WEBVTT 00:00:00.000 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.730 Commons license. 00:00:03.730 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.060 continue to offer high quality educational resources for free. 00:00:10.060 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.560 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.560 --> 00:00:17.904 at ocw.mit.edu. 00:00:21.332 --> 00:00:22.040 DUANE BONING: OK. 00:00:22.040 --> 00:00:23.930 As you'll see with today's lecture, 00:00:23.930 --> 00:00:27.350 I made a small change in the lecture sequence. 00:00:27.350 --> 00:00:32.060 I basically combined the last part of Tuesday's lecture 00:00:32.060 --> 00:00:35.540 with what I was going to talk about next week because they 00:00:35.540 --> 00:00:38.360 really go together better anyway, which is multivariate 00:00:38.360 --> 00:00:41.810 SPC together with a couple of the advanced control charting. 00:00:41.810 --> 00:00:47.210 And so next week we'll do the yield modeling lecture 00:00:47.210 --> 00:00:51.840 rather than try to squish that in today. 00:00:51.840 --> 00:00:55.050 So what I want to do today is first 00:00:55.050 --> 00:00:56.212 have a very brief warm up. 00:00:56.212 --> 00:00:57.920 Just get us back into the swing of things 00:00:57.920 --> 00:01:00.800 with conventional control charts and then talk a little bit 00:01:00.800 --> 00:01:03.080 about these alternative control charts 00:01:03.080 --> 00:01:07.520 with a little bit of increased sensitivity for rapid detection 00:01:07.520 --> 00:01:12.020 of shifts, as well as drifts. 00:01:12.020 --> 00:01:14.360 But primarily targeted at shifts. 00:01:14.360 --> 00:01:16.790 And these are the moving average-- exponentially 00:01:16.790 --> 00:01:19.700 weighted moving average and the cumulative sum chart. 00:01:19.700 --> 00:01:24.260 And then we'll move into multivariate SPC, which 00:01:24.260 --> 00:01:28.940 is a little bit of a tutorial on multivariate statistics 00:01:28.940 --> 00:01:30.900 in general. 00:01:30.900 --> 00:01:31.910 So that's the plan. 00:01:31.910 --> 00:01:36.770 So to get us warmed up we'll do a rapid design of a control 00:01:36.770 --> 00:01:38.750 chart. 00:01:38.750 --> 00:01:40.880 How to do this relatively quickly. 00:01:40.880 --> 00:01:43.400 I will tell you that we have a model based 00:01:43.400 --> 00:01:46.340 on lots of historical data of this process 00:01:46.340 --> 00:01:49.490 that when it is in control it's operating 00:01:49.490 --> 00:01:52.760 as a normally distributed process with a mean of 1 00:01:52.760 --> 00:01:54.380 and a variance of 1. 00:01:54.380 --> 00:01:58.280 We're going to be taking samples every nine parts. 00:01:58.280 --> 00:02:00.720 And then the next 9 and so on. 00:02:00.720 --> 00:02:02.290 So what should the-- 00:02:02.290 --> 00:02:03.200 here's the easy one. 00:02:03.200 --> 00:02:04.920 Wake you up in the morning. 00:02:04.920 --> 00:02:07.682 What should the center line be? 00:02:07.682 --> 00:02:08.273 AUDIENCE: 5. 00:02:08.273 --> 00:02:08.940 DUANE BONING: 5. 00:02:08.940 --> 00:02:09.710 OK. 00:02:09.710 --> 00:02:14.700 Everybody-- how about the plus minus 3 sigma? 00:02:14.700 --> 00:02:20.910 We're quite happy with the 1 in 370 ARL 00:02:20.910 --> 00:02:24.990 with normal typical plus minus 3 sigma control limits. 00:02:24.990 --> 00:02:27.750 Where should the plus minus 3 sigma control limits be? 00:02:31.840 --> 00:02:33.190 I hear a 6 and 4. 00:02:33.190 --> 00:02:35.548 Why? 00:02:35.548 --> 00:02:37.940 AUDIENCE: [INAUDIBLE]. 00:02:37.940 --> 00:02:38.975 DUANE BONING: I'm sorry? 00:02:38.975 --> 00:02:40.082 AUDIENCE: [INAUDIBLE] 00:02:40.082 --> 00:02:40.790 DUANE BONING: OK. 00:02:40.790 --> 00:02:42.396 Which is? 00:02:42.396 --> 00:02:48.252 AUDIENCE: Mu plus sigma [INAUDIBLE].. 00:02:48.252 --> 00:02:53.130 DUANE BONING: Mu plus sigma plus minus 3 sigma. 00:02:53.130 --> 00:02:56.080 3 sigma over square root of sample size. 00:02:56.080 --> 00:02:59.835 And so the 3 sigma and the 3, which comes from the root 00:02:59.835 --> 00:03:04.170 [? end ?] make it a nice plus minus 1. 00:03:04.170 --> 00:03:06.040 And there are some data. 00:03:06.040 --> 00:03:06.570 OK. 00:03:06.570 --> 00:03:09.720 So I plotted 99. 00:03:09.720 --> 00:03:12.840 We ran the process 99 runs. 00:03:12.840 --> 00:03:14.707 We have 11 samples. 00:03:14.707 --> 00:03:15.540 Here's the question. 00:03:15.540 --> 00:03:16.920 Is the process in control? 00:03:21.340 --> 00:03:23.740 We have a yes. 00:03:23.740 --> 00:03:24.790 We have another yes. 00:03:31.510 --> 00:03:32.920 I contend you don't know. 00:03:36.960 --> 00:03:39.010 It's a little bit tricky here. 00:03:39.010 --> 00:03:40.730 AUDIENCE: Not the 1 [INAUDIBLE]. 00:03:43.323 --> 00:03:44.490 DUANE BONING: Yeah, I guess. 00:03:44.490 --> 00:03:46.157 That's actually not what I have in mind. 00:03:46.157 --> 00:03:50.650 You could go in and start down the path of looking and saying, 00:03:50.650 --> 00:03:54.450 OK, what's the probability of having, for example three 00:03:54.450 --> 00:03:55.890 rising points in a row? 00:03:55.890 --> 00:03:57.570 And all the other [INAUDIBLE]. 00:03:57.570 --> 00:04:04.060 But I'll say none of that's happening. 00:04:04.060 --> 00:04:06.960 So I will say that the mean-- 00:04:06.960 --> 00:04:09.480 there's nothing too strange happening with the mean 00:04:09.480 --> 00:04:11.220 that you can tell with this chart. 00:04:11.220 --> 00:04:14.010 Is the process in control? 00:04:14.010 --> 00:04:15.977 AUDIENCE: [INAUDIBLE] variance [INAUDIBLE]---- 00:04:15.977 --> 00:04:16.769 DUANE BONING: Yeah. 00:04:16.769 --> 00:04:18.686 AUDIENCE: [INAUDIBLE] variance is getting huge 00:04:18.686 --> 00:04:20.172 compared to earlier points. 00:04:20.172 --> 00:04:20.880 DUANE BONING: Oh. 00:04:20.880 --> 00:04:22.995 So you're saying the variance is getting larger. 00:04:22.995 --> 00:04:24.210 AUDIENCE: [INAUDIBLE] variance-- yeah. 00:04:24.210 --> 00:04:25.043 DUANE BONING: Maybe. 00:04:25.043 --> 00:04:26.250 It's hard to tell with this. 00:04:26.250 --> 00:04:28.290 This Is a mean chart. 00:04:28.290 --> 00:04:31.500 All it's telling you is the tracking of the mean. 00:04:31.500 --> 00:04:34.410 We said that the process model is that we have 00:04:34.410 --> 00:04:35.940 this mean and this variance. 00:04:35.940 --> 00:04:38.550 And so I guess the main point here 00:04:38.550 --> 00:04:41.670 is you don't really know if you're just tracking the mean. 00:04:41.670 --> 00:04:46.300 What's happening also with the variance within your sample? 00:04:46.300 --> 00:04:48.730 And in fact, on the next page this 00:04:48.730 --> 00:04:53.830 is the raw data-- all 99 points where we grouped 9 by 9 by 9 00:04:53.830 --> 00:04:56.120 to get the sample. 00:04:56.120 --> 00:05:00.970 And if you saw this, you might be a little less comfortable. 00:05:00.970 --> 00:05:03.250 Now you could always go back and look at raw data. 00:05:03.250 --> 00:05:05.720 And raw data is always really good to do. 00:05:05.720 --> 00:05:07.390 But what you would also want to do 00:05:07.390 --> 00:05:11.930 is a typical S chart or a range chart or something like that. 00:05:11.930 --> 00:05:13.570 And in this case, if you design the S 00:05:13.570 --> 00:05:19.030 chart with those apriori pieces of data-- 00:05:19.030 --> 00:05:25.150 sample size is 9 and the process variance is 1-- 00:05:25.150 --> 00:05:29.080 with the C4 correction factor and so on-- all of that just 00:05:29.080 --> 00:05:32.030 out of the formula-- 00:05:32.030 --> 00:05:34.210 you get these control limits. 00:05:34.210 --> 00:05:37.630 And sure enough, at about sample 10 00:05:37.630 --> 00:05:39.070 we're above the control limit. 00:05:39.070 --> 00:05:42.310 Sample 11 is even more above the control limit. 00:05:42.310 --> 00:05:46.630 And that's indicative of essentially 00:05:46.630 --> 00:05:49.400 the increased variance of those two groups. 00:05:49.400 --> 00:05:52.000 And I will tell you, I generated this data with JMP. 00:05:52.000 --> 00:05:56.470 And sure enough, right at the whatever it was-- 00:05:56.470 --> 00:05:59.020 the 77th point-- 00:05:59.020 --> 00:06:03.220 I re-sampled from a normal distribution with something 00:06:03.220 --> 00:06:06.350 like a variance of 1.9 or 2 or something like that. 00:06:06.350 --> 00:06:12.550 So the main point here was you need to monitor both aspects-- 00:06:12.550 --> 00:06:16.720 both the mean and the variance to know whether your process 00:06:16.720 --> 00:06:23.080 model is still operative. 00:06:23.080 --> 00:06:25.060 We're saying that for a normal distribution 00:06:25.060 --> 00:06:28.600 the mean and variance are really telling you everything. 00:06:28.600 --> 00:06:30.520 Now, the other sub point in here is 00:06:30.520 --> 00:06:33.700 that if you were actually looking at this run data, 00:06:33.700 --> 00:06:34.900 there is a lot of noise. 00:06:34.900 --> 00:06:36.970 There is inherent process noise. 00:06:36.970 --> 00:06:38.860 And part of the goal here is you're 00:06:38.860 --> 00:06:43.630 trying to detect a mean in the face of that noise. 00:06:43.630 --> 00:06:47.770 And so one way of thinking about sampling and calculating x bar 00:06:47.770 --> 00:06:50.510 is that, indeed, you are filtering. 00:06:50.510 --> 00:06:51.370 OK. 00:06:51.370 --> 00:06:55.030 So you want to filter to suppress 00:06:55.030 --> 00:06:58.120 the additional variance effect when you're 00:06:58.120 --> 00:07:00.880 looking for the mean phrase. 00:07:00.880 --> 00:07:04.930 But again, you've got to also monitor for the variance. 00:07:04.930 --> 00:07:08.170 But this noise makes it a little bit tricky 00:07:08.170 --> 00:07:11.400 in terms of being able to detect the means. 00:07:11.400 --> 00:07:12.790 So you want to filter. 00:07:12.790 --> 00:07:17.020 And one of the ideas of these alternative control charts 00:07:17.020 --> 00:07:20.860 is explicitly think about the filtering action 00:07:20.860 --> 00:07:22.030 that you're doing-- 00:07:22.030 --> 00:07:25.390 a little bit almost from a more signal processing 00:07:25.390 --> 00:07:27.970 point of view, but not necessarily 00:07:27.970 --> 00:07:33.670 have to wait for an additional nine runs 00:07:33.670 --> 00:07:36.670 before you call that a sample, aggregate it, 00:07:36.670 --> 00:07:40.630 and then plot it, and then wait for another nine runs. 00:07:40.630 --> 00:07:45.990 One alternative approach here is to take a running average. 00:07:45.990 --> 00:07:51.010 So I take a window, say, of 9 points and measurements, 00:07:51.010 --> 00:07:54.400 calculate my statistics, and then for each new run 00:07:54.400 --> 00:07:58.000 move my window over and recalculate my statistics 00:07:58.000 --> 00:08:00.070 so that I'm in some sense updating 00:08:00.070 --> 00:08:04.480 what my best guess of the current state of the process 00:08:04.480 --> 00:08:08.380 is after every run. 00:08:08.380 --> 00:08:10.780 And the advantages of this are that you can get 00:08:10.780 --> 00:08:13.860 more averages per unit of data. 00:08:13.860 --> 00:08:16.000 OK. 00:08:16.000 --> 00:08:23.750 You can go ahead and use the run data alone if you wanted to. 00:08:23.750 --> 00:08:27.140 You could actually plot for example, 00:08:27.140 --> 00:08:30.070 the individual measurements and even do a hybrid sort of thing 00:08:30.070 --> 00:08:32.110 where you did a running average just 00:08:32.110 --> 00:08:35.210 for the standard deviation. 00:08:35.210 --> 00:08:38.720 But the real advantage of these is they essentially improve 00:08:38.720 --> 00:08:40.610 your ability to detect. 00:08:40.610 --> 00:08:43.159 They improve the filtering action 00:08:43.159 --> 00:08:47.480 potentially because you can play with that window size 00:08:47.480 --> 00:08:53.095 to filter more but also be able to react potentially faster. 00:08:53.095 --> 00:08:54.470 So I'm going to describe a couple 00:08:54.470 --> 00:08:57.527 of these alternative charts. 00:08:57.527 --> 00:08:59.360 And I'm going to focus really on the running 00:08:59.360 --> 00:09:01.770 average detection of the mean. 00:09:01.770 --> 00:09:06.860 But the same idea also applies to filtering of the variance 00:09:06.860 --> 00:09:08.360 across multiple runs. 00:09:08.360 --> 00:09:17.810 So there's also running S charts or moving range charts 00:09:17.810 --> 00:09:20.340 and those sorts of things. 00:09:20.340 --> 00:09:23.330 So don't forget the lesson we just learned a minute ago. 00:09:23.330 --> 00:09:25.310 You got to monitor both mean and variance. 00:09:25.310 --> 00:09:29.030 But for the next 10, 15 slides I'm only 00:09:29.030 --> 00:09:33.300 going to talk about running averages on the average. 00:09:33.300 --> 00:09:35.730 So one idea of the running average 00:09:35.730 --> 00:09:38.450 is a simple running average or moving average 00:09:38.450 --> 00:09:43.880 where we simply take some n values of the past 00:09:43.880 --> 00:09:46.130 and calculate say a simple average. 00:09:46.130 --> 00:09:48.830 And similarly, we can do that for the variance. 00:09:48.830 --> 00:09:52.340 So in that case, this simplest picture, 00:09:52.340 --> 00:09:54.500 this simple moving average is just 00:09:54.500 --> 00:09:57.080 an equally weighted window. 00:09:57.080 --> 00:09:58.640 You pick your window size. 00:09:58.640 --> 00:10:02.300 Maybe it's a window size of 9 and you 00:10:02.300 --> 00:10:04.760 calculate a new moving average statistic, which 00:10:04.760 --> 00:10:06.800 I've labeled m sub i here. 00:10:06.800 --> 00:10:10.520 And so that's pretty trivial formula. 00:10:10.520 --> 00:10:12.140 Now how do you set the control limits 00:10:12.140 --> 00:10:15.890 on a moving average chart? 00:10:15.890 --> 00:10:19.970 Well, you want plus minus 3 sigma control 00:10:19.970 --> 00:10:23.990 limits based on the variance that you 00:10:23.990 --> 00:10:27.080 expect in your statistic. 00:10:27.080 --> 00:10:28.220 Right? 00:10:28.220 --> 00:10:32.750 So previously we'd have sample size of 9. 00:10:32.750 --> 00:10:35.180 And we'd say, well, what's the variance? 00:10:35.180 --> 00:10:39.920 It's 1 over-- it's sigma over root n 00:10:39.920 --> 00:10:42.980 is the standard deviation associated 00:10:42.980 --> 00:10:45.110 with our normal sample size. 00:10:45.110 --> 00:10:47.090 And that same formula ends up working here 00:10:47.090 --> 00:10:49.280 because we've got equal weighting. 00:10:49.280 --> 00:10:50.360 Right? 00:10:50.360 --> 00:10:57.590 So if we simply apply, again, assuming each of the points 00:10:57.590 --> 00:11:00.950 are truly in process control and I do not have correlation 00:11:00.950 --> 00:11:04.130 in time between them-- 00:11:04.130 --> 00:11:07.010 it's very easy to see what the variance is. 00:11:07.010 --> 00:11:10.640 Now you'll notice something a little strange 00:11:10.640 --> 00:11:13.380 right here at the beginning. 00:11:13.380 --> 00:11:17.420 This is kind of an artifact of the startup of the chart. 00:11:17.420 --> 00:11:20.660 Let's say I have just started the use of the control chart 00:11:20.660 --> 00:11:23.370 and I've only got five samples so far, 00:11:23.370 --> 00:11:26.960 but I've got a moving average window of 9. 00:11:26.960 --> 00:11:28.790 Well, I don't have nine data points yet. 00:11:28.790 --> 00:11:32.690 I have to form my moving average really with a window size 00:11:32.690 --> 00:11:34.590 only of 5 so far. 00:11:34.590 --> 00:11:37.670 And because I'm taking the variance divided 00:11:37.670 --> 00:11:41.930 by 5 instead of divided by 9, the variance 00:11:41.930 --> 00:11:45.050 of my average of a five point sample is larger. 00:11:45.050 --> 00:11:48.500 And you can see essentially each sample 00:11:48.500 --> 00:11:50.840 until I get to my full window size of 9, 00:11:50.840 --> 00:11:53.610 has slightly larger control chart limits. 00:11:53.610 --> 00:11:54.110 OK. 00:11:54.110 --> 00:11:57.630 So that's what's going on there. 00:11:57.630 --> 00:11:58.130 OK. 00:11:58.130 --> 00:12:00.320 And you can also sort of see here now 00:12:00.320 --> 00:12:02.700 the filtering action at work. 00:12:02.700 --> 00:12:10.140 Notice that each of these samples 00:12:10.140 --> 00:12:13.770 don't vary dramatically from the previous one. 00:12:13.770 --> 00:12:18.540 They are no longer uncorrelated from one sample to the next. 00:12:18.540 --> 00:12:21.270 In fact, with this equal weighting, 00:12:21.270 --> 00:12:27.720 every new time point shares 88% of the data 00:12:27.720 --> 00:12:31.340 with the previous data point. 00:12:31.340 --> 00:12:34.410 So there's a huge amount of correlation from one point 00:12:34.410 --> 00:12:35.740 in time in the next. 00:12:35.740 --> 00:12:40.470 So you should not try to use things 00:12:40.470 --> 00:12:44.070 like the WECO rules associated with three data 00:12:44.070 --> 00:12:47.760 points in a row or four or five data points in a row rising. 00:12:47.760 --> 00:12:51.690 Those things are right out because this 00:12:51.690 --> 00:12:54.990 is a running control chart or moving average control chart. 00:12:54.990 --> 00:12:57.240 And you've now got correlation. 00:12:57.240 --> 00:13:01.200 It's benefit though is that it is doing that filtering. 00:13:01.200 --> 00:13:05.850 And you can take whatever current point 00:13:05.850 --> 00:13:08.670 that you get as a sort of your best 00:13:08.670 --> 00:13:13.440 guess of what the current state of the overall mean is. 00:13:13.440 --> 00:13:16.560 And so you would trigger on alarms 00:13:16.560 --> 00:13:18.140 above or below the control limits. 00:13:21.920 --> 00:13:25.520 So simple moving average-- this is relatively simple. 00:13:25.520 --> 00:13:30.620 Any questions on this before we do a unequally 00:13:30.620 --> 00:13:33.790 weighted moving average? 00:13:33.790 --> 00:13:35.135 Pretty clear. 00:13:35.135 --> 00:13:36.760 AUDIENCE: I have a question, Professor. 00:13:36.760 --> 00:13:37.510 DUANE BONING: Yes. 00:13:37.510 --> 00:13:38.260 Good. 00:13:38.260 --> 00:13:41.470 These questions have been wonderful by the way. 00:13:41.470 --> 00:13:45.220 The last couple of classes, I think the questions have really 00:13:45.220 --> 00:13:46.310 been right on target. 00:13:46.310 --> 00:13:47.780 It's been a lot of fun. 00:13:47.780 --> 00:13:50.520 So we have high expectations for your question. 00:13:50.520 --> 00:13:52.658 [LAUGHTER] 00:13:53.838 --> 00:13:54.380 AUDIENCE: OK. 00:13:54.380 --> 00:13:57.110 So that's a lot of pressure, but my question 00:13:57.110 --> 00:13:59.300 is that for the window size on 9, 00:13:59.300 --> 00:14:02.840 so do the two windows next to each other-- they 00:14:02.840 --> 00:14:05.630 overlap by 8 data points? 00:14:05.630 --> 00:14:07.490 DUANE BONING: Right. 00:14:07.490 --> 00:14:08.040 AUDIENCE: OK. 00:14:08.040 --> 00:14:08.540 That's it. 00:14:08.540 --> 00:14:11.710 DUANE BONING: That's exactly right. 00:14:11.710 --> 00:14:12.710 AUDIENCE: OK, thank you. 00:14:15.320 --> 00:14:18.500 DUANE BONING: Now, one problem with that overlap of eight data 00:14:18.500 --> 00:14:24.290 points is, by the time you get to 8 points in the past, 00:14:24.290 --> 00:14:26.762 it may be-- 00:14:26.762 --> 00:14:30.260 that's a long time ago, depending on how rapidly you're 00:14:30.260 --> 00:14:31.670 sampling or whatever. 00:14:31.670 --> 00:14:35.480 And compared to the data point that you just took, 00:14:35.480 --> 00:14:37.340 you might think that the data point you just 00:14:37.340 --> 00:14:39.950 took represents the current state of the process 00:14:39.950 --> 00:14:45.240 a lot better than the 0.789 time points ago. 00:14:45.240 --> 00:14:47.720 And so there is a more general approach here, 00:14:47.720 --> 00:14:51.140 where you can pick some weighting for how much you 00:14:51.140 --> 00:14:57.910 want to fold in data from different points in time. 00:14:57.910 --> 00:15:01.690 And a classic way to do the weighting 00:15:01.690 --> 00:15:05.230 is to use an exponentially weighted moving 00:15:05.230 --> 00:15:08.260 average, where you might define the weighting function-- 00:15:08.260 --> 00:15:10.900 these a sub i's here-- 00:15:10.900 --> 00:15:15.070 I guess I've switched notation to w here-- 00:15:15.070 --> 00:15:18.010 when I'm at the current sample point or the current time point 00:15:18.010 --> 00:15:23.860 t, I might weight points backwards in time by index i, 00:15:23.860 --> 00:15:28.210 i going from 1 to 9 to 1,000-- whatever-- 00:15:28.210 --> 00:15:30.820 by a function looking something like this, 00:15:30.820 --> 00:15:37.880 with some r factor, where r is weighting factor. 00:15:37.880 --> 00:15:40.570 So what happens here, as I might think 00:15:40.570 --> 00:15:46.600 of r as being how much I believe my current data point-- 00:15:46.600 --> 00:15:50.410 might be I believe my current data point by 90%, 00:15:50.410 --> 00:15:54.640 and then the next data point, 1 minus r, is only 10%. 00:15:54.640 --> 00:15:57.200 So I'm only mixing in 10% of that. 00:15:57.200 --> 00:15:59.290 And then the next data point beyond that-- 00:15:59.290 --> 00:16:02.290 it's maybe 0.1 times 0.1. 00:16:02.290 --> 00:16:05.200 It's only 0.01, so I'm only mixing 00:16:05.200 --> 00:16:07.660 in 1% of the previous data point. 00:16:07.660 --> 00:16:15.190 So very rapidly, I have this drop-off as a function of data 00:16:15.190 --> 00:16:16.375 points in the past-- 00:16:18.970 --> 00:16:20.340 this is my i index-- 00:16:24.120 --> 00:16:28.710 so that I much more heavily weight recent history, 00:16:28.710 --> 00:16:32.910 and then exponentially decay into the past. 00:16:35.513 --> 00:16:37.180 But I still have this degree of freedom. 00:16:40.305 --> 00:16:41.430 I can pick r. 00:16:41.430 --> 00:16:45.960 I can pick the relative weighting of more recent data 00:16:45.960 --> 00:16:49.910 compared to old data. 00:16:49.910 --> 00:16:52.710 Now, one way of looking at this is, 00:16:52.710 --> 00:16:58.790 actually, I have access to all past data, 00:16:58.790 --> 00:17:02.750 and I would apply this out to 10 data 00:17:02.750 --> 00:17:08.420 points, or 50, or 100 with the appropriate weighting function. 00:17:08.420 --> 00:17:10.849 And I guess, with computer systems these days, 00:17:10.849 --> 00:17:12.980 it would be no problem to actually apply it 00:17:12.980 --> 00:17:14.930 in that fashion. 00:17:14.930 --> 00:17:18.410 There's a beautiful thing about the exponentially weighted 00:17:18.410 --> 00:17:21.470 moving average, which is I actually 00:17:21.470 --> 00:17:25.300 don't have to keep track of all of the raw data. 00:17:25.300 --> 00:17:27.050 There's another way of thinking about what 00:17:27.050 --> 00:17:30.320 the EWMA, this exponentially weighted moving average 00:17:30.320 --> 00:17:31.830 is doing. 00:17:31.830 --> 00:17:35.540 And that's that I have my previous best estimate 00:17:35.540 --> 00:17:39.590 of the state of my variable-- my previous best estimate 00:17:39.590 --> 00:17:40.880 of the mean-- 00:17:40.880 --> 00:17:42.860 and now I have one new data point, 00:17:42.860 --> 00:17:44.660 and I use that one new data point 00:17:44.660 --> 00:17:47.690 to update my best estimate. 00:17:47.690 --> 00:17:52.170 My best estimate had embedded in it all of the past history, 00:17:52.170 --> 00:17:54.020 so now, with the additional new weight, 00:17:54.020 --> 00:17:58.820 I can just do a simple update of my best state estimate. 00:17:58.820 --> 00:18:03.050 And there is essentially a recursive formula 00:18:03.050 --> 00:18:07.280 that makes this explicit, and it's exactly the same, thing 00:18:07.280 --> 00:18:10.730 if you now apply it iteratively out to all history, 00:18:10.730 --> 00:18:14.690 as actually performing that computation of applying 00:18:14.690 --> 00:18:17.210 the weights on all past data. 00:18:17.210 --> 00:18:20.630 So that is to say, let's say, a sub i 00:18:20.630 --> 00:18:25.250 is my best estimate of whatever my statistic is-- my mean. 00:18:25.250 --> 00:18:28.640 Again, I might weight my current data point, 00:18:28.640 --> 00:18:31.950 my most recent data point, say, 90%, 00:18:31.950 --> 00:18:34.760 which would be really rapid update. 00:18:34.760 --> 00:18:37.430 I really believe my most recent data. 00:18:37.430 --> 00:18:42.200 And then my 1 minus r applies to my previous-- 00:18:42.200 --> 00:18:47.430 oops-- to my previous estimate. 00:18:54.350 --> 00:18:57.230 So I'm believing only 10% my previous estimate, 00:18:57.230 --> 00:19:00.223 and 90% my new estimate. 00:19:00.223 --> 00:19:01.640 So it makes it a little bit easier 00:19:01.640 --> 00:19:03.980 in terms of the computation, but I 00:19:03.980 --> 00:19:06.140 think it's also conceptually nice in terms 00:19:06.140 --> 00:19:08.870 of what the action here is. 00:19:08.870 --> 00:19:14.310 Now, one can go through and do a little bit of the mathematics. 00:19:14.310 --> 00:19:18.980 It's actually a little easier to do it on the full summation 00:19:18.980 --> 00:19:23.810 version, but you just do basic expectation math 00:19:23.810 --> 00:19:27.230 on the variance of an EWMA-- 00:19:27.230 --> 00:19:30.350 again, drawing from a normal distribution that 00:19:30.350 --> 00:19:32.120 is in control. 00:19:32.120 --> 00:19:35.750 And one can estimate the standard deviation 00:19:35.750 --> 00:19:38.030 of this a statistic. 00:19:38.030 --> 00:19:40.470 And it's kind of a little bit messy. 00:19:40.470 --> 00:19:45.020 It's got the underlying process variation in it, the fact 00:19:45.020 --> 00:19:47.300 that you are still-- 00:19:47.300 --> 00:19:51.080 you can still be doing sampling. 00:19:51.080 --> 00:19:56.300 For each x sub i, that might be a sample of n data points. 00:19:56.300 --> 00:19:59.530 Very often, you might simply be doing it one data point, 00:19:59.530 --> 00:20:02.240 so n, very often, is just 1-- 00:20:02.240 --> 00:20:07.220 and then this formula with r in it, including this 1 minus 1 00:20:07.220 --> 00:20:11.690 minus r to the 2t, where t is how far along you are. 00:20:11.690 --> 00:20:14.150 So it's kind of nasty, but one thing that's nice 00:20:14.150 --> 00:20:16.400 is, once you've start started up your control chart 00:20:16.400 --> 00:20:18.590 and you're away from that little artifact 00:20:18.590 --> 00:20:20.330 that we saw with the moving average, 00:20:20.330 --> 00:20:24.350 where the variance limits change-- 00:20:24.350 --> 00:20:28.100 once you get far enough away from that start up, 00:20:28.100 --> 00:20:33.530 your t is large, and that this whole term just drops out-- 00:20:33.530 --> 00:20:35.420 drops out more fast when-- 00:20:35.420 --> 00:20:39.830 more rapidly for large values of r, 00:20:39.830 --> 00:20:42.900 because I'm really weighting more towards recent history. 00:20:42.900 --> 00:20:44.990 But even for small or modest values 00:20:44.990 --> 00:20:48.230 of r, after you get out 10, 20 data points, 00:20:48.230 --> 00:20:50.220 it typically drops away. 00:20:50.220 --> 00:20:52.340 So now you've got an estimate for large T 00:20:52.340 --> 00:20:55.610 for the variance, sigma a of your statistic, 00:20:55.610 --> 00:20:58.670 and you can simply plot plus/minus 00:20:58.670 --> 00:21:01.760 3 sigma, or whatever control limit you want, 00:21:01.760 --> 00:21:05.380 depending on your false alarm rate alpha. 00:21:05.380 --> 00:21:08.070 So this is pretty cool. 00:21:08.070 --> 00:21:10.320 One quick observation here is, again, we still 00:21:10.320 --> 00:21:12.720 have a lot of degrees of flexibility 00:21:12.720 --> 00:21:15.000 around the choice of r. 00:21:15.000 --> 00:21:21.105 And if we look at this formula here, and the effect of r-- 00:21:21.105 --> 00:21:25.920 I sort of plotted this out in this next chart. 00:21:25.920 --> 00:21:29.790 This effect of r on the sigma multiplier-- oops-- sorry-- 00:21:29.790 --> 00:21:33.820 is this factor right here. 00:21:33.820 --> 00:21:38.430 So in other words, as r changes, how do my control limits move? 00:21:38.430 --> 00:21:41.580 What is this multiplier of my sigma over root 00:21:41.580 --> 00:21:44.880 n from my normal sampling? 00:21:44.880 --> 00:21:46.410 And you can look at it here. 00:21:46.410 --> 00:21:51.330 And as r gets larger, well, that gets larger-- 00:21:51.330 --> 00:21:53.010 as r approaches 1. 00:21:53.010 --> 00:21:55.750 Typically, you don't ever go above 1. 00:21:55.750 --> 00:21:57.630 You can't believe your most recent data 00:21:57.630 --> 00:21:59.700 point more than 100%. 00:21:59.700 --> 00:22:04.400 It's a ratio, or fraction of the percent. 00:22:04.400 --> 00:22:09.450 You can see that, as r gets larger, 00:22:09.450 --> 00:22:15.040 this multiplier starts to approach some limit. 00:22:15.040 --> 00:22:21.180 And finally, when r is equal to 1, this multiplier is simply 1. 00:22:21.180 --> 00:22:27.140 In the limit, when r is equal to 1, what are we doing? 00:22:27.140 --> 00:22:30.700 We're doing 100% update based on my most recent data point. 00:22:30.700 --> 00:22:33.480 In other words, I'm only looking at my most recent data point. 00:22:33.480 --> 00:22:35.790 My variance is simply the process variance 00:22:35.790 --> 00:22:37.230 coming from sampling. 00:22:37.230 --> 00:22:40.600 This all reduces to [? simply ?] sigma 00:22:40.600 --> 00:22:44.190 a being by sampling variance, whatever 00:22:44.190 --> 00:22:46.770 size sample I was doing within each formulation. 00:22:46.770 --> 00:22:55.080 So in that case, I have no factor or no multiplier 00:22:55.080 --> 00:23:00.630 on that sigma A. But for other values of r, what's happening? 00:23:00.630 --> 00:23:06.430 For smaller values of r, what do we see? 00:23:09.030 --> 00:23:15.690 The sigma A is smaller than the sigma x over root n. 00:23:15.690 --> 00:23:20.310 We're getting reduction in the variance because of filtering, 00:23:20.310 --> 00:23:21.870 the variance of our estimate. 00:23:21.870 --> 00:23:25.110 So that was the whole idea is I can tighten my control 00:23:25.110 --> 00:23:30.450 limits, essentially by the filtering action. 00:23:30.450 --> 00:23:33.960 So as I get smaller and smaller r's, I 00:23:33.960 --> 00:23:36.120 do more and more aggressive filtering 00:23:36.120 --> 00:23:38.190 with lots of past data. 00:23:38.190 --> 00:23:40.980 I get tighter and tighter limits, 00:23:40.980 --> 00:23:45.720 but of course, I'm also suppressing recent data 00:23:45.720 --> 00:23:50.070 little bit more heavily, so there is this trade-off. 00:23:50.070 --> 00:23:53.040 Let's look at some examples. 00:23:53.040 --> 00:23:56.130 Oh, well, I guess I made these points already here. 00:23:56.130 --> 00:24:01.320 Again, the variance of this estimate for the mean-- 00:24:01.320 --> 00:24:03.240 your best guess of the mean-- 00:24:03.240 --> 00:24:07.650 is smaller than you would have had with an x bar chart, 00:24:07.650 --> 00:24:11.370 because the x bar chart, remember, for a sample 00:24:11.370 --> 00:24:14.170 size of n, is simply that. 00:24:14.170 --> 00:24:16.080 So we do get this suppression. 00:24:16.080 --> 00:24:20.460 And it is very nice that it also, in the limit, 00:24:20.460 --> 00:24:24.930 works out for both the sample size of 1 case-- 00:24:24.930 --> 00:24:26.610 these formulas still hold-- 00:24:26.610 --> 00:24:29.302 and also the full update case, or the r 00:24:29.302 --> 00:24:32.280 equals 1, where we have completely unfiltered data. 00:24:32.280 --> 00:24:37.360 You just plot in your pure run data on your chart. 00:24:37.360 --> 00:24:43.530 So what happens if I use much smaller r's, like 0.1? 00:24:43.530 --> 00:24:45.750 I got much tighter control limits, 00:24:45.750 --> 00:24:51.270 but I've got long delays potentially then in detecting-- 00:24:51.270 --> 00:24:52.800 I've got a lot of noise suppression, 00:24:52.800 --> 00:24:54.690 but again, I get back to long delays 00:24:54.690 --> 00:24:58.780 in detecting or responding to a true shift in the mean. 00:24:58.780 --> 00:25:01.890 So I want to look at a few examples here 00:25:01.890 --> 00:25:05.010 to give you a feel of the trade-off between different r's 00:25:05.010 --> 00:25:12.000 for some data, and compare it as well to a typical x bar chart. 00:25:12.000 --> 00:25:19.440 OK, so here's a set of data, and we're seeing a comparison here 00:25:19.440 --> 00:25:21.645 between the x bar chart-- 00:25:24.540 --> 00:25:32.720 so you can see here are the x bar upper and lower control 00:25:32.720 --> 00:25:34.030 limits. 00:25:34.030 --> 00:25:40.970 And sure enough, after 150 runs, where at-- 00:25:40.970 --> 00:25:45.530 I don't know-- about run 90, we make this shift 00:25:45.530 --> 00:25:46.410 in the process-- 00:25:46.410 --> 00:25:50.000 so we shift the whole process by a half 00:25:50.000 --> 00:25:53.090 a standard deviation of the underlying process. 00:25:53.090 --> 00:25:56.460 And sure enough, by about-- 00:25:56.460 --> 00:26:00.470 I don't know-- run 140, we finally got a data point 00:26:00.470 --> 00:26:02.840 in the tail of the mean distribution-- 00:26:02.840 --> 00:26:04.370 the regular x bar-- 00:26:04.370 --> 00:26:06.200 that fell out. 00:26:06.200 --> 00:26:09.110 And we could actually calculate the average run length 00:26:09.110 --> 00:26:13.790 to detect a mean shift of 0.5 using some of the techniques we 00:26:13.790 --> 00:26:16.270 talked about last time. 00:26:16.270 --> 00:26:19.720 And we might not be happy with that very long ARL, 00:26:19.720 --> 00:26:24.280 and the question is, would an EWMA chart help us with this? 00:26:24.280 --> 00:26:26.170 And this is an example where we're 00:26:26.170 --> 00:26:32.020 using about a 30% update based on each new data point, 00:26:32.020 --> 00:26:35.230 and plotted in purple with the blue 00:26:35.230 --> 00:26:38.170 and the yellow are the upper and lower control 00:26:38.170 --> 00:26:40.930 limits for the EWMA. 00:26:40.930 --> 00:26:47.640 And what you see is the effect of averaging. 00:26:47.640 --> 00:26:53.560 Now, I always find it's easiest for me to conceptualize 00:26:53.560 --> 00:26:57.070 the action of the EWMA chart is just kind of like a moving 00:26:57.070 --> 00:27:00.640 average-- the equally weighted moving average, but weighted 00:27:00.640 --> 00:27:03.160 a little bit more towards recent history. 00:27:03.160 --> 00:27:07.690 But the whole idea is I get that aggregation 00:27:07.690 --> 00:27:10.540 from any kind of a moving average so that, 00:27:10.540 --> 00:27:13.180 right after this shift right here, 00:27:13.180 --> 00:27:15.760 now, when I take into account-- 00:27:15.760 --> 00:27:21.400 with an r of 0.3, I'm taking into account three, four, five 00:27:21.400 --> 00:27:22.540 data points-- 00:27:22.540 --> 00:27:25.930 I've got kind of an effective window weighted 00:27:25.930 --> 00:27:29.080 about 30% towards my most recent, but including 00:27:29.080 --> 00:27:30.730 some past history-- 00:27:30.730 --> 00:27:34.450 that all of the plus delta nu data 00:27:34.450 --> 00:27:39.760 points all average, and then my normal process variation 00:27:39.760 --> 00:27:42.440 just cancels out each other. 00:27:42.440 --> 00:27:46.510 And so it's very intuitive, I think, 00:27:46.510 --> 00:27:51.820 that this EWMA is going to aggregate or respond 00:27:51.820 --> 00:27:56.230 to or sense the delta shift in the mean more 00:27:56.230 --> 00:28:03.580 than an individual x bar chart would have been doing-- 00:28:03.580 --> 00:28:05.770 so that it responds in this case, 00:28:05.770 --> 00:28:10.330 I don't know, after about 15 or 20 data points. 00:28:10.330 --> 00:28:15.690 You can start to see excursions beyond the control limit. 00:28:15.690 --> 00:28:16.723 Question-- 00:28:16.723 --> 00:28:18.390 AUDIENCE: At the beginning of the chart, 00:28:18.390 --> 00:28:20.803 why are the control limits [INAUDIBLE] 00:28:20.803 --> 00:28:21.720 spreading [INAUDIBLE]? 00:28:21.720 --> 00:28:27.100 The last time, they were [INAUDIBLE] 00:28:27.100 --> 00:28:28.790 DUANE BONING: Yes, yes-- good point. 00:28:31.650 --> 00:28:35.400 I think you can go back to this formula and see if that works. 00:28:42.720 --> 00:28:45.150 I bet, if you look at the effective the t 00:28:45.150 --> 00:28:53.510 action, formulaically or analytically, 00:28:53.510 --> 00:28:57.130 I think that, as t starts out being very small-- 00:29:04.980 --> 00:29:08.130 so if t is 0, this is essentially 1. 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. 00:29:18.240 --> 00:29:19.750 I've got a reduction. 00:29:19.750 --> 00:29:23.950 So I think, for small t, I've only 00:29:23.950 --> 00:29:26.890 got a limited amount of data, and so I'm not-- 00:29:26.890 --> 00:29:29.770 I don't have the long tail in the EWMA. 00:29:29.770 --> 00:29:32.500 So I think I've only got a small amount of data 00:29:32.500 --> 00:29:34.750 with a larger weight in my control limits 00:29:34.750 --> 00:29:36.110 I think are smaller. 00:29:36.110 --> 00:29:41.200 But then, as t gets larger, I've got a little bit more 00:29:41.200 --> 00:29:43.150 and more data until t-- 00:29:43.150 --> 00:29:45.310 so I think if I were to plot this 1 minus 1 00:29:45.310 --> 00:29:47.200 minus r to the 2t-- 00:29:47.200 --> 00:29:54.220 kind of has a funky shape that dies out. 00:29:54.220 --> 00:29:56.980 But I wouldn't worry too much about the startup times. 00:29:56.980 --> 00:29:59.620 There are artifacts when you start up a control chart, 00:29:59.620 --> 00:30:01.810 but once you just start a control chart anyway, 00:30:01.810 --> 00:30:04.720 you're still feeling out the process, 00:30:04.720 --> 00:30:07.930 so I wouldn't worry too much about the startup 00:30:07.930 --> 00:30:08.900 times on these. 00:30:08.900 --> 00:30:09.400 Yeah? 00:30:09.400 --> 00:30:18.768 AUDIENCE: [INAUDIBLE] How do you get this side [INAUDIBLE] 00:30:18.768 --> 00:30:19.560 DUANE BONING: Yeah. 00:30:19.560 --> 00:30:22.920 So the question is-- if you didn't hear that in Singapore-- 00:30:22.920 --> 00:30:25.110 was, when would you use an EWMA or what 00:30:25.110 --> 00:30:29.170 are the factors in using an EWMA from an x bar chart? 00:30:29.170 --> 00:30:34.920 And I think, if you've got a lot of noise 00:30:34.920 --> 00:30:39.600 and you're really worried about detecting rapidly a shift 00:30:39.600 --> 00:30:42.510 in the process, an EWMA-- 00:30:42.510 --> 00:30:44.790 and we'll see the CUSUM in a minute-- 00:30:44.790 --> 00:30:50.160 has improved sensitivity to rapid detection of the shift. 00:30:50.160 --> 00:30:53.410 But it comes at one really important cost, 00:30:53.410 --> 00:30:59.280 which is a loss in transparency in what's going on. 00:30:59.280 --> 00:31:01.320 The x bar, I think, conceptually, 00:31:01.320 --> 00:31:03.930 is a lot easier to understand, have 00:31:03.930 --> 00:31:07.290 everybody on the line understand what it is. 00:31:07.290 --> 00:31:11.280 The data, if you also explode out and show the full data-- 00:31:11.280 --> 00:31:14.610 the x bar and the s-- 00:31:14.610 --> 00:31:17.610 gives you a good feel directly to the statistics 00:31:17.610 --> 00:31:19.020 of the process. 00:31:19.020 --> 00:31:20.940 Here you look at this and you lose touch, 00:31:20.940 --> 00:31:27.240 I think, a little bit with the statistics of-- 00:31:27.240 --> 00:31:29.550 and the relationship to your raw data. 00:31:29.550 --> 00:31:31.440 It's a little harder to get your mind around. 00:31:36.660 --> 00:31:40.530 I don't know-- 10, 15 years ago, I fell in love with the EWMA 00:31:40.530 --> 00:31:42.870 chart and approach-- 00:31:42.870 --> 00:31:47.080 not only for charting, but also for doing 00:31:47.080 --> 00:31:49.950 simple run-by-run control. 00:31:49.950 --> 00:31:52.410 And so I kind of like it, but I can 00:31:52.410 --> 00:31:55.980 empathize with it being a little bit harder to explain 00:31:55.980 --> 00:31:59.250 and maintain on the line. 00:31:59.250 --> 00:32:02.580 It's also got a little bit more calculation, so back 00:32:02.580 --> 00:32:05.010 in the days of-- 00:32:05.010 --> 00:32:07.380 and it's still true in some of the manufacturing lines. 00:32:07.380 --> 00:32:10.050 I know, when we went on the tour and some of the plants-- 00:32:10.050 --> 00:32:14.702 very, very up-to-date plants in Singapore 00:32:14.702 --> 00:32:16.410 and other places around the world, people 00:32:16.410 --> 00:32:18.510 are still often taking measurements by hand, 00:32:18.510 --> 00:32:22.890 and doing the hand calculation, and writing it on a chart. 00:32:22.890 --> 00:32:24.270 This has an extra step. 00:32:24.270 --> 00:32:27.390 You've got to do an extra average. 00:32:27.390 --> 00:32:34.300 So in modern day, when almost all the calculations 00:32:34.300 --> 00:32:35.388 are automated. 00:32:35.388 --> 00:32:36.930 And the data collection is automated. 00:32:36.930 --> 00:32:40.350 I think there's even more reason to go with an EWMA. 00:32:40.350 --> 00:32:45.510 So the thing I like about the best about the EWMA is I 00:32:45.510 --> 00:32:50.040 think of it as giving me my best current estimate based 00:32:50.040 --> 00:32:53.670 not only on just my last data point, but a little bit 00:32:53.670 --> 00:32:55.650 of filtering, including past history. 00:32:55.650 --> 00:33:01.770 It's giving my best educated state of the current process. 00:33:01.770 --> 00:33:05.790 And why I like that for then run-by-run control is, 00:33:05.790 --> 00:33:08.880 based on my current model, my updated-- 00:33:08.880 --> 00:33:11.100 on a recursive basis, updated model 00:33:11.100 --> 00:33:14.740 of the current state of the process I can make a decision. 00:33:14.740 --> 00:33:19.620 So here the decision in SPC is the process in control or not 00:33:19.620 --> 00:33:24.720 'But if you think of this being not just the EWMA not-- 00:33:24.720 --> 00:33:26.140 giving you more information. 00:33:26.140 --> 00:33:30.810 It's not just giving you, am I in control or not, 00:33:30.810 --> 00:33:32.620 given the random spread? 00:33:32.620 --> 00:33:35.190 But if you interpret this little purple data 00:33:35.190 --> 00:33:39.030 point here as my best estimate of where 00:33:39.030 --> 00:33:42.420 the mean in the recent past has been, 00:33:42.420 --> 00:33:45.930 if there is a little bit of drift or movement maybe. 00:33:45.930 --> 00:33:51.120 I do active control and not pure SPC control. 00:33:51.120 --> 00:33:53.850 I'm not making any longer just the decision. 00:33:53.850 --> 00:33:55.080 Nothing has changed. 00:33:55.080 --> 00:33:57.030 I'm saying there are small changes, 00:33:57.030 --> 00:34:00.180 and maybe I want to go and make a small modification 00:34:00.180 --> 00:34:02.860 or an adjustment on the equipment. 00:34:02.860 --> 00:34:03.360 Now. 00:34:03.360 --> 00:34:07.050 If, you are in true statistical process control, 00:34:07.050 --> 00:34:08.739 that would be the wrong thing to do. 00:34:08.739 --> 00:34:12.040 You'd be over-adjusting, and we'll talk about that later. 00:34:12.040 --> 00:34:16.230 But that's another thing I like about the EWMA is very often, 00:34:16.230 --> 00:34:20.310 there is a little bit of time correlation in the process. 00:34:20.310 --> 00:34:26.679 And the EWMA is tracking those small changes 00:34:26.679 --> 00:34:29.260 and filtering out some of the noise. 00:34:29.260 --> 00:34:30.639 It's a good question. 00:34:30.639 --> 00:34:31.540 I'll take a question. 00:34:31.540 --> 00:34:35.207 There was somebody had a question in Singapore. 00:34:35.207 --> 00:34:35.790 AUDIENCE: Yes. 00:34:35.790 --> 00:34:40.230 So under smaller r will lead to the narrow control limit, 00:34:40.230 --> 00:34:41.715 right? 00:34:41.715 --> 00:34:42.840 DUANE BONING: That's right. 00:34:42.840 --> 00:34:45.380 AUDIENCE: That's so the narrow control limit-- 00:34:45.380 --> 00:34:48.480 you will have a smaller beta? 00:34:53.389 --> 00:34:56.449 DUANE BONING: So a smaller beta would be less chance-- 00:34:56.449 --> 00:34:57.580 AUDIENCE: Smaller beta? 00:34:57.580 --> 00:34:58.330 DUANE BONING: Yes. 00:34:58.330 --> 00:35:00.630 AUDIENCE: Yeah, [INAUDIBLE] type II error. 00:35:00.630 --> 00:35:03.960 DUANE BONING: Right-- but its power is then larger, 00:35:03.960 --> 00:35:08.790 so the chance of detecting a true out of control point 00:35:08.790 --> 00:35:11.430 increases. 00:35:11.430 --> 00:35:14.850 But there's a very important additional factor, which 00:35:14.850 --> 00:35:17.540 is there's additional time lag. 00:35:17.540 --> 00:35:19.290 So I'm going to show a couple of examples, 00:35:19.290 --> 00:35:22.410 because you can see in other examples 00:35:22.410 --> 00:35:25.890 the effective the filtering and the time response that it takes 00:35:25.890 --> 00:35:28.710 to detect can go up. 00:35:28.710 --> 00:35:30.773 AUDIENCE: OK. 00:35:30.773 --> 00:35:32.440 DUANE BONING: So another question here-- 00:35:32.440 --> 00:35:33.120 AUDIENCE: I'm was just curious. 00:35:33.120 --> 00:35:35.640 What sample size were you using for the x bar chart? 00:35:35.640 --> 00:35:36.950 DUANE BONING: I don't know. 00:35:36.950 --> 00:35:40.710 I'm not sure what was used in the-- 00:35:40.710 --> 00:35:41.790 as a sample size. 00:35:41.790 --> 00:35:44.350 I think you're just seeing the plot of the sample, 00:35:44.350 --> 00:35:47.025 so the sample size is actually the same in both cases. 00:35:52.370 --> 00:35:56.390 But I don't what was being used. 00:35:56.390 --> 00:36:00.590 In some sense, it doesn't actually matter too much. 00:36:00.590 --> 00:36:02.780 The EWMA is looking-- 00:36:02.780 --> 00:36:05.500 is an EWMA over sample. 00:36:09.020 --> 00:36:10.970 I might have 1,000 parts an hour, 00:36:10.970 --> 00:36:16.400 and every 15 minutes, I might do a sample size of 10. 00:36:16.400 --> 00:36:18.820 So I'm only getting sample size 10, lots more. 00:36:18.820 --> 00:36:23.960 And what's being plotted here, in some sense, is the sample. 00:36:23.960 --> 00:36:27.070 So I have an x bar where I might plot those samples. 00:36:27.070 --> 00:36:28.900 Here what I'm doing-- 00:36:28.900 --> 00:36:34.240 the EWMA has enough-- it has the flexibility to extend to n 00:36:34.240 --> 00:36:36.700 equals 1 sampling, if that's what I'm doing. 00:36:36.700 --> 00:36:39.340 But in some sense, it could also be 00:36:39.340 --> 00:36:45.160 used where, instead of each of those n equals 10 samples was 00:36:45.160 --> 00:36:46.570 the only data I looked at. 00:36:46.570 --> 00:36:48.370 Each one of those is my t-- 00:36:48.370 --> 00:36:53.220 point in t, and my previous sample of size 10 is t minus 1. 00:36:56.260 --> 00:37:00.520 So in some sense, it's doing the whole sample-by-sample 00:37:00.520 --> 00:37:02.270 comparison. 00:37:02.270 --> 00:37:06.550 So my point here was both the EWMA 00:37:06.550 --> 00:37:12.650 and the x bar are using, in this derivation, the same sample 00:37:12.650 --> 00:37:13.150 size. 00:37:13.150 --> 00:37:15.100 I'm not differentiating in that. 00:37:15.100 --> 00:37:18.640 All I'm doing in this case really, with the EWMA 00:37:18.640 --> 00:37:21.880 is using multiple samples back in time, 00:37:21.880 --> 00:37:25.770 and the x bar only uses one. 00:37:25.770 --> 00:37:32.070 So in fact, if I go to r equals 1, the two converge. 00:37:32.070 --> 00:37:33.240 They are the same. 00:37:33.240 --> 00:37:38.190 The r equals EWMA is exactly the same as the x bar. 00:37:38.190 --> 00:37:39.860 AUDIENCE: But the larger my sample size 00:37:39.860 --> 00:37:44.160 is the less of a spread I get between my two-- 00:37:44.160 --> 00:37:45.486 [INAUDIBLE] 00:37:45.486 --> 00:37:47.640 DUANE BONING: They would both go together, though, 00:37:47.640 --> 00:37:49.015 because I had to have the factor. 00:37:49.015 --> 00:37:52.350 So back here on slide 12, if I change my sample size, 00:37:52.350 --> 00:37:56.520 both sigma A and my upper control limits 00:37:56.520 --> 00:38:00.550 and lower federal limits for the x bar shrink exactly the same. 00:38:00.550 --> 00:38:04.840 So I would scale both of them, if I increase. 00:38:04.840 --> 00:38:10.600 So what I was talking about here is with different values of r. 00:38:10.600 --> 00:38:14.840 So this was an r equals 0.3. 00:38:14.840 --> 00:38:17.740 Here's a smaller r, r equals 0.1. 00:38:17.740 --> 00:38:19.240 So what's happening here? 00:38:19.240 --> 00:38:22.600 I'm only accepting my most recent data point 10%, 00:38:22.600 --> 00:38:25.870 and mixing past history and 90%. 00:38:25.870 --> 00:38:28.900 That's more averaging of the past. 00:38:28.900 --> 00:38:31.570 I'm using more past history, and so 00:38:31.570 --> 00:38:38.030 in this case, what you see is kind of a more slowly drifting, 00:38:38.030 --> 00:38:39.750 more heavily-- 00:38:39.750 --> 00:38:43.190 excuse me-- more heavily filtered version. 00:38:43.190 --> 00:38:47.830 And here, with the mean shift occurring about here, 00:38:47.830 --> 00:38:50.890 you can start to see in this case 00:38:50.890 --> 00:38:59.017 a big mean shift at 0.5 with, I guess, a sample size of 5. 00:38:59.017 --> 00:39:00.850 You do need to know the sample size in order 00:39:00.850 --> 00:39:05.260 to know the effect of the mean shift and ability 00:39:05.260 --> 00:39:08.660 to detect with either chart. 00:39:08.660 --> 00:39:10.000 So somewhere in here-- 00:39:10.000 --> 00:39:11.260 I guess around there-- 00:39:11.260 --> 00:39:15.730 I got I got lucky, and I'm starting to sample now 00:39:15.730 --> 00:39:19.960 with the x bar picking up that detection. 00:39:19.960 --> 00:39:24.760 But with the EWMA, in some sense, I've still got 90%. 00:39:24.760 --> 00:39:29.770 It takes 9 or 10 data points for the more recent data 00:39:29.770 --> 00:39:33.310 to outweigh the past data, and so I've actually 00:39:33.310 --> 00:39:37.450 got, in this case, a time lag in detecting 00:39:37.450 --> 00:39:43.880 the mean shift with the EWMA compared to the x bar. 00:39:43.880 --> 00:39:46.620 So a question here first-- 00:39:46.620 --> 00:39:48.584 AUDIENCE: I don't see how we can compare them 00:39:48.584 --> 00:39:52.680 because they actually are different time scales. 00:39:52.680 --> 00:39:54.810 All the x bar will effectively-- let 00:39:54.810 --> 00:40:04.390 us say the sample size is [INAUDIBLE] 00:40:04.390 --> 00:40:06.970 DUANE BONING: So the question was, how do you compare 00:40:06.970 --> 00:40:08.740 the time scales on these? 00:40:08.740 --> 00:40:12.080 So here's what I think is going on. 00:40:12.080 --> 00:40:14.650 And this is sort of what I was talking about a second m 00:40:14.650 --> 00:40:17.320 but it's probably still a little confusing. 00:40:17.320 --> 00:40:22.240 I believe that this n equals 5 sample size means 00:40:22.240 --> 00:40:27.100 that, inside of this data point, if I were to explode that, 00:40:27.100 --> 00:40:32.090 it actually has five data points inside of it. 00:40:32.090 --> 00:40:37.420 And I'm only plotting the average of those five on here. 00:40:37.420 --> 00:40:42.160 This data point also has those same five data points, 00:40:42.160 --> 00:40:45.310 but then it also grabbed the data 00:40:45.310 --> 00:40:48.100 that was lurking in the previous one. 00:40:48.100 --> 00:40:53.800 And it has its five data points and its five-- 00:40:53.800 --> 00:40:55.040 whoops. 00:40:55.040 --> 00:40:55.540 Yah! 00:41:01.900 --> 00:41:02.830 Don't want to-- 00:41:08.340 --> 00:41:11.400 I think it was trying to exit me out here. 00:41:14.200 --> 00:41:15.960 So while it's doing that, the point 00:41:15.960 --> 00:41:21.840 here was that the n equals 5 is not looking back 00:41:21.840 --> 00:41:23.910 five data points in the x bar. 00:41:23.910 --> 00:41:26.800 It's buried in each of the individual points-- 00:41:26.800 --> 00:41:38.330 so that both EWMA and x bar are not 00:41:38.330 --> 00:41:42.440 affected by this n equals 5 sampling differently. 00:41:42.440 --> 00:41:47.510 But then the time scale is correct in terms of-- 00:41:47.510 --> 00:41:50.300 x bar only looks at my most recent sample. 00:41:50.300 --> 00:41:54.050 The EWMA does look back in history, 00:41:54.050 --> 00:41:57.170 and that's why you've got this additional time lag. 00:41:57.170 --> 00:42:00.170 Was that getting at your question or your observation, 00:42:00.170 --> 00:42:02.370 or is your question a little different? 00:42:02.370 --> 00:42:03.850 AUDIENCE: [INAUDIBLE] 00:42:03.850 --> 00:42:05.060 DUANE BONING: OK. 00:42:05.060 --> 00:42:05.960 OK. 00:42:05.960 --> 00:42:08.617 And there was a question in Singapore. 00:42:08.617 --> 00:42:09.200 AUDIENCE: Yes. 00:42:09.200 --> 00:42:11.630 So my question is sort of built on what 00:42:11.630 --> 00:42:13.110 [? Stephen ?] was asking. 00:42:13.110 --> 00:42:17.510 So for a smaller r, you have a higher probability 00:42:17.510 --> 00:42:19.152 of detecting a mean shift. 00:42:19.152 --> 00:42:21.697 But at same time, for a smaller r, 00:42:21.697 --> 00:42:23.780 the time it takes for you to detect the mean shift 00:42:23.780 --> 00:42:25.070 is longer. 00:42:25.070 --> 00:42:27.800 How does these two coexist? 00:42:27.800 --> 00:42:30.530 Well, you have a higher chance of detecting it, 00:42:30.530 --> 00:42:32.840 but it takes longer as well. 00:42:32.840 --> 00:42:35.210 DUANE BONING: Yes. 00:42:35.210 --> 00:42:36.980 They're a trade-off. 00:42:36.980 --> 00:42:41.660 And so what you are basically given is a degree of freedom 00:42:41.660 --> 00:42:44.030 that you get to play with, or experiment with, 00:42:44.030 --> 00:42:46.650 or adapt to your process. 00:42:46.650 --> 00:42:49.940 So what are the factors in adapting to your process? 00:42:49.940 --> 00:42:53.000 If your process is really noisy and you 00:42:53.000 --> 00:42:54.980 think it's going to be hard to detect 00:42:54.980 --> 00:43:00.020 a mean shift in the phase of that noise, 00:43:00.020 --> 00:43:01.970 then you do more averaging. 00:43:01.970 --> 00:43:03.620 You'd have a smaller r. 00:43:03.620 --> 00:43:05.990 If you're trying to detect-- 00:43:05.990 --> 00:43:08.600 maybe with a higher risk of a false alarm, 00:43:08.600 --> 00:43:12.230 but you're trying to detect more quickly, 00:43:12.230 --> 00:43:16.670 then I'd have a larger r, so it responds faster, 00:43:16.670 --> 00:43:18.720 but doesn't filter as much. 00:43:18.720 --> 00:43:22.610 So you can see here, with the r equals 0.1 versus the r 00:43:22.610 --> 00:43:23.990 equals 0.3-- 00:43:23.990 --> 00:43:25.730 this is the same data, I believe. 00:43:25.730 --> 00:43:27.110 Is this the same data? 00:43:27.110 --> 00:43:29.250 I guess not-- it's just another example. 00:43:29.250 --> 00:43:38.240 Here's an r equals 0.3 kind of case, where, for this data, 00:43:38.240 --> 00:43:40.400 this looks like about the right trade-off. 00:43:40.400 --> 00:43:43.160 I'm getting reasonable filtering. 00:43:43.160 --> 00:43:45.980 The black dots are my EWMA. 00:43:45.980 --> 00:43:50.480 And for a mean shift here, I've got very rapid-- 00:43:50.480 --> 00:43:54.980 I've got rapid detection, much more rapid than, 00:43:54.980 --> 00:43:58.730 in fact, the x bar, which doesn't really 00:43:58.730 --> 00:44:00.890 trigger finally until way-- 00:44:00.890 --> 00:44:05.790 another 20, 25 runs later. 00:44:05.790 --> 00:44:10.070 So I think what you'd have to do is, again make the trade-offs 00:44:10.070 --> 00:44:13.340 between responsivity, the size of the shift 00:44:13.340 --> 00:44:15.440 that you're trying to detect, and how 00:44:15.440 --> 00:44:20.960 much cost you to have different delays in detecting the shift. 00:44:20.960 --> 00:44:23.540 So we've got two questions here-- one here first. 00:44:23.540 --> 00:44:26.490 AUDIENCE: [INAUDIBLE] It's almost 00:44:26.490 --> 00:44:28.808 like you are taking a larger sample size. 00:44:28.808 --> 00:44:34.290 [INAUDIBLE] You're taking the 5 samples for-- 00:44:34.290 --> 00:44:39.180 5 measurements per sample, plus 5 previously or slightly lower 00:44:39.180 --> 00:44:42.510 [INAUDIBLE] plus 5 [INAUDIBLE] plus 5 [INAUDIBLE] So 00:44:42.510 --> 00:44:44.850 you're just taking more samples per-- 00:44:44.850 --> 00:44:46.890 more measurements per data point, right? 00:44:46.890 --> 00:44:47.640 DUANE BONING: Yes. 00:44:47.640 --> 00:44:49.342 So that's a fair way of looking at it. 00:44:49.342 --> 00:44:50.800 That's a good way of looking at it. 00:44:50.800 --> 00:44:53.820 So just so everybody could hear, with the x bar, 00:44:53.820 --> 00:44:58.590 I take my n equals whatever sample, but with the EWMA, 00:44:58.590 --> 00:45:00.780 I still got that from my most recent update, 00:45:00.780 --> 00:45:02.550 but I'm doing additional filtering 00:45:02.550 --> 00:45:04.770 or averaging of past history in. 00:45:04.770 --> 00:45:07.590 And the net effective that is I've got a net aggregate-- 00:45:07.590 --> 00:45:10.800 more number of samples, if you will-- 00:45:10.800 --> 00:45:13.020 unequally weighted, but a more-- 00:45:13.020 --> 00:45:15.360 and therefore, that makes perfect sense 00:45:15.360 --> 00:45:19.440 that your control limits come down. 00:45:19.440 --> 00:45:21.720 That's a very good way of looking at it. 00:45:21.720 --> 00:45:23.740 AUDIENCE: So in the previous example, 00:45:23.740 --> 00:45:26.868 does EWMA also trigger before x bar? 00:45:26.868 --> 00:45:27.910 Because x bar [INAUDIBLE] 00:45:27.910 --> 00:45:30.118 DUANE BONING: Oh, yes-- good point, good point-- yes. 00:45:32.740 --> 00:45:35.445 AUDIENCE: Is there ever a case when x bar will trigger 00:45:35.445 --> 00:45:37.537 [INAUDIBLE] 00:45:37.537 --> 00:45:39.370 DUANE BONING: Good question-- are there ever 00:45:39.370 --> 00:45:42.100 situations where x bar might trigger still 00:45:42.100 --> 00:45:47.470 before an EWMA for an x bar chart? 00:45:47.470 --> 00:45:49.420 I don't quite know the trade-offs 00:45:49.420 --> 00:45:53.140 and where you get in the limits of the trade-off. 00:45:56.200 --> 00:45:58.510 You've always got chance at work, 00:45:58.510 --> 00:46:02.050 and I suspect that, with a large enough-- 00:46:02.050 --> 00:46:06.970 so the following scenario probably comes closest. 00:46:06.970 --> 00:46:13.030 Let's say you had a very large mean shift, but a very small r. 00:46:13.030 --> 00:46:17.920 With an x bar, I react entirely to a sample drawn 00:46:17.920 --> 00:46:19.810 from the very large mean shift. 00:46:19.810 --> 00:46:23.590 Let's say it was a 2 sigma or a 3 sigma means shift. 00:46:23.590 --> 00:46:25.690 But I have a very small r. 00:46:25.690 --> 00:46:29.020 Maybe I'm only calling that 0.05. 00:46:29.020 --> 00:46:32.550 That 5% mix in of that most recent data point 00:46:32.550 --> 00:46:35.620 is probably swamped out by my past data still, 00:46:35.620 --> 00:46:38.920 and so in that case, I think my EWMA would likely not 00:46:38.920 --> 00:46:42.140 detect the mean shift. 00:46:42.140 --> 00:46:44.380 So I think of the EWMA as especially good 00:46:44.380 --> 00:46:46.420 for improving responsivity and detection 00:46:46.420 --> 00:46:49.900 ability for small effects, because it's aggregating, 00:46:49.900 --> 00:46:52.370 just like the CUSUM that we'll see in a second. 00:46:52.370 --> 00:46:56.020 But I think, if there are really big-- bigger effects, 00:46:56.020 --> 00:46:59.710 you lose by the filtering in the time response. 00:46:59.710 --> 00:47:02.005 Was that an example you're going to suggest? 00:47:02.005 --> 00:47:07.430 AUDIENCE: [INAUDIBLE] 00:47:07.430 --> 00:47:08.680 DUANE BONING: Oh, good point-- 00:47:08.680 --> 00:47:12.128 AUDIENCE: [INAUDIBLE] 00:47:12.128 --> 00:47:12.920 DUANE BONING: Yeah. 00:47:12.920 --> 00:47:19.118 AUDIENCE: [INAUDIBLE] 00:47:19.118 --> 00:47:19.910 DUANE BONING: Yeah. 00:47:19.910 --> 00:47:20.600 Yeah. 00:47:20.600 --> 00:47:25.150 So other rules here would help-- yeah, absolutely. 00:47:25.150 --> 00:47:32.370 OK, so I guess this is sort of saying a little bit 00:47:32.370 --> 00:47:33.060 the same thing. 00:47:33.060 --> 00:47:37.610 This, I think, is really good for small mean shifts-- 00:47:37.610 --> 00:47:38.630 that are persistent. 00:47:38.630 --> 00:47:42.950 It's not a one-time shot, that I've got a delta mu mean shift, 00:47:42.950 --> 00:47:45.060 and this allows us to aggregate. 00:47:45.060 --> 00:47:48.300 There are additional ideas we can use. 00:47:48.300 --> 00:47:49.800 Yeah-- question [INAUDIBLE] 00:47:49.800 --> 00:47:53.630 AUDIENCE: [INAUDIBLE] 00:47:53.630 --> 00:47:55.940 DUANE BONING: No, you would not want to-- 00:47:55.940 --> 00:47:58.130 so I think Hayden's point here was 00:47:58.130 --> 00:48:01.430 on the purple beta points you might apply, 00:48:01.430 --> 00:48:03.230 which was just an x bar. 00:48:03.230 --> 00:48:06.860 You might apply the WECO but you should not-- 00:48:06.860 --> 00:48:09.560 there might be variants you could derive 00:48:09.560 --> 00:48:14.100 some of the other WECO rules, but I don't 00:48:14.100 --> 00:48:15.600 know how you would do that. 00:48:15.600 --> 00:48:16.500 It's not intuitive. 00:48:16.500 --> 00:48:18.930 I think, for the EWMA, you pretty much 00:48:18.930 --> 00:48:23.220 are just trying to track the mean, 00:48:23.220 --> 00:48:28.350 and not additional trends or-- 00:48:28.350 --> 00:48:31.320 I would not use any of the additional WECO rules 00:48:31.320 --> 00:48:32.440 with an EWMA chart. 00:48:35.060 --> 00:48:37.370 So there's another chart that is very often used. 00:48:37.370 --> 00:48:39.500 It's called a CUSUM I just want to give you 00:48:39.500 --> 00:48:41.150 a quick feel for that, and then move 00:48:41.150 --> 00:48:47.840 to some multivariate issues, which is also targeted 00:48:47.840 --> 00:48:50.780 at this idea that, when I have a small mean, 00:48:50.780 --> 00:48:54.650 shift what I'd like to do is use the fact that it is 00:48:54.650 --> 00:48:58.010 a persistent mean shift that I'm then sampling around 00:48:58.010 --> 00:48:59.820 within the process noise. 00:48:59.820 --> 00:49:02.000 And if there were a simple way to aggregate 00:49:02.000 --> 00:49:06.500 the mean shift algorithm many data points in the past, 00:49:06.500 --> 00:49:11.480 I'd build up a signal that's larger and might improve 00:49:11.480 --> 00:49:17.780 my ability to detect that effect and reduce the average run 00:49:17.780 --> 00:49:20.780 length for detecting that effect. 00:49:20.780 --> 00:49:24.200 And this CUSUM, or cumulative sum, 00:49:24.200 --> 00:49:28.200 is another way of thinking of it as a filter-- 00:49:28.200 --> 00:49:30.140 and in this case, just a discrete time 00:49:30.140 --> 00:49:34.730 integrator-- which is looking at the sum of deviations 00:49:34.730 --> 00:49:39.300 of each of my sample point from the assumed grand mean. 00:49:39.300 --> 00:49:42.350 So I'm not just filtering and plotting the average, 00:49:42.350 --> 00:49:43.880 but I'm saying-- 00:49:43.880 --> 00:49:48.837 asking the question, am I still in control at my assumed x bar 00:49:48.837 --> 00:49:51.800 bar, my mu 0? 00:49:51.800 --> 00:49:54.560 And if I'm not, then this statistic, 00:49:54.560 --> 00:49:59.780 this sum of being to one side of it, the c sub j 00:49:59.780 --> 00:50:03.140 here will grow if I've got a plus mean shift, 00:50:03.140 --> 00:50:05.900 or become more and more negative and grow if I've 00:50:05.900 --> 00:50:08.780 got a negative mean shift-- 00:50:08.780 --> 00:50:12.740 so that I'll convert a mean shift to an integration 00:50:12.740 --> 00:50:14.570 of a mean shift. 00:50:14.570 --> 00:50:16.430 The mean shift is a step. 00:50:16.430 --> 00:50:21.350 The integration of a step response is a ramp. 00:50:21.350 --> 00:50:25.940 So what we would hope to do with a CUSUM chart 00:50:25.940 --> 00:50:28.970 is improve our mean shift sensitivity, 00:50:28.970 --> 00:50:32.060 and basically have charts that look like this. 00:50:32.060 --> 00:50:39.310 When I'm in control, I might wander around the mean. 00:50:39.310 --> 00:50:41.510 Sometimes I'll be above, sometimes below, 00:50:41.510 --> 00:50:45.460 so this c sub j, this cumulative statistic, 00:50:45.460 --> 00:50:47.110 does wander around a bit. 00:50:47.110 --> 00:50:50.050 But if I do have a mean shift, on average, 00:50:50.050 --> 00:50:53.560 I keep adding to that C sub J. 00:50:53.560 --> 00:50:55.330 Sometimes I might go down a little bit, 00:50:55.330 --> 00:50:58.580 because I still have something a little bit below the mean. 00:50:58.580 --> 00:51:02.530 And so what you create is this ramp. 00:51:02.530 --> 00:51:07.880 You create a slope caused by that mean shift. 00:51:07.880 --> 00:51:11.260 Now, one can also now apply some statistics 00:51:11.260 --> 00:51:19.330 and design, control limits for that C sub J statistic, 00:51:19.330 --> 00:51:22.360 or use a couple of other approaches. 00:51:22.360 --> 00:51:28.430 And one approach is this so-called V-Mask that says it-- 00:51:28.430 --> 00:51:34.060 what I would like to do is, if, in fact, I've got a mean shift, 00:51:34.060 --> 00:51:35.980 this slope-- 00:51:35.980 --> 00:51:37.690 let me erase a little bit here-- 00:51:40.840 --> 00:51:48.410 whoops-- so there are two different approaches-- 00:51:48.410 --> 00:51:50.260 and I just want to briefly mention 00:51:50.260 --> 00:51:54.460 both of those-- for detecting things on these charts. 00:51:54.460 --> 00:51:59.026 One is draw a control limit for the aggregate of the C sub 00:51:59.026 --> 00:52:07.120 J. The other is to look and say, once I've got a mean shift, I 00:52:07.120 --> 00:52:11.980 start building data points with, on average, some slope that 00:52:11.980 --> 00:52:18.700 is different than a slope that I would expect by random chance 00:52:18.700 --> 00:52:24.460 alone, just forming the sum statistic from a normally 00:52:24.460 --> 00:52:28.440 sampled process. 00:52:28.440 --> 00:52:33.150 The tabular CUSUM-- this idea of actually looking 00:52:33.150 --> 00:52:35.890 at the statistics of-- 00:52:35.890 --> 00:52:38.490 let me skip this for a second-- 00:52:38.490 --> 00:52:41.490 this normalized statistic z and s, 00:52:41.490 --> 00:52:46.157 and looking and deciding on what control limits are, and so on-- 00:52:46.157 --> 00:52:47.490 I'm actually going to skip that. 00:52:47.490 --> 00:52:49.080 There's about five slides in here, 00:52:49.080 --> 00:52:53.520 but it's described very nicely in, I think, chapter 8.1 00:52:53.520 --> 00:52:55.320 in Montgomery. 00:52:55.320 --> 00:52:56.830 And I think that's pretty natural. 00:52:56.830 --> 00:53:04.500 It's just sort of a statistic-- a little bit confusing 00:53:04.500 --> 00:53:06.870 derivation of the statistics that-- 00:53:06.870 --> 00:53:11.760 I just trust the statisticians who have done right. 00:53:11.760 --> 00:53:14.740 But there's also another picture here, 00:53:14.740 --> 00:53:17.280 which is based on the slope idea. 00:53:17.280 --> 00:53:21.150 And you can similarly derive some formulas 00:53:21.150 --> 00:53:27.060 based on alpha and beta risks, what size of a delta 00:53:27.060 --> 00:53:31.260 shift in the mean you would hope to detect, 00:53:31.260 --> 00:53:35.250 and what kind of a slope you would expect to see, 00:53:35.250 --> 00:53:36.790 in those cases. 00:53:36.790 --> 00:53:39.900 And so you've got these two parameters that 00:53:39.900 --> 00:53:43.920 go towards the formulation of this V-Mask 00:53:43.920 --> 00:53:48.350 that you overlay on your data points 00:53:48.350 --> 00:53:52.820 based on an expected slope, and you need a certain number 00:53:52.820 --> 00:53:54.770 of data points in order to be able to build up 00:53:54.770 --> 00:53:56.000 your estimate of the slope. 00:53:56.000 --> 00:53:57.650 That's the dead zone. 00:53:57.650 --> 00:54:02.090 And what you do for each data point here is overlay 00:54:02.090 --> 00:54:05.600 the V-Mask and say, look back in time and see, 00:54:05.600 --> 00:54:12.140 did any of these points intersect with the mask-- 00:54:12.140 --> 00:54:15.230 have a higher slope than you would have expected? 00:54:15.230 --> 00:54:17.580 And then you update that on a running basis. 00:54:17.580 --> 00:54:20.300 So in this case, here at time point 00:54:20.300 --> 00:54:23.960 18 or 19-- whatever that is-- everything looks fine. 00:54:23.960 --> 00:54:25.260 I go up. 00:54:25.260 --> 00:54:27.530 Everything is still looking fine. 00:54:27.530 --> 00:54:29.960 Still here, I'm starting to get some evidence maybe 00:54:29.960 --> 00:54:33.290 that things have changed, but statistically speaking, 00:54:33.290 --> 00:54:35.210 that's still within the range that I would 00:54:35.210 --> 00:54:38.340 expect with some alpha error. 00:54:38.340 --> 00:54:43.400 But then finally, I get to a new data point here at time 23, 00:54:43.400 --> 00:54:47.540 where, in aggregate, I've got some indication 00:54:47.540 --> 00:54:49.860 that the slope is too large. 00:54:49.860 --> 00:54:53.120 So this is kind of a intuitive graphical approach that's 00:54:53.120 --> 00:54:55.160 talking about the slope associated 00:54:55.160 --> 00:55:00.770 with a mean shift in the CUSUM statistic. 00:55:00.770 --> 00:55:01.700 Yes? 00:55:01.700 --> 00:55:05.210 AUDIENCE: Would you also perform a linear regression 00:55:05.210 --> 00:55:07.852 for the last three or four parts and check the [INAUDIBLE]?? 00:55:07.852 --> 00:55:09.560 DUANE BONING: So the question is, can you 00:55:09.560 --> 00:55:14.580 perform a linear regression and calculate the slope of that? 00:55:14.580 --> 00:55:17.900 And I think that's identical to what this is doing graphically. 00:55:17.900 --> 00:55:20.300 So in fact, if you were numerically-- 00:55:20.300 --> 00:55:24.200 or implementing an automatic version of this, 00:55:24.200 --> 00:55:28.790 that would probably be how you would implement it. 00:55:28.790 --> 00:55:31.430 That might even give you some more flexibility 00:55:31.430 --> 00:55:35.420 in terms of looking at the goodness of that fit, 00:55:35.420 --> 00:55:39.440 but I think it's essentially identical to this. 00:55:39.440 --> 00:55:41.120 OK. 00:55:41.120 --> 00:55:43.580 So just to summarize these alternative charts, 00:55:43.580 --> 00:55:46.250 there's a few simple messages. 00:55:46.250 --> 00:55:48.830 Noisy data-- if you're trying to detect 00:55:48.830 --> 00:55:50.930 a mean in the noisy data, you might 00:55:50.930 --> 00:55:52.580 want to use some filtering. 00:55:52.580 --> 00:55:55.100 We have the sample size that does some filtering, 00:55:55.100 --> 00:55:57.050 but I might want to use more data 00:55:57.050 --> 00:56:02.300 in the past and this linear discrete filtering, 00:56:02.300 --> 00:56:08.210 either with the CUSUM as a running integrator or an EWMA-- 00:56:08.210 --> 00:56:11.990 which is an iterative discrete filter-- 00:56:11.990 --> 00:56:13.170 can be applied. 00:56:13.170 --> 00:56:15.200 And I think, conceptually, they're 00:56:15.200 --> 00:56:17.330 fairly nice in terms of that noise suppression 00:56:17.330 --> 00:56:20.240 for detecting of the mean-- 00:56:20.240 --> 00:56:22.610 got to remember that your choice of factors, 00:56:22.610 --> 00:56:26.480 like r and the EWMA, depends on your process, and the mix 00:56:26.480 --> 00:56:29.480 of noise, and how fast you want to respond to be 00:56:29.480 --> 00:56:31.520 able to detect a mean shift. 00:56:31.520 --> 00:56:34.500 And then, going back to what we started with our little example 00:56:34.500 --> 00:56:37.430 at the very beginning. 00:56:37.430 --> 00:56:39.350 Noisy data need some filtering. 00:56:39.350 --> 00:56:41.480 We've talked about it with respect to the mean-- 00:56:41.480 --> 00:56:42.860 that, if I'm filtering that data, 00:56:42.860 --> 00:56:45.980 you also still want to look back and monitor the variance 00:56:45.980 --> 00:56:46.890 as well. 00:56:46.890 --> 00:56:52.610 So you can apply these charts also to moving, S bar charts, 00:56:52.610 --> 00:56:54.870 and so on. 00:56:54.870 --> 00:56:56.270 So what I want to do next is move 00:56:56.270 --> 00:56:59.510 a little bit to multivariate process control, 00:56:59.510 --> 00:57:03.860 control charting when I've got more than one parameter that I 00:57:03.860 --> 00:57:07.370 may be monitoring on the unit process or on the product. 00:57:07.370 --> 00:57:09.800 And what's really interesting here 00:57:09.800 --> 00:57:15.110 is there are essentially two classic mistakes that 00:57:15.110 --> 00:57:17.282 are made again and again. 00:57:17.282 --> 00:57:19.490 And I want to give you a feel for those two mistakes, 00:57:19.490 --> 00:57:21.880 and strategies for avoiding them. 00:57:21.880 --> 00:57:23.630 OK? 00:57:23.630 --> 00:57:27.650 One classic problem is you've got more than one parameter 00:57:27.650 --> 00:57:30.530 you're monitoring-- you implement more than one control 00:57:30.530 --> 00:57:33.020 chart-- 00:57:33.020 --> 00:57:34.830 seems natural. 00:57:34.830 --> 00:57:37.730 But the problem is you can get many, many, many more false 00:57:37.730 --> 00:57:43.610 alarms than you might have expected, or believed 00:57:43.610 --> 00:57:46.190 you are going to get, because you're just 00:57:46.190 --> 00:57:48.290 monitoring more parameters. 00:57:48.290 --> 00:57:51.120 And so I want to talk a little bit about that. 00:57:51.120 --> 00:57:53.540 That's even if there's no correlation 00:57:53.540 --> 00:57:56.540 between your parameters-- 00:57:56.540 --> 00:57:58.580 or especially if there's no correlation 00:57:58.580 --> 00:58:00.320 between your parameters, you can get-- 00:58:00.320 --> 00:58:05.060 just by the fact that you're monitoring more than one thing, 00:58:05.060 --> 00:58:07.490 you've got to think about false alarm rate a little bit 00:58:07.490 --> 00:58:08.580 differently. 00:58:08.580 --> 00:58:10.130 And then the second classic problem 00:58:10.130 --> 00:58:14.330 is, well, there is correlation among the parameters-- 00:58:14.330 --> 00:58:15.830 the multiple parameters. 00:58:15.830 --> 00:58:17.580 How do you deal with that? 00:58:17.580 --> 00:58:20.420 And in fact, in this case, you often 00:58:20.420 --> 00:58:23.500 run into the opposite risk. 00:58:23.500 --> 00:58:25.600 Instead of too many false alarms, 00:58:25.600 --> 00:58:28.810 you may have points of data or product 00:58:28.810 --> 00:58:33.070 that is truly out of normal operation 00:58:33.070 --> 00:58:34.990 that you may not detect. 00:58:34.990 --> 00:58:38.830 It's a highly unusual point that may in 00:58:38.830 --> 00:58:41.800 indicate something has changed in your process, 00:58:41.800 --> 00:58:44.740 but if you're just monitoring with univariate control charts, 00:58:44.740 --> 00:58:45.670 you'll never see it. 00:58:48.940 --> 00:58:53.020 That is a common mistake number two. 00:58:53.020 --> 00:58:56.620 Multiple charts is common mistake number one first. 00:58:56.620 --> 00:58:59.770 If I truly have independent parameters being modeled-- 00:58:59.770 --> 00:59:03.250 monitored at my process step, and I set each one of them 00:59:03.250 --> 00:59:07.270 up to have some acceptable alpha rate-- 00:59:07.270 --> 00:59:08.440 false alarm rate-- 00:59:08.440 --> 00:59:13.630 1 out of every 370 process steps going with a plus/minus 00:59:13.630 --> 00:59:17.260 three-sigma-- 00:59:17.260 --> 00:59:21.730 three-sigma control limits, alpha's about 0.0027-- 00:59:21.730 --> 00:59:26.110 sometimes talked about 0.003 as an approximation. 00:59:26.110 --> 00:59:29.260 But now I've got p of those separate charts. 00:59:29.260 --> 00:59:32.740 What is the aggregate false alarm probability? 00:59:32.740 --> 00:59:35.020 Well to be careful about it, if they are, in fact, 00:59:35.020 --> 00:59:38.590 independent, that alpha prime, might aggregate false alarm, 00:59:38.590 --> 00:59:43.010 is, 1 minus 1 minus alpha to the p. 00:59:43.010 --> 00:59:47.710 This is the probability that I don't 00:59:47.710 --> 00:59:52.210 get a false alarm for each of-- for all of the other p control 00:59:52.210 --> 00:59:55.180 charts all at the same time. 00:59:55.180 --> 00:59:57.040 And then 1 minus that is the chance 00:59:57.040 --> 00:59:58.960 that I did get one of them. 00:59:58.960 --> 01:00:03.250 Now for very small alpha, you do that expansion, the 1 minus x 01:00:03.250 --> 01:00:07.780 for small x to the p is simply 1 minus x So you do that 01:00:07.780 --> 01:00:10.960 and you get sort of the nice intuitive result 01:00:10.960 --> 01:00:14.740 that if I've got 10 independent control charts, 01:00:14.740 --> 01:00:18.850 I've got about 10 times the rate of false alarm 01:00:18.850 --> 01:00:19.990 that I would have from one. 01:00:22.990 --> 01:00:23.800 So what do you do? 01:00:32.560 --> 01:00:35.120 What would you do? 01:00:35.120 --> 01:00:38.570 Let me say it this way. 01:00:38.570 --> 01:00:42.500 You've decided, based on your cost analysis or whatever, 01:00:42.500 --> 01:00:48.490 that you're willing about once every 370 runs 01:00:48.490 --> 01:00:52.820 to have a false alarm somewhere on this process, 01:00:52.820 --> 01:00:54.533 that you want to at least-- look at maybe 01:00:54.533 --> 01:00:55.700 you look at it real quickly. 01:00:55.700 --> 01:00:56.867 It doesn't take a long time. 01:00:56.867 --> 01:01:02.090 But that's about how often you want the operator to stop 01:01:02.090 --> 01:01:06.590 and take a look, on average, even when the process is 01:01:06.590 --> 01:01:07.400 in control. 01:01:07.400 --> 01:01:10.340 So that's your false alarm rate that you're willing to accept, 01:01:10.340 --> 01:01:12.110 but you're monitoring 10 parameters. 01:01:17.600 --> 01:01:19.350 [INAUDIBLE] some approaches you would use? 01:01:23.145 --> 01:01:26.040 AUDIENCE: You have to increase your control limits. 01:01:26.040 --> 01:01:27.700 DUANE BONING: Right, perfect-- 01:01:27.700 --> 01:01:29.700 you would increase your control limits. 01:01:29.700 --> 01:01:32.790 If I expand my control limits out a little bit, 01:01:32.790 --> 01:01:38.850 I have smaller false alarm rate on each chart. 01:01:38.850 --> 01:01:50.880 Well, I can basically decide what my aggregate false alarm 01:01:50.880 --> 01:01:53.040 rate would be-- 01:01:53.040 --> 01:01:54.930 1 out of 370. 01:01:54.930 --> 01:01:59.640 And to set my false alarm rate on each individual chart, 01:01:59.640 --> 01:02:03.420 just divide by how many charts I have. 01:02:03.420 --> 01:02:10.080 Now, I might have a 1 in 3,700 false alarm rate on each chart, 01:02:10.080 --> 01:02:15.180 and I use that new alpha to set my control limit. 01:02:15.180 --> 01:02:19.380 They might end up being, in that case, plus minus 3.4 01:02:19.380 --> 01:02:23.410 sigma control limits. 01:02:23.410 --> 01:02:29.170 So the individual false alarm rate is lower on each one, 01:02:29.170 --> 01:02:31.540 because really, the question you are asking-- 01:02:31.540 --> 01:02:37.840 the key question was not that individual parameter, 01:02:37.840 --> 01:02:40.840 but rather, the question, is my process in control? 01:02:40.840 --> 01:02:48.320 In aggregate, do I have an event that looks unlikely-- 01:02:48.320 --> 01:02:54.790 so unlikely that I have 1 minus alpha confidence, 1 minus 01:02:54.790 --> 01:02:59.020 alpha prime confidence, that, in fact, something unusual 01:02:59.020 --> 01:03:02.020 is going on in the process? 01:03:02.020 --> 01:03:03.790 So that's a strategy. 01:03:03.790 --> 01:03:07.180 I want to also point out-- and this is kind of a neat little 01:03:07.180 --> 01:03:09.850 article that I found last year-- 01:03:09.850 --> 01:03:13.390 that this is a more generic problem, 01:03:13.390 --> 01:03:17.020 a more generic common mistake than just control charting. 01:03:17.020 --> 01:03:21.820 It applies all the time, when people are doing significance 01:03:21.820 --> 01:03:25.510 tests or hypothesis tests, but doing more than one of them 01:03:25.510 --> 01:03:28.090 at a time. 01:03:28.090 --> 01:03:30.890 Control charge is just a running hypothesis test. 01:03:30.890 --> 01:03:33.280 Now, if I'm checking 10 hypotheses, 01:03:33.280 --> 01:03:37.090 10 control charts, and 10 parameters. 01:03:37.090 --> 01:03:41.320 My aggregate false alarm rate is higher. 01:03:41.320 --> 01:03:43.570 And there's this neat little article 01:03:43.570 --> 01:03:47.530 pulled from-- that I found in The Economist last February 01:03:47.530 --> 01:03:51.490 that's called "Medical Statistics Signs of the Times." 01:03:51.490 --> 01:03:53.020 The subtitle here-- you may not be 01:03:53.020 --> 01:03:56.710 able to read it-- is, "Why So Much Medical Research is Rot." 01:03:59.830 --> 01:04:02.200 I'll read you the first couple of sentences here. 01:04:02.200 --> 01:04:05.140 People born under the astrological sign of Leo 01:04:05.140 --> 01:04:07.870 are 15% more likely to be admitted 01:04:07.870 --> 01:04:10.090 to hospital with gastric bleeding 01:04:10.090 --> 01:04:13.240 than those born with the other 11 signs. 01:04:13.240 --> 01:04:16.990 Sagittarians are 38% more likely than others to land up 01:04:16.990 --> 01:04:19.720 there because of a broken arm. 01:04:19.720 --> 01:04:22.390 Those are the conclusions that many medical researchers 01:04:22.390 --> 01:04:24.880 would be forced to make from a set of data presented 01:04:24.880 --> 01:04:28.480 to the American Association for the Advancement of Science. 01:04:28.480 --> 01:04:30.400 At least they would be forced to draw them, 01:04:30.400 --> 01:04:33.370 if they applied the lax statistical methods 01:04:33.370 --> 01:04:36.310 of their own work to the records of hospital admissions 01:04:36.310 --> 01:04:38.840 in Ontario, Canada. 01:04:38.840 --> 01:04:41.860 And the basic confusion drawn here in the red 01:04:41.860 --> 01:04:44.980 arises because each result is tested separately 01:04:44.980 --> 01:04:47.410 to see how likely in statistical terms 01:04:47.410 --> 01:04:49.460 it was to have happen by chance. 01:04:49.460 --> 01:04:51.460 If the likelihood is below a certain threshold-- 01:04:51.460 --> 01:04:53.548 typically 95%-- 01:04:53.548 --> 01:04:59.170 so we've got a 95% confidence, which says, 1 in 20 times, 01:04:59.170 --> 01:05:02.890 by chance alone, I might see a signal that big-- 01:05:02.890 --> 01:05:04.640 by chance alone. 01:05:04.640 --> 01:05:06.550 And now I check 12-- 01:05:06.550 --> 01:05:09.700 maybe astrological sign is your 12 things 01:05:09.700 --> 01:05:12.290 I'm checking for as factors-- 01:05:12.290 --> 01:05:17.680 and I've got a 1 in 20 chance, and I check 12 of them-- 01:05:17.680 --> 01:05:20.200 probably I'm going to pop up and say, oh, 01:05:20.200 --> 01:05:23.290 that one's significant. 01:05:23.290 --> 01:05:27.190 And so that's exactly the same common mistake 01:05:27.190 --> 01:05:29.920 that you can see with multiple control charts. 01:05:29.920 --> 01:05:33.490 And so the point here would simply be you should-- 01:05:33.490 --> 01:05:35.230 if you're checking multiple factors, 01:05:35.230 --> 01:05:38.410 you got to have a stronger indication at any one factor-- 01:05:38.410 --> 01:05:41.470 otherwise, you're just fishing, and by chance alone, you're 01:05:41.470 --> 01:05:44.230 going to find stuff. 01:05:44.230 --> 01:05:45.950 So that's kind of a neat example. 01:05:45.950 --> 01:05:48.340 And so here was the point I made about how you fix it. 01:05:48.340 --> 01:05:51.010 You just fix your aggregate alpha prime, 01:05:51.010 --> 01:05:52.720 you have your individual alpha, and then 01:05:52.720 --> 01:05:54.880 you pick your upper and lower control limits 01:05:54.880 --> 01:05:57.430 based on that alpha. 01:05:57.430 --> 01:06:00.340 There's a second common problem, and as I said, 01:06:00.340 --> 01:06:06.010 it's the obverse of this, which is you very often assume 01:06:06.010 --> 01:06:09.370 that, if I'm monitoring 10 process control parameters, 01:06:09.370 --> 01:06:11.290 I assume they're all independent. 01:06:11.290 --> 01:06:13.130 But very often, they are not. 01:06:13.130 --> 01:06:15.702 There is correlation among those parameters, 01:06:15.702 --> 01:06:18.160 and I want to give you a little bit of a feel for what that 01:06:18.160 --> 01:06:21.640 looks like, and what the risk is, 01:06:21.640 --> 01:06:25.810 and what approaches are for control charting and particular 01:06:25.810 --> 01:06:27.360 of this. 01:06:27.360 --> 01:06:30.670 So here's an example drawn from some data that 01:06:30.670 --> 01:06:33.370 came from an LFM thesis a few years ago, 01:06:33.370 --> 01:06:35.200 a body and white assembly. 01:06:35.200 --> 01:06:39.070 So body and white-- this is when they are basically putting 01:06:39.070 --> 01:06:45.390 together all of the different panels on the car and measuring 01:06:45.390 --> 01:06:49.120 many, many points with all kinds of laser, 01:06:49.120 --> 01:06:53.590 automated apparatus to be able to look at dimensional control 01:06:53.590 --> 01:06:56.320 of these panels in the assembly-- 01:06:56.320 --> 01:06:58.150 of some of these panels. 01:06:58.150 --> 01:07:01.030 Other correlated data comes up all over the place-- 01:07:01.030 --> 01:07:05.260 injection molding-- I think we're missing-- 01:07:05.260 --> 01:07:06.850 I can't remember his name. 01:07:06.850 --> 01:07:09.610 He asked a question last time about critical dimensions 01:07:09.610 --> 01:07:10.810 on a semiconductor wafer. 01:07:10.810 --> 01:07:14.110 Yes, they are often correlated. 01:07:14.110 --> 01:07:16.070 But here's the body and white example, 01:07:16.070 --> 01:07:18.610 and what I'm sort of showing are these little coordinate 01:07:18.610 --> 01:07:22.760 measuring machine, CMM kinds of measurements. 01:07:22.760 --> 01:07:26.720 I have a few of these different data points. 01:07:26.720 --> 01:07:29.600 And the point is, if I think of this as a metal 01:07:29.600 --> 01:07:33.320 panel in some way, and I'm measuring the position, say, 01:07:33.320 --> 01:07:39.890 of these four corner points, you would expect there 01:07:39.890 --> 01:07:42.710 to be correlation among them. 01:07:42.710 --> 01:07:46.160 And certain kinds of distortion might change 01:07:46.160 --> 01:07:48.060 relationships between them. 01:07:48.060 --> 01:07:50.630 And in fact, you might even come up with other statistics, 01:07:50.630 --> 01:07:53.450 like a y1, and a y3, and a y4, that 01:07:53.450 --> 01:07:56.720 try to detect or look at tracking 01:07:56.720 --> 01:07:59.630 of a couple of these parameters and deviations. 01:07:59.630 --> 01:08:02.420 But the point is I wouldn't necessarily 01:08:02.420 --> 01:08:05.090 want to monitor those four corner locations 01:08:05.090 --> 01:08:09.250 as independent parameters. 01:08:09.250 --> 01:08:10.250 Here's another example. 01:08:10.250 --> 01:08:14.560 This comes from an ultrasonic welding set of data. 01:08:14.560 --> 01:08:16.689 And you may not be able to see this. 01:08:16.689 --> 01:08:21.529 This is a wonderful correlation plot of three, four, five, six, 01:08:21.529 --> 01:08:25.450 seven, eight, 10 different parameters against each other. 01:08:25.450 --> 01:08:29.740 There are different things, like P2 weld energy, P2 peak power. 01:08:29.740 --> 01:08:31.660 These are simply scatter plots of data 01:08:31.660 --> 01:08:33.300 drawn from that process. 01:08:33.300 --> 01:08:36.710 You can see some of these parameters that track down-- 01:08:36.710 --> 01:08:41.622 that's total weld energy versus P2 weld energy. 01:08:41.622 --> 01:08:43.330 I don't even know what those things mean, 01:08:43.330 --> 01:08:45.663 but they sure sound like they're going to be correlated, 01:08:45.663 --> 01:08:46.670 don't they? 01:08:46.670 --> 01:08:49.240 And sure enough, there's extremely strong correlation 01:08:49.240 --> 01:08:50.710 in that data. 01:08:50.710 --> 01:08:53.350 There's other cases where there's also 01:08:53.350 --> 01:08:55.340 fairly strong correlation. 01:08:55.340 --> 01:08:59.290 So again, these kinds of correlated data, 01:08:59.290 --> 01:09:01.779 where knowing something about one of the measurements 01:09:01.779 --> 01:09:06.939 tells you about the likelihood of the other measurement, 01:09:06.939 --> 01:09:09.319 occur all the time. 01:09:09.319 --> 01:09:12.550 So what if you have truly random independent variables? 01:09:12.550 --> 01:09:14.600 What would a scatter plot look like, 01:09:14.600 --> 01:09:17.350 and how do we go about setting the control limits? 01:09:17.350 --> 01:09:18.580 That's pretty natural, right? 01:09:18.580 --> 01:09:22.420 So here's an example where we had x1 and x2. 01:09:22.420 --> 01:09:27.229 The scatterplot basically looks like a cloud of data-- 01:09:27.229 --> 01:09:30.729 and if I added more data, you start to-- 01:09:30.729 --> 01:09:32.710 depending on the scaling, if I were simply 01:09:32.710 --> 01:09:38.200 to draw, say, a joint 95% confidence integral, 01:09:38.200 --> 01:09:40.660 it's just kind of a cloud of data, 01:09:40.660 --> 01:09:46.790 but there's no cross information between x1 and x2. 01:09:46.790 --> 01:09:49.970 In other words, if I told you that x1 was right here-- 01:09:49.970 --> 01:09:54.170 the value of x1 was this value-- 01:09:54.170 --> 01:09:58.250 that really doesn't tell me very much about what value of x2 01:09:58.250 --> 01:09:59.360 could be. 01:09:59.360 --> 01:10:04.280 It still falls anywhere within its distribution. 01:10:04.280 --> 01:10:07.310 These are two independent random variables. 01:10:07.310 --> 01:10:09.530 And you can decide what the proper limits are. 01:10:09.530 --> 01:10:13.730 Now you know enough about picking x1 and x2, 01:10:13.730 --> 01:10:16.520 and I would do the alpha prime idea 01:10:16.520 --> 01:10:20.510 to pick independent limits on those two cases. 01:10:20.510 --> 01:10:23.030 This is an example scatterplot for 01:10:23.030 --> 01:10:25.700 correlated random variables. 01:10:25.700 --> 01:10:31.010 And in this case, it starts to look like an oval or a football 01:10:31.010 --> 01:10:35.540 kind of shape, where there is statistical correlation 01:10:35.540 --> 01:10:37.640 between these two parameters-- 01:10:37.640 --> 01:10:43.760 by which I mean, if I tell you that x1 is in this range, 01:10:43.760 --> 01:10:46.760 that tells you something about where the value of x2 01:10:46.760 --> 01:10:48.200 is likely to fall. 01:10:48.200 --> 01:10:51.380 It might not fall within its full natural range. 01:10:51.380 --> 01:10:53.930 There is cross information or cross-correlation 01:10:53.930 --> 01:10:55.470 between the two. 01:10:55.470 --> 01:10:59.900 Now, if I were to plot x1 and x2 separately, 01:10:59.900 --> 01:11:03.580 it looks an awful lot like the uncorrelated case. 01:11:03.580 --> 01:11:08.550 So if I had independent control charts for the two variables, 01:11:08.550 --> 01:11:10.010 essentially what I'm doing is just 01:11:10.010 --> 01:11:13.520 projecting all of these data points. 01:11:13.520 --> 01:11:19.010 I project all of these data points to get my range for x2, 01:11:19.010 --> 01:11:22.010 and I project all of these data points. 01:11:22.010 --> 01:11:25.730 And in fact, if I plot both of those, x1 and x2, 01:11:25.730 --> 01:11:27.846 I would likely get-- 01:11:27.846 --> 01:11:30.800 yikes-- don't want to do that-- 01:11:30.800 --> 01:11:37.220 I would likely get two separate, normally distributed variables, 01:11:37.220 --> 01:11:39.770 and I might be inclined to simply plot them 01:11:39.770 --> 01:11:48.050 both as x bar and S kinds of control charts. 01:11:48.050 --> 01:11:51.230 The risk here, though, when you do 01:11:51.230 --> 01:11:56.180 that is you might miss outliers, data 01:11:56.180 --> 01:12:02.660 points that do not fall in the normal or the natural variation 01:12:02.660 --> 01:12:04.220 of the process-- 01:12:04.220 --> 01:12:07.220 this being the natural variation of the process 01:12:07.220 --> 01:12:10.460 that I will not detect. 01:12:10.460 --> 01:12:14.340 This data point still looks like it's within that distribution. 01:12:14.340 --> 01:12:17.450 In, fact, it's very close to the mean of x2. 01:12:17.450 --> 01:12:22.020 And it's still within this distribution. 01:12:22.020 --> 01:12:25.310 But within the natural variation of the process, 01:12:25.310 --> 01:12:30.260 within that oval, its distance, in some sense, 01:12:30.260 --> 01:12:34.760 from the central moments of that distribution-- 01:12:34.760 --> 01:12:36.110 it's pretty distant. 01:12:36.110 --> 01:12:38.990 It's not indicative of the normal operation. 01:12:38.990 --> 01:12:41.660 It's a highly unlikely, by chance alone, data 01:12:41.660 --> 01:12:46.510 point, and I would never see it on the control chart. 01:12:46.510 --> 01:12:51.960 So there are multivariate charts that 01:12:51.960 --> 01:12:57.510 let us get at and chart on a single chart-- 01:12:57.510 --> 01:13:01.560 multiple factors that are correlated together-- 01:13:01.560 --> 01:13:05.400 and detect those kinds of problems. 01:13:05.400 --> 01:13:08.790 What I want to give you is a notion first 01:13:08.790 --> 01:13:14.760 of a statistic that aggregates, in some sense-- 01:13:14.760 --> 01:13:16.950 numerically aggregates exactly what 01:13:16.950 --> 01:13:19.350 I was talking about qualitatively here, 01:13:19.350 --> 01:13:21.720 that tells you this notion of, what 01:13:21.720 --> 01:13:27.120 is the distance from the correlated variation 01:13:27.120 --> 01:13:30.030 cloud of each of my measurement points? 01:13:30.030 --> 01:13:33.720 And if I have a statistic and I know it's PDF, 01:13:33.720 --> 01:13:37.410 then I can set control limits on that statistic. 01:13:37.410 --> 01:13:40.680 And that statistic is the Hotelling T squared, 01:13:40.680 --> 01:13:43.375 is one very important one. 01:13:43.375 --> 01:13:44.250 We can also monitor-- 01:13:48.060 --> 01:13:51.000 have EWMA and CUSUM extensions of these. 01:13:51.000 --> 01:13:53.850 Which I'm not really going to talk much about. 01:13:53.850 --> 01:13:57.810 So what we have to do here is have a slight extension 01:13:57.810 --> 01:14:00.360 to our notion of a probability density function 01:14:00.360 --> 01:14:04.133 to have joint probability density functions that tell me 01:14:04.133 --> 01:14:06.300 if I've got, say, two variables-- or three, or four, 01:14:06.300 --> 01:14:09.300 whatever-- what is the probability density associated 01:14:09.300 --> 01:14:10.510 with those? 01:14:10.510 --> 01:14:13.110 And what we want is a single scale control 01:14:13.110 --> 01:14:14.610 chart that aggregates these. 01:14:17.260 --> 01:14:20.620 Now, I've got a little bit of mathematical notation 01:14:20.620 --> 01:14:24.430 in here that I'm not going to go through the careful full 01:14:24.430 --> 01:14:26.140 explanation or derivation of. 01:14:26.140 --> 01:14:28.600 What I want to simply show you is 01:14:28.600 --> 01:14:34.000 how a multivariate expression of the same variables we've 01:14:34.000 --> 01:14:38.830 been using all along gives you a nice qualitative feel for what 01:14:38.830 --> 01:14:40.970 this statistic looks like. 01:14:40.970 --> 01:14:43.390 So if we have a vector of measurements-- 01:14:43.390 --> 01:14:44.740 let's have p parameters-- 01:14:44.740 --> 01:14:46.720 I'm measuring 10 different parameters-- 01:14:46.720 --> 01:14:52.150 body in white or whatever it is for that particular process-- 01:14:52.150 --> 01:14:56.650 and put those in an x-- underscore, an x vector. 01:14:56.650 --> 01:14:59.190 And then I can define a vector of means. 01:14:59.190 --> 01:15:02.590 So I have a set of means for all 10 parameters. 01:15:02.590 --> 01:15:08.230 And I can also calculate, if I have lots and lots 01:15:08.230 --> 01:15:09.190 of historical data. 01:15:09.190 --> 01:15:13.330 There isn't a true underlying covariance matrix 01:15:13.330 --> 01:15:17.020 that talks about or characterizes how these data go 01:15:17.020 --> 01:15:19.040 together with each other. 01:15:19.040 --> 01:15:21.220 Now, I can estimate that covariance matrix 01:15:21.220 --> 01:15:24.340 based on sample data, so I similarly 01:15:24.340 --> 01:15:28.210 have a true mean and a true covariance. 01:15:28.210 --> 01:15:35.260 Then I might estimate with an x bar and an S-- 01:15:35.260 --> 01:15:37.180 here I'm using a capital S-- 01:15:37.180 --> 01:15:39.610 to be a sample covariance matrix. 01:15:39.610 --> 01:15:42.430 And there are formulas in the book using 01:15:42.430 --> 01:15:46.990 a very simple linear, matrix-- 01:15:46.990 --> 01:15:49.960 linear algebra for how you would do that, 01:15:49.960 --> 01:15:53.500 given a particular set of data. 01:15:53.500 --> 01:15:57.550 And they look a lot like the univariate example. 01:15:57.550 --> 01:16:00.100 But I did want to point out that there's 01:16:00.100 --> 01:16:03.910 both embedded in this the notion of variance-- 01:16:03.910 --> 01:16:08.200 how variable 2 varies together with variable 2-- 01:16:08.200 --> 01:16:12.070 that's just our normal notion of variance. 01:16:12.070 --> 01:16:14.110 And there's also covariance-- 01:16:14.110 --> 01:16:18.265 that is, a measure of how these two variables move together. 01:16:21.150 --> 01:16:26.540 We can actually write out what the PDFs probability density 01:16:26.540 --> 01:16:28.830 functions-- are in these joint cases. 01:16:28.830 --> 01:16:31.730 And if you write it in matrix form using 01:16:31.730 --> 01:16:35.720 this covariance matrix and the mu vector, 01:16:35.720 --> 01:16:39.930 it looks almost exactly the same as the univariate case. 01:16:39.930 --> 01:16:44.840 Notice that, instead of variance in the univariate case, 01:16:44.840 --> 01:16:51.230 now we have the covariance matrix. 01:16:51.230 --> 01:16:58.820 And similarly, instead of just x and u being univariate values, 01:16:58.820 --> 01:17:01.640 I've got those as vectors. 01:17:01.640 --> 01:17:06.800 But the PDF looks almost exactly the same. 01:17:06.800 --> 01:17:12.650 And qualitatively, you're getting an exponential 01:17:12.650 --> 01:17:18.080 drop-off, this e drop-off with the square of the distance 01:17:18.080 --> 01:17:21.080 of a point from the mean-- 01:17:21.080 --> 01:17:23.130 the normalized distance from the point. 01:17:23.130 --> 01:17:24.830 So if we look back at the univariate, 01:17:24.830 --> 01:17:28.400 the whole idea is I subtract off the mean when I standardize, 01:17:28.400 --> 01:17:31.610 divide by the natural variation, and that tells me 01:17:31.610 --> 01:17:33.470 the shape of the bell curve-- 01:17:33.470 --> 01:17:36.650 shape of the normal curve, how fast things drop off. 01:17:36.650 --> 01:17:42.020 And I can also, for any particular multivariate u-- 01:17:42.020 --> 01:17:46.880 multivariate data point x that has its 10 parameters in it-- 01:17:46.880 --> 01:17:51.290 what's really cool is this formulation here 01:17:51.290 --> 01:17:57.710 of the distance of that x vector from the mean point-- 01:17:57.710 --> 01:18:00.740 the main vector squared-- 01:18:00.740 --> 01:18:05.000 that squared distance, but weighted then or normalized 01:18:05.000 --> 01:18:07.250 to the natural spread-- 01:18:07.250 --> 01:18:10.370 that natural spread or width of that cloud-- 01:18:10.370 --> 01:18:13.190 is just a distance measure-- 01:18:13.190 --> 01:18:15.710 squared distance measure of each data 01:18:15.710 --> 01:18:18.800 point from the central moment of the central point 01:18:18.800 --> 01:18:22.160 of the distribution, scaled to how much natural spread 01:18:22.160 --> 01:18:24.770 or scatter there is in the data. 01:18:24.770 --> 01:18:26.075 OK, so that's the PDF. 01:18:28.670 --> 01:18:32.450 Those are just the formulas for the sample use cases. 01:18:32.450 --> 01:18:38.130 What I can now go in and do is ask the question, 01:18:38.130 --> 01:18:43.400 what is the distribution associated with that distance 01:18:43.400 --> 01:18:44.900 metric? 01:18:44.900 --> 01:18:48.500 And that distance metric is essentially-- 01:18:48.500 --> 01:18:54.440 if I expand out that distance formula, 01:18:54.440 --> 01:19:02.430 it is a sum of squared distances normalized to the correlation, 01:19:02.430 --> 01:19:09.230 which starts to sound like sum of squared unit normal 01:19:09.230 --> 01:19:14.240 variables, which was our chi-squared distribution-- 01:19:14.240 --> 01:19:17.790 sum of squared normal-- 01:19:17.790 --> 01:19:22.490 so what's cool here is that the chi-squared distribution with p 01:19:22.490 --> 01:19:23.960 parameters-- 01:19:23.960 --> 01:19:25.340 p degrees of freedom-- 01:19:25.340 --> 01:19:29.390 is the right statistic for measuring how likely it 01:19:29.390 --> 01:19:32.870 is to see those distance-- the T squared, 01:19:32.870 --> 01:19:33.890 those distance metrics. 01:19:36.500 --> 01:19:39.230 So one can actually have the PDF. 01:19:39.230 --> 01:19:43.370 One can have the distribution for that PDF, 01:19:43.370 --> 01:19:49.610 and based on that, you can formulate or form a-- 01:19:49.610 --> 01:19:53.270 that statistic based on my actual data x. 01:19:53.270 --> 01:19:55.790 and then I can have an alpha squared 01:19:55.790 --> 01:20:00.680 that I can pick my cutoff points from appropriately. 01:20:00.680 --> 01:20:04.710 Oops-- I thought I had a PDF in here somewhere. 01:20:04.710 --> 01:20:09.170 So when I'm making a control chart, 01:20:09.170 --> 01:20:11.810 I use the same ideas that we saw before. 01:20:11.810 --> 01:20:13.670 I have a false alarm rate. 01:20:13.670 --> 01:20:16.130 In this case, I'm going to aggregate all my data 01:20:16.130 --> 01:20:18.170 onto a single control chart so I can just 01:20:18.170 --> 01:20:20.480 use my aggregate alpha. 01:20:20.480 --> 01:20:26.545 And I pick the tail points on the chi-squared distribution. 01:20:26.545 --> 01:20:28.670 Remember, a chi-squared might have looked something 01:20:28.670 --> 01:20:35.280 like this, and I might pick a way out here as my chi-squared 01:20:35.280 --> 01:20:38.300 so that I've got my alpha out there in the tail. 01:20:38.300 --> 01:20:44.180 That picks my cutoff point for my upper control limit. 01:20:44.180 --> 01:20:50.360 Generally, you can also do that for a lower control limit. 01:20:50.360 --> 01:20:53.570 Very often, if you do some-- apply some of these formulas, 01:20:53.570 --> 01:20:56.360 the lower control limit is negative. 01:20:56.360 --> 01:20:58.220 And you can't have negative square, 01:20:58.220 --> 01:21:01.340 so you pick 0 as your lower control limit. 01:21:01.340 --> 01:21:04.910 And you're really only monitoring four data points 01:21:04.910 --> 01:21:07.550 that are larger than-- 01:21:07.550 --> 01:21:13.610 I've got more variation in my system than I expected. 01:21:13.610 --> 01:21:18.200 So just to close up on this univariate versus chi-squared 01:21:18.200 --> 01:21:22.220 chart, here's an example-- 01:21:22.220 --> 01:21:28.430 comparing the univariate case up here to the multivariate case 01:21:28.430 --> 01:21:30.090 down here. 01:21:30.090 --> 01:21:35.570 And what I'm plotting here is, again, the scatter of data. 01:21:35.570 --> 01:21:38.840 You might see there's a slight shaded oval case. 01:21:38.840 --> 01:21:40.610 And all the blue data points are what 01:21:40.610 --> 01:21:44.840 I would do if I just projected those down to an x1 and x2, 01:21:44.840 --> 01:21:47.720 and did univariate control charting. 01:21:47.720 --> 01:21:52.640 If I then had this little red dot that fell outside 01:21:52.640 --> 01:21:55.760 of the joint confidence interval-- 01:21:55.760 --> 01:21:58.640 on my individual charts, notice where it falls-- 01:21:58.640 --> 01:22:00.950 perfectly within the control limits-- 01:22:00.950 --> 01:22:04.280 nothing that tells me that that point is unusual. 01:22:04.280 --> 01:22:06.470 But if I then plot that data point-- 01:22:06.470 --> 01:22:11.720 that red data point there and on the chi-squared chart-- 01:22:11.720 --> 01:22:15.830 I calculated the chi statistic, that distance statistic-- 01:22:15.830 --> 01:22:21.140 that would trigger, because its distance from the mean 01:22:21.140 --> 01:22:21.935 is unusual. 01:22:26.700 --> 01:22:32.460 Now, if I actually didn't know my true covariance matrix, 01:22:32.460 --> 01:22:40.056 but I had to estimate it, I have to estimate sigma with an s. 01:22:40.056 --> 01:22:42.420 I'm right back to this whole difference 01:22:42.420 --> 01:22:45.180 between a t and a unit normal. 01:22:45.180 --> 01:22:47.730 So just to close this out we often 01:22:47.730 --> 01:22:51.930 talk about these as t-squared, Hotelling t-squared statistics, 01:22:51.930 --> 01:22:55.660 but it's exactly the same idea. 01:22:55.660 --> 01:22:58.950 And it's got its own cut-off points. 01:22:58.950 --> 01:23:04.410 OK, so with that, I think we've pulled to a close on control 01:23:04.410 --> 01:23:06.000 charting for the-- 01:23:06.000 --> 01:23:07.860 at least for the time being. 01:23:07.860 --> 01:23:11.460 Next time, we'll start diving in on yield modeling 01:23:11.460 --> 01:23:13.733 and different uses of some statistics, 01:23:13.733 --> 01:23:15.150 but I think what you've got now is 01:23:15.150 --> 01:23:17.880 some exposure to EWMA, CUSUM. 01:23:17.880 --> 01:23:21.450 You'll play around with those with the three or four problems 01:23:21.450 --> 01:23:23.940 on this week's problem set. 01:23:23.940 --> 01:23:25.860 And start looking at the quizzes. 01:23:25.860 --> 01:23:29.370 We'll have the first quiz a week from today-- 01:23:29.370 --> 01:23:31.955 in class. 01:23:31.955 --> 01:23:33.580 AUDIENCE: Professor, I have a question. 01:23:33.580 --> 01:23:34.020 DUANE BONING: Yeah. 01:23:34.020 --> 01:23:34.860 Go ahead and break. 01:23:34.860 --> 01:23:38.580 I'll take your question while people are dispersing. 01:23:38.580 --> 01:23:41.120 AUDIENCE: So for the [INAUDIBLE] you taught us today, 01:23:41.120 --> 01:23:42.870 are they going to be included in the quiz? 01:23:45.660 --> 01:23:49.140 DUANE BONING: It's possible you'll see an EWMA or a CUSUM, 01:23:49.140 --> 01:23:54.810 but they would only be fairly simple questions for those. 01:23:54.810 --> 01:23:59.400 It's not likely you'll see a multivariate problem. 01:23:59.400 --> 01:24:01.580 AUDIENCE: OK, thank you.