WEBVTT 00:00:15.088 --> 00:00:16.880 DAVID SONTAG: So I'll begin today's lecture 00:00:16.880 --> 00:00:20.860 by giving a brief recap of risk stratification. 00:00:20.860 --> 00:00:24.048 We didn't get to finish talking survival modeling on Thursday, 00:00:24.048 --> 00:00:25.840 and so I'll go a little bit more into that, 00:00:25.840 --> 00:00:27.590 and I'll answer some of the questions that 00:00:27.590 --> 00:00:30.735 arose during our discussions and on Piazza since. 00:00:30.735 --> 00:00:32.860 And then the vast majority of today's lecture we'll 00:00:32.860 --> 00:00:35.100 be talking about a new topic-- 00:00:35.100 --> 00:00:37.600 in particular, physiological time series modeling. 00:00:37.600 --> 00:00:40.400 I'll give two examples of physiological time series 00:00:40.400 --> 00:00:43.810 modeling-- the first one coming from monitoring patients 00:00:43.810 --> 00:00:46.050 in intensive care units, and the second one 00:00:46.050 --> 00:00:47.800 asking a very different type of question-- 00:00:47.800 --> 00:00:53.620 that of diagnosing patients' heart conditions using EKGs. 00:00:53.620 --> 00:00:55.390 And both of these correspond to readings 00:00:55.390 --> 00:00:57.040 that you had for today's lecture, 00:00:57.040 --> 00:00:59.200 and we'll go into much more depth in these-- 00:00:59.200 --> 00:01:01.640 of those papers today, and I'll provide much more color 00:01:01.640 --> 00:01:02.140 around them. 00:01:05.379 --> 00:01:07.862 So just to briefly remind you where we were on Thursday, 00:01:07.862 --> 00:01:10.320 we talked about how one could formalize risk stratification 00:01:10.320 --> 00:01:12.840 instead of as a classification problem of what would happen, 00:01:12.840 --> 00:01:15.570 let's say, in some predefined time period, 00:01:15.570 --> 00:01:17.640 rather thinking about risk stratification 00:01:17.640 --> 00:01:21.390 as a regression question, or regression task. 00:01:21.390 --> 00:01:23.910 Given what you know about a patient at time zero, 00:01:23.910 --> 00:01:26.380 predicting time to event-- 00:01:26.380 --> 00:01:29.790 so for example, here the event might be death, divorce, 00:01:29.790 --> 00:01:31.150 college graduation. 00:01:31.150 --> 00:01:35.850 And patient one-- that event happened at time step nine. 00:01:35.850 --> 00:01:38.340 Patient two, that event happened at time step 12. 00:01:38.340 --> 00:01:42.510 And for patient four, we don't know when that event happened, 00:01:42.510 --> 00:01:45.240 because it was censored. 00:01:45.240 --> 00:01:47.710 In particular, after time step seven, 00:01:47.710 --> 00:01:50.340 we no longer get to view any of the patients' data, 00:01:50.340 --> 00:01:52.960 and so we don't know when that red dot would be-- 00:01:52.960 --> 00:01:55.360 sometime in the future or never. 00:01:55.360 --> 00:01:57.550 So this is what we mean by right censor data, which 00:01:57.550 --> 00:02:01.443 is precisely what survival modeling is aiming to solve. 00:02:01.443 --> 00:02:03.235 Are there questions about this setup first? 00:02:06.358 --> 00:02:08.259 AUDIENCE: You flipped the x on-- 00:02:08.259 --> 00:02:09.759 DAVID SONTAG: Yeah, I realized that. 00:02:09.759 --> 00:02:11.860 I flipped the x and the o in today's presentation, 00:02:11.860 --> 00:02:14.720 but that's not relevant. 00:02:14.720 --> 00:02:18.370 So f of t is the probability of death, 00:02:18.370 --> 00:02:20.742 or the event occurring at time step t. 00:02:20.742 --> 00:02:22.450 And although in this slide I'm showing it 00:02:22.450 --> 00:02:24.300 as an unconditional model, in general, 00:02:24.300 --> 00:02:25.875 you should think about this as a conditional density. 00:02:25.875 --> 00:02:28.333 So you might be conditioning on some covariates or features 00:02:28.333 --> 00:02:31.810 that you have for that patient at baseline. 00:02:31.810 --> 00:02:34.515 And very important for survival modeling 00:02:34.515 --> 00:02:35.890 and for the next things I'll tell 00:02:35.890 --> 00:02:39.790 you are the survival function, to note it as capital S of t. 00:02:39.790 --> 00:02:45.170 And that's simply 1 minus the cumulative density function. 00:02:45.170 --> 00:02:48.040 So it's the probability that the event occurring, 00:02:48.040 --> 00:02:49.120 which is time-- 00:02:49.120 --> 00:02:51.640 which is denoted here as capital T, 00:02:51.640 --> 00:02:54.430 occurs greater than some little t. 00:02:54.430 --> 00:02:56.770 So it's this function, which is simply 00:02:56.770 --> 00:02:58.450 given to you by the integral from 0 00:02:58.450 --> 00:03:01.370 to infinity of the density. 00:03:01.370 --> 00:03:04.520 So in pictures, this is the density. 00:03:04.520 --> 00:03:06.230 On the x-axis is time. 00:03:06.230 --> 00:03:08.040 The y-axis is the density function. 00:03:08.040 --> 00:03:11.660 And this black curve is what I'm denoting as f of t. 00:03:11.660 --> 00:03:18.390 And this white area is capital s of c, the survival probability, 00:03:18.390 --> 00:03:21.250 or survival function. 00:03:21.250 --> 00:03:21.990 Yes? 00:03:21.990 --> 00:03:23.532 AUDIENCE: So I just want to be clear. 00:03:23.532 --> 00:03:26.446 So if you were to integrate the entire curve, 00:03:26.446 --> 00:03:31.717 [INAUDIBLE] by infinity you're going to be [INAUDIBLE].. 00:03:31.717 --> 00:03:33.550 DAVID SONTAG: In the way that I described it 00:03:33.550 --> 00:03:38.950 to here, yes, because we're talking about the time 00:03:38.950 --> 00:03:41.050 to event. 00:03:41.050 --> 00:03:44.680 But often we might be in scenarios where the event may 00:03:44.680 --> 00:03:47.662 never occur, and so that-- 00:03:47.662 --> 00:03:49.870 you can formalize that in a couple of different ways. 00:03:49.870 --> 00:03:52.060 You could put that at point mass at s of infinity, 00:03:52.060 --> 00:03:55.270 or you could simply say that the integral from 0 to infinity 00:03:55.270 --> 00:03:57.730 is some quantity less than 1. 00:03:57.730 --> 00:03:59.440 And in the readings that I'm referencing 00:03:59.440 --> 00:04:00.850 in the very bottom of those slides-- it shows you 00:04:00.850 --> 00:04:03.292 how you can very easily modify all of the frameworks 00:04:03.292 --> 00:04:05.500 I'm telling you about here to deal with that scenario 00:04:05.500 --> 00:04:07.120 where the event may never occur. 00:04:07.120 --> 00:04:09.702 But for the purposes of my presentation, 00:04:09.702 --> 00:04:11.410 you can assume that the event will always 00:04:11.410 --> 00:04:13.000 occur at some point. 00:04:13.000 --> 00:04:17.350 It's a very minor modification where you, in essence, divide 00:04:17.350 --> 00:04:21.700 the densities by a constant, which accounts for the fact 00:04:21.700 --> 00:04:26.170 that it wouldn't integrate to one otherwise. 00:04:26.170 --> 00:04:30.100 Now, a key question that has to be solved 00:04:30.100 --> 00:04:33.400 when trying to use a parametric approach to survivor modeling 00:04:33.400 --> 00:04:35.240 is, what should that f of t look like? 00:04:35.240 --> 00:04:38.070 What should that density function look like? 00:04:38.070 --> 00:04:42.040 And what I'm showing you here is a table of some very commonly 00:04:42.040 --> 00:04:43.990 used density functions. 00:04:43.990 --> 00:04:46.060 What you see in these two columns-- 00:04:46.060 --> 00:04:49.600 on the right hand column is the density function f of t itself. 00:04:49.600 --> 00:04:52.180 Lambda denotes some parameter of the model. 00:04:52.180 --> 00:04:54.520 t is the time. 00:04:54.520 --> 00:04:57.940 And on this second middle column is the survival function. 00:04:57.940 --> 00:05:01.510 So this is obtained for these particular parametric forms 00:05:01.510 --> 00:05:06.790 by an analytical solution solving that integral from t 00:05:06.790 --> 00:05:08.320 to infinity. 00:05:08.320 --> 00:05:11.090 This is the analytic solution for that. 00:05:11.090 --> 00:05:13.600 And so these go by common names of exponential, 00:05:13.600 --> 00:05:15.880 weeble, log-normal, and so on. 00:05:15.880 --> 00:05:17.950 And critically, all of these have support only 00:05:17.950 --> 00:05:21.192 on the positive real numbers, because the event can ever 00:05:21.192 --> 00:05:22.150 occur at negative time. 00:05:24.690 --> 00:05:27.900 Now, we live in a day and age where 00:05:27.900 --> 00:05:32.340 we no longer have to make standard parametric assumptions 00:05:32.340 --> 00:05:33.600 for densities. 00:05:33.600 --> 00:05:36.990 We could, for example, try to formalize the density 00:05:36.990 --> 00:05:41.110 as some output of some deep neural network. 00:05:41.110 --> 00:05:44.833 But if we don't use a parametric approach, 00:05:44.833 --> 00:05:46.500 so there are two ways to try to do that. 00:05:46.500 --> 00:05:48.470 One way to do that would be to say that we're 00:05:48.470 --> 00:05:50.120 going to model the post-- 00:05:50.120 --> 00:05:56.660 the distribution, f of t, as one of these things, where lambda 00:05:56.660 --> 00:05:58.640 or whatever the parameters of distribution 00:05:58.640 --> 00:06:00.470 are given to by the output of, let's 00:06:00.470 --> 00:06:03.890 say, a deep neural network on the covariate x. 00:06:03.890 --> 00:06:05.413 So that would be one approach. 00:06:05.413 --> 00:06:06.830 A very different approach would be 00:06:06.830 --> 00:06:10.130 a non-parametric distribution where you say, OK, I'm 00:06:10.130 --> 00:06:12.200 going to define f of t extremely flexibly, 00:06:12.200 --> 00:06:15.980 not as one of these forms. 00:06:15.980 --> 00:06:18.500 And there one runs into a slightly different challenge, 00:06:18.500 --> 00:06:20.720 because as I'll show you in the next slide, 00:06:20.720 --> 00:06:22.520 to do maximum likelihood estimation 00:06:22.520 --> 00:06:24.860 of these distributions from censor data, 00:06:24.860 --> 00:06:27.680 one needs to get-- one needs to make use of this survival 00:06:27.680 --> 00:06:29.820 function, s of t. 00:06:29.820 --> 00:06:33.060 And so if you're f if t is complex, 00:06:33.060 --> 00:06:37.153 and you don't have a nice analytic solution for s of t, 00:06:37.153 --> 00:06:38.820 then you're going to have to somehow use 00:06:38.820 --> 00:06:41.400 a numerical approximation of s of t during limiting. 00:06:41.400 --> 00:06:43.200 So it's definitely possible, but it's going 00:06:43.200 --> 00:06:44.492 to be a little bit more effort. 00:06:46.510 --> 00:06:49.010 So now here's where I'm going to get into maximum likelihood 00:06:49.010 --> 00:06:51.530 estimation of these distributions, 00:06:51.530 --> 00:06:54.350 and to define for you the likelihood function, 00:06:54.350 --> 00:06:57.150 I'm going to break it down into two different settings. 00:06:57.150 --> 00:06:59.300 The first setting is an observation 00:06:59.300 --> 00:07:03.530 which is uncensored, meaning we do observe when the event-- 00:07:03.530 --> 00:07:05.710 death, for example-- occurs. 00:07:05.710 --> 00:07:08.720 And in that case, the probability of the event-- 00:07:08.720 --> 00:07:09.450 it's very simple. 00:07:09.450 --> 00:07:13.345 It's just probability of the event occurring at capital-- 00:07:13.345 --> 00:07:15.230 at capital T, random variable T, equals 00:07:15.230 --> 00:07:16.580 a little t-- is just f or t. 00:07:16.580 --> 00:07:17.080 Done. 00:07:19.600 --> 00:07:22.870 However, what happens if, for this data point, 00:07:22.870 --> 00:07:26.650 you don't observe when the event occurred because of censoring? 00:07:26.650 --> 00:07:30.120 Well, of course, you could just throw away that data point, 00:07:30.120 --> 00:07:32.380 not use it in your estimation, but that's 00:07:32.380 --> 00:07:34.930 precisely what we mentioned at the very beginning 00:07:34.930 --> 00:07:38.625 of last week's lecture-- was the goal of survival modeling 00:07:38.625 --> 00:07:40.000 to not do that, because if we did 00:07:40.000 --> 00:07:44.440 that, it would introduce bias into our estimation procedure. 00:07:44.440 --> 00:07:48.110 So we would like to be able to use that observation 00:07:48.110 --> 00:07:50.770 that this data point was censored, 00:07:50.770 --> 00:07:53.530 but the only information we can get from that observation 00:07:53.530 --> 00:07:57.040 is that capital T, the event time, 00:07:57.040 --> 00:08:00.490 must have occurred some time larger 00:08:00.490 --> 00:08:03.040 than the observed-- the time of censoring, which 00:08:03.040 --> 00:08:05.410 is little t here. 00:08:05.410 --> 00:08:09.130 So we don't know precisely when capital T was, but we 00:08:09.130 --> 00:08:12.560 know it's something larger than the observed centering time 00:08:12.560 --> 00:08:13.660 little of t. 00:08:13.660 --> 00:08:17.830 And that, remember, is precisely what the survival function 00:08:17.830 --> 00:08:19.000 is capturing. 00:08:19.000 --> 00:08:20.680 So for a censored observation, we're 00:08:20.680 --> 00:08:24.253 going to use capital S of t within the likelihood. 00:08:24.253 --> 00:08:26.920 So now we can then combine these two for censored and uncensored 00:08:26.920 --> 00:08:30.820 data, and what we get is the following likelihood objective. 00:08:30.820 --> 00:08:33.880 This is-- I'm showing you here the log likelihood objective. 00:08:33.880 --> 00:08:38.320 Recall from last week that little b of i simply denotes 00:08:38.320 --> 00:08:40.570 is this observation censored or not? 00:08:40.570 --> 00:08:44.290 So if bi is 1, it means the time that you're given 00:08:44.290 --> 00:08:47.200 is the time of the censoring event. 00:08:47.200 --> 00:08:49.540 And if bi is 0, it means the time you're given 00:08:49.540 --> 00:08:51.787 is the time that the event occurs. 00:08:51.787 --> 00:08:54.370 So here what we're going to do is now sum over all of the data 00:08:54.370 --> 00:08:56.320 points in your data set from little i 00:08:56.320 --> 00:09:02.500 equals 1 to little n of bi times log of probability 00:09:02.500 --> 00:09:06.660 under the censored model plus 1 minus bi times log 00:09:06.660 --> 00:09:08.410 of probability under the uncensored model. 00:09:08.410 --> 00:09:10.510 And so this bi is just going to switch on which of these two 00:09:10.510 --> 00:09:12.760 you're going to use for that given data point. 00:09:12.760 --> 00:09:15.550 So the learning objective for maximum likelihood estimation 00:09:15.550 --> 00:09:18.070 here is very similar to what you're used to 00:09:18.070 --> 00:09:21.640 in learning distributions with the big difference 00:09:21.640 --> 00:09:23.470 that, for censored data, we're going 00:09:23.470 --> 00:09:29.080 to use the survival function to estimate its probability. 00:09:29.080 --> 00:09:30.313 Are there any questions? 00:09:35.150 --> 00:09:37.270 And this, of course, could then be 00:09:37.270 --> 00:09:39.760 optimized via your favorite algorithm, 00:09:39.760 --> 00:09:42.400 whether it be stochastic gradient descent, 00:09:42.400 --> 00:09:43.920 or second order method, and so on. 00:09:43.920 --> 00:09:44.420 Yep? 00:09:44.420 --> 00:09:45.785 AUDIENCE: I have a question about the a kind 00:09:45.785 --> 00:09:46.452 of side project. 00:09:46.452 --> 00:09:48.620 You mentioned that we could use [INAUDIBLE].. 00:09:48.620 --> 00:09:49.370 DAVID SONTAG: Yes. 00:09:49.370 --> 00:09:51.310 AUDIENCE: And then combine it with the parametric approach. 00:09:51.310 --> 00:09:51.730 DAVID SONTAG: Yes. 00:09:51.730 --> 00:09:53.563 AUDIENCE: So is that true that we just still 00:09:53.563 --> 00:09:55.705 have the parametric assumption that we kind of map 00:09:55.705 --> 00:09:57.250 the input to the parameters? 00:09:57.250 --> 00:09:58.385 DAVID SONTAG: Exactly. 00:09:58.385 --> 00:09:59.260 That's exactly right. 00:09:59.260 --> 00:10:04.980 So consider the following picture where for-- 00:10:04.980 --> 00:10:08.500 this is time, t. 00:10:08.500 --> 00:10:11.812 And this is f of t. 00:10:11.812 --> 00:10:13.670 You can imagine for any one patient 00:10:13.670 --> 00:10:15.907 you might have a different function. 00:10:15.907 --> 00:10:18.490 You might-- but they might all be of the same parametric form. 00:10:18.490 --> 00:10:21.332 So they might be like that, or maybe 00:10:21.332 --> 00:10:22.540 they're shifted a little bit. 00:10:25.130 --> 00:10:27.070 So you think about each of these three 00:10:27.070 --> 00:10:30.355 things as being from the same parametric family 00:10:30.355 --> 00:10:33.310 of distributions, but with different means. 00:10:33.310 --> 00:10:35.380 And in this case, then the mean is 00:10:35.380 --> 00:10:37.603 given to as the output of the deep neural network. 00:10:37.603 --> 00:10:39.520 And so that would be the way it would be used, 00:10:39.520 --> 00:10:41.812 and then one could just back propagate in the usual way 00:10:41.812 --> 00:10:43.106 to do learning. 00:10:43.106 --> 00:10:43.692 Yep? 00:10:43.692 --> 00:10:45.400 AUDIENCE: Can you repeat what b sub i is? 00:10:45.400 --> 00:10:45.780 DAVID SONTAG: Excuse me? 00:10:45.780 --> 00:10:47.572 AUDIENCE: Could you repeat what b sub i is? 00:10:47.572 --> 00:10:50.740 DAVID SONTAG: b sub i is just an indicator whether the i-th data 00:10:50.740 --> 00:10:54.510 point was censored or not censored. 00:10:54.510 --> 00:10:55.020 Yes? 00:10:55.020 --> 00:10:59.200 AUDIENCE: So [INAUDIBLE] equal it's more a probability density 00:10:59.200 --> 00:11:00.130 function [INAUDIBLE]. 00:11:00.130 --> 00:11:01.880 DAVID SONTAG: Cumulative density function. 00:11:01.880 --> 00:11:05.750 AUDIENCE: Yeah, but [INAUDIBLE] probability. 00:11:05.750 --> 00:11:10.350 No, for the [INAUDIBLE] it's probability density function. 00:11:10.350 --> 00:11:11.820 DAVID SONTAG Yes, so just to-- 00:11:11.820 --> 00:11:13.200 AUDIENCE: [INAUDIBLE] 00:11:13.200 --> 00:11:15.113 DAVID SONTAG: Excuse me? 00:11:15.113 --> 00:11:16.530 AUDIENCE: Will that be any problem 00:11:16.530 --> 00:11:18.440 to combine those two types there? 00:11:18.440 --> 00:11:20.190 DAVID SONTAG: That's a very good question. 00:11:20.190 --> 00:11:24.550 So the observation was that you have two different types 00:11:24.550 --> 00:11:27.412 of probabilities used here. 00:11:27.412 --> 00:11:28.870 In this case, we're using something 00:11:28.870 --> 00:11:32.200 like the cumulative density, whereas here we're 00:11:32.200 --> 00:11:35.380 using the probability density function. 00:11:35.380 --> 00:11:38.890 The question was, are these two on different scales? 00:11:38.890 --> 00:11:40.540 Does it make sense to combine them 00:11:40.540 --> 00:11:43.360 in this type of linear fashion with the same weighting? 00:11:43.360 --> 00:11:45.490 And I think it does make sense. 00:11:45.490 --> 00:11:59.430 So think about a setting where you have a very small time 00:11:59.430 --> 00:12:00.210 range. 00:12:00.210 --> 00:12:02.273 You're not exactly sure when this event occurs. 00:12:02.273 --> 00:12:03.690 It's something in this time range. 00:12:06.690 --> 00:12:10.440 In the setting of the censored data, 00:12:10.440 --> 00:12:14.180 where that time range could potentially be very large, 00:12:14.180 --> 00:12:17.650 your model is providing-- 00:12:21.150 --> 00:12:23.670 your log probability is somehow going 00:12:23.670 --> 00:12:28.590 to be much more flat, because you're covering 00:12:28.590 --> 00:12:29.715 much more probability mass. 00:12:32.930 --> 00:12:34.830 And so that observation, I think, 00:12:34.830 --> 00:12:37.200 intuitively is likely to have a much-- 00:12:37.200 --> 00:12:41.640 a bit of a smaller effect on the overall learning algorithm. 00:12:41.640 --> 00:12:44.590 These observations-- you know precisely where they are, 00:12:44.590 --> 00:12:51.280 and so as you deviate from that, you incur the corresponding log 00:12:51.280 --> 00:12:53.740 loss penalty. 00:12:53.740 --> 00:12:55.918 But I do think that it makes sense 00:12:55.918 --> 00:12:57.210 to have them in the same scale. 00:12:57.210 --> 00:12:59.977 If anyone in the room has done work with [INAUDIBLE] modeling 00:12:59.977 --> 00:13:02.310 and has a different answer to that, I'd love to hear it. 00:13:06.450 --> 00:13:09.510 Not today, but maybe someone in the future 00:13:09.510 --> 00:13:11.260 will answer this question differently. 00:13:11.260 --> 00:13:14.250 I'm going to move on for now. 00:13:14.250 --> 00:13:18.480 So the remaining question that I want to talk about today 00:13:18.480 --> 00:13:22.020 is how one evaluates survival models. 00:13:22.020 --> 00:13:27.240 So we talked about binary classification a lot 00:13:27.240 --> 00:13:29.090 in the context of risk stratification 00:13:29.090 --> 00:13:31.590 in the beginning, and we talked about how area under the ROC 00:13:31.590 --> 00:13:34.900 curve is one measure of classification performance, 00:13:34.900 --> 00:13:36.630 but here we're doing more-- 00:13:36.630 --> 00:13:40.120 something more akin to regression, not classification. 00:13:40.120 --> 00:13:43.180 A standard measure that's used to measure performance 00:13:43.180 --> 00:13:46.240 is known as the C-statistic, or concordance index. 00:13:46.240 --> 00:13:48.130 Those are one in the same-- 00:13:48.130 --> 00:13:49.630 and is defined as follows. 00:13:49.630 --> 00:13:52.050 And it has a very intuitive definition. 00:13:52.050 --> 00:13:55.300 It sums over pairs of data points 00:13:55.300 --> 00:13:59.120 that can be compared to one another, 00:13:59.120 --> 00:14:04.860 and it says, OK, what is the likelihood 00:14:04.860 --> 00:14:08.550 of the event happening for an event that 00:14:08.550 --> 00:14:11.100 occurs before an event-- 00:14:11.100 --> 00:14:11.820 another event. 00:14:11.820 --> 00:14:16.020 And what you want is that the likelihood of the event that, 00:14:16.020 --> 00:14:18.060 on average, in essence, should occur later 00:14:18.060 --> 00:14:21.215 should be larger than the event that should occur earlier. 00:14:21.215 --> 00:14:23.340 I'm going to first illustrate it with this picture, 00:14:23.340 --> 00:14:24.840 and then I'll work through the math. 00:14:24.840 --> 00:14:28.510 So here's the picture, and then we'll talk about the math. 00:14:28.510 --> 00:14:31.960 So what I'm showing you here are every single observation 00:14:31.960 --> 00:14:34.130 in your data set, and they're sorted 00:14:34.130 --> 00:14:40.900 by either the censoring time or the event time. 00:14:40.900 --> 00:14:45.130 So by black, I'm illustrating uncensored data points. 00:14:45.130 --> 00:14:50.010 And by red, I'm denoting censored data points. 00:14:50.010 --> 00:14:54.140 Now, here we see that this data point-- 00:14:54.140 --> 00:14:58.593 the event happened before this data point's censoring event. 00:14:58.593 --> 00:15:00.260 Now, since this data point was censored, 00:15:00.260 --> 00:15:03.030 it means it's true event time you could think about as 00:15:03.030 --> 00:15:05.700 sometime into the far future. 00:15:05.700 --> 00:15:11.330 So what we would want is that the model 00:15:11.330 --> 00:15:20.050 gives that the probability that this event happens by this time 00:15:20.050 --> 00:15:24.010 should be larger than the probability 00:15:24.010 --> 00:15:29.190 that this event happens by this time, 00:15:29.190 --> 00:15:31.320 because this actually occurred first. 00:15:31.320 --> 00:15:33.842 And these two are comparable together-- to each other. 00:15:33.842 --> 00:15:35.550 On the other hand, it wouldn't make sense 00:15:35.550 --> 00:15:39.450 to compare y2 and y4, because both of these 00:15:39.450 --> 00:15:41.610 were censored data points, and we don't know 00:15:41.610 --> 00:15:43.090 precisely when they occurred. 00:15:43.090 --> 00:15:45.090 So for example, it could have very well happened 00:15:45.090 --> 00:15:50.280 that the event 2 happened after event 4. 00:15:50.280 --> 00:15:53.250 So what I'm showing you here with each of these lines 00:15:53.250 --> 00:15:54.750 are the pairwise comparisons that 00:15:54.750 --> 00:15:56.325 are actually possible to make. 00:15:56.325 --> 00:15:58.200 You can make pairwise comparisons, of course, 00:15:58.200 --> 00:16:00.730 between any pair of events that actually did occur, 00:16:00.730 --> 00:16:02.272 and you can make pairwise comparisons 00:16:02.272 --> 00:16:06.690 between censored events and events that occurred before it. 00:16:06.690 --> 00:16:11.500 Now, if you now look at this formula, the formula in this 00:16:11.500 --> 00:16:15.100 indicate-- this is looking at an indicator of survival 00:16:15.100 --> 00:16:18.080 functions between pairs of data points, and which pairs of data 00:16:18.080 --> 00:16:18.580 points? 00:16:18.580 --> 00:16:21.170 It was precisely those pairs of data points, 00:16:21.170 --> 00:16:24.700 which I'm showing comparisons of with these blue lines here. 00:16:24.700 --> 00:16:28.210 So we're going to sum over i such that bi is equal to 0, 00:16:28.210 --> 00:16:32.830 and remember that means it is an uncensored data point. 00:16:32.830 --> 00:16:35.650 And then we look at-- 00:16:35.650 --> 00:16:41.050 we look at yi compared to all other yj that's great-- 00:16:41.050 --> 00:16:45.520 that has a value greater than-- both censored and uncensored. 00:16:45.520 --> 00:16:51.860 Now, if your data had no sensor data points in it, 00:16:51.860 --> 00:16:56.310 then you can verify that, in fact, this corresponds-- 00:16:56.310 --> 00:16:58.310 so there's one other assumption one has to make, 00:16:58.310 --> 00:16:59.610 which is that-- 00:16:59.610 --> 00:17:02.817 suppose that your outcome is binary. 00:17:02.817 --> 00:17:05.109 And so if you might wonder how you get a binary outcome 00:17:05.109 --> 00:17:09.760 from this, imagine that your density function 00:17:09.760 --> 00:17:13.359 looked a little bit like this, where it could occur either 00:17:13.359 --> 00:17:18.490 at time 1 or time 2. 00:17:18.490 --> 00:17:21.160 So something like that. 00:17:24.130 --> 00:17:29.300 So if the event can occur at only two times, 00:17:29.300 --> 00:17:31.770 not a whole range of times, then this 00:17:31.770 --> 00:17:35.570 is analogous to a binary outcome. 00:17:35.570 --> 00:17:37.210 And so if you have a binary outcome 00:17:37.210 --> 00:17:40.630 like this and no censoring, then, in fact, that C-statistic 00:17:40.630 --> 00:17:42.763 is exactly equal to the area under the ROC curve. 00:17:42.763 --> 00:17:44.930 So that just connects it a little bit back to things 00:17:44.930 --> 00:17:45.400 we're used to. 00:17:45.400 --> 00:17:45.850 Yep? 00:17:45.850 --> 00:17:47.767 AUDIENCE: Just to make sure that I understand. 00:17:47.767 --> 00:17:50.370 So y1 is going to be we observed an event, 00:17:50.370 --> 00:17:53.920 and y2 is going to be we know that no event occurred 00:17:53.920 --> 00:17:55.210 until that day? 00:17:55.210 --> 00:17:58.320 DAVID SONTAG: Every dot corresponds to one event, 00:17:58.320 --> 00:17:59.370 either censored or not. 00:17:59.370 --> 00:18:00.070 AUDIENCE: Thank you. 00:18:00.070 --> 00:18:01.445 DAVID SONTAG: And they're sorted. 00:18:01.445 --> 00:18:04.110 In this figure, they're sorted by the time 00:18:04.110 --> 00:18:08.475 of either the censoring or the event occurring. 00:18:14.420 --> 00:18:16.570 So I talked to-- 00:18:16.570 --> 00:18:18.190 when I talked about C-statistic, it-- 00:18:18.190 --> 00:18:21.730 that's one way to measure performance of your survival 00:18:21.730 --> 00:18:23.780 modeling, but you might remember that I-- 00:18:23.780 --> 00:18:25.780 that when we talked about binary classification, 00:18:25.780 --> 00:18:27.363 we said how area under there ROC curve 00:18:27.363 --> 00:18:29.080 in itself is very limiting, and so we 00:18:29.080 --> 00:18:30.340 should think through other performance 00:18:30.340 --> 00:18:31.215 metrics of relevance. 00:18:31.215 --> 00:18:33.652 So here are a few other things that you could do. 00:18:33.652 --> 00:18:35.110 One thing you could do is you could 00:18:35.110 --> 00:18:38.330 use the mean squared error. 00:18:38.330 --> 00:18:41.410 So again, thinking about this as a regression problem. 00:18:41.410 --> 00:18:43.090 But of course, that only makes sense 00:18:43.090 --> 00:18:45.430 for uncensored data points. 00:18:45.430 --> 00:18:47.563 So focus just in the uncensored data points, 00:18:47.563 --> 00:18:49.480 look to see how well we're doing at predicting 00:18:49.480 --> 00:18:51.410 when the event occurs. 00:18:51.410 --> 00:18:55.270 The second thing one could do, since you have the ability 00:18:55.270 --> 00:18:58.490 to define the likelihood of an observation, 00:18:58.490 --> 00:19:02.260 censored or not censored, one could hold out data, 00:19:02.260 --> 00:19:05.080 and look at the held-out likelihood or log likelihood 00:19:05.080 --> 00:19:07.170 of that held-out data. 00:19:07.170 --> 00:19:08.760 And the third thing you could do is 00:19:08.760 --> 00:19:12.270 you can-- after learning using this survival modeling 00:19:12.270 --> 00:19:17.250 framework, one could then turn it into a binary classification 00:19:17.250 --> 00:19:19.380 problem by, for example, artificially 00:19:19.380 --> 00:19:24.060 choosing time ranges, like greater than three months is 1. 00:19:24.060 --> 00:19:25.830 Less than three months is 0. 00:19:25.830 --> 00:19:27.460 That would be one crude definition. 00:19:27.460 --> 00:19:29.010 And then once you've done a reduction 00:19:29.010 --> 00:19:30.468 to a binary classification problem, 00:19:30.468 --> 00:19:32.763 you could use all of the existing performance 00:19:32.763 --> 00:19:35.430 metrics they're used to thinking about for binary classification 00:19:35.430 --> 00:19:37.020 to evaluate the performance there-- 00:19:37.020 --> 00:19:40.550 things like positive predictive value, for example. 00:19:40.550 --> 00:19:42.990 And you could, of course, choose different reductions 00:19:42.990 --> 00:19:44.970 and get different performance statistics out. 00:19:44.970 --> 00:19:47.700 So this is just a small subset of ways 00:19:47.700 --> 00:19:49.710 to try to evaluate survivor modeling, 00:19:49.710 --> 00:19:51.812 but it's a very, very rich literature. 00:19:51.812 --> 00:19:53.520 And again, on the bottom of these slides, 00:19:53.520 --> 00:19:54.978 I pointed you to several references 00:19:54.978 --> 00:19:57.640 that you could go to to learn more. 00:19:57.640 --> 00:19:59.710 The final comment I wanted to make 00:19:59.710 --> 00:20:02.470 is that I only told you about one estimator 00:20:02.470 --> 00:20:05.830 in today's lecture, and that's known as the likelihood based 00:20:05.830 --> 00:20:07.030 estimator. 00:20:07.030 --> 00:20:09.307 But there is a whole other estimation approach 00:20:09.307 --> 00:20:11.890 for survival modelings, which is very important to know about, 00:20:11.890 --> 00:20:14.218 that are called partial likelihood estimators. 00:20:14.218 --> 00:20:16.510 And for those of you who have heard of Cox proportional 00:20:16.510 --> 00:20:18.468 hazards models-- and I know they were discussed 00:20:18.468 --> 00:20:19.750 in Friday's recitation-- 00:20:19.750 --> 00:20:21.700 that's an example of a class of model 00:20:21.700 --> 00:20:26.040 that's commonly used within this partial likelihood estimator. 00:20:26.040 --> 00:20:28.540 Now, at a very intuitive level, what this partial likelihood 00:20:28.540 --> 00:20:31.600 estimator is doing is it's working with something 00:20:31.600 --> 00:20:33.100 like the C-statistic. 00:20:33.100 --> 00:20:38.890 So notice how the C-statistic only looks at relative 00:20:38.890 --> 00:20:40.810 orderings of events-- 00:20:40.810 --> 00:20:44.130 of their event occurrences. 00:20:44.130 --> 00:20:47.330 It doesn't care about exactly when the event occurred or not. 00:20:47.330 --> 00:20:52.100 In some sense, there's a constant. 00:20:52.100 --> 00:20:55.910 There's-- in this survival function, 00:20:55.910 --> 00:21:01.400 which could be divided out from both sides of this inequality, 00:21:01.400 --> 00:21:05.330 and it wouldn't affect anything about the statistic. 00:21:05.330 --> 00:21:07.488 And so one could think about other ways of learning 00:21:07.488 --> 00:21:09.030 these models by saying, well, we want 00:21:09.030 --> 00:21:10.770 to learn a survival function such 00:21:10.770 --> 00:21:14.620 that it gets the ordering correct between data points. 00:21:14.620 --> 00:21:17.760 Now, such a survival function wouldn't do a very good job. 00:21:17.760 --> 00:21:20.400 There's no reason it would do any good at getting 00:21:20.400 --> 00:21:23.700 the precise time of when an event occurs, 00:21:23.700 --> 00:21:28.980 but if your goal were to just figure out 00:21:28.980 --> 00:21:31.405 what is the sorted order of patients by risk 00:21:31.405 --> 00:21:33.780 so that you're going to do an intervention on the 10 most 00:21:33.780 --> 00:21:37.710 risky people, then getting that order incorrect is going to be 00:21:37.710 --> 00:21:40.290 enough, and that's precisely the intuition used 00:21:40.290 --> 00:21:42.190 behind these partial likelihood estimators-- 00:21:42.190 --> 00:21:44.940 so they focus on something which is a little bit less 00:21:44.940 --> 00:21:47.040 than the original goal, but in doing 00:21:47.040 --> 00:21:49.570 so, they can have much better statistical complexity, 00:21:49.570 --> 00:21:51.445 meaning the amount of data they need in order 00:21:51.445 --> 00:21:52.780 to fit this models well. 00:21:52.780 --> 00:21:54.390 And again, this is a very rich topic. 00:21:54.390 --> 00:21:56.100 All I wanted to do is give you a pointer to it 00:21:56.100 --> 00:21:57.940 so that you can go read more about it if this is something 00:21:57.940 --> 00:21:58.732 of interest to you. 00:22:01.910 --> 00:22:06.590 So now moving on into the recap, one 00:22:06.590 --> 00:22:09.110 of the most important points that we discussed last week 00:22:09.110 --> 00:22:10.910 was about non-stationarity. 00:22:10.910 --> 00:22:13.250 And there was a question posted to Piazza, 00:22:13.250 --> 00:22:14.990 which was really interesting, which is how do you actually 00:22:14.990 --> 00:22:16.115 deal with non-stationarity. 00:22:16.115 --> 00:22:18.080 And I spoke a lot about it existing, 00:22:18.080 --> 00:22:19.700 and I talked about how to test for it, 00:22:19.700 --> 00:22:23.527 but I didn't say what to do if you have it. 00:22:23.527 --> 00:22:25.610 So I thought this was such an interesting question 00:22:25.610 --> 00:22:28.190 that I would also talk about it a bit during lecture. 00:22:28.190 --> 00:22:32.540 So the short answer is, if you have to have a solution 00:22:32.540 --> 00:22:36.280 that you deploy tomorrow, then here's 00:22:36.280 --> 00:22:38.442 the hack that sometimes works. 00:22:38.442 --> 00:22:40.900 You take your most recent data, like the last three months' 00:22:40.900 --> 00:22:42.358 data, and you hope that there's not 00:22:42.358 --> 00:22:45.490 much non-stationarity within last three months. 00:22:45.490 --> 00:22:47.800 You throw out all the historical data, 00:22:47.800 --> 00:22:51.410 and you just train using the most recent data. 00:22:51.410 --> 00:22:55.110 So a bit unsatisfying, because you 00:22:55.110 --> 00:22:57.950 might have now extremely little data left to learn with, 00:22:57.950 --> 00:23:02.885 but if you have enough volume, it might be good enough. 00:23:02.885 --> 00:23:04.260 But the real interesting question 00:23:04.260 --> 00:23:06.460 from a research perspective is how could you optimally use 00:23:06.460 --> 00:23:07.540 that historical data. 00:23:07.540 --> 00:23:10.150 So here are three different ways. 00:23:10.150 --> 00:23:14.420 So one way has to do with imputation. 00:23:14.420 --> 00:23:17.713 Imagine that the way in which your data was non-stationary 00:23:17.713 --> 00:23:19.130 was because there were, let's say, 00:23:19.130 --> 00:23:24.110 parts of time when certain features were just unavailable. 00:23:24.110 --> 00:23:27.452 I gave you this example last week of laboratory test results 00:23:27.452 --> 00:23:29.660 across time, and I showed you how there are sometimes 00:23:29.660 --> 00:23:31.202 these really big blocks of time where 00:23:31.202 --> 00:23:34.810 no lab tests are available, or very few are available. 00:23:34.810 --> 00:23:37.387 Well, luckily we live in a world with high dimensional data, 00:23:37.387 --> 00:23:39.720 and what that means is there's often a lot of redundancy 00:23:39.720 --> 00:23:40.930 in the data. 00:23:40.930 --> 00:23:45.840 So what you could imagine doing is imputing features 00:23:45.840 --> 00:23:48.390 that you observed to be missing, such 00:23:48.390 --> 00:23:50.520 that the missingness properties, in fact, 00:23:50.520 --> 00:23:54.177 aren't changing as much across time after imputation. 00:23:54.177 --> 00:23:56.010 And if you do that as a pre-processing step, 00:23:56.010 --> 00:23:57.810 it may allow you to make use of much 00:23:57.810 --> 00:24:00.690 more of the historical data. 00:24:00.690 --> 00:24:03.570 A different approach, which is intimately tied to that, 00:24:03.570 --> 00:24:05.490 has to do with transforming the data. 00:24:05.490 --> 00:24:07.770 Instead of imputing it, transforming it 00:24:07.770 --> 00:24:10.710 into another representation altogether, such that 00:24:10.710 --> 00:24:15.102 that presentation is invariant across time. 00:24:15.102 --> 00:24:16.560 And here I'm giving you a reference 00:24:16.560 --> 00:24:19.380 to this paper by Ganin et al from the Journal of Machine 00:24:19.380 --> 00:24:21.660 Learning Research 2016, which talks about how 00:24:21.660 --> 00:24:24.815 to do domain and variant learning of neural networks, 00:24:24.815 --> 00:24:26.190 and that's one approach to do so. 00:24:26.190 --> 00:24:28.482 And I view those two as being very similar-- imputation 00:24:28.482 --> 00:24:30.210 and transformations. 00:24:30.210 --> 00:24:32.970 A second approach is to re-weight the data 00:24:32.970 --> 00:24:36.230 to look like the current data. 00:24:36.230 --> 00:24:38.170 So imagine that you go back in time, 00:24:38.170 --> 00:24:39.670 and you say, you know what? 00:24:39.670 --> 00:24:43.050 I ICD-10 codes, for some very weird reason-- 00:24:43.050 --> 00:24:44.530 this is not true, by the way-- 00:24:44.530 --> 00:24:47.260 ICD-10 codes in this untrue world 00:24:47.260 --> 00:24:51.870 happen to be used between March and April of 2003. 00:24:51.870 --> 00:24:57.190 And then they weren't used again until 2015. 00:24:57.190 --> 00:24:59.630 So instead of throwing away all of the previous data, 00:24:59.630 --> 00:25:02.630 we're going to recognize that those-- 00:25:02.630 --> 00:25:04.740 that three month interval 10 years ago 00:25:04.740 --> 00:25:07.680 was actually drawn from a very similar distribution as what 00:25:07.680 --> 00:25:09.200 we're going to be testing on today. 00:25:09.200 --> 00:25:12.230 So we're going to weight those data points up very much, 00:25:12.230 --> 00:25:14.330 and down weight the data points that are 00:25:14.330 --> 00:25:16.760 less like the ones from today. 00:25:16.760 --> 00:25:19.790 That's the intuition behind these re-weighting approaches, 00:25:19.790 --> 00:25:22.010 and we're going to talk much more about that 00:25:22.010 --> 00:25:23.900 in the context of causal inference, 00:25:23.900 --> 00:25:25.953 not because these two have to do with each other, 00:25:25.953 --> 00:25:28.370 but they have-- they end up using a very similar technique 00:25:28.370 --> 00:25:32.600 for how to deal with datas that shift, or covariate shift. 00:25:32.600 --> 00:25:34.910 And the final technique that I'll mention 00:25:34.910 --> 00:25:37.710 is based on online learning algorithms. 00:25:37.710 --> 00:25:44.420 So the idea there is that there might be cut points, change 00:25:44.420 --> 00:25:47.760 points across time. 00:25:47.760 --> 00:25:52.275 So maybe the data looks one way up until this change point, 00:25:52.275 --> 00:25:53.900 and then suddenly the data looks really 00:25:53.900 --> 00:25:55.590 different until this change point, 00:25:55.590 --> 00:25:57.132 and then suddenly the data looks very 00:25:57.132 --> 00:25:59.440 different on into the future. 00:25:59.440 --> 00:26:01.940 So here I'm showing you there are two change points in which 00:26:01.940 --> 00:26:04.610 data set shift happens. 00:26:04.610 --> 00:26:06.900 What these online learning algorithms do is they say, 00:26:06.900 --> 00:26:09.350 OK, suppose we were forced to make predictions 00:26:09.350 --> 00:26:11.360 throughout this time period using 00:26:11.360 --> 00:26:13.400 only the historical data to make predictions 00:26:13.400 --> 00:26:15.200 at each point in time. 00:26:15.200 --> 00:26:18.650 Well, if we could somehow recognize 00:26:18.650 --> 00:26:21.172 that there might be these shifts, 00:26:21.172 --> 00:26:22.880 we could design algorithms that are going 00:26:22.880 --> 00:26:25.910 to be robust to those shifts. 00:26:25.910 --> 00:26:28.040 And then one could try to analyze-- mathematically 00:26:28.040 --> 00:26:30.350 analyze those algorithms based on the amount of regret 00:26:30.350 --> 00:26:33.770 they would have to, for example, an algorithm that knew exactly 00:26:33.770 --> 00:26:35.098 when those changes were. 00:26:35.098 --> 00:26:36.890 And of course, we don't know precisely when 00:26:36.890 --> 00:26:38.970 those changes were. 00:26:38.970 --> 00:26:41.660 And so there's a whole field of algorithms trying to do that, 00:26:41.660 --> 00:26:44.930 and here I'm just give me one citation for a recent work. 00:26:47.680 --> 00:26:49.240 So to conclude risk stratification-- 00:26:49.240 --> 00:26:51.970 this is the last slide here. 00:26:51.970 --> 00:26:55.080 Maybe ask your question after class. 00:26:55.080 --> 00:26:56.490 We've talked about two approaches 00:26:56.490 --> 00:26:58.890 for formalizing risk stratification-- first 00:26:58.890 --> 00:27:00.000 as binary classification. 00:27:00.000 --> 00:27:02.430 Second as regression. 00:27:02.430 --> 00:27:04.950 And in the regression framework, one 00:27:04.950 --> 00:27:06.772 has to think about censoring, which is why 00:27:06.772 --> 00:27:07.980 we call it survival modeling. 00:27:11.090 --> 00:27:16.550 Second, in our examples, and again in your homework 00:27:16.550 --> 00:27:20.990 assignment that's coming up next week, 00:27:20.990 --> 00:27:25.010 we'll see that often the variables, 00:27:25.010 --> 00:27:29.850 the features that are most predictive make a lot of sense. 00:27:29.850 --> 00:27:32.480 In the diabetes case, we said-- 00:27:32.480 --> 00:27:36.740 we saw how patients having comorbidities of diabetes, 00:27:36.740 --> 00:27:39.080 like hypertension, or patients being obese 00:27:39.080 --> 00:27:42.370 were very predictive of patients getting diabetes. 00:27:42.370 --> 00:27:46.120 So you might ask yourself, is there something causal there? 00:27:46.120 --> 00:27:49.870 Are those features that are very predictive in fact causing-- 00:27:49.870 --> 00:27:52.180 what's causing the patient to develop type 2 diabetes? 00:27:52.180 --> 00:27:55.580 Like, for example, obesity causing diabetes. 00:27:55.580 --> 00:27:58.290 And this is where I want to caution you. 00:27:58.290 --> 00:28:02.190 You shouldn't interpret these very predictive features 00:28:02.190 --> 00:28:04.950 in a causal fashion, particularly 00:28:04.950 --> 00:28:07.650 not when one starts to work with high dimensional data, 00:28:07.650 --> 00:28:12.680 as we do in this course. 00:28:12.680 --> 00:28:15.290 The reason for that is very subtle, 00:28:15.290 --> 00:28:18.200 and we'll talk about that in the causal inference lectures, 00:28:18.200 --> 00:28:20.180 but I just wanted to give you a pointer 00:28:20.180 --> 00:28:22.500 now that you shouldn't think about it in that way. 00:28:22.500 --> 00:28:26.540 And you'll understand why in just a few weeks. 00:28:26.540 --> 00:28:31.820 And finally we talked about ways of dealing with missing data. 00:28:31.820 --> 00:28:35.620 I gave you one feature representation 00:28:35.620 --> 00:28:39.407 for the diabetes case, which was designed 00:28:39.407 --> 00:28:40.490 to deal with missing data. 00:28:40.490 --> 00:28:46.890 It said, was there any diagnosis code 250.01 00:28:46.890 --> 00:28:49.280 in the last three months? 00:28:49.280 --> 00:28:50.650 And if there was, you have a 1. 00:28:50.650 --> 00:28:51.317 If you don't, 0. 00:28:51.317 --> 00:28:53.178 So it's designed to recognize that you 00:28:53.178 --> 00:28:55.720 don't have information, perhaps, for some large chunk of time 00:28:55.720 --> 00:28:58.100 in that window. 00:28:58.100 --> 00:29:01.520 But that missing data could also be dangerous 00:29:01.520 --> 00:29:05.690 if that missingness itself has caused you to non-stationarity, 00:29:05.690 --> 00:29:09.290 which is then going to result in your test distribution looking 00:29:09.290 --> 00:29:11.490 different from your training distribution. 00:29:11.490 --> 00:29:14.450 And that's where approaches that are based on imputation 00:29:14.450 --> 00:29:17.840 could actually be very valuable, not because they improve 00:29:17.840 --> 00:29:20.240 your predictive accuracy when everything goes right, 00:29:20.240 --> 00:29:22.820 but because they might improve your predictive accuracy when 00:29:22.820 --> 00:29:24.565 things go wrong. 00:29:24.565 --> 00:29:26.690 And so one of your readings for last week's lecture 00:29:26.690 --> 00:29:29.510 was actually an example of that, where they used a Gaussian 00:29:29.510 --> 00:29:34.190 process model to impute much of the missing data in a patient's 00:29:34.190 --> 00:29:36.200 continuous vital signs, and then they 00:29:36.200 --> 00:29:38.210 used a recurrent neural network to predict 00:29:38.210 --> 00:29:41.480 based on that imputed data. 00:29:41.480 --> 00:29:46.050 So in that case, there are really two things going on. 00:29:46.050 --> 00:29:48.345 First is this robustness to data set shift, 00:29:48.345 --> 00:29:49.720 but there's a second thing, which 00:29:49.720 --> 00:29:51.220 is going on as well, which has to do 00:29:51.220 --> 00:29:54.370 with a trade-off between the amount of data you have 00:29:54.370 --> 00:29:58.960 and the complexity of the prediction problem. 00:29:58.960 --> 00:30:00.520 By doing imputations, sometimes you 00:30:00.520 --> 00:30:02.320 make your problem look a bit simpler, 00:30:02.320 --> 00:30:05.170 and simpler algorithms might succeed where otherwise they 00:30:05.170 --> 00:30:07.665 would fail because not having enough data. 00:30:07.665 --> 00:30:09.040 And that's something that you saw 00:30:09.040 --> 00:30:12.150 in that last week's reading. 00:30:12.150 --> 00:30:14.730 So I'm done with risk stratification. 00:30:14.730 --> 00:30:18.030 I'll take a one minute breather for everyone in the room, 00:30:18.030 --> 00:30:20.765 and then we'll start with the main topic 00:30:20.765 --> 00:30:22.890 of this lecture, which is physiological time-series 00:30:22.890 --> 00:30:23.390 modeling. 00:30:27.870 --> 00:30:28.720 Let's say started. 00:30:37.047 --> 00:30:38.880 So here's a baby that's not doing very well. 00:30:42.050 --> 00:30:44.180 This baby is in the intensive care unit. 00:30:48.050 --> 00:30:51.230 Maybe it was a premature infant. 00:30:51.230 --> 00:30:56.360 Maybe it's a baby who has some chronic disease, 00:30:56.360 --> 00:30:59.510 and, of course, parents are very worried. 00:30:59.510 --> 00:31:02.410 This baby is getting very close monitoring. 00:31:02.410 --> 00:31:04.220 It's connected to lots of different probes. 00:31:07.031 --> 00:31:10.160 In number one here, it's illustrating a three probe-- 00:31:10.160 --> 00:31:13.670 three lead ECG, which we'll be talking about much more, which 00:31:13.670 --> 00:31:17.270 is measuring its heart, how the baby's heart is doing. 00:31:17.270 --> 00:31:21.170 Over here, this number three is something 00:31:21.170 --> 00:31:24.206 attached to the baby's foot, which is measuring its-- 00:31:24.206 --> 00:31:27.563 it's a pulse oximeter, which is measuring the baby's oxygen 00:31:27.563 --> 00:31:29.480 saturation, the amount of oxygen in the blood. 00:31:32.780 --> 00:31:35.900 Number four is a probe which is measuring the baby's 00:31:35.900 --> 00:31:37.040 temperature and so on. 00:31:37.040 --> 00:31:40.168 And so we're really taking really close measurements 00:31:40.168 --> 00:31:41.960 of this baby, because we want to understand 00:31:41.960 --> 00:31:43.400 how is this baby doing. 00:31:43.400 --> 00:31:47.120 We recognize that there might be really sudden changes 00:31:47.120 --> 00:31:49.010 in the baby's state of health that we 00:31:49.010 --> 00:31:52.650 want to be able to recognize as early as possible. 00:31:52.650 --> 00:31:56.240 And so behind the scenes, next to this baby, 00:31:56.240 --> 00:31:58.790 you'll, of course, have a huge number of monitors, 00:31:58.790 --> 00:32:00.915 each of the monitors showing the readouts from each 00:32:00.915 --> 00:32:03.200 of these different signals. 00:32:03.200 --> 00:32:07.870 And this type of data is really prevalent in intensive care 00:32:07.870 --> 00:32:10.600 units, but you'll also see in today's lecture 00:32:10.600 --> 00:32:12.760 how some aspects of this data are now 00:32:12.760 --> 00:32:15.040 starting to make its way to the home, as well. 00:32:15.040 --> 00:32:20.590 So for example, EKGs are now available on Apple and Samsung 00:32:20.590 --> 00:32:24.250 watches to help understand-- 00:32:24.250 --> 00:32:27.010 help to help with diagnosis of arrhythmias, 00:32:27.010 --> 00:32:29.290 even for people at home. 00:32:29.290 --> 00:32:30.790 And so from this type of data, there 00:32:30.790 --> 00:32:34.210 are a number of really important use cases to think about. 00:32:34.210 --> 00:32:36.210 The first one is to recognize that often we're 00:32:36.210 --> 00:32:39.030 getting really noisy data, and we want to try 00:32:39.030 --> 00:32:40.710 to infer the true signal. 00:32:40.710 --> 00:32:43.170 So imagine, for example, the temperature probe. 00:32:43.170 --> 00:32:47.100 The baby's true temperature might be 98.5, 00:32:47.100 --> 00:32:50.640 but for whatever reason-- we'll see a few reasons here today-- 00:32:50.640 --> 00:32:53.197 maybe you're getting an observation of 93. 00:32:53.197 --> 00:32:54.030 And you didn't know. 00:32:54.030 --> 00:32:56.190 Is that actually the true baby temperature? 00:32:56.190 --> 00:32:57.300 In which case we-- 00:32:57.300 --> 00:32:59.250 it would be in a lot of trouble. 00:32:59.250 --> 00:33:01.080 Or is that an anomalous reading? 00:33:01.080 --> 00:33:03.288 So we like t be able to distinguish between those two 00:33:03.288 --> 00:33:04.440 things. 00:33:04.440 --> 00:33:09.090 And in other cases, we are interested in not necessarily 00:33:09.090 --> 00:33:12.030 fully understanding what's going on with the baby along each 00:33:12.030 --> 00:33:15.660 of those axes, but we just want to use that data 00:33:15.660 --> 00:33:17.880 for predictive purposes, for risk stratification, 00:33:17.880 --> 00:33:19.367 for example. 00:33:19.367 --> 00:33:21.200 And so the type of machine learning approach 00:33:21.200 --> 00:33:25.520 that we'll take here will depend on the following three factors. 00:33:25.520 --> 00:33:28.350 First, do we have label data available? 00:33:28.350 --> 00:33:30.470 For example, do we know the ground truth 00:33:30.470 --> 00:33:34.130 of what the baby's true temperature was, 00:33:34.130 --> 00:33:38.630 at least for a few of the babies in the training set? 00:33:38.630 --> 00:33:39.680 Second. 00:33:39.680 --> 00:33:43.310 Do we have a good mechanistic or statistical model 00:33:43.310 --> 00:33:46.113 of how this data might evolve across time? 00:33:46.113 --> 00:33:47.780 We know a lot about hearts, for example. 00:33:47.780 --> 00:33:49.655 Cardiology is one of those fields of medicine 00:33:49.655 --> 00:33:51.500 where it's really well studied. 00:33:51.500 --> 00:33:53.360 There are good simulators of hearts, 00:33:53.360 --> 00:33:54.950 and how they beat across time, and how 00:33:54.950 --> 00:34:01.150 that affects the electrical stimulation across the body. 00:34:01.150 --> 00:34:03.970 And if we have these good mechanistic 00:34:03.970 --> 00:34:05.770 or statistical models, that can often 00:34:05.770 --> 00:34:08.800 allow one to trade off not having much label data, 00:34:08.800 --> 00:34:11.540 or just not having much data period. 00:34:11.540 --> 00:34:13.850 And it's really these three points 00:34:13.850 --> 00:34:16.429 which I want to illustrate the extremes of in today's 00:34:16.429 --> 00:34:16.955 lecture-- 00:34:16.955 --> 00:34:18.830 what do you do when you don't have much data, 00:34:18.830 --> 00:34:20.000 and what you do when-- what you can 00:34:20.000 --> 00:34:21.395 do when you have a ton of data. 00:34:21.395 --> 00:34:24.054 And I think it's going to be really informative for us 00:34:24.054 --> 00:34:26.179 as we go out into the world and will have to tackle 00:34:26.179 --> 00:34:27.304 each of those two settings. 00:34:30.159 --> 00:34:33.500 So here's an example of two different babies with very 00:34:33.500 --> 00:34:35.150 different trajectories. 00:34:35.150 --> 00:34:38.449 One in the x-axis here is time in seconds. 00:34:38.449 --> 00:34:41.688 The y-axis here-- 00:34:41.688 --> 00:34:42.980 I think seconds, maybe minutes. 00:34:42.980 --> 00:34:46.130 The y-axis here is beats per minute of the baby's heart 00:34:46.130 --> 00:34:50.630 rate, and you see in some cases it's really 00:34:50.630 --> 00:34:51.938 fluctuating a lot up and down. 00:34:51.938 --> 00:34:54.230 In some cases, it's sort of going in a similar-- in one 00:34:54.230 --> 00:34:58.700 direction, and in all cases, the short term observations 00:34:58.700 --> 00:35:03.562 are very different from the long range trajectories. 00:35:03.562 --> 00:35:05.020 So the first problem that I want us 00:35:05.020 --> 00:35:10.450 to think about is one of trying to understand, 00:35:10.450 --> 00:35:13.972 how do we deconvolve between the truth of what's going on with, 00:35:13.972 --> 00:35:15.680 for example, the patient's blood pressure 00:35:15.680 --> 00:35:20.163 or oxygen versus interventions that are happening to them? 00:35:20.163 --> 00:35:21.580 So on the bottom here, I'm showing 00:35:21.580 --> 00:35:24.750 examples of interventions. 00:35:24.750 --> 00:35:27.810 Here in this oxygen uptake, we notice 00:35:27.810 --> 00:35:31.047 how between roughly 1,000 and 2,000 seconds suddenly 00:35:31.047 --> 00:35:32.255 there's no signal whatsoever. 00:35:32.255 --> 00:35:34.213 And that's an example of what's called dropout. 00:35:36.520 --> 00:35:39.650 Over here, we see a different type of-- 00:35:39.650 --> 00:35:42.430 the effect of a different intervention, 00:35:42.430 --> 00:35:44.770 which is due to a probe recalibration. 00:35:44.770 --> 00:35:46.870 Now, at that time, there was a drop out 00:35:46.870 --> 00:35:50.170 followed by a sudden change in the values, 00:35:50.170 --> 00:35:52.720 and that's really happening due to a recalibration step. 00:35:52.720 --> 00:35:55.710 And in both of these cases, what's 00:35:55.710 --> 00:35:58.132 going on with the individual might be relatively 00:35:58.132 --> 00:36:00.090 constant across time, but what's being observed 00:36:00.090 --> 00:36:04.240 is dramatically affected by those interventions. 00:36:04.240 --> 00:36:06.070 So we want to ask the question, can we 00:36:06.070 --> 00:36:08.788 identify those artifactual processes? 00:36:08.788 --> 00:36:11.080 Can we identify that these interventions were happening 00:36:11.080 --> 00:36:12.080 at those points in time? 00:36:15.680 --> 00:36:18.000 And then, if we could identify them, 00:36:18.000 --> 00:36:21.120 then we could potentially subtract their effect out. 00:36:21.120 --> 00:36:27.210 So we could impute the data, which we know-- now 00:36:27.210 --> 00:36:30.390 know to be missing, and then have this much higher quality 00:36:30.390 --> 00:36:33.130 signal used for some downstream predictive purpose, 00:36:33.130 --> 00:36:34.910 for example. 00:36:34.910 --> 00:36:37.510 And the second reason why this can be really important 00:36:37.510 --> 00:36:40.660 is to tackle this problem called alarm fatigue. 00:36:43.370 --> 00:36:47.030 Alarm fatigue is one of the most important challenges facing 00:36:47.030 --> 00:36:48.500 medicine today. 00:36:48.500 --> 00:36:52.370 As we get better and better in doing risk stratification, 00:36:52.370 --> 00:36:58.700 as we come up with more and more diagnostic tools and tests, 00:36:58.700 --> 00:37:02.090 that means these red flags are being raised more and more 00:37:02.090 --> 00:37:03.690 often. 00:37:03.690 --> 00:37:08.170 And each one of these has some associated false positive rate 00:37:08.170 --> 00:37:09.800 for it. 00:37:09.800 --> 00:37:13.510 And so the more tests you have-- 00:37:13.510 --> 00:37:15.250 suppose the false positive rate is 00:37:15.250 --> 00:37:18.160 kept constant-- the more tests you have, the more likely 00:37:18.160 --> 00:37:20.140 it is that the union of all of those 00:37:20.140 --> 00:37:24.568 is going to be some error. 00:37:24.568 --> 00:37:27.540 And so when you're in an intensive care unit, 00:37:27.540 --> 00:37:29.500 there are alarms going off all the time. 00:37:29.500 --> 00:37:31.630 And something that happens is that nurses end up 00:37:31.630 --> 00:37:35.110 starting to ignore those alarms, because so often 00:37:35.110 --> 00:37:37.480 those alarms are false positives, 00:37:37.480 --> 00:37:39.700 are due to, for example, artifacts 00:37:39.700 --> 00:37:41.835 like what I'm showing you here. 00:37:41.835 --> 00:37:43.960 And so if we had techniques, such as the ones we'll 00:37:43.960 --> 00:37:47.680 talk about right now, which could recognize when, 00:37:47.680 --> 00:37:50.470 for example, the sudden drop in a patient's heart rate 00:37:50.470 --> 00:37:54.940 is due to an artifact and not due to the patient's true heart 00:37:54.940 --> 00:37:56.958 rate dropping-- 00:37:56.958 --> 00:37:58.500 if we had enough confidence in that-- 00:37:58.500 --> 00:37:59.958 in distinguishing those two things, 00:37:59.958 --> 00:38:03.150 then we might not decide to raise that red flag. 00:38:03.150 --> 00:38:06.430 And that might reduce the amount of false alarms, 00:38:06.430 --> 00:38:09.150 and that then might reduce the amount of alarm fatigue. 00:38:09.150 --> 00:38:11.850 And that could have a very big impact on health care. 00:38:15.980 --> 00:38:19.150 So the technique which we'll talk about today 00:38:19.150 --> 00:38:24.170 goes by the name of switching linear dynamical systems. 00:38:24.170 --> 00:38:25.820 Who here has seen a picture like this 00:38:25.820 --> 00:38:29.630 on-- this picture on the bottom before. 00:38:29.630 --> 00:38:32.173 About half of the room. 00:38:32.173 --> 00:38:33.590 So for the other half of the room, 00:38:33.590 --> 00:38:36.620 I'm going to give a bit of a recap 00:38:36.620 --> 00:38:38.960 into probabilistic modeling. 00:38:38.960 --> 00:38:43.830 All of you are now familiar with general probabilities. 00:38:43.830 --> 00:38:48.230 So you're used to thinking about, for example, 00:38:48.230 --> 00:38:51.230 univariate Gaussian distributions. 00:38:51.230 --> 00:38:54.050 We talked about how one could model survival, which 00:38:54.050 --> 00:38:57.440 was an example of such a distribution, 00:38:57.440 --> 00:38:59.088 but for today's lecture, we're going 00:38:59.088 --> 00:39:01.130 to be thinking now about multivariate probability 00:39:01.130 --> 00:39:01.820 distributions. 00:39:01.820 --> 00:39:05.870 In particular, we'll be thinking about how a patient's state-- 00:39:05.870 --> 00:39:08.120 let's say their true blood pressure-- 00:39:08.120 --> 00:39:09.990 evolves across time. 00:39:09.990 --> 00:39:14.570 And so now we're interested in not just the random variable 00:39:14.570 --> 00:39:16.740 at one point in time, but that same random variable 00:39:16.740 --> 00:39:18.782 at the second point in time, third point in time, 00:39:18.782 --> 00:39:21.488 fourth point in time, fifth point in time, and so on. 00:39:21.488 --> 00:39:23.030 So what I'm showing you here is known 00:39:23.030 --> 00:39:26.270 as a graphical model, also known as a Bayesian network. 00:39:26.270 --> 00:39:29.050 And it's one way of illustrating a multivariate probability 00:39:29.050 --> 00:39:31.460 distribution that has particular conditional independence 00:39:31.460 --> 00:39:33.490 properties. 00:39:33.490 --> 00:39:40.690 Specifically, in this model, one node 00:39:40.690 --> 00:39:42.260 corresponds to one random variable. 00:39:42.260 --> 00:39:46.840 So this is describing a joint distribution on x1 00:39:46.840 --> 00:39:55.117 through x6, y1 through y6. 00:39:55.117 --> 00:39:56.700 So it's this multivariate distribution 00:39:56.700 --> 00:40:00.570 on 12 random variables. 00:40:00.570 --> 00:40:03.600 The fact that this is shaded in simply 00:40:03.600 --> 00:40:07.110 denotes that, at test time, when we use these models, typically 00:40:07.110 --> 00:40:09.780 these y variables are observed. 00:40:09.780 --> 00:40:13.410 Whereas our goal is usually to infer the x variables. 00:40:13.410 --> 00:40:16.950 Those are typically unobserved, meaning that our typical task 00:40:16.950 --> 00:40:20.340 is one of doing posterior inference to infer 00:40:20.340 --> 00:40:22.725 the x's given the y's. 00:40:25.470 --> 00:40:28.860 Now, associated with this graph, I already 00:40:28.860 --> 00:40:31.740 told you the nodes correspond to random variables. 00:40:31.740 --> 00:40:36.330 The graph tells us how is this joint distribution factorized. 00:40:36.330 --> 00:40:41.130 In particular, it's going to be factorized 00:40:41.130 --> 00:40:42.240 in the following way-- 00:40:42.240 --> 00:40:45.210 as the product over random variables 00:40:45.210 --> 00:40:49.000 of the probability of the i-th random variable. 00:40:49.000 --> 00:40:51.840 I'm going to use z to just denote a random variable. 00:40:51.840 --> 00:40:55.680 Think of z as the union of x and y. 00:40:55.680 --> 00:40:59.610 zi conditioned on the parents-- 00:40:59.610 --> 00:41:01.800 the values of the parents of zi. 00:41:05.820 --> 00:41:10.080 So I'm going to assume this factorization, 00:41:10.080 --> 00:41:13.800 and in particular for this graphical model, which 00:41:13.800 --> 00:41:15.870 goes by the name of a Markov model, 00:41:15.870 --> 00:41:18.810 it has a very specific factorization. 00:41:18.810 --> 00:41:22.180 And we're just going to read it off from this definition. 00:41:22.180 --> 00:41:26.340 So we're going to go in order-- first x1, then y1, 00:41:26.340 --> 00:41:28.410 then x2, then y2, and so on, which 00:41:28.410 --> 00:41:36.630 is going based on a root to children 00:41:36.630 --> 00:41:39.340 transversal of this graph. 00:41:39.340 --> 00:41:44.410 So the first random variable is x1. 00:41:44.410 --> 00:41:50.230 Second variable is y2, and what are the parents of y-- 00:41:50.230 --> 00:41:51.757 sorry, what are the parents of y1. 00:41:51.757 --> 00:41:52.840 Everyone can say out loud. 00:41:52.840 --> 00:41:54.070 AUDIENCE: x1. 00:41:54.070 --> 00:41:55.090 DAVID SONTAG: x1. 00:41:55.090 --> 00:42:01.450 So y1 in this factorization is only going to depend on x1. 00:42:01.450 --> 00:42:02.740 Next we have x2. 00:42:02.740 --> 00:42:03.940 What are the parents of x2? 00:42:03.940 --> 00:42:05.390 Everyone say out loud? 00:42:05.390 --> 00:42:06.370 AUDIENCE: x1. 00:42:06.370 --> 00:42:07.840 DAVID SONTAG: x1. 00:42:07.840 --> 00:42:09.790 Then we have y2. 00:42:09.790 --> 00:42:11.633 What are the parents of y2. 00:42:11.633 --> 00:42:12.550 Everyone say out loud. 00:42:12.550 --> 00:42:14.080 AUDIENCE: x2. 00:42:14.080 --> 00:42:16.960 DAVID SONTAG: x2 and so on. 00:42:16.960 --> 00:42:20.920 So this joint distribution is going 00:42:20.920 --> 00:42:23.560 to have a particularly simple form, which 00:42:23.560 --> 00:42:26.280 is given to by this factorization shown here. 00:42:26.280 --> 00:42:28.420 And this factorization corresponds one to one 00:42:28.420 --> 00:42:32.400 with the particular graph in the way that I just told you. 00:42:32.400 --> 00:42:35.760 And in this way, we can define a very complex probability 00:42:35.760 --> 00:42:39.900 distribution by a number of much simpler conditional probability 00:42:39.900 --> 00:42:41.220 distributions. 00:42:41.220 --> 00:42:44.740 For example, if each of the random variables were binary, 00:42:44.740 --> 00:42:48.840 then to describe probability of y1 given x1, 00:42:48.840 --> 00:42:50.250 we only need two numbers. 00:42:50.250 --> 00:42:52.840 For each value of x1, either 0 or 1, 00:42:52.840 --> 00:42:55.290 we give the probability of y1 equals 1. 00:42:55.290 --> 00:42:59.530 And then, of course, probably y1 equals 0 is just 1 minus that. 00:42:59.530 --> 00:43:02.290 So we can describe that very complicated joint distribution 00:43:02.290 --> 00:43:07.200 by a number of much smaller distributions. 00:43:07.200 --> 00:43:10.700 Now, the reason why I'm drawing it in this way 00:43:10.700 --> 00:43:13.940 is because we're making some really strong assumptions 00:43:13.940 --> 00:43:18.020 about the temporal dynamics in this problem. 00:43:18.020 --> 00:43:23.360 In particular, the fact that x3 only 00:43:23.360 --> 00:43:27.720 has an arrow from x2 and not from x1 00:43:27.720 --> 00:43:32.540 implies that x3 is conditionally independent of x1. 00:43:32.540 --> 00:43:34.400 If you knew x2's value. 00:43:34.400 --> 00:43:37.970 So in some sense, think about this as cutting. 00:43:37.970 --> 00:43:40.700 If you're to take x2 out of the model 00:43:40.700 --> 00:43:43.040 and remove all edges incident on it, 00:43:43.040 --> 00:43:46.490 then x1 and x3 are now separated from one another. 00:43:46.490 --> 00:43:48.110 They're independent. 00:43:48.110 --> 00:43:51.740 Now, for those of you who do know graphical models, 00:43:51.740 --> 00:43:54.770 you'll recognize that that type of independent statement that I 00:43:54.770 --> 00:43:56.480 made is only true for Markov models, 00:43:56.480 --> 00:43:58.605 and the semantics for Bayesian networks 00:43:58.605 --> 00:43:59.730 are a little bit different. 00:43:59.730 --> 00:44:02.058 But actually for this model, it's-- they're one 00:44:02.058 --> 00:44:02.600 and the same. 00:44:05.910 --> 00:44:08.990 So we're going to make the following assumptions 00:44:08.990 --> 00:44:12.890 for the conditional distributions shown here. 00:44:12.890 --> 00:44:16.850 First, we're going to suppose that xt is given to you 00:44:16.850 --> 00:44:19.490 by a Gaussian distribution. 00:44:19.490 --> 00:44:23.570 Remember xt-- t is denoting a time step. 00:44:23.570 --> 00:44:26.815 Let's say 3-- it only depends in this picture-- 00:44:26.815 --> 00:44:28.190 the conditional distribution only 00:44:28.190 --> 00:44:30.650 depends on the previous time step's value, x2, 00:44:30.650 --> 00:44:32.310 or xt minus 1. 00:44:32.310 --> 00:44:34.850 So you'll notice how I'm going to say here 00:44:34.850 --> 00:44:36.620 xt is going to distribute as something, 00:44:36.620 --> 00:44:38.690 but the only random variables in this something 00:44:38.690 --> 00:44:42.680 can be xt minus 1, according to these assumptions. 00:44:42.680 --> 00:44:44.180 In particular, we're going to assume 00:44:44.180 --> 00:44:47.930 that it's some Gaussian distribution, whose mean is 00:44:47.930 --> 00:44:51.020 some linear transformation of xt minus 1, 00:44:51.020 --> 00:44:55.240 and which has a fixed covariance matrix q. 00:44:55.240 --> 00:45:00.310 So at each step of this process, the next random variable 00:45:00.310 --> 00:45:03.700 is some random walk from the previous random variable 00:45:03.700 --> 00:45:07.833 where you're moving according to some Gaussian distribution. 00:45:07.833 --> 00:45:09.250 In a very similar way, we're going 00:45:09.250 --> 00:45:17.410 to assume that yt is drawn also as a Gaussian distribution, 00:45:17.410 --> 00:45:20.550 but now depending on xt. 00:45:20.550 --> 00:45:24.120 So I want you to think about xt as the true state 00:45:24.120 --> 00:45:25.410 of the patient. 00:45:25.410 --> 00:45:28.590 It's a vector that's summarizing their blood 00:45:28.590 --> 00:45:31.200 pressure, their oxygen saturation, 00:45:31.200 --> 00:45:33.150 a whole bunch of other parameters, 00:45:33.150 --> 00:45:35.460 or maybe even just one of those. 00:45:35.460 --> 00:45:39.300 And y1 are the observations that you do observe. 00:45:39.300 --> 00:45:41.890 So let's say x1 is the patient's true blood pressure. 00:45:41.890 --> 00:45:43.980 y1 is the observed blood pressure, 00:45:43.980 --> 00:45:47.010 what comes from your monitor. 00:45:47.010 --> 00:45:48.660 So then a reasonable assumption would 00:45:48.660 --> 00:45:52.350 be that, well, if all this were equal, 00:45:52.350 --> 00:45:53.910 if it was a true observation, then 00:45:53.910 --> 00:45:55.750 y1 should be very close to x1. 00:45:55.750 --> 00:45:58.680 So you might assume that this covariance matrix is-- 00:45:58.680 --> 00:46:01.460 the covariance is-- the variance is very, very small. 00:46:01.460 --> 00:46:07.280 y1 should be very close to x1 if it's a good observation. 00:46:07.280 --> 00:46:10.100 And of course, if it's a noisy observation-- 00:46:10.100 --> 00:46:15.680 like, for example, if the probe was disconnected from the baby, 00:46:15.680 --> 00:46:19.790 then y1 should have no relationship to x1. 00:46:19.790 --> 00:46:23.460 And that dependence on the actual state of the world 00:46:23.460 --> 00:46:26.730 I'm denoting here by these superscripts, s of t. 00:46:26.730 --> 00:46:28.730 I'm ignoring that right now, and I'll bring that 00:46:28.730 --> 00:46:31.910 in in the next slide. 00:46:31.910 --> 00:46:36.230 Similarly, the relationship between x2 and x1 00:46:36.230 --> 00:46:38.510 should be one which captures some of the dynamics 00:46:38.510 --> 00:46:42.140 that I showed in the previous slides, where I showed over 00:46:42.140 --> 00:46:46.040 here now this is the patient's true heart rate evolving 00:46:46.040 --> 00:46:48.080 across time, let's say. 00:46:48.080 --> 00:46:51.800 Notice how, if you look very locally, 00:46:51.800 --> 00:46:56.720 it looks like there are some very, very big local dynamics. 00:46:56.720 --> 00:46:58.790 Whereas if you look more globally, 00:46:58.790 --> 00:47:01.340 again, there's some smoothness, but there are some-- again, 00:47:01.340 --> 00:47:03.590 it looks like some random changes across time. 00:47:03.590 --> 00:47:10.070 And so those-- that drift has to somehow 00:47:10.070 --> 00:47:13.550 be summarized in this model by that A random variable. 00:47:13.550 --> 00:47:16.130 And I'll get into more detail about that in just a moment. 00:47:18.750 --> 00:47:20.990 So what I just showed you was an example 00:47:20.990 --> 00:47:23.360 of a linear dynamical system, but it 00:47:23.360 --> 00:47:27.170 was assuming that there were none of these events happening, 00:47:27.170 --> 00:47:30.082 none of these artifacts happening. 00:47:30.082 --> 00:47:31.540 The actual model that we were going 00:47:31.540 --> 00:47:33.040 to want to be able to use then is 00:47:33.040 --> 00:47:34.330 going to also incorporate the fact 00:47:34.330 --> 00:47:35.320 that there might be artifacts. 00:47:35.320 --> 00:47:36.640 And to model that, we need to introduce 00:47:36.640 --> 00:47:38.473 additional random variables corresponding to 00:47:38.473 --> 00:47:40.250 whether those artifacts occurred or not. 00:47:40.250 --> 00:47:42.290 And so that's now this model. 00:47:42.290 --> 00:47:45.370 So I'm going to let these S's-- 00:47:45.370 --> 00:47:47.850 these are other random variables, 00:47:47.850 --> 00:47:51.310 which are denoting artifactual events. 00:47:51.310 --> 00:47:52.970 They are also evolving with time. 00:47:52.970 --> 00:47:55.420 For example, if there's artifactual factual event 00:47:55.420 --> 00:47:57.875 at three seconds, maybe there's also an artifactual event 00:47:57.875 --> 00:47:58.720 at four seconds. 00:47:58.720 --> 00:48:00.887 And we like to model the relationship between those. 00:48:00.887 --> 00:48:02.600 That's why you have these arrows. 00:48:02.600 --> 00:48:08.180 And then the way that we interpret the observations 00:48:08.180 --> 00:48:12.620 that we do get depends on both the true value 00:48:12.620 --> 00:48:14.340 of what's going on with the patient 00:48:14.340 --> 00:48:17.612 and whether there was an artifactual event or not. 00:48:17.612 --> 00:48:19.070 And you'll notice that there's also 00:48:19.070 --> 00:48:20.780 an edge going from the artifactual events 00:48:20.780 --> 00:48:23.270 to the true values to note the fact 00:48:23.270 --> 00:48:27.680 that those interventions might actually 00:48:27.680 --> 00:48:29.030 be affecting the patient. 00:48:29.030 --> 00:48:31.040 For example, if you give them a medication 00:48:31.040 --> 00:48:36.800 to change their blood pressure, then that procedure 00:48:36.800 --> 00:48:39.895 is going to affect the next time step's value of the patient's 00:48:39.895 --> 00:48:40.520 blood pressure. 00:48:44.360 --> 00:48:47.917 So when one wants to learn this model, 00:48:47.917 --> 00:48:49.750 you have to ask yourself, what types of data 00:48:49.750 --> 00:48:51.167 do you have available? 00:48:54.370 --> 00:48:59.680 Unfortunately, it's very hard to get data on both the ground 00:48:59.680 --> 00:49:02.210 truth, what's going on with the patient, 00:49:02.210 --> 00:49:06.530 and whether these artifacts truly occurred or not. 00:49:06.530 --> 00:49:09.530 Instead, what we actually have are just these observations. 00:49:09.530 --> 00:49:13.450 We get these very noisy blood pressure draws across time. 00:49:13.450 --> 00:49:16.500 So what this paper does is it uses a maximum likelihood 00:49:16.500 --> 00:49:18.797 estimation approach, where it recognizes 00:49:18.797 --> 00:49:20.880 that we're going to be learning from missing data. 00:49:20.880 --> 00:49:23.940 We're going to explicitly think of these x's and the s's 00:49:23.940 --> 00:49:25.875 as latent variables. 00:49:25.875 --> 00:49:27.990 And we're going to maximize the likelihood 00:49:27.990 --> 00:49:31.820 of the whole entire model, marginalizing over x and s. 00:49:31.820 --> 00:49:34.485 So just maximizing the marginal likelihood over the y's. 00:49:37.240 --> 00:49:39.740 Now, for those of you who have studied unsupervised learning 00:49:39.740 --> 00:49:43.570 before, you might recognize that as a very hard learning 00:49:43.570 --> 00:49:44.070 problem. 00:49:44.070 --> 00:49:47.780 In fact, it's-- that likelihood is non-convex. 00:49:47.780 --> 00:49:51.990 And one could imagine all sorts of a heuristics for learning, 00:49:51.990 --> 00:49:55.460 such as gradient descent, or, as this paper uses, 00:49:55.460 --> 00:49:59.180 expectation maximization, and because of that non-convexity, 00:49:59.180 --> 00:50:00.750 each of these algorithms typically 00:50:00.750 --> 00:50:04.040 will only reach a local maxima of the likelihood. 00:50:04.040 --> 00:50:08.420 So this paper uses EM, which intuitively iterates 00:50:08.420 --> 00:50:14.420 between inferring those missing variables-- so imputing the x's 00:50:14.420 --> 00:50:17.210 and the s's given the current model, 00:50:17.210 --> 00:50:20.300 and doing posterior inference to infer the missing 00:50:20.300 --> 00:50:22.760 variables given the observed variables, using 00:50:22.760 --> 00:50:24.140 the current model. 00:50:24.140 --> 00:50:27.020 And then, once you've imputed those variables, 00:50:27.020 --> 00:50:28.910 attempting to refit the model. 00:50:28.910 --> 00:50:30.920 So that's called the m-step for maximization, 00:50:30.920 --> 00:50:32.900 which updates the model and just iterates between those two 00:50:32.900 --> 00:50:33.400 things. 00:50:33.400 --> 00:50:36.590 That's one learning algorithm which 00:50:36.590 --> 00:50:39.650 is guaranteed to reach a local maxima of the likelihood 00:50:39.650 --> 00:50:42.830 under some regularity assumptions. 00:50:42.830 --> 00:50:44.690 And so this paper uses that algorithm, 00:50:44.690 --> 00:50:46.520 but you need to be asking yourself, 00:50:46.520 --> 00:50:50.270 if all you ever observe are the y's, 00:50:50.270 --> 00:50:54.830 then will this algorithm ever recover anything 00:50:54.830 --> 00:50:56.600 close to the true model? 00:50:56.600 --> 00:50:58.310 For example, there might be large amounts 00:50:58.310 --> 00:51:00.080 of non-identifiability here. 00:51:00.080 --> 00:51:04.490 It could be that you could swap the meaning 00:51:04.490 --> 00:51:10.170 of the s's, and you'd get a similar likelihood on the y's. 00:51:10.170 --> 00:51:14.010 That's where bringing in domain knowledge becomes critical. 00:51:14.010 --> 00:51:17.670 So this is going to be an example where we have no label 00:51:17.670 --> 00:51:22.948 data or very little label data. 00:51:22.948 --> 00:51:24.740 And we're going to do unsupervised learning 00:51:24.740 --> 00:51:26.282 of this model, but we're going to use 00:51:26.282 --> 00:51:28.790 a ton of domain knowledge in order to constrain 00:51:28.790 --> 00:51:31.050 the model as much as possible. 00:51:31.050 --> 00:51:33.490 So what is that domain knowledge? 00:51:33.490 --> 00:51:37.730 Well, first we're going to use the fact 00:51:37.730 --> 00:51:47.200 that we know that a true heart rate evolves in a fashion that 00:51:47.200 --> 00:51:53.530 can be very well modeled by an autoregressive process. 00:51:53.530 --> 00:51:56.260 So the autoregressive process that's used in this paper 00:51:56.260 --> 00:51:58.630 is used to model the normal heart rate dynamics. 00:51:58.630 --> 00:52:01.060 In a moment, I'll tell you how to model the abnormal heart 00:52:01.060 --> 00:52:03.370 rate observations. 00:52:03.370 --> 00:52:05.530 And intuitively-- I'll first go over the intuition, 00:52:05.530 --> 00:52:06.850 then I'll give you the math. 00:52:06.850 --> 00:52:08.650 Intuitively what it does is it recognizes 00:52:08.650 --> 00:52:14.060 that this complicated signal can be decomposed into two pieces. 00:52:14.060 --> 00:52:18.020 The first piece shown here is called a baseline signal, 00:52:18.020 --> 00:52:20.315 and that, if you squint your eyes 00:52:20.315 --> 00:52:22.700 and you sort or ignore the very local fluctuations, 00:52:22.700 --> 00:52:24.860 this is what you get out. 00:52:24.860 --> 00:52:27.230 And then you can look at the residual 00:52:27.230 --> 00:52:32.330 of subtracting this signal, subtracting this baseline 00:52:32.330 --> 00:52:33.710 from the signal. 00:52:33.710 --> 00:52:36.250 And what you get out looks like this. 00:52:36.250 --> 00:52:39.770 Notice here it's around 0 mean. 00:52:39.770 --> 00:52:42.585 So it's a 0 mean signal with some random fluctuations, 00:52:42.585 --> 00:52:44.210 and the fluctuations are happening here 00:52:44.210 --> 00:52:47.210 at a much faster rate than-- 00:52:47.210 --> 00:52:49.830 and for the original baseline. 00:52:49.830 --> 00:52:56.910 And so the sum of bt and this residual is a very-- 00:52:56.910 --> 00:53:00.200 it looks-- is exactly equal to the true heart rate. 00:53:00.200 --> 00:53:03.290 And each of these two things we can model very well. 00:53:03.290 --> 00:53:08.210 This we can model by a random walk with-- 00:53:08.210 --> 00:53:10.970 which goes very slowly, and this we 00:53:10.970 --> 00:53:15.297 can model by a random walk which goes very quickly. 00:53:15.297 --> 00:53:17.630 And that is exactly what I'm now going to show over here 00:53:17.630 --> 00:53:19.180 on the left hand side. 00:53:19.180 --> 00:53:22.880 bt, this baseline signal, we're going 00:53:22.880 --> 00:53:26.540 to model as a Gaussian distribution, which 00:53:26.540 --> 00:53:29.600 is parameterized as a function of not just bt minus 1, 00:53:29.600 --> 00:53:32.480 but also bt minus 2, and bt minus 3. 00:53:32.480 --> 00:53:34.940 And so we're going to be taking a weighted average 00:53:34.940 --> 00:53:39.560 of the previous few time steps, where we're smoothing out, 00:53:39.560 --> 00:53:45.220 in essence, the observation-- the previous few observations. 00:53:45.220 --> 00:53:47.970 If you were to-- 00:53:47.970 --> 00:53:50.310 if you're being a keen observer, you'll 00:53:50.310 --> 00:53:53.790 notice that this is no longer a Markov model. 00:54:04.870 --> 00:54:11.460 For example, if this p1 and p2 are equal to 2, 00:54:11.460 --> 00:54:14.790 this then corresponds to a second order Markov model, 00:54:14.790 --> 00:54:18.600 because each random variable depends on the previous two 00:54:18.600 --> 00:54:24.530 time steps of the Markov chain. 00:54:24.530 --> 00:54:31.790 And so after-- so you would model now bt by this process, 00:54:31.790 --> 00:54:34.880 and you would probably be averaging 00:54:34.880 --> 00:54:36.920 over a large number of previous time steps 00:54:36.920 --> 00:54:39.020 to get this smooth property. 00:54:39.020 --> 00:54:45.620 And then you'd model xt minus bt by this autoregressive process, 00:54:45.620 --> 00:54:47.780 where you might, for example, just 00:54:47.780 --> 00:54:50.313 be looking at just the previous couple of time steps. 00:54:50.313 --> 00:54:51.980 And you recognize that you're just doing 00:54:51.980 --> 00:54:55.600 much more random fluctuations. 00:54:55.600 --> 00:54:59.480 And then-- so that's how one would now model normal heart 00:54:59.480 --> 00:55:00.650 rate dynamics. 00:55:00.650 --> 00:55:02.900 And again, it's just-- 00:55:02.900 --> 00:55:04.730 this is an example of a statistical model. 00:55:04.730 --> 00:55:06.110 There is no mechanistic knowledge 00:55:06.110 --> 00:55:08.540 of hearts being used here, but we 00:55:08.540 --> 00:55:13.710 can fit the data of normal hearts pretty well using this. 00:55:13.710 --> 00:55:15.960 But the next question and the most interesting one 00:55:15.960 --> 00:55:20.510 is, how does one now model artifactual events? 00:55:20.510 --> 00:55:26.120 So for that, that's where some mechanistic knowledge comes in. 00:55:26.120 --> 00:55:30.180 So one models that the probe dropouts 00:55:30.180 --> 00:55:35.120 are given by recognizing that, if a probe 00:55:35.120 --> 00:55:39.020 is removed from the baby, then there should no longer be-- 00:55:39.020 --> 00:55:41.253 or at least if you-- after a small amount of time, 00:55:41.253 --> 00:55:42.920 there should no longer be any dependence 00:55:42.920 --> 00:55:44.450 on the true value of the baby. 00:55:44.450 --> 00:55:48.080 For example, the blood pressure, once the blood pressure probe 00:55:48.080 --> 00:55:50.870 is removed, is no longer related to the baby's true blood 00:55:50.870 --> 00:55:52.910 pressure. 00:55:52.910 --> 00:55:57.130 But there might be some delay to that lack of dependence. 00:55:57.130 --> 00:55:59.450 And so-- and that is going to be encoded in some domain 00:55:59.450 --> 00:55:59.950 knowledge. 00:55:59.950 --> 00:56:01.840 So for example, in the temperature probe, 00:56:01.840 --> 00:56:04.480 when you remove the temperature probe from the baby, 00:56:04.480 --> 00:56:07.682 it starts heating up again-- or it starts cooling, so 00:56:07.682 --> 00:56:09.640 assuming that the ambient temperature is cooler 00:56:09.640 --> 00:56:11.280 than the baby's temperature. 00:56:11.280 --> 00:56:12.790 So you take it off the baby. 00:56:12.790 --> 00:56:14.170 It starts cooling down. 00:56:14.170 --> 00:56:15.692 How fast does it cool down? 00:56:15.692 --> 00:56:17.400 Well, you could assume that it cools down 00:56:17.400 --> 00:56:20.320 with some exponential decay from the baby's temperature. 00:56:20.320 --> 00:56:22.750 And this is something that is very reasonable, 00:56:22.750 --> 00:56:24.490 and you could imagine, maybe if you 00:56:24.490 --> 00:56:26.530 had label data for just a few of the babies, 00:56:26.530 --> 00:56:28.780 you could try to fit the parameters of the exponential 00:56:28.780 --> 00:56:30.840 very quickly. 00:56:30.840 --> 00:56:33.160 And in this way, now, we parameterize the conditional 00:56:33.160 --> 00:56:39.040 distribution of the temperature probe, given both the state 00:56:39.040 --> 00:56:42.220 and whether the artifact occurred or not, 00:56:42.220 --> 00:56:45.710 using this very simple exponential decay. 00:56:45.710 --> 00:56:49.957 And in this paper, they give a very similar type of-- 00:56:49.957 --> 00:56:51.790 they make similar types of-- analogous types 00:56:51.790 --> 00:56:54.588 of assumptions for all of the other artifactual probes. 00:56:54.588 --> 00:56:56.380 You should think about this as constraining 00:56:56.380 --> 00:56:58.757 these conditional distributions I showed you here. 00:56:58.757 --> 00:57:01.090 They're no longer allowed to be arbitrary distributions, 00:57:01.090 --> 00:57:03.910 and so that, when one does now expectation maximization 00:57:03.910 --> 00:57:06.573 to try to maximize the marginal likelihood of the data, 00:57:06.573 --> 00:57:07.990 you've now constrained it in a way 00:57:07.990 --> 00:57:10.073 that you hopefully are moved on to identifyability 00:57:10.073 --> 00:57:11.310 of the learning problem. 00:57:11.310 --> 00:57:13.330 It makes all of the difference in learning here. 00:57:18.130 --> 00:57:21.730 So in this paper, their evaluation 00:57:21.730 --> 00:57:23.830 did a little bit of fine tuning for each baby. 00:57:23.830 --> 00:57:26.650 In particular, they assumed that the first 30 minutes 00:57:26.650 --> 00:57:31.150 near the start consists of normal dynamics 00:57:31.150 --> 00:57:33.190 so that's there are no artifacts. 00:57:33.190 --> 00:57:34.750 That's, of course, a big assumption, 00:57:34.750 --> 00:57:39.100 but they use that to try to fine tune the dynamic model 00:57:39.100 --> 00:57:43.540 to fine tune it for each baby and for themselves. 00:57:43.540 --> 00:57:45.070 And then they looked at the ability 00:57:45.070 --> 00:57:47.357 to try to identify artifactual processes. 00:57:47.357 --> 00:57:49.690 Now, I want to go a little bit slowly through this plot, 00:57:49.690 --> 00:57:52.350 because it's quite interesting. 00:57:52.350 --> 00:57:57.990 So what I'm showing you here is a ROC curve 00:57:57.990 --> 00:58:00.292 of the ability to predict each of the four 00:58:00.292 --> 00:58:01.500 different types of artifacts. 00:58:01.500 --> 00:58:03.810 For example, at any one point in time, 00:58:03.810 --> 00:58:05.990 was there a blood sample being taken or not? 00:58:05.990 --> 00:58:07.890 At any one point in time, was there 00:58:07.890 --> 00:58:12.270 a core temperature disconnect of the core temperature probe? 00:58:12.270 --> 00:58:13.770 And to evaluate it, they're assuming 00:58:13.770 --> 00:58:18.850 that they have some label data for evaluation purposes only. 00:58:18.850 --> 00:58:22.110 And of course, you want to be at the very far top left corner 00:58:22.110 --> 00:58:23.866 up here. 00:58:23.866 --> 00:58:27.820 And what we're showing here are three different curves-- 00:58:27.820 --> 00:58:31.120 the very faint dotted line, which 00:58:31.120 --> 00:58:34.780 I'm going to trace out with my cursor, is the baseline. 00:58:34.780 --> 00:58:39.068 Think of that as a much worse algorithm. 00:58:41.640 --> 00:58:42.140 Sorry. 00:58:42.140 --> 00:58:44.523 That's that line over there. 00:58:44.523 --> 00:58:45.190 Everyone see it? 00:58:49.030 --> 00:58:52.110 And this approach are the other two lines. 00:58:52.110 --> 00:58:54.800 Now, what's differentiating those other two lines 00:58:54.800 --> 00:58:57.940 corresponds to the particular type of approximate inference 00:58:57.940 --> 00:59:00.120 algorithm that's used. 00:59:00.120 --> 00:59:05.640 To do this posterior inference, to infer 00:59:05.640 --> 00:59:10.290 the true value of the x's given your noisy observations 00:59:10.290 --> 00:59:14.160 in the model given here is actually a very hard inference 00:59:14.160 --> 00:59:15.920 problem. 00:59:15.920 --> 00:59:18.330 Mathematically, I think one can show 00:59:18.330 --> 00:59:21.692 that it's an NP-hard computational problem. 00:59:21.692 --> 00:59:23.650 And so they have to approximate it in some way, 00:59:23.650 --> 00:59:26.010 and they use two different approximations here. 00:59:26.010 --> 00:59:28.400 The first approximation is based on what they're 00:59:28.400 --> 00:59:31.110 calling a Gaussian sum approximation, 00:59:31.110 --> 00:59:33.420 and it's a deterministic approximation. 00:59:33.420 --> 00:59:37.240 The second approximation is based on a Monte Carlo method. 00:59:37.240 --> 00:59:40.290 And what you see here is that the Gaussian sum approximation 00:59:40.290 --> 00:59:41.970 is actually dramatically better. 00:59:41.970 --> 00:59:43.920 So for example, in this blood sample one, 00:59:43.920 --> 00:59:48.750 that the ROC curve looks like this for the Gaussian sum 00:59:48.750 --> 00:59:49.640 approximation. 00:59:49.640 --> 00:59:51.390 Whereas for the Monte Carlo approximation, 00:59:51.390 --> 00:59:54.510 it's actually significantly lower. 00:59:54.510 --> 00:59:56.400 And this is just to point out that, even 00:59:56.400 --> 01:00:03.660 in this setting, where we have very little data, 01:00:03.660 --> 01:00:06.780 we're using a lot of domain knowledge, the actual details 01:00:06.780 --> 01:00:09.053 of how one does the math-- in particular, 01:00:09.053 --> 01:00:10.470 the proximate inference-- can make 01:00:10.470 --> 01:00:13.047 a really big difference in the performance of this system. 01:00:13.047 --> 01:00:14.880 And so it's something that one should really 01:00:14.880 --> 01:00:16.047 think deeply about, as well. 01:00:18.666 --> 01:00:21.700 I'm going to skip that slide, and then just mention 01:00:21.700 --> 01:00:23.170 very briefly this one. 01:00:23.170 --> 01:00:28.640 This is showing an inference of the events. 01:00:28.640 --> 01:00:34.600 So here I'm showing you three different observations. 01:00:34.600 --> 01:00:39.130 And on the bottom here, I'm showing the prediction 01:00:39.130 --> 01:00:43.950 of when artifact-- two different artifactual events happened. 01:00:43.950 --> 01:00:46.020 And these predictions were actually quite good, 01:00:46.020 --> 01:00:48.180 using this model. 01:00:48.180 --> 01:00:52.210 So I'm done with that first example, and-- 01:00:52.210 --> 01:00:55.380 and the-- just to recap the important points 01:00:55.380 --> 01:01:01.300 of that example, it was that we had almost no label data. 01:01:01.300 --> 01:01:05.470 We're tackling this problem using a cleverly chosen 01:01:05.470 --> 01:01:08.780 statistical model with some domain knowledge built in, 01:01:08.780 --> 01:01:12.040 and that can go really far. 01:01:12.040 --> 01:01:14.500 So now we'll shift gears to talk about a different type 01:01:14.500 --> 01:01:18.340 of problem involving physiological data, 01:01:18.340 --> 01:01:22.570 and that's of detecting atrial fibrillation. 01:01:22.570 --> 01:01:26.280 So what I'm showing you here is an AliveCore device. 01:01:26.280 --> 01:01:27.850 I own one of these. 01:01:27.850 --> 01:01:30.540 So if you want to drop by my E25 545 office, 01:01:30.540 --> 01:01:32.860 you can-- you can play around with it. 01:01:32.860 --> 01:01:35.930 And if you attach it to your mobile phone, 01:01:35.930 --> 01:01:43.800 it'll show you your electric conductance through your heart 01:01:43.800 --> 01:01:46.710 as measured through your two fingers 01:01:46.710 --> 01:01:48.670 touching this device shown over here. 01:01:48.670 --> 01:01:51.270 And from that, one can try to detect whether the patient has 01:01:51.270 --> 01:01:52.990 atrial fibrillation. 01:01:52.990 --> 01:01:54.941 So what is atrial fibrillation? 01:01:58.617 --> 01:01:59.200 Good question. 01:01:59.200 --> 01:02:00.284 It's [INAUDIBLE]. 01:02:04.240 --> 01:02:10.270 So this is from the American Heart Association. 01:02:10.270 --> 01:02:13.810 They defined atrial fibrillation as a quivering or irregular 01:02:13.810 --> 01:02:16.450 heartbeat, also known as arrhythmia. 01:02:16.450 --> 01:02:18.220 And one of the big challenges is that it 01:02:18.220 --> 01:02:21.030 could lead to blood clot, stroke, heart failure, and so 01:02:21.030 --> 01:02:21.530 on. 01:02:21.530 --> 01:02:23.980 So here is how a patient might describe 01:02:23.980 --> 01:02:26.020 having atrial fibrillation. 01:02:26.020 --> 01:02:28.180 My heart flip-flops, skips beats, 01:02:28.180 --> 01:02:31.150 feels like it's banging against my chest wall, 01:02:31.150 --> 01:02:33.790 particularly when I'm carrying stuff up my stairs 01:02:33.790 --> 01:02:35.542 or bending down. 01:02:35.542 --> 01:02:37.250 Now let's try to look at a picture of it. 01:02:48.040 --> 01:02:55.330 So this is a normal heartbeat. 01:02:55.330 --> 01:02:59.860 Hearts move-- pumping like this. 01:02:59.860 --> 01:03:03.130 And if you were to look at the signal 01:03:03.130 --> 01:03:04.810 output of the EKG of a normal heartbeat, 01:03:04.810 --> 01:03:05.620 it would look like this. 01:03:05.620 --> 01:03:07.735 And it's roughly corresponding to the different-- 01:03:07.735 --> 01:03:09.840 the signal is corresponding to different cycles 01:03:09.840 --> 01:03:12.420 of the heartbeat. 01:03:12.420 --> 01:03:15.000 Now for a patient who has atrial fibrillation, 01:03:15.000 --> 01:03:16.290 it looks more like this. 01:03:21.650 --> 01:03:25.677 So much more obviously abnormal, at least in this figure. 01:03:25.677 --> 01:03:27.510 And if you look at the corresponding signal, 01:03:27.510 --> 01:03:29.382 it also looks very different. 01:03:29.382 --> 01:03:31.590 So this is just to give you some intuition about what 01:03:31.590 --> 01:03:33.577 I mean by atrial fibrillation. 01:03:36.990 --> 01:03:39.930 So what we're going to try to do now is to detect it. 01:03:39.930 --> 01:03:44.090 So we're going to take data like that 01:03:44.090 --> 01:03:48.580 and try to classify it into a number of different categories. 01:03:48.580 --> 01:03:52.630 Now this is something which has been studied for decades, 01:03:52.630 --> 01:03:57.430 and last year, 2017, there was a competition 01:03:57.430 --> 01:04:01.450 run by Professor Roger Mark, who is here 01:04:01.450 --> 01:04:04.390 at MIT, which is trying to see, well, how could-- 01:04:04.390 --> 01:04:06.460 how good are we at trying to figure out 01:04:06.460 --> 01:04:09.940 which patients have different types of heart rhythms 01:04:09.940 --> 01:04:11.780 based on data that looks like this? 01:04:11.780 --> 01:04:13.300 So this is a normal rhythm, which 01:04:13.300 --> 01:04:16.700 is also called a sinus rhythm. 01:04:16.700 --> 01:04:18.750 And over here it's atrial-- 01:04:18.750 --> 01:04:22.120 this is an example one patient who has atrial fibrillation. 01:04:22.120 --> 01:04:25.200 This is another type of rhythm that's not atrial fibrillation, 01:04:25.200 --> 01:04:26.590 but is abnormal. 01:04:26.590 --> 01:04:29.670 And this is a noisy recording-- for example, if a patient's-- 01:04:29.670 --> 01:04:32.220 doesn't really have their two fingers very well put 01:04:32.220 --> 01:04:35.180 on to the two leads of the device. 01:04:35.180 --> 01:04:41.040 So given one of these categories, can we predict-- 01:04:41.040 --> 01:04:42.760 one of these signals, could predict 01:04:42.760 --> 01:04:45.355 which category it came from? 01:04:45.355 --> 01:04:47.230 So if you looked at this, you might recognize 01:04:47.230 --> 01:04:48.970 that they look a bit different. 01:04:48.970 --> 01:04:53.380 So could some of you guess what might 01:04:53.380 --> 01:04:55.780 be predictive features that differentiate 01:04:55.780 --> 01:04:59.440 one of these signals from the other? 01:04:59.440 --> 01:05:00.303 In the back? 01:05:00.303 --> 01:05:01.720 AUDIENCE: The presence and absence 01:05:01.720 --> 01:05:07.065 of one of the peaks the QRS complex are [INAUDIBLE].. 01:05:07.065 --> 01:05:08.440 DAVID SONTAG: So speak in English 01:05:08.440 --> 01:05:10.722 for people who don't know what these terms mean. 01:05:10.722 --> 01:05:12.680 AUDIENCE: There is one large piece, which can-- 01:05:12.680 --> 01:05:16.730 probably we can consider one mV and there is another peak, 01:05:16.730 --> 01:05:18.520 which is sort of like-- 01:05:18.520 --> 01:05:20.630 they have reverse polarity between normal rhythm 01:05:20.630 --> 01:05:21.310 and [INAUDIBLE]. 01:05:21.310 --> 01:05:22.102 DAVID SONTAG: Good. 01:05:22.102 --> 01:05:23.820 So are you a cardiologist? 01:05:23.820 --> 01:05:24.710 AUDIENCE: No. 01:05:24.710 --> 01:05:26.440 DAVID SONTAG: No, OK. 01:05:26.440 --> 01:05:29.050 So what the student suggested is one 01:05:29.050 --> 01:05:31.660 could look for sort of these inversions 01:05:31.660 --> 01:05:34.670 to try to describe it a little bit differently. 01:05:34.670 --> 01:05:41.290 So here you're suggesting the lack of those inversions 01:05:41.290 --> 01:05:45.430 is predictive of an abnormal rhythm. 01:05:45.430 --> 01:05:47.655 What about another feature that could be predictive? 01:05:47.655 --> 01:05:48.155 Yep? 01:05:48.155 --> 01:05:49.840 AUDIENCE: The spacing between the peaks 01:05:49.840 --> 01:05:52.030 is more irregular with the AF. 01:05:52.030 --> 01:05:53.740 DAVID SONTAG: The spacing between beats 01:05:53.740 --> 01:05:56.853 is more irregular with the AF rhythm. 01:05:56.853 --> 01:05:58.270 So you're sort of looking at this. 01:05:58.270 --> 01:06:00.160 You see how here this spacing is very 01:06:00.160 --> 01:06:01.538 different from this spacing. 01:06:01.538 --> 01:06:03.580 Whereas in the normal rhythm, sort of the spacing 01:06:03.580 --> 01:06:05.690 looks pretty darn regular. 01:06:05.690 --> 01:06:07.060 All right, good. 01:06:07.060 --> 01:06:11.050 So if I was to show you 40 examples of these 01:06:11.050 --> 01:06:12.940 and then ask you to classify some new ones, 01:06:12.940 --> 01:06:15.280 how well do you think you'll be able to do? 01:06:15.280 --> 01:06:15.780 Pretty well? 01:06:20.970 --> 01:06:23.550 I would be surprised if you couldn't do reasonably 01:06:23.550 --> 01:06:26.250 well at least distinguishing between normal rhythm and AF 01:06:26.250 --> 01:06:30.510 rhythm, because there seem to be some pretty clear signals here. 01:06:30.510 --> 01:06:32.580 Of course, as you get into alternatives, 01:06:32.580 --> 01:06:34.848 then the story gets much more complex. 01:06:34.848 --> 01:06:36.390 But let me dig in a little bit deeper 01:06:36.390 --> 01:06:37.980 into what I mean by this. 01:06:37.980 --> 01:06:39.600 So let's define some of these terms. 01:06:39.600 --> 01:06:44.430 Well, cardiologists have studied this for a really long time, 01:06:44.430 --> 01:06:46.530 and they have-- so what I'm showing 01:06:46.530 --> 01:06:49.380 you here is one heart cycle. 01:06:49.380 --> 01:06:53.220 And they've-- you can put names to each of the peaks that you 01:06:53.220 --> 01:06:55.860 would see in a regular heart cycle-- so that-- for example, 01:06:55.860 --> 01:06:59.250 that very high peak is known as the R peak. 01:06:59.250 --> 01:07:03.060 And you could look at, for example, the interval-- 01:07:03.060 --> 01:07:06.720 so this is one beat. 01:07:06.720 --> 01:07:10.320 You could look at the interval between the R peak of one beat 01:07:10.320 --> 01:07:13.050 and the R peak of another peak, and define 01:07:13.050 --> 01:07:15.440 that to be the RR interval. 01:07:15.440 --> 01:07:18.050 In a similar way, one could take-- 01:07:18.050 --> 01:07:21.060 one could find different distinctive elements 01:07:21.060 --> 01:07:22.140 of the signal-- 01:07:22.140 --> 01:07:23.032 by the way, each-- 01:07:25.680 --> 01:07:28.110 each time step corresponds to the heart 01:07:28.110 --> 01:07:30.410 being in a different position. 01:07:30.410 --> 01:07:33.860 For a healthy heart, these are relatively deterministic. 01:07:33.860 --> 01:07:36.330 And so you could look at other distances and derive 01:07:36.330 --> 01:07:38.010 features from those distances, as well, 01:07:38.010 --> 01:07:40.160 just like we were talking about, both within a beat 01:07:40.160 --> 01:07:42.220 and across beats. 01:07:42.220 --> 01:07:42.895 Yep? 01:07:42.895 --> 01:07:44.312 AUDIENCE: So what's the difference 01:07:44.312 --> 01:07:46.090 between a segment and an interval again? 01:07:48.333 --> 01:07:50.250 DAVID SONTAG: I don't know what the difference 01:07:50.250 --> 01:07:51.420 between a segment and an interval is. 01:07:51.420 --> 01:07:52.070 Does anyone else know? 01:07:52.070 --> 01:07:54.070 I mean, I guess the interval is between probably 01:07:54.070 --> 01:07:56.490 the heads of peaks, whereas segments might refer to 01:07:56.490 --> 01:07:59.193 within a interval. 01:07:59.193 --> 01:07:59.860 That's my guess. 01:07:59.860 --> 01:08:00.902 Does someone know better? 01:08:04.190 --> 01:08:05.630 For the purpose of today's class, 01:08:05.630 --> 01:08:07.366 that's a good enough understanding. 01:08:10.940 --> 01:08:14.060 The point is this is well understood. 01:08:14.060 --> 01:08:16.093 One could derive features from this. 01:08:16.093 --> 01:08:16.776 AUDIENCE: By us. 01:08:16.776 --> 01:08:17.609 DAVID SONTAG: By us. 01:08:20.180 --> 01:08:23.399 So what would a traditional approach be to this problem? 01:08:23.399 --> 01:08:24.020 So this is-- 01:08:24.020 --> 01:08:27.050 I'm pulling this figure from a paper from 2002. 01:08:27.050 --> 01:08:30.200 What it'll do is it'll take in that signal. 01:08:30.200 --> 01:08:32.960 It'll do some filtering of it. 01:08:32.960 --> 01:08:35.750 Then it'll run a peak detection logic, which 01:08:35.750 --> 01:08:38.840 will find these peaks, and then it'll 01:08:38.840 --> 01:08:43.939 measure intervals between these peaks and within a beat. 01:08:43.939 --> 01:08:48.069 And it'll take those computations 01:08:48.069 --> 01:08:49.760 or make some decision based on it. 01:08:49.760 --> 01:08:51.590 So that's a traditional algorithm, 01:08:51.590 --> 01:08:54.310 and they work pretty reasonably. 01:08:54.310 --> 01:08:56.560 And so what do I mean by signal processing? 01:08:56.560 --> 01:08:58.790 Well, this is an example of that. 01:08:58.790 --> 01:09:01.880 I encourage any of you to go home today and try to code up 01:09:01.880 --> 01:09:03.140 a peaked finding algorithm. 01:09:03.140 --> 01:09:06.819 It's not that hard, at least not to get an OK one. 01:09:06.819 --> 01:09:11.149 You might imagine keeping a running tab 01:09:11.149 --> 01:09:13.811 of what's the highest signal you've seen so far. 01:09:13.811 --> 01:09:16.019 Then you look to see what is the first time it drops, 01:09:16.019 --> 01:09:18.394 and the second time-- and the next time it goes up larger 01:09:18.394 --> 01:09:22.064 than, let's say, the previous-- 01:09:22.064 --> 01:09:22.939 suppose that one of-- 01:09:22.939 --> 01:09:26.689 you want to look for when the drop is-- the maximum value-- 01:09:26.689 --> 01:09:28.790 recent maximum value divided by 2. 01:09:28.790 --> 01:09:31.279 And then you-- then you reset. 01:09:31.279 --> 01:09:33.800 And you can imagine in this way very quickly coding up 01:09:33.800 --> 01:09:37.755 a peak finding algorithm. 01:09:37.755 --> 01:09:39.380 And so this is just, again, to give you 01:09:39.380 --> 01:09:43.130 some intuition behind what a traditional approach would be. 01:09:43.130 --> 01:09:46.790 And then you can very quickly see that that-- 01:09:46.790 --> 01:09:49.729 once you start to look at some intervals between peaks, 01:09:49.729 --> 01:09:52.880 that alone is often good enough for predicting 01:09:52.880 --> 01:09:55.050 whether a patient has atrial fibrillation. 01:09:55.050 --> 01:09:58.940 So this is a figure taken from paper in 2001 01:09:58.940 --> 01:10:01.310 showing a single patient's time series. 01:10:01.310 --> 01:10:04.940 So the x-axis is for that single patient, 01:10:04.940 --> 01:10:07.250 their heart beats across time. 01:10:07.250 --> 01:10:09.830 The y-axis is just showing the RR interval 01:10:09.830 --> 01:10:14.300 between the previous beat and the current beat. 01:10:14.300 --> 01:10:18.080 And down here in the bottom is the ground truth 01:10:18.080 --> 01:10:20.990 of whether the patient is assessed to have-- 01:10:20.990 --> 01:10:27.650 to be in-- to have a normal rhythm or atrial fibrillation, 01:10:27.650 --> 01:10:30.630 which is noted as this higher value here. 01:10:30.630 --> 01:10:33.830 So these are AF rhythms. 01:10:33.830 --> 01:10:34.710 This is normal. 01:10:34.710 --> 01:10:36.800 This is AF again. 01:10:36.800 --> 01:10:40.670 And what you can see is that the RR interval actually 01:10:40.670 --> 01:10:41.640 gets you pretty far. 01:10:41.640 --> 01:10:44.210 You notice how it's pretty high up here. 01:10:44.210 --> 01:10:46.130 Suddenly it drops. 01:10:46.130 --> 01:10:47.930 The RR interval drops for a while, 01:10:47.930 --> 01:10:50.450 and that's when the patient has AF. 01:10:50.450 --> 01:10:51.860 Then it goes up again. 01:10:51.860 --> 01:10:54.780 Then it drops again, and so on. 01:10:54.780 --> 01:10:56.780 And so it's not deterministic, the relationship, 01:10:56.780 --> 01:10:59.143 but there's definitely a lot of signal just from that. 01:10:59.143 --> 01:11:00.560 So you might say, OK, well, what's 01:11:00.560 --> 01:11:02.480 the next thing we could do to try to clean up the signal 01:11:02.480 --> 01:11:03.230 a little bit more? 01:11:03.230 --> 01:11:11.210 So flash backwards from 2001 to 1970 here at MIT, studied by-- 01:11:11.210 --> 01:11:13.760 actually, no, this is not MIT. 01:11:13.760 --> 01:11:16.070 This is somewhere else, sorry. 01:11:16.070 --> 01:11:21.398 But still 1970-- where they used a Markov model very 01:11:21.398 --> 01:11:23.690 similar to the Markov models we were just talking about 01:11:23.690 --> 01:11:30.410 in the previous example to model what a sequence of normal RR 01:11:30.410 --> 01:11:34.310 intervals looks like versus what a sequence of abnormal, 01:11:34.310 --> 01:11:37.370 for example, AF RR intervals looks like. 01:11:37.370 --> 01:11:39.590 And in that way, one can recognize 01:11:39.590 --> 01:11:42.980 that, for any one observation of an RR interval 01:11:42.980 --> 01:11:45.540 might not by itself be perfectly predictive, 01:11:45.540 --> 01:11:47.480 but if you look at sort of a sequence of them 01:11:47.480 --> 01:11:50.480 for a patient with atrial fibrillation, 01:11:50.480 --> 01:11:53.420 there is some common pattern to it. 01:11:53.420 --> 01:11:56.090 And you can-- one can detect it by just looking at likelihood 01:11:56.090 --> 01:11:59.450 of that sequence under each of these two different models, 01:11:59.450 --> 01:12:01.230 normal and abnormal. 01:12:01.230 --> 01:12:04.070 And that did pretty well-- even better than the previous 01:12:04.070 --> 01:12:05.310 approaches for-- 01:12:05.310 --> 01:12:08.370 for predicting atrial fibrillation. 01:12:08.370 --> 01:12:11.790 This is the paper I wanted to say from MIT. 01:12:11.790 --> 01:12:15.880 Now 1991, this is also from Roger Mark's group. 01:12:15.880 --> 01:12:19.480 Now this is a neural network based approach, where it says, 01:12:19.480 --> 01:12:22.108 OK, we're going to take a bunch of these things. 01:12:22.108 --> 01:12:24.150 We're going to derive a bunch of these intervals, 01:12:24.150 --> 01:12:25.890 and then we're going to throw that through a black box 01:12:25.890 --> 01:12:27.432 supervised machine learning algorithm 01:12:27.432 --> 01:12:30.240 to predict whether a patient has AF or not. 01:12:30.240 --> 01:12:32.220 So these are very-- 01:12:32.220 --> 01:12:34.890 first of all, there are some simple approaches here 01:12:34.890 --> 01:12:36.540 that work reasonably well. 01:12:36.540 --> 01:12:42.280 Using neural networks in this domain is not a new thing, 01:12:42.280 --> 01:12:44.140 but where are we as a field? 01:12:44.140 --> 01:12:46.920 So as I mentioned, there was this competition last year, 01:12:46.920 --> 01:12:48.887 and what I'm showing you here-- the citation 01:12:48.887 --> 01:12:50.470 is from one of the winning approaches. 01:12:50.470 --> 01:12:52.845 And this winning approach really brings the two paradigms 01:12:52.845 --> 01:12:53.910 together. 01:12:53.910 --> 01:12:57.600 It extracts a large number of expert derived features-- 01:12:57.600 --> 01:12:59.342 so shown here. 01:12:59.342 --> 01:13:01.050 And these are exactly the types of things 01:13:01.050 --> 01:13:06.390 you might think, like proportion, median RR 01:13:06.390 --> 01:13:11.417 interval of regular rhythms, max RR irregularity measure. 01:13:11.417 --> 01:13:13.500 And there's just a whole range of different things 01:13:13.500 --> 01:13:16.160 that you can imagine manually deriving from the data. 01:13:16.160 --> 01:13:17.910 And you throw all of these features 01:13:17.910 --> 01:13:21.840 into a machine learning algorithm, 01:13:21.840 --> 01:13:25.040 maybe a random forest, maybe a neural network, doesn't matter. 01:13:25.040 --> 01:13:27.180 And what you get out is a slightly better algorithm 01:13:27.180 --> 01:13:28.555 than what if you had just come up 01:13:28.555 --> 01:13:30.510 with a simple rule on your own. 01:13:30.510 --> 01:13:33.470 That was the winning algorithm then. 01:13:33.470 --> 01:13:36.970 And in the summary paper, they conjectured that, well, maybe 01:13:36.970 --> 01:13:39.357 it's the case that they were-- 01:13:39.357 --> 01:13:41.440 they'd expected that convolutional neural networks 01:13:41.440 --> 01:13:42.443 would win. 01:13:42.443 --> 01:13:44.860 And they were surprised that none of the winning solutions 01:13:44.860 --> 01:13:47.070 involved convolution neural networks. 01:13:47.070 --> 01:13:50.297 And they conjectured that may be the reason why is because maybe 01:13:50.297 --> 01:13:52.630 with these 8,000 patients that they had [INAUDIBLE] that 01:13:52.630 --> 01:13:56.590 just wasn't enough to give the more complex models advantage. 01:13:56.590 --> 01:14:00.370 So flip forward now to this year and the article 01:14:00.370 --> 01:14:05.840 that you read in your readings in Nature Medicine, 01:14:05.840 --> 01:14:07.420 where the Stanford group now showed 01:14:07.420 --> 01:14:10.540 how a convolutional neural network approach, which 01:14:10.540 --> 01:14:13.960 is, in many ways, extremely naive-- all it does 01:14:13.960 --> 01:14:17.870 is it takes the sequence data in. 01:14:17.870 --> 01:14:20.710 It makes no attempt at trying to understand the underlying 01:14:20.710 --> 01:14:23.800 physiology, and just predicts from that-- 01:14:23.800 --> 01:14:25.647 can do really, really well. 01:14:25.647 --> 01:14:27.230 And so there are couple of differences 01:14:27.230 --> 01:14:29.590 that I want to emphasize to the previous work. 01:14:29.590 --> 01:14:31.360 First, the censor is different. 01:14:31.360 --> 01:14:35.580 Whereas the previous work used this alive core censor, 01:14:35.580 --> 01:14:37.420 in this paper from Stanford, they're 01:14:37.420 --> 01:14:40.870 using a different censor called the Zio patch, which 01:14:40.870 --> 01:14:44.110 is attached to the human body and conceivably 01:14:44.110 --> 01:14:45.580 much less noisy. 01:14:45.580 --> 01:14:47.560 So that's one big difference. 01:14:47.560 --> 01:14:49.810 The second big difference is that there's dramatically 01:14:49.810 --> 01:14:50.770 more data. 01:14:50.770 --> 01:14:52.510 Instead of 8,000 patients to train from, 01:14:52.510 --> 01:14:54.790 now they have over 90,000 records 01:14:54.790 --> 01:14:58.060 from 50,000 different patients to train from. 01:14:58.060 --> 01:14:59.740 The third major difference is that now, 01:14:59.740 --> 01:15:02.740 rather than just trying to classify into four categories-- 01:15:02.740 --> 01:15:06.723 normal, abnormal, other, or noisy-- 01:15:06.723 --> 01:15:08.140 now we're going to try to classify 01:15:08.140 --> 01:15:09.880 into 14 different categories. 01:15:09.880 --> 01:15:12.850 We're, in essence, breaking apart that other class 01:15:12.850 --> 01:15:15.610 into much finer grain detail of different types 01:15:15.610 --> 01:15:17.780 of abnormal rhythms. 01:15:17.780 --> 01:15:20.110 And so here are some of those other abnormal rhythms, 01:15:20.110 --> 01:15:28.140 things like complete heart block, 01:15:28.140 --> 01:15:31.650 and a bunch of other names I can't pronounce. 01:15:31.650 --> 01:15:34.472 And from each one of these, they gathered a lot of data. 01:15:34.472 --> 01:15:35.430 And that actually did-- 01:15:35.430 --> 01:15:36.870 so it's not described in the paper, 01:15:36.870 --> 01:15:38.160 but I've talked to the authors, and they 01:15:38.160 --> 01:15:40.690 did-- they gathered this data in a very interesting way. 01:15:40.690 --> 01:15:42.720 So they sort of-- they did their training iteratively. 01:15:42.720 --> 01:15:44.460 They looked to see where their errors were, 01:15:44.460 --> 01:15:46.752 and then they went and gathered more data from patients 01:15:46.752 --> 01:15:48.180 with that subcategory. 01:15:48.180 --> 01:15:51.930 So many of these other categories 01:15:51.930 --> 01:15:54.267 are very under-- might be underrepresented 01:15:54.267 --> 01:15:56.100 in the general population, but they actually 01:15:56.100 --> 01:15:57.810 gather a lot of patients of that type 01:15:57.810 --> 01:16:00.520 in their data set for training purposes. 01:16:00.520 --> 01:16:02.700 And so I think those three things ended up 01:16:02.700 --> 01:16:05.320 making a very big difference. 01:16:05.320 --> 01:16:07.050 So what is their convolutional network? 01:16:07.050 --> 01:16:10.180 Well, first of all, it's a 1-D signal. 01:16:10.180 --> 01:16:12.180 So it's a little bit different from the con nets 01:16:12.180 --> 01:16:13.380 you typically see in computer vision, 01:16:13.380 --> 01:16:15.088 and I'll show you an illustration of that 01:16:15.088 --> 01:16:16.080 in the next slide. 01:16:16.080 --> 01:16:17.430 It's a very deep model. 01:16:17.430 --> 01:16:20.100 So it's 34 layers. 01:16:20.100 --> 01:16:23.010 So the input comes in on the very top in this picture. 01:16:23.010 --> 01:16:26.730 It's passed through a number of layers. 01:16:26.730 --> 01:16:30.210 Each layer consists of convolution followed 01:16:30.210 --> 01:16:33.600 by rectified linear units, and there is sub 01:16:33.600 --> 01:16:35.790 sampling at every other layer so that you 01:16:35.790 --> 01:16:38.010 go from a very wide signal-- 01:16:38.010 --> 01:16:39.645 so a very long-- 01:16:39.645 --> 01:16:40.770 I can't remember how long-- 01:16:40.770 --> 01:16:43.830 1 second long signal summarized down 01:16:43.830 --> 01:16:47.165 into sort of much-- just many smaller number of dimensions, 01:16:47.165 --> 01:16:49.290 which you then have a sort of fully connected layer 01:16:49.290 --> 01:16:52.770 at the bottom to do for your predictions. 01:16:52.770 --> 01:16:55.590 And then they also have these shortcut connections, 01:16:55.590 --> 01:16:58.770 which allow you to pass information from earlier layers 01:16:58.770 --> 01:17:00.630 down to the very end of the network, 01:17:00.630 --> 01:17:02.255 or even into intermediate layers. 01:17:02.255 --> 01:17:04.380 And for those of you who are familiar with residual 01:17:04.380 --> 01:17:06.850 networks, it's the same idea. 01:17:06.850 --> 01:17:08.340 So what is a 1D convolution? 01:17:08.340 --> 01:17:10.270 Well, it looks a little bit like this. 01:17:10.270 --> 01:17:12.960 So this is the signal. 01:17:12.960 --> 01:17:15.570 I'm going to just approximate it by a bunch of 1's and 0's. 01:17:15.570 --> 01:17:16.560 I'll say this is a 1. 01:17:16.560 --> 01:17:17.360 This is a 0. 01:17:17.360 --> 01:17:18.480 This is a 1, 1, so on. 01:17:21.620 --> 01:17:25.280 A convolutional network has a filter associated with it. 01:17:25.280 --> 01:17:28.070 That filter is then applied in a 1D model. 01:17:28.070 --> 01:17:29.630 It's applied in a linear fashion. 01:17:29.630 --> 01:17:32.240 It's just taken a dot product with the filter's values, 01:17:32.240 --> 01:17:35.150 with the values of the signal at each point in time. 01:17:35.150 --> 01:17:38.130 So it looks a little bit like this, 01:17:38.130 --> 01:17:39.450 and this is what you get out. 01:17:39.450 --> 01:17:42.330 So this is the convolution of a single filter 01:17:42.330 --> 01:17:44.760 with the whole signal. 01:17:44.760 --> 01:17:47.140 And the computation I did there-- so for example, 01:17:47.140 --> 01:17:49.860 this first number came from taking the dot product 01:17:49.860 --> 01:17:51.360 of the first three numbers-- 01:17:51.360 --> 01:17:53.370 1, 0, 1-- with the filter. 01:17:53.370 --> 01:18:01.548 So it's 1 times 2 plus 3 times 0 plus 1 times 1, which is 3. 01:18:01.548 --> 01:18:03.090 And so each of the subsequent numbers 01:18:03.090 --> 01:18:04.900 was computed in the same way. 01:18:04.900 --> 01:18:09.060 And I usually have you figure out what this last one is, 01:18:09.060 --> 01:18:12.440 but I'll leave that for you to do at home. 01:18:12.440 --> 01:18:14.097 And that's what a 1D convolution is. 01:18:14.097 --> 01:18:16.680 And so they have-- they do this for lots of different filters. 01:18:16.680 --> 01:18:19.155 Each of those filters might be of varying lengths, 01:18:19.155 --> 01:18:21.030 and each of those will detect different types 01:18:21.030 --> 01:18:23.040 of signal patterns. 01:18:23.040 --> 01:18:25.800 And in this way, after having many layers of these, 01:18:25.800 --> 01:18:28.320 one can, in an automatic fashion, 01:18:28.320 --> 01:18:31.080 extract many of the same types of signals used in that earlier 01:18:31.080 --> 01:18:32.997 work, but also be much more flexible to detect 01:18:32.997 --> 01:18:34.420 some new ones, as well. 01:18:34.420 --> 01:18:37.120 Hold your question, because I need to wrap up. 01:18:37.120 --> 01:18:38.710 So in the paper that you read, they 01:18:38.710 --> 01:18:41.902 talked about how they evaluated this. 01:18:41.902 --> 01:18:44.110 And so I'm not going to go into much depth in it now. 01:18:44.110 --> 01:18:46.330 I just want to point out two different metrics 01:18:46.330 --> 01:18:47.320 that they used. 01:18:47.320 --> 01:18:48.910 So the first metric they used was 01:18:48.910 --> 01:18:52.690 what they called a sequential error metric. 01:18:52.690 --> 01:18:55.990 What that looked at is you had this very long sequence 01:18:55.990 --> 01:19:00.670 for each patient, and they labeled different one 01:19:00.670 --> 01:19:02.350 second intervals of that sequence 01:19:02.350 --> 01:19:05.690 into abnormal, normal, and so on. 01:19:05.690 --> 01:19:07.113 So you could ask, how good are we 01:19:07.113 --> 01:19:08.780 at labeling each of the different points 01:19:08.780 --> 01:19:09.600 along the sequence? 01:19:09.600 --> 01:19:11.720 And that's the sequence metric. 01:19:11.720 --> 01:19:14.510 The different-- the second metric is the set metric, 01:19:14.510 --> 01:19:16.520 and that looks at, if the patient has 01:19:16.520 --> 01:19:19.730 something that's abnormal anywhere, did you detect it? 01:19:19.730 --> 01:19:22.040 So that's, in essence, taking an or of 01:19:22.040 --> 01:19:23.510 each of those 1 second intervals, 01:19:23.510 --> 01:19:25.310 and then looking across patients. 01:19:25.310 --> 01:19:27.410 And from a clinical diagnostic perspective, 01:19:27.410 --> 01:19:29.510 the set metric might be most useful, but then 01:19:29.510 --> 01:19:31.340 when you want to introspect and understand 01:19:31.340 --> 01:19:34.370 where is that happening, then the sequential metric is 01:19:34.370 --> 01:19:35.600 important. 01:19:35.600 --> 01:19:38.300 And the key take home message from the paper is that, 01:19:38.300 --> 01:19:41.240 if you compared the model's predictions-- this is, I think, 01:19:41.240 --> 01:19:44.990 using an f1 metric-- 01:19:44.990 --> 01:19:49.790 to what you would get from a panel of cardiologists, 01:19:49.790 --> 01:19:53.510 these models are doing as well, if not better than these panels 01:19:53.510 --> 01:19:54.500 of cardiologists. 01:19:54.500 --> 01:19:56.930 So this is extremely exciting. 01:19:56.930 --> 01:19:58.700 This is technology-- or variance of this 01:19:58.700 --> 01:20:02.240 is technology that you're going to see deployed now. 01:20:02.240 --> 01:20:04.760 So for those of you who have purchased these Apple watches, 01:20:04.760 --> 01:20:07.220 these Samsung watches, I don't know exactly what they're 01:20:07.220 --> 01:20:08.637 using, but I wouldn't be surprised 01:20:08.637 --> 01:20:10.580 if they're using techniques similar to this. 01:20:10.580 --> 01:20:12.390 And you're going to see much more of that in the future. 01:20:12.390 --> 01:20:14.030 So this is going to be really the first example 01:20:14.030 --> 01:20:15.447 in this course so far of something 01:20:15.447 --> 01:20:18.280 that's really been deployed. 01:20:18.280 --> 01:20:20.660 And so in summary, we're very often 01:20:20.660 --> 01:20:22.450 in the realm of not enough data. 01:20:22.450 --> 01:20:24.860 And in this lecture today, we gave two examples 01:20:24.860 --> 01:20:26.030 how you can deal with that. 01:20:26.030 --> 01:20:31.340 First, you can try to use mechanistic and statistical 01:20:31.340 --> 01:20:38.150 models to try to work in settings where 01:20:38.150 --> 01:20:39.590 you don't have much data. 01:20:39.590 --> 01:20:42.333 And in other extremes, you do have a lot of data, 01:20:42.333 --> 01:20:44.000 and you can try to ignore that, and just 01:20:44.000 --> 01:20:45.292 use these black box approaches. 01:20:45.292 --> 01:20:46.930 That's all for today.