:49,080 --> 00:52:53,520
but it's described very nicely
in, I think, chapter 8.1

949
00:52:53,520 --> 00:52:55,320
in Montgomery.

950
00:52:55,320 --> 00:52:56,830
And I think that's
pretty natural.

951
00:52:56,830 --> 00:53:04,500
It's just sort of a statistic--
a little bit confusing

952
00:53:04,500 --> 00:53:06,870
derivation of the
statistics that--

953
00:53:06,870 --> 00:53:11,760
I just trust the statisticians
who have done right.

954
00:53:11,760 --> 00:53:14,740
But there's also
another picture here,

955
00:53:14,740 --> 00:53:17,280
which is based on
the slope idea.

956
00:53:17,280 --> 00:53:21,150
And you can similarly
derive some formulas

957
00:53:21,150 --> 00:53:27,060
based on alpha and beta
risks, what size of a delta

958
00:53:27,060 --> 00:53:31,260
shift in the mean you
would hope to detect,

959
00:53:31,260 --> 00:53:35,250
and what kind of a slope
you would expect to see,

960
00:53:35,250 --> 00:53:36,790
in those cases.

961
00:53:36,790 --> 00:53:39,900
And so you've got these
two parameters that

962
00:53:39,900 --> 00:53:43,920
go towards the
formulation of this V-Mask

963
00:53:43,920 --> 00:53:48,350
that you overlay
on your data points

964
00:53:48,350 --> 00:53:52,820
based on an expected slope,
and you need a certain number

965
00:53:52,820 --> 00:53:54,770
of data points in order
to be able to build up

966
00:53:54,770 --> 00:53:56,000
your estimate of the slope.

967
00:53:56,000 --> 00:53:57,650
That's the dead zone.

968
00:53:57,650 --> 00:54:02,090
And what you do for each
data point here is overlay

969
00:54:02,090 --> 00:54:05,600
the V-Mask and say, look
back in time and see,

970
00:54:05,600 --> 00:54:12,140
did any of these points
intersect with the mask--

971
00:54:12,140 --> 00:54:15,230
have a higher slope than
you would have expected?

972
00:54:15,230 --> 00:54:17,580
And then you update
that on a running basis.

973
00:54:17,580 --> 00:54:20,300
So in this case,
here at time point

974
00:54:20,300 --> 00:54:23,960
18 or 19-- whatever that
is-- everything looks fine.

975
00:54:23,960 --> 00:54:25,260
I go up.

976
00:54:25,260 --> 00:54:27,530
Everything is
still looking fine.

977
00:54:27,530 --> 00:54:29,960
Still here, I'm starting
to get some evidence maybe

978
00:54:29,960 --> 00:54:33,290
that things have changed,
but statistically speaking,

979
00:54:33,290 --> 00:54:35,210
that's still within
the range that I would

980
00:54:35,210 --> 00:54:38,340
expect with some alpha error.

981
00:54:38,340 --> 00:54:43,400
But then finally, I get to a
new data point here at time 23,

982
00:54:43,400 --> 00:54:47,540
where, in aggregate,
I've got some indication

983
00:54:47,540 --> 00:54:49,860
that the slope is too large.

984
00:54:49,860 --> 00:54:53,120
So this is kind of a intuitive
graphical approach that's

985
00:54:53,120 --> 00:54:55,160
talking about the
slope associated

986
00:54:55,160 --> 00:55:00,770
with a mean shift in
the CUSUM statistic.

987
00:55:00,770 --> 00:55:01,700
Yes?

988
00:55:01,700 --> 00:55:05,210
AUDIENCE: Would you also
perform a linear regression

989
00:55:05,210 --> 00:55:07,852
for the last three or four
parts and check the [INAUDIBLE]??

990
00:55:07,852 --> 00:55:09,560
DUANE BONING: So the
question is, can you

991
00:55:09,560 --> 00:55:14,580
perform a linear regression and
calculate the slope of that?

992
00:55:14,580 --> 00:55:17,900
And I think that's identical to
what this is doing graphically.

993
00:55:17,900 --> 00:55:20,300
So in fact, if you
were numerically--

994
00:55:20,300 --> 00:55:24,200
or implementing an
automatic version of this,

995
00:55:24,200 --> 00:55:28,790
that would probably be how
you would implement it.

996
00:55:28,790 --> 00:55:31,430
That might even give you
some more flexibility

997
00:55:31,430 --> 00:55:35,420
in terms of looking at
the goodness of that fit,

998
00:55:35,420 --> 00:55:39,440
but I think it's essentially
identical to this.

999
00:55:39,440 --> 00:55:41,120
OK.

1000
00:55:41,120 --> 00:55:43,580
So just to summarize
these alternative charts,

1001
00:55:43,580 --> 00:55:46,250
there's a few simple messages.

1002
00:55:46,250 --> 00:55:48,830
Noisy data-- if you're
trying to detect

1003
00:55:48,830 --> 00:55:50,930
a mean in the noisy
data, you might

1004
00:55:50,930 --> 00:55:52,580
want to use some filtering.

1005
00:55:52,580 --> 00:55:55,100
We have the sample size
that does some filtering,

1006
00:55:55,100 --> 00:55:57,050
but I might want
to use more data

1007
00:55:57,050 --> 00:56:02,300
in the past and this
linear discrete filtering,

1008
00:56:02,300 --> 00:56:08,210
either with the CUSUM as a
running integrator or an EWMA--

1009
00:56:08,210 --> 00:56:11,990
which is an iterative
discrete filter--

1010
00:56:11,990 --> 00:56:13,170
can be applied.

1011
00:56:13,170 --> 00:56:15,200
And I think,
conceptually, they're

1012
00:56:15,200 --> 00:56:17,330
fairly nice in terms of
that noise suppression

1013
00:56:17,330 --> 00:56:20,240
for detecting of the mean--

1014
00:56:20,240 --> 00:56:22,610
got to remember that
your choice of factors,

1015
00:56:22,610 --> 00:56:26,480
like r and the EWMA, depends
on your process, and the mix

1016
00:56:26,480 --> 00:56:29,480
of noise, and how fast
you want to respond to be

1017
00:56:29,480 --> 00:56:31,520
able to detect a mean shift.

1018
00:56:31,520 --> 00:56:34,500
And then, going back to what we
started with our little example

1019
00:56:34,500 --> 00:56:37,430
at the very beginning.

1020
00:56:37,430 --> 00:56:39,350
Noisy data need some filtering.

1021
00:56:39,350 --> 00:56:41,480
We've talked about it
with respect to the mean--

1022
00:56:41,480 --> 00:56:42,860
that, if I'm
filtering that data,

1023
00:56:42,860 --> 00:56:45,980
you also still want to look
back and monitor the variance

1024
00:56:45,980 --> 00:56:46,890
as well.

1025
00:56:46,890 --> 00:56:52,610
So you can apply these charts
also to moving, S bar charts,

1026
00:56:52,610 --> 00:56:54,870
and so on.

1027
00:56:54,870 --> 00:56:56,270
So what I want to
do next is move

1028
00:56:56,270 --> 00:56:59,510
a little bit to multivariate
process control,

1029
00:56:59,510 --> 00:57:03,860
control charting when I've got
more than one parameter that I

1030
00:57:03,860 --> 00:57:07,370
may be monitoring on the unit
process or on the product.

1031
00:57:07,370 --> 00:57:09,800
And what's really
interesting here

1032
00:57:09,800 --> 00:57:15,110
is there are essentially
two classic mistakes that

1033
00:57:15,110 --> 00:57:17,282
are made again and again.

1034
00:57:17,282 --> 00:57:19,490
And I want to give you a
feel for those two mistakes,

1035
00:57:19,490 --> 00:57:21,880
and strategies
for avoiding them.

1036
00:57:21,880 --> 00:57:23,630
OK?

1037
00:57:23,630 --> 00:57:27,650
One classic problem is you've
got more than one parameter

1038
00:57:27,650 --> 00:57:30,530
you're monitoring-- you
implement more than one control

1039
00:57:30,530 --> 00:57:33,020
chart--

1040
00:57:33,020 --> 00:57:34,830
seems natural.

1041
00:57:34,830 --> 00:57:37,730
But the problem is you can get
many, many, many more false

1042
00:57:37,730 --> 00:57:43,610
alarms than you might
have expected, or believed

1043
00:57:43,610 --> 00:57:46,190
you are going to get,
because you're just

1044
00:57:46,190 --> 00:57:48,290
monitoring more parameters.

1045
00:57:48,290 --> 00:57:51,120
And so I want to talk a
little bit about that.

1046
00:57:51,120 --> 00:57:53,540
That's even if
there's no correlation

1047
00:57:53,540 --> 00:57:56,540
between your parameters--

1048
00:57:56,540 --> 00:57:58,580
or especially if
there's no correlation

1049
00:57:58,580 --> 00:58:00,320
between your parameters,
you can get--

1050
00:58:00,320 --> 00:58:05,060
just by the fact that you're
monitoring more than one thing,

1051
00:58:05,060 --> 00:58:07,490
you've got to think about
false alarm rate a little bit

1052
00:58:07,490 --> 00:58:08,580
differently.

1053
00:58:08,580 --> 00:58:10,130
And then the second
classic problem

1054
00:58:10,130 --> 00:58:14,330
is, well, there is correlation
among the parameters--

1055
00:58:14,330 --> 00:58:15,830
the multiple parameters.

1056
00:58:15,830 --> 00:58:17,580
How do you deal with that?

1057
00:58:17,580 --> 00:58:20,420
And in fact, in
this case, you often

1058
00:58:20,420 --> 00:58:23,500
run into the opposite risk.

1059
00:58:23,500 --> 00:58:25,600
Instead of too
many false alarms,

1060
00:58:25,600 --> 00:58:28,810
you may have points
of data or product

1061
00:58:28,810 --> 00:58:33,070
that is truly out
of normal operation

1062
00:58:33,070 --> 00:58:34,990
that you may not detect.

1063
00:58:34,990 --> 00:58:38,830
It's a highly unusual
point that may in

1064
00:58:38,830 --> 00:58:41,800
indicate something has
changed in your process,

1065
00:58:41,800 --> 00:58:44,740
but if you're just monitoring
with univariate control charts,

1066
00:58:44,740 --> 00:58:45,670
you'll never see it.

1067
00:58:48,940 --> 00:58:53,020
That is a common
mistake number two.

1068
00:58:53,020 --> 00:58:56,620
Multiple charts is common
mistake number one first.

1069
00:58:56,620 --> 00:58:59,770
If I truly have independent
parameters being modeled--

1070
00:58:59,770 --> 00:59:03,250
monitored at my process step,
and I set each one of them

1071
00:59:03,250 --> 00:59:07,270
up to have some
acceptable alpha rate--

1072
00:59:07,270 --> 00:59:08,440
false alarm rate--

1073
00:59:08,440 --> 00:59:13,630
1 out of every 370 process
steps going with a plus/minus

1074
00:59:13,630 --> 00:59:17,260
three-sigma--

1075
00:59:17,260 --> 00:59:21,730
three-sigma control limits,
alpha's about 0.0027--

1076
00:59:21,730 --> 00:59:26,110
sometimes talked about
0.003 as an approximation.

1077
00:59:26,110 --> 00:59:29,260
But now I've got p of
those separate charts.

1078
00:59:29,260 --> 00:59:32,740
What is the aggregate
false alarm probability?

1079
00:59:32,740 --> 00:59:35,020
Well to be careful about
it, if they are, in fact,

1080
00:59:35,020 --> 00:59:38,590
independent, that alpha prime,
might aggregate false alarm,

1081
00:59:38,590 --> 00:59:43,010
is, 1 minus 1 minus
alpha to the p.

1082
00:59:43,010 --> 00:59:47,710
This is the probability
that I don't

1083
00:59:47,710 --> 00:59:52,210
get a false alarm for each of--
for all of the other p control

1084
00:59:52,210 --> 00:59:55,180
charts all at the same time.

1085
00:59:55,180 --> 00:59:57,040
And then 1 minus
that is the chance

1086
00:59:57,040 --> 00:59:58,960
that I did get one of them.

1087
00:59:58,960 --> 01:00:03,250
Now for very small alpha, you
do that expansion, the 1 minus x

1088
01:00:03,250 --> 01:00:07,780
for small x to the p is simply
1 minus x So you do that

1089
01:00:07,780 --> 01:00:10,960
and you get sort of the
nice intuitive result

1090
01:00:10,960 --> 01:00:14,740
that if I've got 10
independent control charts,

1091
01:00:14,740 --> 01:00:18,850
I've got about 10 times
the rate of false alarm

1092
01:00:18,850 --> 01:00:19,990
that I would have from one.

1093
01:00:22,990 --> 01:00:23,800
So what do you do?

1094
01:00:32,560 --> 01:00:35,120
What would you do?

1095
01:00:35,120 --> 01:00:38,570
Let me say it this way.

1096
01:00:38,570 --> 01:00:42,500
You've decided, based on your
cost analysis or whatever,

1097
01:00:42,500 --> 01:00:48,490
that you're willing
about once every 370 runs

1098
01:00:48,490 --> 01:00:52,820
to have a false alarm
somewhere on this process,

1099
01:00:52,820 --> 01:00:54,533
that you want to at
least-- look at maybe

1100
01:00:54,533 --> 01:00:55,700
you look at it real quickly.

1101
01:00:55,700 --> 01:00:56,867
It doesn't take a long time.

1102
01:00:56,867 --> 01:01:02,090
But that's about how often
you want the operator to stop

1103
01:01:02,090 --> 01:01:06,590
and take a look, on average,
even when the process is

1104
01:01:06,590 --> 01:01:07,400
in control.

1105
01:01:07,400 --> 01:01:10,340
So that's your false alarm rate
that you're willing to accept,

1106
01:01:10,340 --> 01:01:12,110
but you're monitoring
10 parameters.

1107
01:01:17,600 --> 01:01:19,350
[INAUDIBLE] some
approaches you would use?

1108
01:01:23,145 --> 01:01:26,040
AUDIENCE: You have to
increase your control limits.

1109
01:01:26,040 --> 01:01:27,700
DUANE BONING: Right, perfect--

1110
01:01:27,700 --> 01:01:29,700
you would increase
your control limits.

1111
01:01:29,700 --> 01:01:32,790
If I expand my control
limits out a little bit,

1112
01:01:32,790 --> 01:01:38,850
I have smaller false
alarm rate on each chart.

1113
01:01:38,850 --> 01:01:50,880
Well, I can basically decide
what my aggregate false alarm

1114
01:01:50,880 --> 01:01:53,040
rate would be--

1115
01:01:53,040 --> 01:01:54,930
1 out of 370.

1116
01:01:54,930 --> 01:01:59,640
And to set my false alarm
rate on each individual chart,

1117
01:01:59,640 --> 01:02:03,420
just divide by how
many charts I have.

1118
01:02:03,420 --> 01:02:10,080
Now, I might have a 1 in 3,700
false alarm rate on each chart,

1119
01:02:10,080 --> 01:02:15,180
and I use that new alpha
to set my control limit.

1120
01:02:15,180 --> 01:02:19,380
They might end up being, in
that case, plus minus 3.4

1121
01:02:19,380 --> 01:02:23,410
sigma control limits.

1122
01:02:23,410 --> 01:02:29,170
So the individual false alarm
rate is lower on each one,

1123
01:02:29,170 --> 01:02:31,540
because really, the
question you are asking--

1124
01:02:31,540 --> 01:02:37,840
the key question was not
that individual parameter,

1125
01:02:37,840 --> 01:02:40,840
but rather, the question,
is my process in control?

1126
01:02:40,840 --> 01:02:48,320
In aggregate, do I have an
event that looks unlikely--

1127
01:02:48,320 --> 01:02:54,790
so unlikely that I have 1
minus alpha confidence, 1 minus

1128
01:02:54,790 --> 01:02:59,020
alpha prime confidence, that,
in fact, something unusual

1129
01:02:59,020 --> 01:03:02,020
is going on in the process?

1130
01:03:02,020 --> 01:03:03,790
So that's a strategy.

1131
01:03:03,790 --> 01:03:07,180
I want to also point out-- and
this is kind of a neat little

1132
01:03:07,180 --> 01:03:09,850
article that I found last year--

1133
01:03:09,850 --> 01:03:13,390
that this is a more
generic problem,

1134
01:03:13,390 --> 01:03:17,020
a more generic common mistake
than just control charting.

1135
01:03:17,020 --> 01:03:21,820
It applies all the time, when
people are doing significance

1136
01:03:21,820 --> 01:03:25,510
tests or hypothesis tests, but
doing more than one of them

1137
01:03:25,510 --> 01:03:28,090
at a time.

1138
01:03:28,090 --> 01:03:30,890
Control charge is just a
running hypothesis test.

1139
01:03:30,890 --> 01:03:33,280
Now, if I'm checking
10 hypotheses,

1140
01:03:33,280 --> 01:03:37,090
10 control charts,
and 10 parameters.

1141
01:03:37,090 --> 01:03:41,320
My aggregate false
alarm rate is higher.

1142
01:03:41,320 --> 01:03:43,570
And there's this
neat little article

1143
01:03:43,570 --> 01:03:47,530
pulled from-- that I found in
The Economist last February

1144
01:03:47,530 --> 01:03:51,490
that's called "Medical
Statistics Signs of the Times."

1145
01:03:51,490 --> 01:03:53,020
The subtitle here--
you may not be

1146
01:03:53,020 --> 01:03:56,710
able to read it-- is, "Why So
Much Medical Research is Rot."

1147
01:03:59,830 --> 01:04:02,200
I'll read you the first
couple of sentences here.

1148
01:04:02,200 --> 01:04:05,140
People born under the
astrological sign of Leo

1149
01:04:05,140 --> 01:04:07,870
are 15% more likely
to be admitted

1150
01:04:07,870 --> 01:04:10,090
to hospital with
gastric bleeding

1151
01:04:10,090 --> 01:04:13,240
than those born with
the other 11 signs.

1152
01:04:13,240 --> 01:04:16,990
Sagittarians are 38% more
likely than others to land up

1153
01:04:16,990 --> 01:04:19,720
there because of a broken arm.

1154
01:04:19,720 --> 01:04:22,390
Those are the conclusions
that many medical researchers

1155
01:04:22,390 --> 01:04:24,880
would be forced to make
from a set of data presented

1156
01:04:24,880 --> 01:04:28,480
to the American Association
for the Advancement of Science.

1157
01:04:28,480 --> 01:04:30,400
At least they would be
forced to draw them,

1158
01:04:30,400 --> 01:04:33,370
if they applied the
lax statistical methods

1159
01:04:33,370 --> 01:04:36,310
of their own work to the
records of hospital admissions

1160
01:04:36,310 --> 01:04:38,840
in Ontario, Canada.

1161
01:04:38,840 --> 01:04:41,860
And the basic confusion
drawn here in the red

1162
01:04:41,860 --> 01:04:44,980
arises because each result
is tested separately

1163
01:04:44,980 --> 01:04:47,410
to see how likely
in statistical terms

1164
01:04:47,410 --> 01:04:49,460
it was to have happen by chance.

1165
01:04:49,460 --> 01:04:51,460
If the likelihood is below
a certain threshold--

1166
01:04:51,460 --> 01:04:53,548
typically 95%--

1167
01:04:53,548 --> 01:04:59,170
so we've got a 95% confidence,
which says, 1 in 20 times,

1168
01:04:59,170 --> 01:05:02,890
by chance alone, I might
see a signal that big--

1169
01:05:02,890 --> 01:05:04,640
by chance alone.

1170
01:05:04,640 --> 01:05:06,550
And now I check 12--

1171
01:05:06,550 --> 01:05:09,700
maybe astrological
sign is your 12 things

1172
01:05:09,700 --> 01:05:12,290
I'm checking for as factors--

1173
01:05:12,290 --> 01:05:17,680
and I've got a 1 in 20 chance,
and I check 12 of them--

1174
01:05:17,680 --> 01:05:20,200
probably I'm going to
pop up and say, oh,

1175
01:05:20,200 --> 01:05:23,290
that one's significant.

1176
01:05:23,290 --> 01:05:27,190
And so that's exactly
the same common mistake

1177
01:05:27,190 --> 01:05:29,920
that you can see with
multiple control charts.

1178
01:05:29,920 --> 01:05:33,490
And so the point here would
simply be you should--

1179
01:05:33,490 --> 01:05:35,230
if you're checking
multiple factors,

1180
01:05:35,230 --> 01:05:38,410
you got to have a stronger
indication at any one factor--

1181
01:05:38,410 --> 01:05:41,470
otherwise, you're just fishing,
and by chance alone, you're

1182
01:05:41,470 --> 01:05:44,230
going to find stuff.

1183
01:05:44,230 --> 01:05:45,950
So that's kind of
a neat example.

1184
01:05:45,950 --> 01:05:48,340
And so here was the point I
made about how you fix it.

1185
01:05:48,340 --> 01:05:51,010
You just fix your
aggregate alpha prime,

1186
01:05:51,010 --> 01:05:52,720
you have your individual
alpha, and then

1187
01:05:52,720 --> 01:05:54,880
you pick your upper and
lower control limits

1188
01:05:54,880 --> 01:05:57,430
based on that alpha.

1189
01:05:57,430 --> 01:06:00,340
There's a second common
problem, and as I said,

1190
01:06:00,340 --> 01:06:06,010
it's the obverse of this,
which is you very often assume

1191
01:06:06,010 --> 01:06:09,370
that, if I'm monitoring 10
process control parameters,

1192
01:06:09,370 --> 01:06:11,290
I assume they're
all independent.

1193
01:06:11,290 --> 01:06:13,130
But very often, they are not.

1194
01:06:13,130 --> 01:06:15,702
There is correlation
among those parameters,

1195
01:06:15,702 --> 01:06:18,160
and I want to give you a little
bit of a feel for what that

1196
01:06:18,160 --> 01:06:21,640
looks like, and
what the risk is,

1197
01:06:21,640 --> 01:06:25,810
and what approaches are for
control charting and particular

1198
01:06:25,810 --> 01:06:27,360
of this.

1199
01:06:27,360 --> 01:06:30,670
So here's an example
drawn from some data that

1200
01:06:30,670 --> 01:06:33,370
came from an LFM
thesis a few years ago,

1201
01:06:33,370 --> 01:06:35,200
a body and white assembly.

1202
01:06:35,200 --> 01:06:39,070
So body and white-- this is
when they are basically putting

1203
01:06:39,070 --> 01:06:45,390
together all of the different
panels on the car and measuring

1204
01:06:45,390 --> 01:06:49,120
many, many points with
all kinds of laser,

1205
01:06:49,120 --> 01:06:53,590
automated apparatus to be able
to look at dimensional control

1206
01:06:53,590 --> 01:06:56,320
of these panels
in the assembly--

1207
01:06:56,320 --> 01:06:58,150
of some of these panels.

1208
01:06:58,150 --> 01:07:01,030
Other correlated data comes
up all over the place--

1209
01:07:01,030 --> 01:07:05,260
injection molding-- I
think we're missing--

1210
01:07:05,260 --> 01:07:06,850
I can't remember his name.

1211
01:07:06,850 --> 01:07:09,610
He asked a question last time
about critical dimensions

1212
01:07:09,610 --> 01:07:10,810
on a semiconductor wafer.

1213
01:07:10,810 --> 01:07:14,110
Yes, they are often correlated.

1214
01:07:14,110 --> 01:07:16,070
But here's the body
and white example,

1215
01:07:16,070 --> 01:07:18,610
and what I'm sort of showing
are these little coordinate

1216
01:07:18,610 --> 01:07:22,760
measuring machine, CMM
kinds of measurements.

1217
01:07:22,760 --> 01:07:26,720
I have a few of these
different data points.

1218
01:07:26,720 --> 01:07:29,600
And the point is, if I
think of this as a metal

1219
01:07:29,600 --> 01:07:33,320
panel in some way, and I'm
measuring the position, say,

1220
01:07:33,320 --> 01:07:39,890
of these four corner points,
you would expect there

1221
01:07:39,890 --> 01:07:42,710
to be correlation among them.

1222
01:07:42,710 --> 01:07:46,160
And certain kinds of
distortion might change

1223
01:07:46,160 --> 01:07:48,060
relationships between them.

1224
01:07:48,060 --> 01:07:50,630
And in fact, you might even
come up with other statistics,

1225
01:07:50,630 --> 01:07:53,450
like a y1, and a
y3, and a y4, that

1226
01:07:53,450 --> 01:07:56,720
try to detect or
look at tracking

1227
01:07:56,720 --> 01:07:59,630
of a couple of these
parameters and deviations.

1228
01:07:59,630 --> 01:08:02,420
But the point is I
wouldn't necessarily

1229
01:08:02,420 --> 01:08:05,090
want to monitor those
four corner locations

1230
01:08:05,090 --> 01:08:09,250
as independent parameters.

1231
01:08:09,250 --> 01:08:10,250
Here's another example.

1232
01:08:10,250 --> 01:08:14,560
This comes from an ultrasonic
welding set of data.

1233
01:08:14,560 --> 01:08:16,689
And you may not be
able to see this.

1234
01:08:16,689 --> 01:08:21,529
This is a wonderful correlation
plot of three, four, five, six,

1235
01:08:21,529 --> 01:08:25,450
seven, eight, 10 different
parameters against each other.

1236
01:08:25,450 --> 01:08:29,740
There are different things, like
P2 weld energy, P2 peak power.

1237
01:08:29,740 --> 01:08:31,660
These are simply
scatter plots of data

1238
01:08:31,660 --> 01:08:33,300
drawn from that process.

1239
01:08:33,300 --> 01:08:36,710
You can see some of these
parameters that track down--

1240
01:08:36,710 --> 01:08:41,622
that's total weld energy
versus P2 weld energy.

1241
01:08:41,622 --> 01:08:43,330
I don't even know what
those things mean,

1242
01:08:43,330 --> 01:08:45,663
but they sure sound like
they're going to be correlated,

1243
01:08:45,663 --> 01:08:46,670
don't they?

1244
01:08:46,670 --> 01:08:49,240
And sure enough, there's
extremely strong correlation

1245
01:08:49,240 --> 01:08:50,710
in that data.

1246
01:08:50,710 --> 01:08:53,350
There's other cases
where there's also

1247
01:08:53,350 --> 01:08:55,340
fairly strong correlation.

1248
01:08:55,340 --> 01:08:59,290
So again, these kinds
of correlated data,

1249
01:08:59,290 --> 01:09:01,779
where knowing something
about one of the measurements

1250
01:09:01,779 --> 01:09:06,939
tells you about the likelihood
of the other measurement,

1251
01:09:06,939 --> 01:09:09,319
occur all the time.

1252
01:09:09,319 --> 01:09:12,550
So what if you have truly
random independent variables?

1253
01:09:12,550 --> 01:09:14,600
What would a scatter
plot look like,

1254
01:09:14,600 --> 01:09:17,350
and how do we go about
setting the control limits?

1255
01:09:17,350 --> 01:09:18,580
That's pretty natural, right?

1256
01:09:18,580 --> 01:09:22,420
So here's an example
where we had x1 and x2.

1257
01:09:22,420 --> 01:09:27,229
The scatterplot basically
looks like a cloud of data--

1258
01:09:27,229 --> 01:09:30,729
and if I added more
data, you start to--

1259
01:09:30,729 --> 01:09:32,710
depending on the
scaling, if I were simply

1260
01:09:32,710 --> 01:09:38,200
to draw, say, a joint
95% confidence integral,

1261
01:09:38,200 --> 01:09:40,660
it's just kind of
a cloud of data,

1262
01:09:40,660 --> 01:09:46,790
but there's no cross
information between x1 and x2.

1263
01:09:46,790 --> 01:09:49,970
In other words, if I told
you that x1 was right here--

1264
01:09:49,970 --> 01:09:54,170
the value of x1 was this value--

1265
01:09:54,170 --> 01:09:58,250
that really doesn't tell me
very much about what value of x2

1266
01:09:58,250 --> 01:09:59,360
could be.

1267
01:09:59,360 --> 01:10:04,280
It still falls anywhere
within its distribution.

1268
01:10:04,280 --> 01:10:07,310
These are two independent
random variables.

1269
01:10:07,310 --> 01:10:09,530
And you can decide what
the proper limits are.

1270
01:10:09,530 --> 01:10:13,730
Now you know enough
about picking x1 and x2,

1271
01:10:13,730 --> 01:10:16,520
and I would do the
alpha prime idea

1272
01:10:16,520 --> 01:10:20,510
to pick independent
limits on those two cases.

1273
01:10:20,510 --> 01:10:23,030
This is an example
scatterplot for

1274
01:10:23,030 --> 01:10:25,700
correlated random variables.

1275
01:10:25,700 --> 01:10:31,010
And in this case, it starts to
look like an oval or a football

1276
01:10:31,010 --> 01:10:35,540
kind of shape, where there
is statistical correlation

1277
01:10:35,540 --> 01:10:37,640
between these two parameters--

1278
01:10:37,640 --> 01:10:43,760
by which I mean, if I tell
you that x1 is in this range,

1279
01:10:43,760 --> 01:10:46,760
that tells you something
about where the value of x2

1280
01:10:46,760 --> 01:10:48,200
is likely to fall.

1281
01:10:48,200 --> 01:10:51,380
It might not fall within
its full natural range.

1282
01:10:51,380 --> 01:10:53,930
There is cross information
or cross-correlation

1283
01:10:53,930 --> 01:10:55,470
between the two.

1284
01:10:55,470 --> 01:10:59,900
Now, if I were to plot
x1 and x2 separately,

1285
01:10:59,900 --> 01:11:03,580
it looks an awful lot like
the uncorrelated case.

1286
01:11:03,580 --> 01:11:08,550
So if I had independent control
charts for the two variables,

1287
01:11:08,550 --> 01:11:10,010
essentially what
I'm doing is just

1288
01:11:10,010 --> 01:11:13,520
projecting all of
these data points.

1289
01:11:13,520 --> 01:11:19,010
I project all of these data
points to get my range for x2,

1290
01:11:19,010 --> 01:11:22,010
and I project all of
these data points.

1291
01:11:22,010 --> 01:11:25,730
And in fact, if I plot
both of those, x1 and x2,

1292
01:11:25,730 --> 01:11:27,846
I would likely get--

1293
01:11:27,846 --> 01:11:30,800
yikes-- don't want to do that--

1294
01:11:30,800 --> 01:11:37,220
I would likely get two separate,
normally distributed variables,

1295
01:11:37,220 --> 01:11:39,770
and I might be inclined
to simply plot them

1296
01:11:39,770 --> 01:11:48,050
both as x bar and S
kinds of control charts.

1297
01:11:48,050 --> 01:11:51,230
The risk here,
though, when you do

1298
01:11:51,230 --> 01:11:56,180
that is you might
miss outliers, data

1299
01:11:56,180 --> 01:12:02,660
points that do not fall in the
normal or the natural variation

1300
01:12:02,660 --> 01:12:04,220
of the process--

1301
01:12:04,220 --> 01:12:07,220
this being the natural
variation of the process

1302
01:12:07,220 --> 01:12:10,460
that I will not detect.

1303
01:12:10,460 --> 01:12:14,340
This data point still looks like
it's within that distribution.

1304
01:12:14,340 --> 01:12:17,450
In, fact, it's very
close to the mean of x2.

1305
01:12:17,450 --> 01:12:22,020
And it's still within
this distribution.

1306
01:12:22,020 --> 01:12:25,310
But within the natural
variation of the process,

1307
01:12:25,310 --> 01:12:30,260
within that oval, its
distance, in some sense,

1308
01:12:30,260 --> 01:12:34,760
from the central moments
of that distribution--

1309
01:12:34,760 --> 01:12:36,110
it's pretty distant.

1310
01:12:36,110 --> 01:12:38,990
It's not indicative of
the normal operation.

1311
01:12:38,990 --> 01:12:41,660
It's a highly unlikely,
by chance alone, data

1312
01:12:41,660 --> 01:12:46,510
point, and I would never
see it on the control chart.

1313
01:12:46,510 --> 01:12:51,960
So there are
multivariate charts that

1314
01:12:51,960 --> 01:12:57,510
let us get at and chart
on a single chart--

1315
01:12:57,510 --> 01:13:01,560
multiple factors that
are correlated together--

1316
01:13:01,560 --> 01:13:05,400
and detect those
kinds of problems.

1317
01:13:05,400 --> 01:13:08,790
What I want to give
you is a notion first

1318
01:13:08,790 --> 01:13:14,760
of a statistic that
aggregates, in some sense--

1319
01:13:14,760 --> 01:13:16,950
numerically aggregates
exactly what

1320
01:13:16,950 --> 01:13:19,350
I was talking about
qualitatively here,

1321
01:13:19,350 --> 01:13:21,720
that tells you this
notion of, what

1322
01:13:21,720 --> 01:13:27,120
is the distance from
the correlated variation

1323
01:13:27,120 --> 01:13:30,030
cloud of each of my
measurement points?

1324
01:13:30,030 --> 01:13:33,720
And if I have a statistic
and I know it's PDF,

1325
01:13:33,720 --> 01:13:37,410
then I can set control
limits on that statistic.

1326
01:13:37,410 --> 01:13:40,680
And that statistic is
the Hotelling T squared,

1327
01:13:40,680 --> 01:13:43,375
is one very important one.

1328
01:13:43,375 --> 01:13:44,250
We can also monitor--

1329
01:13:48,060 --> 01:13:51,000
have EWMA and CUSUM
extensions of these.

1330
01:13:51,000 --> 01:13:53,850
Which I'm not really
going to talk much about.

1331
01:13:53,850 --> 01:13:57,810
So what we have to do here
is have a slight extension

1332
01:13:57,810 --> 01:14:00,360
to our notion of a
probability density function

1333
01:14:00,360 --> 01:14:04,133
to have joint probability
density functions that tell me

1334
01:14:04,133 --> 01:14:06,300
if I've got, say, two
variables-- or three, or four,

1335
01:14:06,300 --> 01:14:09,300
whatever-- what is the
probability density associated

1336
01:14:09,300 --> 01:14:10,510
with those?

1337
01:14:10,510 --> 01:14:13,110
And what we want is a
single scale control

1338
01:14:13,110 --> 01:14:14,610
chart that aggregates these.

1339
01:14:17,260 --> 01:14:20,620
Now, I've got a little bit
of mathematical notation

1340
01:14:20,620 --> 01:14:24,430
in here that I'm not going to
go through the careful full

1341
01:14:24,430 --> 01:14:26,140
explanation or derivation of.

1342
01:14:26,140 --> 01:14:28,600
What I want to
simply show you is

1343
01:14:28,600 --> 01:14:34,000
how a multivariate expression
of the same variables we've

1344
01:14:34,000 --> 01:14:38,830
been using all along gives you
a nice qualitative feel for what

1345
01:14:38,830 --> 01:14:40,970
this statistic looks like.

1346
01:14:40,970 --> 01:14:43,390
So if we have a vector
of measurements--

1347
01:14:43,390 --> 01:14:44,740
let's have p parameters--

1348
01:14:44,740 --> 01:14:46,720
I'm measuring 10
different parameters--

1349
01:14:46,720 --> 01:14:52,150
body in white or whatever it is
for that particular process--

1350
01:14:52,150 --> 01:14:56,650
and put those in an x--
underscore, an x vector.

1351
01:14:56,650 --> 01:14:59,190
And then I can define
a vector of means.

1352
01:14:59,190 --> 01:15:02,590
So I have a set of means
for all 10 parameters.

1353
01:15:02,590 --> 01:15:08,230
And I can also calculate,
if I have lots and lots

1354
01:15:08,230 --> 01:15:09,190
of historical data.

1355
01:15:09,190 --> 01:15:13,330
There isn't a true
underlying covariance matrix

1356
01:15:13,330 --> 01:15:17,020
that talks about or
characterizes how these data go

1357
01:15:17,020 --> 01:15:19,040
together with each other.

1358
01:15:19,040 --> 01:15:21,220
Now, I can estimate
that covariance matrix

1359
01:15:21,220 --> 01:15:24,340
based on sample
data, so I similarly

1360
01:15:24,340 --> 01:15:28,210
have a true mean and
a true covariance.

1361
01:15:28,210 --> 01:15:35,260
Then I might estimate
with an x bar and an S--

1362
01:15:35,260 --> 01:15:37,180
here I'm using a capital S--

1363
01:15:37,180 --> 01:15:39,610
to be a sample
covariance matrix.

1364
01:15:39,610 --> 01:15:42,430
And there are formulas
in the book using

1365
01:15:42,430 --> 01:15:46,990
a very simple linear, matrix--

1366
01:15:46,990 --> 01:15:49,960
linear algebra for
how you would do that,

1367
01:15:49,960 --> 01:15:53,500
given a particular set of data.

1368
01:15:53,500 --> 01:15:57,550
And they look a lot like
the univariate example.

1369
01:15:57,550 --> 01:16:00,100
But I did want to
point out that there's

1370
01:16:00,100 --> 01:16:03,910
both embedded in this
the notion of variance--

1371
01:16:03,910 --> 01:16:08,200
how variable 2 varies
together with variable 2--

1372
01:16:08,200 --> 01:16:12,070
that's just our normal
notion of variance.

1373
01:16:12,070 --> 01:16:14,110
And there's also covariance--

1374
01:16:14,110 --> 01:16:18,265
that is, a measure of how these
two variables move together.

1375
01:16:21,150 --> 01:16:26,540
We can actually write out what
the PDFs probability density

1376
01:16:26,540 --> 01:16:28,830
functions-- are in
these joint cases.

1377
01:16:28,830 --> 01:16:31,730
And if you write it
in matrix form using

1378
01:16:31,730 --> 01:16:35,720
this covariance matrix
and the mu vector,

1379
01:16:35,720 --> 01:16:39,930
it looks almost exactly the
same as the univariate case.

1380
01:16:39,930 --> 01:16:44,840
Notice that, instead of
variance in the univariate case,

1381
01:16:44,840 --> 01:16:51,230
now we have the
covariance matrix.

1382
01:16:51,230 --> 01:16:58,820
And similarly, instead of just
x and u being univariate values,

1383
01:16:58,820 --> 01:17:01,640
I've got those as vectors.

1384
01:17:01,640 --> 01:17:06,800
But the PDF looks
almost exactly the same.

1385
01:17:06,800 --> 01:17:12,650
And qualitatively, you're
getting an exponential

1386
01:17:12,650 --> 01:17:18,080
drop-off, this e drop-off with
the square of the distance

1387
01:17:18,080 --> 01:17:21,080
of a point from the mean--

1388
01:17:21,080 --> 01:17:23,130
the normalized distance
from the point.

1389
01:17:23,130 --> 01:17:24,830
So if we look back
at the univariate,

1390
01:17:24,830 --> 01:17:28,400
the whole idea is I subtract
off the mean when I standardize,

1391
01:17:28,400 --> 01:17:31,610
divide by the natural
variation, and that tells me

1392
01:17:31,610 --> 01:17:33,470
the shape of the bell curve--

1393
01:17:33,470 --> 01:17:36,650
shape of the normal curve,
how fast things drop off.

1394
01:17:36,650 --> 01:17:42,020
And I can also, for any
particular multivariate u--

1395
01:17:42,020 --> 01:17:46,880
multivariate data point x that
has its 10 parameters in it--

1396
01:17:46,880 --> 01:17:51,290
what's really cool is
this formulation here

1397
01:17:51,290 --> 01:17:57,710
of the distance of that x
vector from the mean point--

1398
01:17:57,710 --> 01:18:00,740
the main vector squared--

1399
01:18:00,740 --> 01:18:05,000
that squared distance, but
weighted then or normalized

1400
01:18:05,000 --> 01:18:07,250
to the natural spread--

1401
01:18:07,250 --> 01:18:10,370
that natural spread or
width of that cloud--

1402
01:18:10,370 --> 01:18:13,190
is just a distance measure--

1403
01:18:13,190 --> 01:18:15,710
squared distance
measure of each data

1404
01:18:15,710 --> 01:18:18,800
point from the central
moment of the central point

1405
01:18:18,800 --> 01:18:22,160
of the distribution, scaled
to how much natural spread

1406
01:18:22,160 --> 01:18:24,770
or scatter there is in the data.

1407
01:18:24,770 --> 01:18:26,075
OK, so that's the PDF.

1408
01:18:28,670 --> 01:18:32,450
Those are just the formulas
for the sample use cases.

1409
01:18:32,450 --> 01:18:38,130
What I can now go in and
do is ask the question,

1410
01:18:38,130 --> 01:18:43,400
what is the distribution
associated with that distance

1411
01:18:43,400 --> 01:18:44,900
metric?

1412
01:18:44,900 --> 01:18:48,500
And that distance
metric is essentially--

1413
01:18:48,500 --> 01:18:54,440
if I expand out that
distance formula,

1414
01:18:54,440 --> 01:19:02,430
it is a sum of squared distances
normalized to the correlation,

1415
01:19:02,430 --> 01:19:09,230
which starts to sound like
sum of squared unit normal

1416
01:19:09,230 --> 01:19:14,240
variables, which was our
chi-squared distribution--

1417
01:19:14,240 --> 01:19:17,790
sum of squared normal--

1418
01:19:17,790 --> 01:19:22,490
so what's cool here is that the
chi-squared distribution with p

1419
01:19:22,490 --> 01:19:23,960
parameters--

1420
01:19:23,960 --> 01:19:25,340
p degrees of freedom--

1421
01:19:25,340 --> 01:19:29,390
is the right statistic for
measuring how likely it

1422
01:19:29,390 --> 01:19:32,870
is to see those
distance-- the T squared,

1423
01:19:32,870 --> 01:19:33,890
those distance metrics.

1424
01:19:36,500 --> 01:19:39,230
So one can actually
have the PDF.

1425
01:19:39,230 --> 01:19:43,370
One can have the
distribution for that PDF,

1426
01:19:43,370 --> 01:19:49,610
and based on that, you
can formulate or form a--

1427
01:19:49,610 --> 01:19:53,270
that statistic based
on my actual data x.

1428
01:19:53,270 --> 01:19:55,790
and then I can have
an alpha squared

1429
01:19:55,790 --> 01:20:00,680
that I can pick my cutoff
points from appropriately.

1430
01:20:00,680 --> 01:20:04,710
Oops-- I thought I had
a PDF in here somewhere.

1431
01:20:04,710 --> 01:20:09,170
So when I'm making
a control chart,

1432
01:20:09,170 --> 01:20:11,810
I use the same ideas
that we saw before.

1433
01:20:11,810 --> 01:20:13,670
I have a false alarm rate.

1434
01:20:13,670 --> 01:20:16,130
In this case, I'm going
to aggregate all my data

1435
01:20:16,130 --> 01:20:18,170
onto a single control
chart so I can just

1436
01:20:18,170 --> 01:20:20,480
use my aggregate alpha.

1437
01:20:20,480 --> 01:20:26,545
And I pick the tail points on
the chi-squared distribution.

1438
01:20:26,545 --> 01:20:28,670
Remember, a chi-squared
might have looked something

1439
01:20:28,670 --> 01:20:35,280
like this, and I might pick a
way out here as my chi-squared

1440
01:20:35,280 --> 01:20:38,300
so that I've got my alpha
out there in the tail.

1441
01:20:38,300 --> 01:20:44,180
That picks my cutoff point
for my upper control limit.

1442
01:20:44,180 --> 01:20:50,360
Generally, you can also do
that for a lower control limit.

1443
01:20:50,360 --> 01:20:53,570
Very often, if you do some--
apply some of these formulas,

1444
01:20:53,570 --> 01:20:56,360
the lower control
limit is negative.

1445
01:20:56,360 --> 01:20:58,220
And you can't have
negative square,

1446
01:20:58,220 --> 01:21:01,340
so you pick 0 as your
lower control limit.

1447
01:21:01,340 --> 01:21:04,910
And you're really only
monitoring four data points

1448
01:21:04,910 --> 01:21:07,550
that are larger than--

1449
01:21:07,550 --> 01:21:13,610
I've got more variation in
my system than I expected.

1450
01:21:13,610 --> 01:21:18,200
So just to close up on this
univariate versus chi-squared

1451
01:21:18,200 --> 01:21:22,220
chart, here's an example--

1452
01:21:22,220 --> 01:21:28,430
comparing the univariate case
up here to the multivariate case

1453
01:21:28,430 --> 01:21:30,090
down here.

1454
01:21:30,090 --> 01:21:35,570
And what I'm plotting here is,
again, the scatter of data.

1455
01:21:35,570 --> 01:21:38,840
You might see there's a
slight shaded oval case.

1456
01:21:38,840 --> 01:21:40,610
And all the blue
data points are what

1457
01:21:40,610 --> 01:21:44,840
I would do if I just projected
those down to an x1 and x2,

1458
01:21:44,840 --> 01:21:47,720
and did univariate
control charting.

1459
01:21:47,720 --> 01:21:52,640
If I then had this little
red dot that fell outside

1460
01:21:52,640 --> 01:21:55,760
of the joint
confidence interval--

1461
01:21:55,760 --> 01:21:58,640
on my individual charts,
notice where it falls--

1462
01:21:58,640 --> 01:22:00,950
perfectly within
the control limits--

1463
01:22:00,950 --> 01:22:04,280
nothing that tells me that
that point is unusual.

1464
01:22:04,280 --> 01:22:06,470
But if I then plot
that data point--

1465
01:22:06,470 --> 01:22:11,720
that red data point there and
on the chi-squared chart--

1466
01:22:11,720 --> 01:22:15,830
I calculated the chi statistic,
that distance statistic--

1467
01:22:15,830 --> 01:22:21,140
that would trigger, because
its distance from the mean

1468
01:22:21,140 --> 01:22:21,935
is unusual.

1469
01:22:26,700 --> 01:22:32,460
Now, if I actually didn't know
my true covariance matrix,

1470
01:22:32,460 --> 01:22:40,056
but I had to estimate it, I have
to estimate sigma with an s.

1471
01:22:40,056 --> 01:22:42,420
I'm right back to
this whole difference

1472
01:22:42,420 --> 01:22:45,180
between a t and a unit normal.

1473
01:22:45,180 --> 01:22:47,730
So just to close
this out we often

1474
01:22:47,730 --> 01:22:51,930
talk about these as t-squared,
Hotelling t-squared statistics,

1475
01:22:51,930 --> 01:22:55,660
but it's exactly the same idea.

1476
01:22:55,660 --> 01:22:58,950
And it's got its
own cut-off points.

1477
01:22:58,950 --> 01:23:04,410
OK, so with that, I think we've
pulled to a close on control

1478
01:23:04,410 --> 01:23:06,000
charting for the--

1479
01:23:06,000 --> 01:23:07,860
at least for the time being.

1480
01:23:07,860 --> 01:23:11,460
Next time, we'll start
diving in on yield modeling

1481
01:23:11,460 --> 01:23:13,733
and different uses
of some statistics,

1482
01:23:13,733 --> 01:23:15,150
but I think what
you've got now is

1483
01:23:15,150 --> 01:23:17,880
some exposure to EWMA, CUSUM.

1484
01:23:17,880 --> 01:23:21,450
You'll play around with those
with the three or four problems

1485
01:23:21,450 --> 01:23:23,940
on this week's problem set.

1486
01:23:23,940 --> 01:23:25,860
And start looking
at the quizzes.

1487
01:23:25,860 --> 01:23:29,370
We'll have the first
quiz a week from today--

1488
01:23:29,370 --> 01:23:31,955
in class.

1489
01:23:31,955 --> 01:23:33,580
AUDIENCE: Professor,
I have a question.

1490
01:23:33,580 --> 01:23:34,020
DUANE BONING: Yeah.

1491
01:23:34,020 --> 01:23:34,860
Go ahead and break.

1492
01:23:34,860 --> 01:23:38,580
I'll take your question
while people are dispersing.

1493
01:23:38,580 --> 01:23:41,120
AUDIENCE: So for the
[INAUDIBLE] you taught us today,

1494
01:23:41,120 --> 01:23:42,870
are they going to be
included in the quiz?

1495
01:23:45,660 --> 01:23:49,140
DUANE BONING: It's possible
you'll see an EWMA or a CUSUM,

1496
01:23:49,140 --> 01:23:54,810
but they would only be fairly
simple questions for those.

1497
01:23:54,810 --> 01:23:59,400
It's not likely you'll see
a multivariate problem.

1498
01:23:59,400 --> 01:24:01,580
AUDIENCE: OK, thank you.