WEBVTT Kind: captions; Language: en-US NOTE Created on 2024-02-07T20:51:27.1993072Z by ClassTranscribe 00:01:22.450 --> 00:01:23.720 Good morning, everybody. 00:01:25.160 --> 00:01:25.690 Morning. 00:01:29.980 --> 00:01:31.830 Alright, so I'm going to get started. 00:01:31.830 --> 00:01:33.590 Just a note. 00:01:33.590 --> 00:01:37.980 So I'll generally start at 9:31 00:01:37.980 --> 00:01:38.590 exactly. 00:01:38.590 --> 00:01:42.600 So I give a minute of slack and. 00:01:43.360 --> 00:01:44.640 At the end of the class, I'll make it 00:01:44.640 --> 00:01:45.056 pretty clear. 00:01:45.056 --> 00:01:46.640 When class is over, just wait till I 00:01:46.640 --> 00:01:47.945 say thank you or something. 00:01:47.945 --> 00:01:49.580 That kind of indicates that class is 00:01:49.580 --> 00:01:50.659 over before you pack up. 00:01:50.660 --> 00:01:52.684 Because otherwise like when students 00:01:52.684 --> 00:01:56.080 start to pack up, like if I get to the 00:01:56.080 --> 00:01:57.507 last slide and then students start to 00:01:57.507 --> 00:01:59.170 pack up, it makes quite a lot of noise 00:01:59.170 --> 00:02:01.590 if like couple 100 people are packing 00:02:01.590 --> 00:02:02.260 up at the same time. 00:02:03.490 --> 00:02:05.930 Right, so by the way these are I forgot 00:02:05.930 --> 00:02:08.210 to mention that brain image is an image 00:02:08.210 --> 00:02:09.600 that's created by Dolly. 00:02:09.600 --> 00:02:10.770 You might have heard of that. 00:02:10.770 --> 00:02:14.580 It's a AI like image generation method 00:02:14.580 --> 00:02:17.440 that can take an image, take a text and 00:02:17.440 --> 00:02:19.210 then generate an image that matches a 00:02:19.210 --> 00:02:19.480 text. 00:02:20.320 --> 00:02:22.470 This is also an image that's created by 00:02:22.470 --> 00:02:22.950 Dolly. 00:02:24.430 --> 00:02:26.280 I forget exactly what the prompt was on 00:02:26.280 --> 00:02:26.720 this one. 00:02:26.720 --> 00:02:28.854 It was it didn't exactly match the 00:02:28.854 --> 00:02:29.028 prompt. 00:02:29.028 --> 00:02:30.660 I think it was like, I think I said 00:02:30.660 --> 00:02:32.170 something like a bunch of animals 00:02:32.170 --> 00:02:32.960 somewhere ring. 00:02:33.970 --> 00:02:36.580 Orange vests and somewhere in green 00:02:36.580 --> 00:02:38.090 vests standing in a line. 00:02:38.090 --> 00:02:39.956 It has some trouble, like associating 00:02:39.956 --> 00:02:41.900 the right words with the right objects, 00:02:41.900 --> 00:02:44.030 but I still think it's pretty fitting 00:02:44.030 --> 00:02:44.780 for nearest neighbor. 00:02:46.130 --> 00:02:47.775 I like how there's like that one guy 00:02:47.775 --> 00:02:49.700 that is like standing out. 00:02:54.930 --> 00:02:58.120 So today I'm going to talk about two 00:02:58.120 --> 00:02:58.860 things really. 00:02:58.860 --> 00:03:01.249 So one is talking a bit more about the 00:03:01.250 --> 00:03:03.540 basic process of supervised machine 00:03:03.540 --> 00:03:06.320 learning, and the other is about the K 00:03:06.320 --> 00:03:07.945 nearest neighbor algorithm, which is 00:03:07.945 --> 00:03:10.330 one of the kind of like fundamental 00:03:10.330 --> 00:03:11.800 algorithms and machine learning. 00:03:12.560 --> 00:03:15.635 And I'll also talk about how we what 00:03:15.635 --> 00:03:17.160 are the sources of error. 00:03:17.160 --> 00:03:18.270 So why is it a? 00:03:18.270 --> 00:03:19.939 What are the different reasons that a 00:03:19.940 --> 00:03:21.460 machine learning algorithm will make 00:03:21.460 --> 00:03:24.390 test error even after it's fit the 00:03:24.390 --> 00:03:24.970 training set? 00:03:25.640 --> 00:03:28.650 And I'll talk about a couple of 00:03:28.650 --> 00:03:29.979 applications, so I'll talk about 00:03:29.980 --> 00:03:32.344 homework, one which has a couple of 00:03:32.344 --> 00:03:32.940 applications in it. 00:03:33.540 --> 00:03:36.410 And I'll also talk about the deep face 00:03:36.410 --> 00:03:37.210 algorithm. 00:03:41.160 --> 00:03:43.620 So a machine learning model is 00:03:43.620 --> 00:03:46.539 something that maps from features to 00:03:46.540 --> 00:03:47.430 prediction. 00:03:47.430 --> 00:03:51.040 So in this notation I've got F of X is 00:03:51.040 --> 00:03:53.700 mapping to YX are the features, F is 00:03:53.700 --> 00:03:55.460 some function that we'll have some 00:03:55.460 --> 00:03:56.150 parameters. 00:03:56.800 --> 00:03:59.120 And why is the prediction? 00:04:00.050 --> 00:04:01.450 So for example you could have a 00:04:01.450 --> 00:04:03.810 classification problem like is this a 00:04:03.810 --> 00:04:04.530 dog or a cat? 00:04:04.530 --> 00:04:06.660 And it might be based on image features 00:04:06.660 --> 00:04:10.263 or image pixels and so then X are the 00:04:10.263 --> 00:04:13.985 image pixels, Y is yes or no, or it 00:04:13.985 --> 00:04:15.460 could be dog or cat depending on how 00:04:15.460 --> 00:04:15.920 you frame it. 00:04:16.940 --> 00:04:20.200 Or if the problem is this e-mail spam 00:04:20.200 --> 00:04:23.210 or not, then the features might be like 00:04:23.210 --> 00:04:25.440 some summary of the words in the 00:04:25.440 --> 00:04:27.680 document and the words in the subject 00:04:27.680 --> 00:04:31.430 and the sender and the output is like 00:04:31.430 --> 00:04:33.420 true or false or one or zero. 00:04:34.600 --> 00:04:36.450 You could also have regression tests, 00:04:36.450 --> 00:04:38.144 for example, what will the stock price 00:04:38.144 --> 00:04:39.572 be of NVIDIA tomorrow? 00:04:39.572 --> 00:04:42.260 And then the features might be the 00:04:42.260 --> 00:04:44.530 historic stock prices, maybe some 00:04:44.530 --> 00:04:45.960 features about what's trending on 00:04:45.960 --> 00:04:47.610 Twitter, I don't know anything you 00:04:47.610 --> 00:04:48.070 want. 00:04:48.070 --> 00:04:50.780 And then the prediction would be the 00:04:50.780 --> 00:04:53.236 numerical value of the stock price 00:04:53.236 --> 00:04:53.619 tomorrow. 00:04:54.360 --> 00:04:55.640 When you're training something like 00:04:55.640 --> 00:04:57.150 that, you've got like a whole bunch of 00:04:57.150 --> 00:04:58.020 historical data. 00:04:58.020 --> 00:04:59.710 So you try to learn a model that can 00:04:59.710 --> 00:05:01.780 predict based, predict the historical 00:05:01.780 --> 00:05:04.140 stock prices given the preceding ones, 00:05:04.140 --> 00:05:05.579 and then you would hope that when you 00:05:05.580 --> 00:05:08.830 apply it to today's data that it would 00:05:08.830 --> 00:05:10.520 be able to predict the price tomorrow. 00:05:12.140 --> 00:05:13.390 Likewise, what will be the high 00:05:13.390 --> 00:05:14.410 temperature tomorrow? 00:05:14.410 --> 00:05:16.297 Features might be other temperatures, 00:05:16.297 --> 00:05:17.900 temperatures in other locations, other 00:05:17.900 --> 00:05:21.410 kinds of barometric data, and the 00:05:21.410 --> 00:05:23.530 output is some temperature. 00:05:24.410 --> 00:05:25.600 Or you could have a structured 00:05:25.600 --> 00:05:27.630 prediction task where you're outputting 00:05:27.630 --> 00:05:29.295 not just one number, but a whole bunch 00:05:29.295 --> 00:05:31.025 of numbers that are somehow related to 00:05:31.025 --> 00:05:31.690 each other. 00:05:31.690 --> 00:05:33.420 For example, what is the pose of this 00:05:33.420 --> 00:05:33.925 person? 00:05:33.925 --> 00:05:36.767 You would output positions of each of 00:05:36.767 --> 00:05:38.520 the key points on the person's body. 00:05:40.140 --> 00:05:40.440 Right. 00:05:40.440 --> 00:05:42.313 All of these though are just mapping a 00:05:42.313 --> 00:05:45.315 set of features to some labeler to some 00:05:45.315 --> 00:05:46.170 set of labels. 00:05:48.630 --> 00:05:50.720 The machine learning has three stages. 00:05:50.720 --> 00:05:52.870 There's a training stage which is when 00:05:52.870 --> 00:05:54.580 you optimize the model parameters. 00:05:55.620 --> 00:05:58.260 There is a validation stage, which is 00:05:58.260 --> 00:06:00.820 when you evaluate some model that's 00:06:00.820 --> 00:06:03.357 been optimized and use the validation 00:06:03.357 --> 00:06:06.317 to select among possible models or to 00:06:06.317 --> 00:06:08.720 select among some parameters that you 00:06:08.720 --> 00:06:09.900 set for those models. 00:06:10.520 --> 00:06:13.380 So the training is purely optimizing 00:06:13.380 --> 00:06:15.530 your some model design that you have on 00:06:15.530 --> 00:06:16.590 the training data. 00:06:16.590 --> 00:06:18.480 The validation is saying whether that 00:06:18.480 --> 00:06:19.800 was a good model design. 00:06:20.420 --> 00:06:23.150 And so you might iterate between the 00:06:23.150 --> 00:06:25.290 training and the validation many times. 00:06:25.290 --> 00:06:27.290 At the end of that, you'll pick what 00:06:27.290 --> 00:06:29.290 you think is the most effective model, 00:06:29.290 --> 00:06:30.680 and then ideally that should be 00:06:30.680 --> 00:06:33.710 evaluated only once on the test data as 00:06:33.710 --> 00:06:35.440 a measure of the final performance. 00:06:39.330 --> 00:06:43.010 So training is fitting the data to 00:06:43.010 --> 00:06:46.190 minimize some loss or maximize some 00:06:46.190 --> 00:06:47.115 objective function. 00:06:47.115 --> 00:06:49.490 So there's kind of a lot to unpack in 00:06:49.490 --> 00:06:51.180 this one little equation. 00:06:51.180 --> 00:06:54.290 So first the Theta here are the 00:06:54.290 --> 00:06:56.405 parameters of the model, so that's what 00:06:56.405 --> 00:06:57.610 would be optimized. 00:06:57.610 --> 00:06:59.566 And here I'm writing it as minimizing 00:06:59.566 --> 00:07:01.650 some loss, which is the most common way 00:07:01.650 --> 00:07:02.280 you would see it. 00:07:03.350 --> 00:07:06.020 Theta star is the Theta that minimizes 00:07:06.020 --> 00:07:06.920 that loss. 00:07:07.290 --> 00:07:10.080 The loss I'll get to it can be 00:07:10.080 --> 00:07:11.440 different different definitions. 00:07:11.440 --> 00:07:13.209 It could be, for example, a 01 00:07:13.210 --> 00:07:15.220 classification loss or a cross entropy 00:07:15.220 --> 00:07:15.620 loss. 00:07:15.620 --> 00:07:18.070 That's evaluating the likelihood of the 00:07:18.070 --> 00:07:19.820 ground truth labels given the data. 00:07:21.330 --> 00:07:23.427 You've got your model F, you've got 00:07:23.427 --> 00:07:24.840 your features X. 00:07:25.850 --> 00:07:28.430 Those errors are slightly off and your 00:07:28.430 --> 00:07:29.510 ground truth prediction. 00:07:29.510 --> 00:07:31.580 So Capital X, capital Y here are the 00:07:31.580 --> 00:07:34.230 training data and they're those are 00:07:34.230 --> 00:07:36.980 pairs of examples or examples, meaning 00:07:36.980 --> 00:07:38.920 that you've got pairs of features and 00:07:38.920 --> 00:07:40.250 then what you're supposed to predict 00:07:40.250 --> 00:07:41.020 from those features. 00:07:43.820 --> 00:07:45.750 So here's one example. 00:07:45.750 --> 00:07:48.040 Let's say that we want to learn to 00:07:48.040 --> 00:07:49.590 predict the next day's temperature 00:07:49.590 --> 00:07:51.410 given the preceding day temperatures. 00:07:51.410 --> 00:07:53.520 So the way that you would commonly 00:07:53.520 --> 00:07:55.000 formulate this is you'd have some 00:07:55.000 --> 00:07:56.810 matrix of features this X. 00:07:56.810 --> 00:08:00.000 So in Python you just have a 2D Numpy 00:08:00.000 --> 00:08:00.400 of A. 00:08:01.110 --> 00:08:04.462 And you would often store it as that 00:08:04.462 --> 00:08:06.330 you have one row per example. 00:08:06.330 --> 00:08:07.970 So each one of these rows. 00:08:07.970 --> 00:08:10.716 Here is a different example, and if you 00:08:10.716 --> 00:08:12.850 have 1000 training examples, you'd have 00:08:12.850 --> 00:08:13.510 1000 rows. 00:08:14.410 --> 00:08:16.090 And then you have one column per 00:08:16.090 --> 00:08:16.840 feature. 00:08:16.840 --> 00:08:19.730 So this might be the temperature of the 00:08:19.730 --> 00:08:21.650 preceding day, the temperature of two 00:08:21.650 --> 00:08:23.415 days ago, three days ago, four days 00:08:23.415 --> 00:08:23.740 ago. 00:08:23.740 --> 00:08:25.530 And this training data would probably 00:08:25.530 --> 00:08:27.390 be based on, like, historical data 00:08:27.390 --> 00:08:28.170 that's available. 00:08:29.840 --> 00:08:32.505 And then Y is what you need to predict. 00:08:32.505 --> 00:08:35.275 So the goal is to predict, for example 00:08:35.275 --> 00:08:38.840 50.5 based on these numbers here, to 00:08:38.840 --> 00:08:41.025 predict 473 from these numbers here, 00:08:41.025 --> 00:08:43.290 and so on South you'll have the same 00:08:43.290 --> 00:08:45.640 number of rows and your Y as you have 00:08:45.640 --> 00:08:48.040 in your X, but X will have a number of 00:08:48.040 --> 00:08:50.677 columns that corresponds to the number 00:08:50.677 --> 00:08:51.570 of features. 00:08:51.570 --> 00:08:53.890 And if Y is just you're just predicting 00:08:53.890 --> 00:08:55.250 a single number, then you will only 00:08:55.250 --> 00:08:56.100 have one column. 00:08:58.790 --> 00:09:00.270 So for this problem, it might be 00:09:00.270 --> 00:09:03.240 natural to use a squared loss, which is 00:09:03.240 --> 00:09:06.620 that we're going to say that the. 00:09:07.360 --> 00:09:09.330 We want to minimize the squared 00:09:09.330 --> 00:09:11.710 difference between each prediction F of 00:09:11.710 --> 00:09:13.580 XI given Theta. 00:09:14.400 --> 00:09:17.630 Is a prediction on the ith training 00:09:17.630 --> 00:09:20.900 features given the parameters Theta. 00:09:21.730 --> 00:09:24.810 And I want to make that as close as 00:09:24.810 --> 00:09:27.387 possible to the correct value Yi and 00:09:27.387 --> 00:09:29.973 I'm going to I'm going to say close as 00:09:29.973 --> 00:09:32.040 possible is defined by a squared 00:09:32.040 --> 00:09:32.990 difference. 00:09:35.410 --> 00:09:37.470 And I might say for this I'm going to 00:09:37.470 --> 00:09:39.720 use a linear model, so we'll talk about 00:09:39.720 --> 00:09:42.720 linear models in more detail next 00:09:42.720 --> 00:09:45.385 Thursday, but it's pretty intuitive. 00:09:45.385 --> 00:09:47.850 You just have a set for each of your 00:09:47.850 --> 00:09:48.105 features. 00:09:48.105 --> 00:09:49.710 You have some coefficient that's 00:09:49.710 --> 00:09:51.800 multiplied by those features, you sum 00:09:51.800 --> 00:09:53.099 them up, and then you have some 00:09:53.100 --> 00:09:53.980 constant term. 00:09:55.170 --> 00:09:56.829 And then if we wanted to optimize this 00:09:56.830 --> 00:09:58.900 model, we could optimize it using 00:09:58.900 --> 00:10:00.390 ordinary least squares regression, 00:10:00.390 --> 00:10:01.610 which again we'll talk about next 00:10:01.610 --> 00:10:02.300 Thursday. 00:10:02.300 --> 00:10:03.980 So the details of this aren't 00:10:03.980 --> 00:10:06.170 important, but the example is just to 00:10:06.170 --> 00:10:08.820 give you a sense of what the training 00:10:08.820 --> 00:10:09.710 process involves. 00:10:09.710 --> 00:10:11.770 You have a feature matrix X. 00:10:11.770 --> 00:10:13.789 You have a prediction vector a matrix 00:10:13.790 --> 00:10:14.120 Y. 00:10:14.950 --> 00:10:16.550 You have to define a loss, define a 00:10:16.550 --> 00:10:18.130 model and figure out how you're going 00:10:18.130 --> 00:10:18.895 to optimize it. 00:10:18.895 --> 00:10:20.350 And then you would actually perform the 00:10:20.350 --> 00:10:22.420 optimization, get the parameters, and 00:10:22.420 --> 00:10:23.110 that's training. 00:10:25.480 --> 00:10:28.050 So often you'll have a bunch of design 00:10:28.050 --> 00:10:29.470 decisions when you're faced with some 00:10:29.470 --> 00:10:30.782 kind of machine learning problem. 00:10:30.782 --> 00:10:33.660 So you might say, well, maybe that 00:10:33.660 --> 00:10:35.520 temperature prediction problem, maybe a 00:10:35.520 --> 00:10:38.450 linear regressor is good enough. 00:10:38.450 --> 00:10:40.493 Maybe I need a neural network. 00:10:40.493 --> 00:10:42.370 Maybe I should use a decision tree. 00:10:42.370 --> 00:10:44.066 So you might have different algorithms 00:10:44.066 --> 00:10:45.610 that you're considering trying. 00:10:46.240 --> 00:10:48.460 And even for each of those algorithms, 00:10:48.460 --> 00:10:50.320 there might be different parameters 00:10:50.320 --> 00:10:51.925 that you're considering, like what's 00:10:51.925 --> 00:10:53.280 the depth of the tree that I should 00:10:53.280 --> 00:10:53.580 use. 00:10:55.190 --> 00:10:58.160 And so you so it's important to have 00:10:58.160 --> 00:11:00.450 some kind of validation set that you 00:11:00.450 --> 00:11:01.960 can use to. 00:11:02.990 --> 00:11:05.022 That you can use to determine how good 00:11:05.022 --> 00:11:07.020 the model is that you chose, or what 00:11:07.020 --> 00:11:08.755 how good the design parameters of that 00:11:08.755 --> 00:11:09.160 model are. 00:11:09.920 --> 00:11:12.766 So for each one of the different kind 00:11:12.766 --> 00:11:13.940 of model designs that you're 00:11:13.940 --> 00:11:15.592 considering, you would train your model 00:11:15.592 --> 00:11:16.980 and then you evaluate it on a 00:11:16.980 --> 00:11:19.300 validation set and then you choose the 00:11:19.300 --> 00:11:20.210 best of those. 00:11:21.100 --> 00:11:23.390 The best of those models as you're like 00:11:23.390 --> 00:11:24.160 final model. 00:11:25.280 --> 00:11:28.296 So in some if you're doing, like if 00:11:28.296 --> 00:11:30.900 you're getting data sets from online, 00:11:30.900 --> 00:11:32.460 sometimes data sets. 00:11:32.460 --> 00:11:35.050 They'll almost always have a train and 00:11:35.050 --> 00:11:37.200 a test set that is designated for you. 00:11:37.200 --> 00:11:38.620 Which means that you can do all the 00:11:38.620 --> 00:11:39.980 training on the train set, but you 00:11:39.980 --> 00:11:41.400 shouldn't look at the test set until 00:11:41.400 --> 00:11:42.500 you're ready to do your final 00:11:42.500 --> 00:11:43.210 evaluation. 00:11:44.090 --> 00:11:45.790 They don't always have a trained and 00:11:45.790 --> 00:11:47.900 Val split, so sometimes you need to 00:11:47.900 --> 00:11:49.680 separate out a portion of the training 00:11:49.680 --> 00:11:51.240 data and use it for validation. 00:11:52.820 --> 00:11:55.750 So the reason that this is important 00:11:55.750 --> 00:11:59.680 because otherwise you will end up over 00:11:59.680 --> 00:12:01.050 optimizing for your test set. 00:12:01.050 --> 00:12:03.120 If you evaluate 1000 different models 00:12:03.120 --> 00:12:04.820 and you choose the best one for 00:12:04.820 --> 00:12:08.401 testing, then you don't really know if 00:12:08.401 --> 00:12:09.759 that test performance is really 00:12:09.760 --> 00:12:12.080 reflecting the performance that you 00:12:12.080 --> 00:12:13.510 would see with another random set of 00:12:13.510 --> 00:12:15.250 test examples because you've optimized 00:12:15.250 --> 00:12:17.120 your model selection for that test set. 00:12:20.220 --> 00:12:22.440 And then the final stages is evaluation 00:12:22.440 --> 00:12:23.570 or testing. 00:12:23.570 --> 00:12:26.090 And here you have some held out test 00:12:26.090 --> 00:12:28.220 held out set of examples that are not 00:12:28.220 --> 00:12:29.340 used in training. 00:12:29.340 --> 00:12:30.890 Because you want to make sure that your 00:12:30.890 --> 00:12:33.370 model does not only work well on the 00:12:33.370 --> 00:12:35.580 things that it fit to, but it will also 00:12:35.580 --> 00:12:36.999 work well if you give it some new 00:12:37.000 --> 00:12:37.610 example. 00:12:37.610 --> 00:12:39.445 Because you're not really interested in 00:12:39.445 --> 00:12:41.045 making predictions for the data where 00:12:41.045 --> 00:12:42.740 you already know the value of the 00:12:42.740 --> 00:12:44.370 prediction, you're interested in making 00:12:44.370 --> 00:12:44.870 new predictions. 00:12:44.870 --> 00:12:46.955 You want to predict tomorrow's 00:12:46.955 --> 00:12:48.650 temperature, even though nobody knows 00:12:48.650 --> 00:12:50.090 tomorrow's temperature or tomorrow's 00:12:50.090 --> 00:12:50.680 stock price. 00:12:52.790 --> 00:12:55.255 Though the term held out means that 00:12:55.255 --> 00:12:57.440 it's not used at all in the training 00:12:57.440 --> 00:13:00.337 process, and that should mean that it's 00:13:00.337 --> 00:13:00.690 not. 00:13:00.690 --> 00:13:02.123 You don't even look at it, you're not 00:13:02.123 --> 00:13:04.380 even aware of what those values are. 00:13:04.380 --> 00:13:07.335 So in the most clean setups, the test, 00:13:07.335 --> 00:13:10.660 the test data is on some evaluation 00:13:10.660 --> 00:13:13.090 server that people cannot access if 00:13:13.090 --> 00:13:13.530 they're doing. 00:13:13.530 --> 00:13:15.830 If there's some kind of benchmark, 00:13:15.830 --> 00:13:16.900 research, benchmark. 00:13:17.610 --> 00:13:19.720 And in many setups you're not allowed 00:13:19.720 --> 00:13:22.690 to even evaluate your method more than 00:13:22.690 --> 00:13:25.185 once a week so that you to make sure 00:13:25.185 --> 00:13:27.325 that people are not like trying out 00:13:27.325 --> 00:13:28.695 many different things and then choosing 00:13:28.695 --> 00:13:30.140 the best one based on the test set. 00:13:31.830 --> 00:13:33.180 So I'm not going to go through these 00:13:33.180 --> 00:13:34.580 performance measures, but there's lots 00:13:34.580 --> 00:13:36.369 of different performance measures that 00:13:36.370 --> 00:13:37.340 people could use. 00:13:37.340 --> 00:13:39.390 The most common for classification is 00:13:39.390 --> 00:13:41.410 just the classification classification 00:13:41.410 --> 00:13:43.740 error, which is the percent of times 00:13:43.740 --> 00:13:46.680 that your classifier is wrong. 00:13:46.680 --> 00:13:48.200 Obviously you want that to be low. 00:13:49.020 --> 00:13:50.850 Accuracy is just one minus the error. 00:13:51.660 --> 00:13:54.835 And then for regression you might use 00:13:54.835 --> 00:13:56.780 like a root mean squared error, which 00:13:56.780 --> 00:13:59.725 is like your average more or less your 00:13:59.725 --> 00:14:02.410 average distance from prediction to. 00:14:03.930 --> 00:14:08.370 To true value or like a residual R2 00:14:08.370 --> 00:14:10.060 which is like how much of the variance 00:14:10.060 --> 00:14:11.380 does your aggressor explain? 00:14:14.400 --> 00:14:15.190 So. 00:14:15.300 --> 00:14:15.900 00:14:16.730 --> 00:14:18.470 If you're doing machine learning, 00:14:18.470 --> 00:14:19.190 research. 00:14:20.060 --> 00:14:21.890 Usually the way the data is collected 00:14:21.890 --> 00:14:23.700 is that the somebody collects like a 00:14:23.700 --> 00:14:26.010 big pool of data and then they randomly 00:14:26.010 --> 00:14:28.750 sample from that one pool of data to 00:14:28.750 --> 00:14:30.520 get their training and test splits. 00:14:31.300 --> 00:14:33.790 And that means that those training and 00:14:33.790 --> 00:14:36.930 test samples are sampled from the same 00:14:36.930 --> 00:14:37.350 distribution. 00:14:37.350 --> 00:14:40.300 They're what's called IID, which means 00:14:40.300 --> 00:14:41.610 independent and identically 00:14:41.610 --> 00:14:43.695 distributed, and it just means that 00:14:43.695 --> 00:14:44.975 they're coming from the same 00:14:44.975 --> 00:14:45.260 distribution. 00:14:46.290 --> 00:14:48.000 In the real world, though, that's often 00:14:48.000 --> 00:14:48.840 not the case. 00:14:48.840 --> 00:14:50.640 So a lot of a lot of machine learning 00:14:50.640 --> 00:14:52.080 theory is predicated. 00:14:52.080 --> 00:14:55.990 It depends on the assumption that the 00:14:55.990 --> 00:14:57.540 training and test data are coming from 00:14:57.540 --> 00:15:00.205 the same distribution but in the real 00:15:00.205 --> 00:15:00.500 world. 00:15:01.550 --> 00:15:03.120 Often they're different distributions. 00:15:03.120 --> 00:15:07.330 For example, you might be, you might, 00:15:07.330 --> 00:15:11.272 you might be trying to like categorize 00:15:11.272 --> 00:15:14.980 images, but the images that you collect 00:15:14.980 --> 00:15:17.680 in your for training are going to be 00:15:17.680 --> 00:15:19.240 different than what the user is provide 00:15:19.240 --> 00:15:20.220 to your system. 00:15:20.220 --> 00:15:21.740 Or you might be trying to recognize 00:15:21.740 --> 00:15:23.650 faces, but you don't have access to all 00:15:23.650 --> 00:15:24.900 the faces in the world. 00:15:24.900 --> 00:15:26.830 You have access to faces of people that 00:15:26.830 --> 00:15:28.490 volunteer to give you your data, which 00:15:28.490 --> 00:15:29.955 may be a different distribution than 00:15:29.955 --> 00:15:30.960 the end users. 00:15:31.190 --> 00:15:32.200 Of your application. 00:15:33.440 --> 00:15:34.760 Or it may be that things change 00:15:34.760 --> 00:15:37.490 overtime and so the distribution 00:15:37.490 --> 00:15:37.970 changes. 00:15:39.350 --> 00:15:41.180 So yes, go ahead. 00:15:47.900 --> 00:15:52.190 So if the distribution changes, the. 00:15:54.170 --> 00:15:55.660 So this is kind of where it gets 00:15:55.660 --> 00:15:57.908 different between research and 00:15:57.908 --> 00:16:00.679 practice, because in practice the 00:16:00.680 --> 00:16:02.580 distribution changes and you don't 00:16:02.580 --> 00:16:02.976 know. 00:16:02.976 --> 00:16:05.570 Like you have to then collect another 00:16:05.570 --> 00:16:08.210 test set based on your users data and 00:16:08.210 --> 00:16:08.980 annotate it. 00:16:08.980 --> 00:16:10.810 And then you could evaluate how you're 00:16:10.810 --> 00:16:12.740 actually doing on user data, but then 00:16:12.740 --> 00:16:14.635 it might change again because things in 00:16:14.635 --> 00:16:16.345 the world change and your users change 00:16:16.345 --> 00:16:16.620 so. 00:16:17.660 --> 00:16:19.270 So you have like this kind of 00:16:19.270 --> 00:16:21.480 intrinsically unknown thing about what 00:16:21.480 --> 00:16:23.460 is the true test distribution in 00:16:23.460 --> 00:16:24.020 practice. 00:16:24.810 --> 00:16:28.835 In an experiment, if somebody if you 00:16:28.835 --> 00:16:30.816 have like some domain what's called a 00:16:30.816 --> 00:16:33.579 domain shift where the test, test, test 00:16:33.580 --> 00:16:34.780 distribution is different than 00:16:34.780 --> 00:16:35.450 training. 00:16:35.450 --> 00:16:37.033 For example, in a driving application 00:16:37.033 --> 00:16:41.014 you could say you have to train it on, 00:16:41.014 --> 00:16:44.575 you have to train it on nice weather 00:16:44.575 --> 00:16:46.240 days, but it could be tested on foggy 00:16:46.240 --> 00:16:46.580 days. 00:16:47.440 --> 00:16:49.970 And then you kind of can know what the 00:16:49.970 --> 00:16:52.230 distribution shift is, and sometimes 00:16:52.230 --> 00:16:54.135 you're allowed to take that test data 00:16:54.135 --> 00:16:56.591 and learn unsupervised to adapt to that 00:16:56.591 --> 00:16:58.970 test data, and you can evaluate how you 00:16:58.970 --> 00:16:59.390 did. 00:16:59.390 --> 00:17:01.800 So in the research world, we're like 00:17:01.800 --> 00:17:03.590 all the tests and training data is 00:17:03.590 --> 00:17:04.470 known up front. 00:17:04.470 --> 00:17:06.070 You still have like a lot more control 00:17:06.070 --> 00:17:07.630 and a lot more knowledge than you often 00:17:07.630 --> 00:17:09.090 do in application scenario. 00:17:16.920 --> 00:17:20.800 So this is a recap of the training and 00:17:20.800 --> 00:17:21.920 evaluation procedure. 00:17:22.700 --> 00:17:25.790 You have you start with some, ideally 00:17:25.790 --> 00:17:27.500 some training data, some validation 00:17:27.500 --> 00:17:28.750 data, some test data. 00:17:29.900 --> 00:17:33.400 You have some model training and design 00:17:33.400 --> 00:17:35.060 phase, so you. 00:17:36.270 --> 00:17:39.670 You have some idea of what kind of what 00:17:39.670 --> 00:17:41.370 different models might be that you want 00:17:41.370 --> 00:17:42.060 to evaluate. 00:17:42.060 --> 00:17:44.260 You have an algorithm to train those 00:17:44.260 --> 00:17:44.790 models. 00:17:44.790 --> 00:17:47.003 So you take the training data, apply it 00:17:47.003 --> 00:17:48.503 to that design, you get some 00:17:48.503 --> 00:17:49.935 parameters, that's your model. 00:17:49.935 --> 00:17:52.180 Evaluate those parameters on the 00:17:52.180 --> 00:17:55.730 validation set and the model validation 00:17:55.730 --> 00:17:55.970 there. 00:17:56.590 --> 00:17:59.160 And then you might look at those 00:17:59.160 --> 00:18:01.160 results and be like, I think I can do 00:18:01.160 --> 00:18:01.510 better. 00:18:01.510 --> 00:18:03.390 So you go back to the drawing board, 00:18:03.390 --> 00:18:05.880 redo your designs and then you repeat 00:18:05.880 --> 00:18:08.642 that process until finally you say now 00:18:08.642 --> 00:18:10.830 I think the best model that I can 00:18:10.830 --> 00:18:12.790 possibly get, and then you evaluate it 00:18:12.790 --> 00:18:13.560 on your test set. 00:18:19.970 --> 00:18:21.640 So any other questions about that 00:18:21.640 --> 00:18:23.170 before I actually get into one of the 00:18:23.170 --> 00:18:25.520 algorithms, the KNN? 00:18:28.100 --> 00:18:30.635 OK, this obviously like this is going 00:18:30.635 --> 00:18:32.530 to feel second nature to you by the end 00:18:32.530 --> 00:18:34.090 of the course because it's what you use 00:18:34.090 --> 00:18:35.440 for every single machine learning 00:18:35.440 --> 00:18:35.890 algorithm. 00:18:35.890 --> 00:18:39.660 So even if it seems like a little 00:18:39.660 --> 00:18:42.030 abstract or foggy right now, I'm sure 00:18:42.030 --> 00:18:42.490 it will not. 00:18:43.290 --> 00:18:44.120 Before too long. 00:18:46.020 --> 00:18:49.050 All right, so first see if you can 00:18:49.050 --> 00:18:52.070 apply your own machine learning, I 00:18:52.070 --> 00:18:52.340 guess. 00:18:53.350 --> 00:18:55.470 So let's say I've got two classes here. 00:18:55.470 --> 00:18:58.704 I've got O's and I've got X's. 00:18:58.704 --> 00:19:01.430 So and plus is a new test sample. 00:19:01.430 --> 00:19:03.930 So what class do you think the black 00:19:03.930 --> 00:19:05.460 plus corresponds to? 00:19:09.830 --> 00:19:11.300 Alright, so I'll do a vote. 00:19:11.300 --> 00:19:13.040 How many people think it's an X? 00:19:14.940 --> 00:19:16.480 How many people think it's a no? 00:19:18.550 --> 00:19:23.014 So it's about 90 maybe like 99.5% think 00:19:23.014 --> 00:19:25.990 it's an X and about .5% think it's a 00:19:25.990 --> 00:19:27.410 no. 00:19:27.410 --> 00:19:27.755 All right. 00:19:27.755 --> 00:19:28.830 So why is it an X? 00:19:29.630 --> 00:19:30.020 Yeah. 00:19:42.250 --> 00:19:45.860 That's like a Matthew way to put it, 00:19:45.860 --> 00:19:46.902 but that's right, yeah. 00:19:46.902 --> 00:19:49.137 So one reason you might think it's an X 00:19:49.137 --> 00:19:51.988 is that it's closest to X. 00:19:51.988 --> 00:19:54.716 That's the closest example to it is an 00:19:54.716 --> 00:19:55.069 X, right? 00:19:55.790 --> 00:19:57.240 Are there any other reasons that you 00:19:57.240 --> 00:19:58.160 think it might be next? 00:19:58.160 --> 00:19:58.360 Yeah. 00:20:01.500 --> 00:20:02.370 It looks like what? 00:20:03.330 --> 00:20:04.360 It looks like an X. 00:20:06.090 --> 00:20:07.220 I guess that's true. 00:20:08.290 --> 00:20:08.630 Yeah. 00:20:09.960 --> 00:20:10.500 Any other? 00:20:24.830 --> 00:20:25.143 OK. 00:20:25.143 --> 00:20:27.410 And then this one was, if you think 00:20:27.410 --> 00:20:29.120 about like drawing, trying to draw a 00:20:29.120 --> 00:20:31.917 line between the X's and the O's, then 00:20:31.917 --> 00:20:34.660 the best line you could draw the plus 00:20:34.660 --> 00:20:36.710 would be on the X side of the line. 00:20:37.940 --> 00:20:39.530 So those are all good answers. 00:20:39.530 --> 00:20:41.150 And actually there, so there's like. 00:20:41.920 --> 00:20:43.840 There's basically like 3 different ways 00:20:43.840 --> 00:20:45.920 that you can solve this problem. 00:20:45.920 --> 00:20:48.220 One is nearest neighbor, which is what 00:20:48.220 --> 00:20:50.440 I'll talk about, which is when you say 00:20:50.440 --> 00:20:52.423 it's closest to the X, so therefore 00:20:52.423 --> 00:20:53.020 it's an X. 00:20:53.020 --> 00:20:55.086 Or most of the points that are. 00:20:55.086 --> 00:20:57.070 Most of the known points that are close 00:20:57.070 --> 00:20:59.299 to it are X's, so therefore it's an X. 00:20:59.300 --> 00:21:01.440 That's an instant space method. 00:21:01.440 --> 00:21:03.990 Another method is a linear method where 00:21:03.990 --> 00:21:06.120 you draw a line and you say, well it's 00:21:06.120 --> 00:21:07.706 on the UX side of the line, so 00:21:07.706 --> 00:21:08.519 therefore it's an X. 00:21:09.230 --> 00:21:11.360 And the third method is a probabilistic 00:21:11.360 --> 00:21:13.056 method where you fit some probabilities 00:21:13.056 --> 00:21:14.935 to the O's into the X's. 00:21:14.935 --> 00:21:16.830 And you say given those probabilities, 00:21:16.830 --> 00:21:18.510 it's more likely to be an X than a no. 00:21:19.170 --> 00:21:21.629 There's a really like all the different 00:21:21.630 --> 00:21:23.833 methods that you can use, and the 00:21:23.833 --> 00:21:25.210 different algorithms are just different 00:21:25.210 --> 00:21:26.520 ways of parameterizing those 00:21:26.520 --> 00:21:27.070 approaches. 00:21:28.610 --> 00:21:30.089 Or different ways of solving them or 00:21:30.090 --> 00:21:31.460 putting constraints on them. 00:21:34.430 --> 00:21:36.990 So this is the key principle of machine 00:21:36.990 --> 00:21:40.460 learning that given some feature target 00:21:40.460 --> 00:21:44.660 pairs X1Y1TO XNN. 00:21:44.660 --> 00:21:49.570 If XI is similar to XJ, then Yi is 00:21:49.570 --> 00:21:50.850 probably similar to YJ. 00:21:51.450 --> 00:21:53.115 In other words, if the features are 00:21:53.115 --> 00:21:55.100 similar, then the targets are also 00:21:55.100 --> 00:21:55.900 probably similar. 00:21:57.020 --> 00:21:57.790 And this is. 00:21:58.440 --> 00:21:59.586 This is kind of the. 00:21:59.586 --> 00:22:01.220 This is, I would say, an assumption of 00:22:01.220 --> 00:22:02.720 every single machine learning algorithm 00:22:02.720 --> 00:22:03.810 that I can think of. 00:22:03.810 --> 00:22:05.500 If it's not the case, things get really 00:22:05.500 --> 00:22:06.010 complicated. 00:22:06.010 --> 00:22:07.210 I don't know how you would possibly 00:22:07.210 --> 00:22:10.250 solve it if XI if there's no. 00:22:11.430 --> 00:22:13.750 If XI being similar to XJ tells you 00:22:13.750 --> 00:22:17.390 nothing about how Yi and YJ relate to 00:22:17.390 --> 00:22:19.310 each other, then it seems like you 00:22:19.310 --> 00:22:20.320 can't do better than chance. 00:22:21.960 --> 00:22:23.920 So with variations on how you define 00:22:23.920 --> 00:22:24.790 the similarity. 00:22:24.790 --> 00:22:26.330 So what does it mean for XI to be 00:22:26.330 --> 00:22:27.570 similar to XJ? 00:22:27.570 --> 00:22:29.650 And also, if you've got a bunch of 00:22:29.650 --> 00:22:31.520 similar points, how you combine those 00:22:31.520 --> 00:22:32.830 similarities to make a final 00:22:32.830 --> 00:22:33.500 prediction. 00:22:33.500 --> 00:22:36.010 Those differences are what distinguish 00:22:36.010 --> 00:22:37.340 the different algorithms from each 00:22:37.340 --> 00:22:39.050 other, but they're all based on this 00:22:39.050 --> 00:22:41.063 idea that if the features are similar, 00:22:41.063 --> 00:22:42.609 the predictions are also similar. 00:22:45.500 --> 00:22:46.940 So this brings us to the nearest 00:22:46.940 --> 00:22:47.810 neighbor algorithm. 00:22:48.780 --> 00:22:50.960 Probably the simplest, but also one of 00:22:50.960 --> 00:22:52.600 the most useful machine learning 00:22:52.600 --> 00:22:53.170 algorithms. 00:22:54.210 --> 00:22:56.760 And it kind of encodes that simple 00:22:56.760 --> 00:22:58.540 intuition most directly. 00:22:58.540 --> 00:23:02.339 So for a given set of test features, 00:23:02.339 --> 00:23:05.365 assign the label or target value to the 00:23:05.365 --> 00:23:07.505 most similar training features. 00:23:07.505 --> 00:23:11.170 And if you say, you can sometimes say 00:23:11.170 --> 00:23:13.910 how many of these similar examples 00:23:13.910 --> 00:23:15.200 you're going to consider. 00:23:15.200 --> 00:23:17.814 The default is often KK equals one. 00:23:17.814 --> 00:23:20.193 So the most similar single example, you 00:23:20.193 --> 00:23:23.460 assign its label to the test data. 00:23:24.140 --> 00:23:25.530 So here's the algorithm. 00:23:25.530 --> 00:23:27.730 It's pretty short. 00:23:28.860 --> 00:23:30.620 You compute the distance of each of 00:23:30.620 --> 00:23:32.030 your training samples to the test 00:23:32.030 --> 00:23:32.530 sample. 00:23:33.510 --> 00:23:35.870 Take the index of the training sample 00:23:35.870 --> 00:23:37.810 with the minimum distance and then you 00:23:37.810 --> 00:23:38.600 get that label. 00:23:38.600 --> 00:23:39.505 That's it. 00:23:39.505 --> 00:23:41.780 I can literally like code it faster 00:23:41.780 --> 00:23:43.830 than I can look up how you would use 00:23:43.830 --> 00:23:45.770 some library to for the nearest 00:23:45.770 --> 00:23:46.440 neighbor algorithm. 00:23:46.440 --> 00:23:47.420 It's like a few lines. 00:23:49.320 --> 00:23:50.290 So. 00:23:51.460 --> 00:23:54.450 And then within this, so there's just a 00:23:54.450 --> 00:23:56.520 couple of designs. 00:23:56.520 --> 00:23:58.720 One is what distance measure do you 00:23:58.720 --> 00:24:00.780 use, another is like how many nearest 00:24:00.780 --> 00:24:01.870 neighbors do you consider? 00:24:02.500 --> 00:24:04.160 And then often if you're applying this 00:24:04.160 --> 00:24:06.390 algorithm, you might want to apply some 00:24:06.390 --> 00:24:08.020 kind of transformation to the input 00:24:08.020 --> 00:24:08.600 features. 00:24:09.380 --> 00:24:11.343 So that they behave better according 00:24:11.343 --> 00:24:13.690 according to your similarity measure. 00:24:14.430 --> 00:24:16.060 The simplest distance function we can 00:24:16.060 --> 00:24:18.946 use is the L2 distance. 00:24:18.946 --> 00:24:24.030 So L2 means like the two norm or the 00:24:24.030 --> 00:24:25.510 Euclidian distance. 00:24:25.510 --> 00:24:28.570 It's the linear distance in like in 00:24:28.570 --> 00:24:29.605 space basically. 00:24:29.605 --> 00:24:31.930 So usually if you think of a distance 00:24:31.930 --> 00:24:33.819 intuitively, you're thinking of the L2. 00:24:37.810 --> 00:24:41.040 So we can try to so K nearest neighbor 00:24:41.040 --> 00:24:42.820 is just the generalization of nearest 00:24:42.820 --> 00:24:44.060 neighbor where you allow there to be 00:24:44.060 --> 00:24:45.996 more than 1 sample, so you can look at 00:24:45.996 --> 00:24:47.340 the K closest samples. 00:24:49.110 --> 00:24:50.500 So we'll try it with these. 00:24:50.500 --> 00:24:53.840 So let's say for this plus up here my 00:24:53.840 --> 00:24:55.632 pointer is not working for this one 00:24:55.632 --> 00:24:55.950 here. 00:24:55.950 --> 00:24:57.700 If you do one nearest neighbor, what 00:24:57.700 --> 00:24:58.920 would be the closest? 00:25:00.360 --> 00:25:03.190 Yeah, I'd say X and for the other one. 00:25:05.760 --> 00:25:06.940 Right. 00:25:06.940 --> 00:25:08.766 So for one nearest neighbor that the 00:25:08.766 --> 00:25:11.010 plus on the left would probably be X 00:25:11.010 --> 00:25:12.610 and the plus on the right would be O. 00:25:13.940 --> 00:25:16.930 And I should clarify here that the plus 00:25:16.930 --> 00:25:19.690 symbol itself is not really relevant, 00:25:19.690 --> 00:25:21.810 it's just the position. 00:25:21.810 --> 00:25:24.251 So here I've got 2 features X1 and X2, 00:25:24.251 --> 00:25:28.880 and I've got two classes O and, but the 00:25:28.880 --> 00:25:31.360 shapes of them are not are just 00:25:31.360 --> 00:25:34.480 abstract ways of representing some 00:25:34.480 --> 00:25:34.990 class. 00:25:36.400 --> 00:25:37.830 In these examples. 00:25:38.740 --> 00:25:40.930 So three nearest neighbor. 00:25:40.930 --> 00:25:42.200 Then you would look at the three 00:25:42.200 --> 00:25:42.600 nearest neighbors. 00:25:42.600 --> 00:25:44.280 So now one of the labels would flip in 00:25:44.280 --> 00:25:44.985 this case. 00:25:44.985 --> 00:25:47.760 So these circles are not meant to 00:25:47.760 --> 00:25:49.800 indicate like the region of influence. 00:25:49.800 --> 00:25:51.953 They're just circling the three nearest 00:25:51.953 --> 00:25:52.346 neighbors. 00:25:52.346 --> 00:25:53.100 They're ovals. 00:25:54.010 --> 00:25:58.520 So this one now has 2O's closer to it 00:25:58.520 --> 00:26:00.405 and so it's label would flip. 00:26:00.405 --> 00:26:02.733 It's most likely label would flip flip 00:26:02.733 --> 00:26:04.840 to O and if you wanted to you could 00:26:04.840 --> 00:26:06.700 output some confidence that says. 00:26:08.030 --> 00:26:10.650 You could say 2/3 of them are close to 00:26:10.650 --> 00:26:12.470 O, so I think it's a 2/3 chance that 00:26:12.470 --> 00:26:13.160 it's a no. 00:26:13.160 --> 00:26:15.130 It would be a pretty crude like 00:26:15.130 --> 00:26:17.440 probability estimate, but maybe better 00:26:17.440 --> 00:26:18.220 than nothing. 00:26:18.220 --> 00:26:20.400 Another way that you could get 00:26:20.400 --> 00:26:21.760 confidence if you were doing one 00:26:21.760 --> 00:26:23.090 nearest neighbor is to look at the 00:26:23.090 --> 00:26:26.025 ratio of the distances between the 00:26:26.025 --> 00:26:28.832 closest example and the closest example 00:26:28.832 --> 00:26:30.310 from the from another class. 00:26:32.310 --> 00:26:33.980 And then likewise I could do 5 nearest 00:26:33.980 --> 00:26:36.430 neighbor, so K could be anything. 00:26:36.430 --> 00:26:38.590 Typically it's not too large though. 00:26:39.350 --> 00:26:40.030 And. 00:26:41.490 --> 00:26:43.940 And classification is the most common 00:26:43.940 --> 00:26:45.530 case is K = 1. 00:26:45.530 --> 00:26:48.130 But you'll see in regression it can be 00:26:48.130 --> 00:26:50.020 kind of helpful to have a larger K. 00:26:52.480 --> 00:26:52.800 Right. 00:26:52.800 --> 00:26:55.080 So then what distance function do we 00:26:55.080 --> 00:26:58.150 use for K&N? 00:26:59.750 --> 00:27:01.990 We we've got a few choices. 00:27:01.990 --> 00:27:03.360 There's actually many choices, of 00:27:03.360 --> 00:27:05.170 course, but these are the most common. 00:27:05.170 --> 00:27:06.980 One is Euclidian, so I just put the 00:27:06.980 --> 00:27:07.722 equation there. 00:27:07.722 --> 00:27:08.870 It's the it's. 00:27:08.870 --> 00:27:11.540 You don't even need root if you're just 00:27:11.540 --> 00:27:14.090 trying to find the closest, because 00:27:14.090 --> 00:27:15.540 square root is monotonic. 00:27:15.540 --> 00:27:15.880 So. 00:27:16.630 --> 00:27:19.790 If a if the squared distance is 00:27:19.790 --> 00:27:21.732 minimized, then the square of the 00:27:21.732 --> 00:27:23.010 square distance is also minimize. 00:27:24.710 --> 00:27:26.910 And but so you've got Euclidian 00:27:26.910 --> 00:27:28.130 distance there, summer squared 00:27:28.130 --> 00:27:30.890 differences, city block which is sum of 00:27:30.890 --> 00:27:32.210 absolute distances. 00:27:33.250 --> 00:27:34.740 Mahalanobis distance. 00:27:34.740 --> 00:27:37.290 This is the most complicated where you 00:27:37.290 --> 00:27:39.080 have where you first like do what's 00:27:39.080 --> 00:27:41.430 called whitening, which is when you 00:27:41.430 --> 00:27:45.630 just put a inverse variance matrix. 00:27:46.400 --> 00:27:50.225 In between the product and. 00:27:50.225 --> 00:27:52.340 So basically this makes it so that if 00:27:52.340 --> 00:27:54.670 some features have a lot more variance, 00:27:54.670 --> 00:27:56.510 a lot more like spread than other 00:27:56.510 --> 00:27:57.070 features. 00:27:57.760 --> 00:28:00.260 Then they 1st at first reduces that 00:28:00.260 --> 00:28:02.280 spread so that they all have about the 00:28:02.280 --> 00:28:03.560 same amount of spreads so that the 00:28:03.560 --> 00:28:05.770 distance functions are like normalized, 00:28:05.770 --> 00:28:06.520 more comparable. 00:28:07.600 --> 00:28:09.687 Between the different features and it 00:28:09.687 --> 00:28:10.925 will also rotate. 00:28:10.925 --> 00:28:13.870 It will also like rotate the data to 00:28:13.870 --> 00:28:15.660 find the major axis. 00:28:15.660 --> 00:28:18.020 We'll talk about that more later. 00:28:18.020 --> 00:28:19.940 I don't want to get too much into the 00:28:19.940 --> 00:28:22.436 distance metric, just be aware of like 00:28:22.436 --> 00:28:23.610 that it's there and what it is. 00:28:25.650 --> 00:28:28.830 So of these measures L2. 00:28:30.060 --> 00:28:32.600 Kind of assumes implicitly assumes that 00:28:32.600 --> 00:28:34.660 all the dimensions are equally scaled, 00:28:34.660 --> 00:28:37.740 because if you have a distance of three 00:28:37.740 --> 00:28:40.140 for one feature and a distance of three 00:28:40.140 --> 00:28:41.579 for another feature, it'll it'll 00:28:41.580 --> 00:28:43.580 contribute the same to the distance. 00:28:43.580 --> 00:28:46.400 But it could be that one feature is 00:28:46.400 --> 00:28:48.400 height and one feature is income, and 00:28:48.400 --> 00:28:49.930 then the scales are totally different. 00:28:50.770 --> 00:28:52.510 And if you were to compute nearest 00:28:52.510 --> 00:28:55.447 neighbor, where your data is like the 00:28:55.447 --> 00:28:57.057 height of a person and their income, 00:28:57.057 --> 00:28:58.396 and you're trying to predict, predict 00:28:58.396 --> 00:29:01.490 their age, then the income is obviously 00:29:01.490 --> 00:29:03.250 going to dominate those distances. 00:29:03.250 --> 00:29:04.850 Because the height distances, if you 00:29:04.850 --> 00:29:06.970 don't normalize, are going to be at 00:29:06.970 --> 00:29:10.570 most like one or two depending on your 00:29:10.570 --> 00:29:10.980 units. 00:29:11.780 --> 00:29:16.120 And the income differences could be in 00:29:16.120 --> 00:29:17.210 the thousands or millions. 00:29:18.980 --> 00:29:23.890 So a city block is kind of similar, you 00:29:23.890 --> 00:29:25.970 just taking the absolute instead of the 00:29:25.970 --> 00:29:26.960 squared differences. 00:29:27.700 --> 00:29:28.870 And the main difference between 00:29:28.870 --> 00:29:30.826 Euclidean and city block is that city 00:29:30.826 --> 00:29:33.937 block will be less sensitive to the 00:29:33.937 --> 00:29:35.880 biggest differences, biggest 00:29:35.880 --> 00:29:37.060 dimensional differences. 00:29:37.930 --> 00:29:41.360 So with Euclidian, if you have say 5 00:29:41.360 --> 00:29:43.601 features and four of them have a 00:29:43.601 --> 00:29:45.895 distance of one and one of them has a 00:29:45.895 --> 00:29:47.926 distance of 1000, then your total 00:29:47.926 --> 00:29:50.990 distance is going to be like a million, 00:29:50.990 --> 00:29:54.649 roughly a million and four your total 00:29:54.650 --> 00:29:55.420 square distance. 00:29:56.120 --> 00:29:58.910 And so that 1000 totally dominates, or 00:29:58.910 --> 00:30:00.480 even if that one is 10. 00:30:00.480 --> 00:30:02.920 Let's say you have 4 distances of 1 and 00:30:02.920 --> 00:30:05.965 a distance of 10, then your total is 00:30:05.965 --> 00:30:08.010 104 once you square them and sum them. 00:30:09.600 --> 00:30:13.250 But with city block, if you have 4 00:30:13.250 --> 00:30:15.564 distances that are one and one distance 00:30:15.564 --> 00:30:17.724 that is 10, then the city block 00:30:17.724 --> 00:30:19.877 distance is 14 because it's one plus 00:30:19.877 --> 00:30:21.340 one 4 * + 10. 00:30:22.010 --> 00:30:24.460 So city block is less sensitive to like 00:30:24.460 --> 00:30:26.916 the biggest feature dimension, the 00:30:26.916 --> 00:30:27.980 biggest feature difference. 00:30:29.730 --> 00:30:32.010 And then Mahalanobis does not assume 00:30:32.010 --> 00:30:33.360 that all the features are already 00:30:33.360 --> 00:30:35.020 scaled for it will rescale them. 00:30:35.020 --> 00:30:37.290 So if you were to do this thing with, 00:30:37.290 --> 00:30:39.260 you're trying to predict somebody's age 00:30:39.260 --> 00:30:41.090 given income and height. 00:30:41.730 --> 00:30:43.770 Then after you apply your inverse 00:30:43.770 --> 00:30:46.420 covariance matrix, it will rescale the 00:30:46.420 --> 00:30:48.970 heights and the ages so that they both 00:30:48.970 --> 00:30:49.750 follow some. 00:30:50.950 --> 00:30:54.000 Unit norm distribution or normalized 00:30:54.000 --> 00:30:57.240 distribution where the variance is now 00:30:57.240 --> 00:30:58.840 one in each of those dimensions. 00:31:05.200 --> 00:31:07.790 So with K&N, if you're doing 00:31:07.790 --> 00:31:10.720 classification, then the prediction is 00:31:10.720 --> 00:31:12.470 usually just the most common class. 00:31:13.430 --> 00:31:15.520 If you're doing regression and you get 00:31:15.520 --> 00:31:17.510 the K nearest neighbors, then the 00:31:17.510 --> 00:31:19.290 prediction is usually the average of 00:31:19.290 --> 00:31:21.406 the labels of those K nearest 00:31:21.406 --> 00:31:21.869 neighbors. 00:31:21.870 --> 00:31:23.820 So for classification, if you're doing 00:31:23.820 --> 00:31:26.026 digit classification and you're 3 00:31:26.026 --> 00:31:29.100 nearest neighbors are 992, you would 00:31:29.100 --> 00:31:29.760 predict 9. 00:31:30.980 --> 00:31:32.210 If your. 00:31:32.630 --> 00:31:38.850 Say trying to how aesthetic people 00:31:38.850 --> 00:31:41.170 would think in images on a score on a 00:31:41.170 --> 00:31:43.680 scale of zero to 10 and your returns 00:31:43.680 --> 00:31:45.850 are 992, then you would take the 00:31:45.850 --> 00:31:47.670 average of those most likely so it 00:31:47.670 --> 00:31:48.769 would be 20 / 3. 00:31:52.440 --> 00:31:54.710 So let's just do another example. 00:31:55.040 --> 00:31:55.700 00:31:56.920 --> 00:31:58.130 So let's say that we're doing 00:31:58.130 --> 00:31:58.960 classification. 00:31:58.960 --> 00:32:00.470 I just kind of randomly found some 00:32:00.470 --> 00:32:03.000 scatter plot on the Internet links down 00:32:03.000 --> 00:32:03.380 there. 00:32:03.380 --> 00:32:05.640 And let's say that we're trying to 00:32:05.640 --> 00:32:07.890 predict the sex, male or female, from 00:32:07.890 --> 00:32:09.370 standing and sitting heights. 00:32:09.370 --> 00:32:11.032 So we've got this standing height on 00:32:11.032 --> 00:32:13.320 the X dimension and the sitting height 00:32:13.320 --> 00:32:14.845 on the Y dimension. 00:32:14.845 --> 00:32:19.035 The circles are female, the males are 00:32:19.035 --> 00:32:19.370 male. 00:32:20.320 --> 00:32:22.590 And let's say that I want to predict 00:32:22.590 --> 00:32:26.240 for the X is it a male or a female and 00:32:26.240 --> 00:32:28.060 I'm doing 1 nearest neighbor. 00:32:28.060 --> 00:32:29.890 So what would what would the answer be? 00:32:31.770 --> 00:32:34.580 Right, the answer would be female 00:32:34.580 --> 00:32:37.290 because the closest circle is a female. 00:32:37.290 --> 00:32:38.580 And what if I do three nearest 00:32:38.580 --> 00:32:38.990 neighbor? 00:32:41.270 --> 00:32:41.540 Right. 00:32:41.540 --> 00:32:42.490 Also female. 00:32:42.490 --> 00:32:46.665 I need to get super large K before it's 00:32:46.665 --> 00:32:48.710 even plausible that it could be male. 00:32:48.710 --> 00:32:50.570 Maybe even like K would have to be the 00:32:50.570 --> 00:32:52.070 whole data set, and that would only 00:32:52.070 --> 00:32:53.180 work if there's more males than 00:32:53.180 --> 00:32:53.600 females. 00:32:54.720 --> 00:32:55.926 And what about the plus? 00:32:55.926 --> 00:32:58.760 If I do if I do 1 N, is it male or 00:32:58.760 --> 00:32:59.190 female? 00:33:00.850 --> 00:33:01.095 OK. 00:33:01.095 --> 00:33:02.560 And what if I do three and north? 00:33:04.950 --> 00:33:08.386 Right, female, because now the out of 00:33:08.386 --> 00:33:10.770 the five closest neighbor out of the 00:33:10.770 --> 00:33:12.600 most relevant out of the three closest 00:33:12.600 --> 00:33:14.060 neighbors, two of them are female and 00:33:14.060 --> 00:33:14.630 one is male. 00:33:15.970 --> 00:33:17.740 What about the circle, male or female? 00:33:19.450 --> 00:33:21.220 Right, it will be mail for. 00:33:22.060 --> 00:33:23.070 Virtually any K. 00:33:24.350 --> 00:33:24.740 All right. 00:33:24.740 --> 00:33:26.010 So that's classification. 00:33:27.880 --> 00:33:29.450 And now let's say we want to do 00:33:29.450 --> 00:33:30.540 regression. 00:33:30.540 --> 00:33:32.530 So we want to predict the sitting 00:33:32.530 --> 00:33:35.104 height given the standing height. 00:33:35.104 --> 00:33:37.360 The standing height is on the X axis. 00:33:38.020 --> 00:33:39.720 And I want to predict this sitting 00:33:39.720 --> 00:33:40.410 height. 00:33:41.670 --> 00:33:43.730 So it might be hard to see if you're 00:33:43.730 --> 00:33:44.060 far away. 00:33:44.060 --> 00:33:47.300 It might be kind of hard to see it very 00:33:47.300 --> 00:33:51.150 clearly but for this height, so that I 00:33:51.150 --> 00:33:52.850 don't know exactly what the value is, 00:33:52.850 --> 00:33:56.360 but whatever, 100 and 4144 or 00:33:56.360 --> 00:33:56.790 something. 00:33:57.530 --> 00:33:59.750 What would be the sitting height? 00:34:00.620 --> 00:34:01.360 Roughly. 00:34:05.400 --> 00:34:08.050 So it would be whatever this is here 00:34:08.050 --> 00:34:10.630 let me use my, I'll use my cursor. 00:34:12.500 --> 00:34:14.760 So it would be whatever this point is 00:34:14.760 --> 00:34:16.200 here it would be the sitting height. 00:34:17.100 --> 00:34:18.716 And notice that if I moved a little bit 00:34:18.716 --> 00:34:20.750 to the left it would drop quite a lot, 00:34:20.750 --> 00:34:22.390 and if I move a little bit to the right 00:34:22.390 --> 00:34:23.685 then this would be the closest point 00:34:23.685 --> 00:34:24.660 and then drop a little. 00:34:25.380 --> 00:34:28.110 So the so it's kind of unstable if I'm 00:34:28.110 --> 00:34:30.677 doing one and what if I were doing 3 00:34:30.677 --> 00:34:33.830 and N then would it be higher than One 00:34:33.830 --> 00:34:35.000 North or lower? 00:34:39.130 --> 00:34:41.030 Yes, it would be lower because if I 00:34:41.030 --> 00:34:42.720 were doing 3 N then it would be the 00:34:42.720 --> 00:34:44.883 average of this point and this point 00:34:44.883 --> 00:34:47.820 and this point which is lower than the 00:34:47.820 --> 00:34:48.310 center point. 00:34:50.130 --> 00:34:51.670 And now let's look at this One South. 00:34:51.670 --> 00:34:54.329 Now this one. 00:34:54.330 --> 00:34:56.090 What is the setting height roughly? 00:34:56.730 --> 00:34:57.890 If I do one and north. 00:35:02.740 --> 00:35:04.570 So it's this guy up here. 00:35:04.570 --> 00:35:07.700 So it would be around 84 and what is it 00:35:07.700 --> 00:35:09.990 roughly if I do three and north? 00:35:17.040 --> 00:35:19.556 So it's probably around here. 00:35:19.556 --> 00:35:22.955 So I'd say around like 81 maybe, but 00:35:22.955 --> 00:35:25.110 it's a big drop because these guys, 00:35:25.110 --> 00:35:27.625 these three points here are the are the 00:35:27.625 --> 00:35:28.820 three nearest neighbors. 00:35:30.010 --> 00:35:32.100 And if I am doing one nearest neighbor 00:35:32.100 --> 00:35:34.020 and I were to plot the regressed 00:35:34.020 --> 00:35:36.390 height, it would be like jumping all 00:35:36.390 --> 00:35:37.280 over the place, right? 00:35:37.280 --> 00:35:38.900 Because every time it only depends on 00:35:38.900 --> 00:35:40.410 that one nearest neighbor. 00:35:40.410 --> 00:35:42.335 So it gives us a really, it can give us 00:35:42.335 --> 00:35:44.580 a really unintuitive, bly jumpy 00:35:44.580 --> 00:35:46.180 regression value. 00:35:46.180 --> 00:35:48.006 But if I do three or five nearest 00:35:48.006 --> 00:35:49.340 neighbor, it's going to end up being 00:35:49.340 --> 00:35:51.230 much smoother as I move from left to 00:35:51.230 --> 00:35:51.380 right. 00:35:52.330 --> 00:35:53.350 And then this is like. 00:35:54.440 --> 00:35:56.080 This happens to be showing a linear 00:35:56.080 --> 00:35:57.970 regression of justice all the data. 00:35:57.970 --> 00:36:00.060 We'll talk about linear regression next 00:36:00.060 --> 00:36:01.920 Thursday, but that's kind of the 00:36:01.920 --> 00:36:02.860 smoothest estimate. 00:36:05.470 --> 00:36:07.830 Alright, I'll show. 00:36:07.830 --> 00:36:09.075 Actually, I want to. 00:36:09.075 --> 00:36:10.380 I know it's kind of. 00:36:11.770 --> 00:36:14.200 Let's see 93935. 00:36:15.450 --> 00:36:17.830 So about in the middle of the class, I 00:36:17.830 --> 00:36:19.480 want to like give everyone a chance to 00:36:19.480 --> 00:36:20.880 like stand up and. 00:36:22.090 --> 00:36:23.625 Check your e-mail or phone or whatever, 00:36:23.625 --> 00:36:24.670 because I think it's hard to 00:36:24.670 --> 00:36:27.040 concentrate for an hour and 15 minutes 00:36:27.040 --> 00:36:27.480 in a row. 00:36:27.480 --> 00:36:29.020 It's easy for me because I'm teaching, 00:36:29.020 --> 00:36:30.120 but harder. 00:36:30.120 --> 00:36:31.400 I would not be able to do it if I were 00:36:31.400 --> 00:36:32.280 sitting in your seats. 00:36:32.280 --> 00:36:33.980 So I'm going to take a break for like 00:36:33.980 --> 00:36:34.580 one minute. 00:36:34.580 --> 00:36:36.660 So feel free to stand up and stretch, 00:36:36.660 --> 00:36:39.500 check your e-mail, whatever you want, 00:36:39.500 --> 00:36:41.640 and then I'll show you these demos. 00:38:28.140 --> 00:38:29.990 Alright, I'm going to pick up again. 00:38:38.340 --> 00:38:39.740 Alright, I'm going to start again. 00:38:41.070 --> 00:38:43.830 Sorry, I know I'm interrupting a lot of 00:38:43.830 --> 00:38:44.860 conversations. 00:38:44.860 --> 00:38:49.488 So here's the first demo here. 00:38:49.488 --> 00:38:50.570 It's kind of simple. 00:38:50.570 --> 00:38:52.600 It's a KCNN demo actually. 00:38:52.600 --> 00:38:53.820 They're both CNN demos. 00:38:53.820 --> 00:38:54.510 Obviously. 00:38:54.510 --> 00:38:57.810 The thing I like about this demo is, I 00:38:57.810 --> 00:38:59.070 guess first I'll explain what it's 00:38:59.070 --> 00:38:59.380 doing. 00:38:59.380 --> 00:39:00.958 So it's got some red points here. 00:39:00.958 --> 00:39:01.881 This is one class. 00:39:01.881 --> 00:39:03.289 It's got some blue points. 00:39:03.290 --> 00:39:04.310 That's another class. 00:39:04.310 --> 00:39:07.035 The red area are all the areas that 00:39:07.035 --> 00:39:09.026 will be classified as red, and the blue 00:39:09.026 --> 00:39:10.614 areas are all the areas that will be 00:39:10.614 --> 00:39:11.209 classified as blue. 00:39:11.930 --> 00:39:15.344 And you can change K and you can change 00:39:15.344 --> 00:39:16.610 the distance measure. 00:39:16.610 --> 00:39:19.090 And then if I click somewhere here, it 00:39:19.090 --> 00:39:21.390 shows me which point is determining the 00:39:21.390 --> 00:39:22.560 classification. 00:39:22.560 --> 00:39:26.073 So I'm clicking on the center point and 00:39:26.073 --> 00:39:28.190 then it's drawing a connecting line and 00:39:28.190 --> 00:39:29.949 radius that correspond to the one 00:39:29.950 --> 00:39:31.465 nearest neighbor because this is set to 00:39:31.465 --> 00:39:31.670 1. 00:39:33.160 --> 00:39:35.640 So one thing I'll note I'll do is just 00:39:35.640 --> 00:39:36.116 change. 00:39:36.116 --> 00:39:38.750 KK is almost always odd because if it's 00:39:38.750 --> 00:39:40.400 even then you have like a split 00:39:40.400 --> 00:39:41.560 decision a lot of times. 00:39:42.770 --> 00:39:45.310 So if I have K = 3, just notice how the 00:39:45.310 --> 00:39:47.790 boundary changes as I increase K. 00:39:50.120 --> 00:39:52.370 It becomes simpler and simpler, right? 00:39:52.370 --> 00:39:54.300 It just becomes like eventually it 00:39:54.300 --> 00:39:55.710 should become well. 00:39:57.440 --> 00:39:59.990 Got got bigger than the data, so in K = 00:39:59.990 --> 00:40:01.770 23 I think there's probably 23 points, 00:40:01.770 --> 00:40:03.250 so it's just the most common class. 00:40:04.790 --> 00:40:07.450 And then it kind of becomes more like a 00:40:07.450 --> 00:40:09.880 straight line with a very high K. 00:40:10.190 --> 00:40:10.720 00:40:16.330 --> 00:40:18.820 Then if I change the distance measure, 00:40:18.820 --> 00:40:19.915 I've got Manhattan. 00:40:19.915 --> 00:40:22.610 Manhattan is that L1 distance, so it 00:40:22.610 --> 00:40:24.800 becomes like a little bit more. 00:40:24.890 --> 00:40:25.470 00:40:26.300 --> 00:40:27.590 A little bit more like. 00:40:28.360 --> 00:40:30.720 Vertical horizontal lines in the 00:40:30.720 --> 00:40:33.410 decision boundary compared to. 00:40:33.530 --> 00:40:34.060 00:40:34.830 --> 00:40:37.120 Compared to the Euclidian distance. 00:40:39.280 --> 00:40:40.780 00:40:41.460 --> 00:40:45.023 And then this is showing this box is 00:40:45.023 --> 00:40:47.945 showing like the box that contains all 00:40:47.945 --> 00:40:51.970 the points within the where K7 the 00:40:51.970 --> 00:40:53.800 seven nearest neighbors according to 00:40:53.800 --> 00:40:55.100 Manhattan distance. 00:40:55.100 --> 00:40:57.504 So you can see that it's kind of like a 00:40:57.504 --> 00:40:59.436 weird in some ways it feels like a 00:40:59.436 --> 00:41:00.440 weird distance measure. 00:41:00.440 --> 00:41:02.910 Another thing that I should bring up. 00:41:02.910 --> 00:41:05.950 I decide not to go into too much detail 00:41:05.950 --> 00:41:07.803 in this today because I think it's like 00:41:07.803 --> 00:41:10.890 a more of a not as central of a point 00:41:10.890 --> 00:41:11.710 as the things that I am. 00:41:11.850 --> 00:41:12.210 Talking about. 00:41:12.920 --> 00:41:16.990 But our intuition for high dimensions 00:41:16.990 --> 00:41:17.730 is really bad. 00:41:18.370 --> 00:41:21.325 So everything I visualize, almost 00:41:21.325 --> 00:41:23.090 everything is in two dimensions because 00:41:23.090 --> 00:41:25.110 that's all I can put on a piece of 00:41:25.110 --> 00:41:26.060 paper or screen. 00:41:27.700 --> 00:41:30.620 I can't visualize 1000 dimensions, but 00:41:30.620 --> 00:41:32.167 things behave kind of differently in 00:41:32.167 --> 00:41:33.790 1000 dimensions in two dimensions. 00:41:33.790 --> 00:41:37.280 So for example, if I randomly sample a 00:41:37.280 --> 00:41:39.197 whole bunch of points in a unit cube 00:41:39.197 --> 00:41:41.944 and 1000 dimensions, almost all the 00:41:41.944 --> 00:41:44.082 points lie like right on the surface of 00:41:44.082 --> 00:41:46.219 that cube, and they'll all lie if I 00:41:46.220 --> 00:41:47.025 have some epsilon. 00:41:47.025 --> 00:41:48.750 If Epsilon is like really really tiny, 00:41:48.750 --> 00:41:50.420 they'll still all be like right on the 00:41:50.420 --> 00:41:51.170 surface of that cube. 00:41:51.880 --> 00:41:54.400 And in high dimensional spaces it takes 00:41:54.400 --> 00:41:56.510 like tons and tons of data to populate 00:41:56.510 --> 00:41:59.320 that space, and so every point tends to 00:41:59.320 --> 00:42:00.890 be pretty far away from every other 00:42:00.890 --> 00:42:02.269 point in a high dimensional space. 00:42:04.440 --> 00:42:06.639 They're just worth being aware of that 00:42:06.640 --> 00:42:08.560 limitation of our minds that we don't 00:42:08.560 --> 00:42:11.200 think well in high dimensions, but I'll 00:42:11.200 --> 00:42:12.680 probably talk about it in more detail 00:42:12.680 --> 00:42:13.910 at some later time. 00:42:14.500 --> 00:42:17.290 So this demo I like even more. 00:42:17.290 --> 00:42:19.260 This is another nearest neighbor demo. 00:42:19.260 --> 00:42:21.280 Again, I get to choose the metric, I'll 00:42:21.280 --> 00:42:22.820 leave it at L2. 00:42:23.550 --> 00:42:25.360 It's that one nearest neighbor I can 00:42:25.360 --> 00:42:26.700 choose the number of points. 00:42:27.470 --> 00:42:31.110 And I'll do three classes. 00:42:32.390 --> 00:42:33.050 So. 00:42:35.540 --> 00:42:36.480 Let's see. 00:42:39.720 --> 00:42:41.800 Alright, so one thing I wanted to point 00:42:41.800 --> 00:42:45.600 out is that one nearest neighbor can be 00:42:45.600 --> 00:42:48.006 pretty sensitive to an individual 00:42:48.006 --> 00:42:48.423 point. 00:42:48.423 --> 00:42:50.670 So let's say I take this one green 00:42:50.670 --> 00:42:52.150 point and I drag it around. 00:42:54.460 --> 00:42:56.770 It can make a really big impact on the 00:42:56.770 --> 00:42:58.810 decision boundary all by itself. 00:43:00.470 --> 00:43:02.200 Right, because only that point matters. 00:43:02.200 --> 00:43:03.920 There's nothing else in this space, so 00:43:03.920 --> 00:43:05.620 it gets to claim the entire space by 00:43:05.620 --> 00:43:06.070 itself. 00:43:07.220 --> 00:43:09.600 Another thing to note about CNN is that 00:43:09.600 --> 00:43:12.660 for one N, if you create a veroni 00:43:12.660 --> 00:43:15.690 diagram which is, you split this into 00:43:15.690 --> 00:43:18.380 different cells where each cell, 00:43:18.380 --> 00:43:20.250 everything within each cell is closest 00:43:20.250 --> 00:43:21.390 to a single point. 00:43:22.160 --> 00:43:23.500 That's kind of that's the decision 00:43:23.500 --> 00:43:24.550 boundary of the cannon. 00:43:26.750 --> 00:43:29.460 So it's pretty sensitive if I change it 00:43:29.460 --> 00:43:30.740 to three and north. 00:43:31.850 --> 00:43:34.760 It's not going to be as sensitive this 00:43:34.760 --> 00:43:36.310 they're making white because it's a 3 00:43:36.310 --> 00:43:36.840 way tie. 00:43:38.430 --> 00:43:40.910 So it's still somewhat sensitive, but 00:43:40.910 --> 00:43:42.960 now if this guy invades the red zone, 00:43:42.960 --> 00:43:45.213 he doesn't really have any impact. 00:43:45.213 --> 00:43:48.220 If he's off by himself, he has a little 00:43:48.220 --> 00:43:49.510 impact, but there has to be like 00:43:49.510 --> 00:43:51.795 another green that is also close. 00:43:51.795 --> 00:43:54.569 So this guy is a supporting guy, so if 00:43:54.570 --> 00:43:55.310 I take him away. 00:43:55.970 --> 00:43:57.400 Then this guy is not going to have too 00:43:57.400 --> 00:43:58.380 much effect out here. 00:43:59.460 --> 00:44:02.280 And obviously as I increase K that. 00:44:02.730 --> 00:44:06.350 Happens even more so now this has 00:44:06.350 --> 00:44:08.240 relatively little influence. 00:44:08.310 --> 00:44:08.890 00:44:10.540 --> 00:44:12.510 A single point by itself can't do too 00:44:12.510 --> 00:44:14.670 much if you have K = 5. 00:44:17.270 --> 00:44:19.740 And then as I change again, you'll see 00:44:19.740 --> 00:44:21.760 that the decision boundary becomes a 00:44:21.760 --> 00:44:22.430 lot smoother. 00:44:22.430 --> 00:44:23.599 So here's K = 1. 00:44:23.600 --> 00:44:24.890 Notice how there's like little blue 00:44:24.890 --> 00:44:25.520 islands. 00:44:26.550 --> 00:44:29.549 K = 3 the islands go away, but it's 00:44:29.550 --> 00:44:30.410 still mostly. 00:44:30.410 --> 00:44:32.630 There's like a little tiny blue area 00:44:32.630 --> 00:44:34.490 here, but it's a kind of jagged 00:44:34.490 --> 00:44:35.490 decision boundary. 00:44:36.110 --> 00:44:39.870 K = 5 Now there's only three regions. 00:44:40.810 --> 00:44:43.510 And K = 7, the boundaries get smoother. 00:44:44.680 --> 00:44:47.200 Also it's worth noting that if K = 1, 00:44:47.200 --> 00:44:48.870 you can never have any training error. 00:44:48.870 --> 00:44:51.890 So obviously like every training point 00:44:51.890 --> 00:44:53.930 will be closest to itself, so therefore 00:44:53.930 --> 00:44:55.163 it will make the correct prediction, it 00:44:55.163 --> 00:44:56.350 will predict its own value. 00:44:57.170 --> 00:44:58.740 Unless you have a bunch of points that 00:44:58.740 --> 00:45:00.720 are right on top of each other, but 00:45:00.720 --> 00:45:02.510 that's kind of a weird edge case. 00:45:03.260 --> 00:45:06.840 And but if K = 7, you can actually have 00:45:06.840 --> 00:45:07.820 misclassifications. 00:45:07.820 --> 00:45:10.970 So there's a green points that would be 00:45:10.970 --> 00:45:12.536 that are in the training data but would 00:45:12.536 --> 00:45:14.200 be classified as blue. 00:45:19.540 --> 00:45:22.166 So some comments on KNN. 00:45:22.166 --> 00:45:26.130 So it's really simple, which is a good 00:45:26.130 --> 00:45:26.410 thing. 00:45:27.200 --> 00:45:29.440 It's an excellent baseline and 00:45:29.440 --> 00:45:30.660 sometimes it's hard to beat. 00:45:30.660 --> 00:45:33.050 For example, we'll look at the digits 00:45:33.050 --> 00:45:36.740 task later the digit cannon with like 00:45:36.740 --> 00:45:39.590 some relatively simple like feature 00:45:39.590 --> 00:45:40.540 transformations. 00:45:41.220 --> 00:45:43.330 Can do as well as any other algorithm 00:45:43.330 --> 00:45:44.500 on digits. 00:45:45.480 --> 00:45:47.220 Even the very simple case that I give 00:45:47.220 --> 00:45:50.080 you gets within a couple percent error 00:45:50.080 --> 00:45:52.040 of the best error that's reported on 00:45:52.040 --> 00:45:52.600 that data set. 00:45:55.640 --> 00:45:56.820 Yeah, so it's simple. 00:45:56.820 --> 00:45:57.540 Hard to be in. 00:45:57.540 --> 00:45:59.408 Naturally scales with the data. 00:45:59.408 --> 00:46:02.659 So if you can apply CNN even if you 00:46:02.660 --> 00:46:04.100 only have one training example per 00:46:04.100 --> 00:46:06.312 class, and you can also apply if you 00:46:06.312 --> 00:46:07.970 have a million training examples per 00:46:07.970 --> 00:46:08.370 class. 00:46:08.370 --> 00:46:10.050 And it will tend to get better the more 00:46:10.050 --> 00:46:11.169 data you have. 00:46:11.760 --> 00:46:13.380 And if you only have one training data 00:46:13.380 --> 00:46:15.160 per class, A lot of other algorithms 00:46:15.160 --> 00:46:16.680 can't be used because there's not 00:46:16.680 --> 00:46:18.980 enough data to fit models to your one 00:46:18.980 --> 00:46:22.040 example, but K and can be used so for 00:46:22.040 --> 00:46:22.970 things like. 00:46:23.720 --> 00:46:26.090 Person like identity verification or 00:46:26.090 --> 00:46:26.330 something? 00:46:26.330 --> 00:46:27.850 You might only have one example of a 00:46:27.850 --> 00:46:29.420 face and you need to match based on 00:46:29.420 --> 00:46:30.560 that example. 00:46:30.560 --> 00:46:31.880 Then you're almost certainly going to 00:46:31.880 --> 00:46:34.510 end up using nearest neighbor as part 00:46:34.510 --> 00:46:35.300 of your algorithm. 00:46:37.250 --> 00:46:40.040 Higher K gives you smoother functions, 00:46:40.040 --> 00:46:42.330 so if you increase K you get a smoother 00:46:42.330 --> 00:46:43.180 prediction function. 00:46:44.630 --> 00:46:47.500 Now 1 disadvantage of K&N is that it 00:46:47.500 --> 00:46:48.440 can be slow. 00:46:48.440 --> 00:46:50.910 So in homework one, if you apply your 00:46:50.910 --> 00:46:52.965 full test set to the full training set, 00:46:52.965 --> 00:46:56.390 it will take 10s of minutes to 00:46:56.390 --> 00:46:57.220 evaluate. 00:46:58.100 --> 00:47:00.080 Maybe 30 minutes or 60 minutes. 00:47:01.660 --> 00:47:03.210 But there's tricks to speed it up. 00:47:03.210 --> 00:47:05.300 So like a simple thing that makes a 00:47:05.300 --> 00:47:07.360 little bit of impact is that when 00:47:07.360 --> 00:47:11.950 you're minimizing the L2 distance of XI 00:47:11.950 --> 00:47:14.780 and XT, you can actually like expand it 00:47:14.780 --> 00:47:16.490 and then notice that some terms don't 00:47:16.490 --> 00:47:17.380 have any impact. 00:47:17.380 --> 00:47:17.880 So. 00:47:18.670 --> 00:47:19.645 XT is the. 00:47:19.645 --> 00:47:21.745 I want to find the minimum training 00:47:21.745 --> 00:47:24.910 image indexed by I that minimizes the 00:47:24.910 --> 00:47:27.930 distance from all my Xis to XT which is 00:47:27.930 --> 00:47:28.890 a test image. 00:47:28.890 --> 00:47:32.905 It doesn't depend on this X t ^2 or the 00:47:32.905 --> 00:47:35.910 XT transpose XT and so I don't need to 00:47:35.910 --> 00:47:36.530 compute that. 00:47:37.170 --> 00:47:39.170 Also, this only needs to be computed 00:47:39.170 --> 00:47:40.460 once per training image. 00:47:41.410 --> 00:47:43.405 Not for every single XT that I'm 00:47:43.405 --> 00:47:45.905 testing, not for every test image that 00:47:45.905 --> 00:47:47.509 test example that I'm testing. 00:47:48.220 --> 00:47:51.460 And so it this is the only thing that 00:47:51.460 --> 00:47:52.860 you have to compute for every pair of 00:47:52.860 --> 00:47:54.060 training and test examples. 00:47:56.600 --> 00:47:59.517 In a GPU you can actually do the. 00:47:59.517 --> 00:48:01.770 You could do the MNIST nearest neighbor 00:48:01.770 --> 00:48:03.595 in sub second. 00:48:03.595 --> 00:48:06.260 It's extremely fast, it's just not fast 00:48:06.260 --> 00:48:06.830 on a CPU. 00:48:08.020 --> 00:48:09.475 There's also approximate nearest 00:48:09.475 --> 00:48:11.560 neighbor methods like flan, or even 00:48:11.560 --> 00:48:13.930 exact nearest neighbor methods that are 00:48:13.930 --> 00:48:15.970 much more efficient than the simple 00:48:15.970 --> 00:48:17.310 method that you would want to use for 00:48:17.310 --> 00:48:17.750 the assignment. 00:48:20.720 --> 00:48:22.010 Another thing that's nice is that 00:48:22.010 --> 00:48:24.020 there's no training time, so there's 00:48:24.020 --> 00:48:25.243 not really any training. 00:48:25.243 --> 00:48:27.800 The training data is your model, so you 00:48:27.800 --> 00:48:29.115 don't have to do anything to train it. 00:48:29.115 --> 00:48:30.760 You just get your data, you input the 00:48:30.760 --> 00:48:30.950 data. 00:48:32.220 --> 00:48:33.680 And last year to learn a distance 00:48:33.680 --> 00:48:34.940 function or learned features or 00:48:34.940 --> 00:48:35.570 something like that. 00:48:37.730 --> 00:48:41.170 Another thing is that with infinite 00:48:41.170 --> 00:48:43.910 examples, one nearest neighbor has 00:48:43.910 --> 00:48:48.030 provable is provably has error that is 00:48:48.030 --> 00:48:50.140 at most twice the Bayes optimal error. 00:48:52.250 --> 00:48:55.640 But that's kind of a useless, somewhat 00:48:55.640 --> 00:48:59.573 useless claim because you never have 00:48:59.573 --> 00:49:02.116 infinite examples, and if you have and 00:49:02.116 --> 00:49:05.550 so I'll explain why that thing works. 00:49:05.550 --> 00:49:07.880 I'm going to have to write on chalk so 00:49:07.880 --> 00:49:09.220 this might not carry over to the 00:49:09.220 --> 00:49:12.101 recording, but basically the idea is 00:49:12.101 --> 00:49:15.949 that if you have if you have infinite 00:49:15.949 --> 00:49:17.509 examples, then what it means is that 00:49:17.510 --> 00:49:19.630 for any possible feature value where 00:49:19.630 --> 00:49:21.280 there's non 0 probability. 00:49:21.380 --> 00:49:23.040 You've got infinite examples on that 00:49:23.040 --> 00:49:24.310 one feature value as well. 00:49:25.210 --> 00:49:28.150 And so when you assign a new test, 00:49:28.150 --> 00:49:30.430 point to that to a label. 00:49:31.130 --> 00:49:34.870 You're randomly choosing one of those 00:49:34.870 --> 00:49:37.010 infinite samples that has the exact 00:49:37.010 --> 00:49:38.770 same features as your test point. 00:49:39.470 --> 00:49:42.140 So if we look at a binary, this is for 00:49:42.140 --> 00:49:43.740 binary classification. 00:49:43.740 --> 00:49:47.570 So let's say that we have like. 00:49:48.850 --> 00:49:52.940 Given some, given some features X, this 00:49:52.940 --> 00:49:54.580 is just like the X of the test that I 00:49:54.580 --> 00:49:55.210 sampled. 00:49:55.850 --> 00:49:59.360 Let's say probability of y = 1 equals 00:49:59.360 --> 00:50:00.050 epsilon. 00:50:00.720 --> 00:50:07.199 And so probability of y = 0 given X = 1 00:50:07.200 --> 00:50:08.330 minus epsilon. 00:50:09.650 --> 00:50:12.380 Then when I sample a test value and 00:50:12.380 --> 00:50:14.000 let's say epsilon is really small. 00:50:16.060 --> 00:50:18.710 When I sample a test value, one thing 00:50:18.710 --> 00:50:21.123 that could happen is that I could 00:50:21.123 --> 00:50:23.335 sample one of these epsilon probability 00:50:23.335 --> 00:50:27.360 test values or test samples, and so the 00:50:27.360 --> 00:50:28.469 true label is 1. 00:50:29.460 --> 00:50:33.010 And then my error will be epsilon times 00:50:33.010 --> 00:50:34.320 1 minus epsilon. 00:50:35.560 --> 00:50:38.520 Or more probably, if Epsilon is small, 00:50:38.520 --> 00:50:40.160 I could sample one of the test samples 00:50:40.160 --> 00:50:41.299 where y = 0. 00:50:42.420 --> 00:50:45.550 And then my probability of sampling 00:50:45.550 --> 00:50:47.634 that is 1 minus epsilon and the 00:50:47.634 --> 00:50:49.180 probability of an error given that I 00:50:49.180 --> 00:50:50.940 sampled it is epsilon. 00:50:50.940 --> 00:50:52.985 So that's the probability that then I 00:50:52.985 --> 00:50:54.149 sample a training sample. 00:50:54.149 --> 00:50:56.080 I randomly choose a training sample of 00:50:56.080 --> 00:50:58.020 all the exact match matching training 00:50:58.020 --> 00:51:00.390 samples that has that class. 00:51:01.330 --> 00:51:02.760 And so the total error. 00:51:03.790 --> 00:51:09.105 Is Epsilon is 2 epsilon minus two 00:51:09.105 --> 00:51:10.480 epsilon squared? 00:51:12.440 --> 00:51:15.130 As Epsilon gets really small, this guy 00:51:15.130 --> 00:51:16.350 goes away, right? 00:51:16.350 --> 00:51:18.540 This will go to zero faster than this. 00:51:19.490 --> 00:51:22.950 And so my error is 2 epsilon. 00:51:23.610 --> 00:51:26.000 But the best thing I could have done 00:51:26.000 --> 00:51:27.680 was just chosen. 00:51:27.680 --> 00:51:30.420 In this case, the optimal decision 00:51:30.420 --> 00:51:33.220 would have been to choose Class 0 every 00:51:33.220 --> 00:51:35.137 time in this scenario, because this is 00:51:35.137 --> 00:51:37.370 the more probable one, and the error 00:51:37.370 --> 00:51:38.970 for this would just be epsilon. 00:51:38.970 --> 00:51:41.014 So my nearest neighbor error is 2 00:51:41.014 --> 00:51:41.385 epsilon. 00:51:41.385 --> 00:51:43.240 The optimal error is epsilon. 00:51:44.950 --> 00:51:46.950 So the reason that I show the 00:51:46.950 --> 00:51:49.540 derivation of that theorem is just 00:51:49.540 --> 00:51:50.180 that. 00:51:50.300 --> 00:51:50.890 00:51:52.000 --> 00:51:54.090 It's like kind of ridiculously 00:51:54.090 --> 00:51:54.606 implausible. 00:51:54.606 --> 00:51:56.910 So the theorem only holds if you 00:51:56.910 --> 00:51:58.626 actually have infinite training samples 00:51:58.626 --> 00:52:00.479 for every single possible value of the 00:52:00.480 --> 00:52:01.050 features. 00:52:01.050 --> 00:52:04.327 So while while theoretically with 00:52:04.327 --> 00:52:06.490 infinite training samples one NN 00:52:06.490 --> 00:52:08.120 has error, that's at most twice the 00:52:08.120 --> 00:52:10.950 Bayes optimal error rate, in practice 00:52:10.950 --> 00:52:12.355 like that tells you absolutely nothing 00:52:12.355 --> 00:52:12.870 at all. 00:52:12.870 --> 00:52:14.650 So I just want to mention that because 00:52:14.650 --> 00:52:16.690 it's an often, it's an often quoted 00:52:16.690 --> 00:52:17.690 thing about nearest neighbor. 00:52:17.690 --> 00:52:18.880 It doesn't mean that it's any good, 00:52:18.880 --> 00:52:21.980 although it is good, just not for that. 00:52:23.180 --> 00:52:24.420 So then. 00:52:24.500 --> 00:52:24.950 00:52:25.830 --> 00:52:27.710 So that was nearest neighbor. 00:52:27.710 --> 00:52:29.570 Now I want to talk a little bit about 00:52:29.570 --> 00:52:31.930 error, how we measure it and what 00:52:31.930 --> 00:52:32.560 causes it. 00:52:33.690 --> 00:52:34.300 So. 00:52:34.950 --> 00:52:36.660 When we measure and analyze 00:52:36.660 --> 00:52:38.080 classification error. 00:52:39.760 --> 00:52:43.060 The most common sounds a little 00:52:43.060 --> 00:52:45.760 redundant, but the most common way to 00:52:45.760 --> 00:52:48.320 measure the error of a classifier is 00:52:48.320 --> 00:52:50.510 with the classification error, which is 00:52:50.510 --> 00:52:51.930 the percent of examples that are 00:52:51.930 --> 00:52:52.440 incorrect. 00:52:53.400 --> 00:52:55.470 So mathematically it's just the sum 00:52:55.470 --> 00:52:56.140 over. 00:52:57.850 --> 00:53:00.229 I'm assuming that this like not equal 00:53:00.230 --> 00:53:02.829 sign just returns A1 or A01 if they're 00:53:02.829 --> 00:53:04.609 not equal, 0 if they're equal. 00:53:05.120 --> 00:53:08.716 And so it's just a count of the number 00:53:08.716 --> 00:53:10.120 of cases where the prediction is 00:53:10.120 --> 00:53:12.390 different than the true value divided 00:53:12.390 --> 00:53:13.610 by the number of cases that are 00:53:13.610 --> 00:53:14.140 evaluated. 00:53:15.550 --> 00:53:17.570 And then if you want to provide more 00:53:17.570 --> 00:53:19.220 insight into the kinds of errors that 00:53:19.220 --> 00:53:21.030 you get, you would use a confusion 00:53:21.030 --> 00:53:21.590 matrix. 00:53:22.400 --> 00:53:24.950 So a confusion matrix is a count of for 00:53:24.950 --> 00:53:26.379 each how many. 00:53:26.380 --> 00:53:27.533 There's two ways of doing it. 00:53:27.533 --> 00:53:29.370 One is just count wise. 00:53:29.370 --> 00:53:32.850 How many examples had a true prediction 00:53:32.850 --> 00:53:35.580 or a true value of 1 label and a 00:53:35.580 --> 00:53:37.200 predicted value of another label. 00:53:37.860 --> 00:53:40.242 So here these are the true labels. 00:53:40.242 --> 00:53:43.210 These are the predicted labels, and 00:53:43.210 --> 00:53:48.520 sometimes you normalize it by the 00:53:48.520 --> 00:53:50.620 fraction of true labels, typically. 00:53:50.620 --> 00:53:53.352 So this means that out of all of the 00:53:53.352 --> 00:53:55.460 true labels that were set, OSA, 00:53:55.460 --> 00:53:58.230 whatever that means of 100% of them, 00:53:58.230 --> 00:53:59.760 were assigned to set OSA. 00:54:01.330 --> 00:54:04.890 Out of all the test samples where the 00:54:04.890 --> 00:54:07.762 true label was versicolor, 62% were 00:54:07.762 --> 00:54:10.740 assigned a versicolor and 38% were 00:54:10.740 --> 00:54:12.210 assigned to VIRGINICA. 00:54:13.150 --> 00:54:15.950 And out of all the test samples where 00:54:15.950 --> 00:54:18.320 the true label is virginica, 100% were 00:54:18.320 --> 00:54:19.650 assigned to virginica. 00:54:19.650 --> 00:54:21.610 So this tells you like a little bit 00:54:21.610 --> 00:54:22.950 more than the classification error, 00:54:22.950 --> 00:54:24.420 because now you can see there's only 00:54:24.420 --> 00:54:26.590 mistakes made on this versa color and 00:54:26.590 --> 00:54:28.250 it only gets confused with virginica. 00:54:30.900 --> 00:54:32.760 So I'll give you an example here. 00:54:34.790 --> 00:54:38.620 So there's no document projector thing, 00:54:38.620 --> 00:54:39.480 unfortunately. 00:54:40.140 --> 00:54:44.077 Which I will try to fix, but I will 00:54:44.077 --> 00:54:45.870 this is simple enough that I can just 00:54:45.870 --> 00:54:48.175 draw on this slide or type on this 00:54:48.175 --> 00:54:48.410 slide. 00:54:50.880 --> 00:54:51.120 Yeah. 00:54:54.590 --> 00:54:55.370 00:54:58.460 --> 00:54:59.040 There. 00:55:05.270 --> 00:55:07.190 OK, I don't want to figure that out. 00:55:07.190 --> 00:55:08.530 So. 00:55:14.470 --> 00:55:14.940 I. 00:55:21.420 --> 00:55:23.060 That sounds good. 00:55:23.060 --> 00:55:23.500 There it goes. 00:55:25.990 --> 00:55:29.845 OK, so I will just verbally do it. 00:55:29.845 --> 00:55:31.770 So let's say so these are the true 00:55:31.770 --> 00:55:32.055 labels. 00:55:32.055 --> 00:55:34.430 These are the predicted labels. 00:55:34.430 --> 00:55:36.020 What is the classification error? 00:55:58.730 --> 00:56:00.082 Yeah, 3 / 7. 00:56:00.082 --> 00:56:04.860 So there's 77 rows right that are other 00:56:04.860 --> 00:56:08.463 than the label row, and there are three 00:56:08.463 --> 00:56:10.023 three times that. 00:56:10.023 --> 00:56:12.300 One of the values is no and one of the 00:56:12.300 --> 00:56:13.090 values is yes. 00:56:13.090 --> 00:56:16.580 So the classification error is 3 / 7. 00:56:17.810 --> 00:56:21.170 And let's do the confusion matrix. 00:56:28.020 --> 00:56:30.960 Right, so the so the true label. 00:56:30.960 --> 00:56:33.060 So how many times do I have a true 00:56:33.060 --> 00:56:35.070 label that's known and a predicted 00:56:35.070 --> 00:56:35.960 label that's no. 00:56:37.520 --> 00:56:38.080 Two. 00:56:38.080 --> 00:56:39.570 OK, how many times do I have a true 00:56:39.570 --> 00:56:41.265 label that's known and predicted label? 00:56:41.265 --> 00:56:41.850 That's yes. 00:56:45.390 --> 00:56:48.026 OK, how many times do I have a true 00:56:48.026 --> 00:56:49.535 label that's yes and predicted label 00:56:49.535 --> 00:56:50.060 that's no? 00:56:51.800 --> 00:56:54.190 One, and I guess I have two of the 00:56:54.190 --> 00:56:54.540 others. 00:56:55.650 --> 00:56:56.329 Is that right? 00:56:56.330 --> 00:56:58.300 I have two times that there's a true 00:56:58.300 --> 00:57:00.420 label yes and a predicted label of no. 00:57:00.420 --> 00:57:01.260 Is that right? 00:57:03.730 --> 00:57:05.049 Or no, yes and yes. 00:57:05.050 --> 00:57:06.050 I'm on yes and yes. 00:57:06.050 --> 00:57:06.920 Two, yes. 00:57:07.890 --> 00:57:08.740 OK, cool. 00:57:08.740 --> 00:57:09.380 All right, good. 00:57:09.380 --> 00:57:09.903 Thumbs up. 00:57:09.903 --> 00:57:11.750 All right, so this sums up to 7. 00:57:11.750 --> 00:57:13.510 So this is a confusion matrix. 00:57:13.510 --> 00:57:14.920 That's just in terms of the total 00:57:14.920 --> 00:57:15.380 counts. 00:57:16.340 --> 00:57:18.510 And then if I want to convert this to. 00:57:19.290 --> 00:57:23.000 A normalized matrix, which is basically 00:57:23.000 --> 00:57:25.330 the probability that I predict a 00:57:25.330 --> 00:57:27.530 particular value given the true label. 00:57:27.530 --> 00:57:29.540 So this will be the probability that I 00:57:29.540 --> 00:57:32.025 predicted no given that the true label 00:57:32.025 --> 00:57:32.720 is no. 00:57:33.360 --> 00:57:35.280 Then I just divide by the total count 00:57:35.280 --> 00:57:37.230 or I divide by the. 00:57:37.880 --> 00:57:40.250 By the number of examples in each row. 00:57:40.250 --> 00:57:42.580 So this one would be what? 00:57:42.580 --> 00:57:44.295 What's the probability that I predict 00:57:44.295 --> 00:57:46.260 no given that the true answer is no? 00:57:48.530 --> 00:57:49.200 F right? 00:57:49.200 --> 00:57:50.760 I just divide this by 4. 00:57:51.790 --> 00:57:53.680 And likewise divide this by 4. 00:57:53.680 --> 00:57:55.360 And what is the probability that I 00:57:55.360 --> 00:57:56.800 predict no given that the true answer 00:57:56.800 --> 00:57:57.340 is yes? 00:57:59.660 --> 00:58:02.440 Right 1 / 3 and this will be 2 / 3. 00:58:03.400 --> 00:58:05.210 So that's how you compute the confusion 00:58:05.210 --> 00:58:07.260 matrix and the classification error. 00:58:12.880 --> 00:58:15.560 All right, so for regression error. 00:58:15.650 --> 00:58:16.380 00:58:17.890 --> 00:58:20.920 You will usually use one of these. 00:58:20.920 --> 00:58:23.316 Root mean squared error is probably one 00:58:23.316 --> 00:58:26.320 of the most common, so that's just 00:58:26.320 --> 00:58:27.280 written there. 00:58:27.280 --> 00:58:30.790 You take this sum of squared values, 00:58:30.790 --> 00:58:33.780 and then you divide it by the total 00:58:33.780 --> 00:58:34.989 number of values. 00:58:34.990 --> 00:58:37.580 N is the range of I. 00:58:38.250 --> 00:58:40.050 And then you take the square root. 00:58:40.050 --> 00:58:42.630 So sometimes the mistake you can make 00:58:42.630 --> 00:58:44.000 on this is to do the order of 00:58:44.000 --> 00:58:44.950 operations wrong. 00:58:45.570 --> 00:58:47.855 Just remember it's in the name root 00:58:47.855 --> 00:58:48.846 mean squared. 00:58:48.846 --> 00:58:53.260 So you take the and then so it's like 00:58:53.260 --> 00:58:55.594 right now as an equation it's the root 00:58:55.594 --> 00:58:58.346 then the mean divided by north and then 00:58:58.346 --> 00:59:00.946 you have this summation squared so you 00:59:00.946 --> 00:59:01.428 take. 00:59:01.428 --> 00:59:02.210 So yeah. 00:59:05.010 --> 00:59:05.500 All right. 00:59:05.500 --> 00:59:08.490 So that's so root squared is kind of 00:59:08.490 --> 00:59:09.960 sensitive to your outliers. 00:59:09.960 --> 00:59:13.850 If you had if you had like some things 00:59:13.850 --> 00:59:15.510 that are mislabeled or just really 00:59:15.510 --> 00:59:17.510 weird examples they could end up 00:59:17.510 --> 00:59:19.260 dominating your RMSE error. 00:59:19.260 --> 00:59:21.560 So if like one of these guys, if I'm 00:59:21.560 --> 00:59:23.490 doing some regression or something and 00:59:23.490 --> 00:59:26.500 one of them is like way, way off, 00:59:26.500 --> 00:59:29.122 that's going to be the that the root 00:59:29.122 --> 00:59:31.580 mean squared error of that one example 00:59:31.580 --> 00:59:33.060 is going to be most of the. 00:59:33.130 --> 00:59:34.010 Mean squared error. 00:59:35.430 --> 00:59:36.930 So you can also sometimes do mean 00:59:36.930 --> 00:59:39.000 absolute error that will be less 00:59:39.000 --> 00:59:40.940 sensitive to outliers, things that have 00:59:40.940 --> 00:59:42.000 extraordinary error. 00:59:43.150 --> 00:59:45.700 And then both of these are sensitive to 00:59:45.700 --> 00:59:46.480 your units. 00:59:46.480 --> 00:59:48.590 So if you're measuring the root mean 00:59:48.590 --> 00:59:51.090 squared error and feet versus meters, 00:59:51.090 --> 00:59:52.740 you'll obviously get different values. 00:59:53.900 --> 00:59:56.120 And so a lot of times sometimes people 00:59:56.120 --> 01:00:01.250 use R2, which is the amount of 01:00:01.250 --> 01:00:02.520 explained variance. 01:00:02.520 --> 01:00:07.329 So you're normalizing so the R2 is 1 01:00:07.330 --> 01:00:09.740 minus this thing here, this ratio. 01:00:10.470 --> 01:00:13.583 And the numerator of this ratio is the 01:00:13.583 --> 01:00:16.890 sum of squared difference between your 01:00:16.890 --> 01:00:18.460 prediction and the true value. 01:00:19.470 --> 01:00:21.535 So if you divide that by N, it's the 01:00:21.535 --> 01:00:21.800 variance. 01:00:21.800 --> 01:00:23.930 It's the conditional variance of the. 01:00:24.860 --> 01:00:27.936 True prediction given your model's 01:00:27.936 --> 01:00:28.819 prediction. 01:00:30.130 --> 01:00:32.746 And then you divide it by the variance 01:00:32.746 --> 01:00:35.854 or the OR you could have a one over 01:00:35.854 --> 01:00:37.402 north here and one over north here and 01:00:37.402 --> 01:00:39.230 then this would be predicted the 01:00:39.230 --> 01:00:40.805 conditional variance and this is the 01:00:40.805 --> 01:00:42.060 variance of the true labels. 01:00:43.280 --> 01:00:46.710 So 1 minus that ratio is the amount of 01:00:46.710 --> 01:00:48.160 the variance that's explained and it 01:00:48.160 --> 01:00:49.340 doesn't have any units. 01:00:49.340 --> 01:00:52.359 If you measure it in feet or meters, 01:00:52.360 --> 01:00:53.770 you're going to get exactly the same 01:00:53.770 --> 01:00:55.440 value because the feet or the meters 01:00:55.440 --> 01:00:57.519 will cancel out and that ratio. 01:01:00.130 --> 01:01:01.520 That we might talk, well, we'll talk 01:01:01.520 --> 01:01:03.120 about that more perhaps when we talk 01:01:03.120 --> 01:01:04.070 about linear regression. 01:01:05.230 --> 01:01:06.360 But just worth knowing. 01:01:07.750 --> 01:01:08.780 At least at a high level. 01:01:10.070 --> 01:01:12.100 All right, so then there's a question 01:01:12.100 --> 01:01:15.620 of why if I fit a model as I can 01:01:15.620 --> 01:01:18.120 possibly fit it, then why do I still 01:01:18.120 --> 01:01:20.230 have error when I evaluate on my test 01:01:20.230 --> 01:01:20.830 samples? 01:01:20.830 --> 01:01:23.060 You'll see in your in your homework 01:01:23.060 --> 01:01:24.670 problem, you're not going to have any 01:01:24.670 --> 01:01:26.180 methods that achieve 0 error in 01:01:26.180 --> 01:01:26.650 testing. 01:01:29.320 --> 01:01:31.050 So there's several possible reasons. 01:01:31.050 --> 01:01:33.280 So one is that there could be an error 01:01:33.280 --> 01:01:34.770 that's intrinsic to the problem. 01:01:34.770 --> 01:01:37.150 It's not possible to have 0 error. 01:01:37.150 --> 01:01:39.020 So if you're trying to predict, for 01:01:39.020 --> 01:01:41.660 example, what the weather is tomorrow, 01:01:41.660 --> 01:01:42.989 then given your features, you're not 01:01:42.990 --> 01:01:44.130 going to have a perfect prediction. 01:01:44.130 --> 01:01:45.666 Nobody knows exactly what the weather 01:01:45.666 --> 01:01:46.139 is tomorrow. 01:01:47.350 --> 01:01:49.350 If you're trying to classify a 01:01:49.350 --> 01:01:51.420 handwritten character again, it might. 01:01:51.420 --> 01:01:53.520 You might not be able to get 0 error 01:01:53.520 --> 01:01:55.630 because somebody might write an A 01:01:55.630 --> 01:01:57.370 exactly the same way that somebody 01:01:57.370 --> 01:02:00.260 wrote a no another time or whatever. 01:02:00.260 --> 01:02:02.190 Sometimes it's just not possible to 01:02:02.190 --> 01:02:04.630 know exact, to be completely confident 01:02:04.630 --> 01:02:07.783 about what the true character of a 01:02:07.783 --> 01:02:08.730 handwritten character is. 01:02:10.160 --> 01:02:11.810 So there's a notion called the Bayes 01:02:11.810 --> 01:02:14.410 optimal error, and that's the error if 01:02:14.410 --> 01:02:16.945 the true function, the probability of 01:02:16.945 --> 01:02:18.770 the label given the data is known. 01:02:18.770 --> 01:02:20.320 So you can't do any better than that. 01:02:23.510 --> 01:02:25.955 Another source of error is called is 01:02:25.955 --> 01:02:28.470 model bias, which means that the model 01:02:28.470 --> 01:02:29.970 doesn't allow you to fit whatever you 01:02:29.970 --> 01:02:30.200 want. 01:02:30.850 --> 01:02:33.600 There's some things that some training 01:02:33.600 --> 01:02:35.500 data can't be fit necessarily. 01:02:36.330 --> 01:02:39.290 And so you can't achieve. 01:02:39.290 --> 01:02:40.890 Even if you had an infinite training 01:02:40.890 --> 01:02:42.530 set, you won't be able to achieve the 01:02:42.530 --> 01:02:43.510 Bayes optimal error. 01:02:44.320 --> 01:02:47.030 So one nearest neighbor, for example, 01:02:47.030 --> 01:02:48.010 has no bias. 01:02:48.010 --> 01:02:50.550 With one nearest neighbor you can fit 01:02:50.550 --> 01:02:52.280 the training set perfectly and if your 01:02:52.280 --> 01:02:53.420 test set comes from the same 01:02:53.420 --> 01:02:54.160 distribution. 01:02:54.780 --> 01:02:56.519 Then you're going to you're going to 01:02:56.520 --> 01:02:57.860 get twice the Bayes optimal error, but. 01:02:59.130 --> 01:03:00.360 You'll get close. 01:03:01.040 --> 01:03:04.695 So the One North has very minimal bias, 01:03:04.695 --> 01:03:06.280 I guess I should say. 01:03:06.280 --> 01:03:08.060 But if you're doing a linear fit, that 01:03:08.060 --> 01:03:10.060 has really high bias, because all you 01:03:10.060 --> 01:03:10.850 can do is fit a line. 01:03:10.850 --> 01:03:12.147 If the data is on a line, you'll still 01:03:12.147 --> 01:03:13.390 fit a line, it won't be a very good 01:03:13.390 --> 01:03:13.540 fit. 01:03:15.390 --> 01:03:18.155 Model variance means that if you were 01:03:18.155 --> 01:03:20.290 to sample different sets of data, 01:03:20.290 --> 01:03:22.190 you're going to come up with different 01:03:22.190 --> 01:03:24.480 predictions on your test data, or 01:03:24.480 --> 01:03:26.870 different parameters for your model. 01:03:27.490 --> 01:03:31.100 So the variance the. 01:03:32.070 --> 01:03:34.070 Bias and variance both have to do with 01:03:34.070 --> 01:03:35.810 the simplicity of your model. 01:03:35.810 --> 01:03:37.780 If you have a really complex model that 01:03:37.780 --> 01:03:39.340 can fit everything, anything. 01:03:39.980 --> 01:03:42.600 Then it's going to have low, then it's 01:03:42.600 --> 01:03:44.892 going to have low bias but high 01:03:44.892 --> 01:03:45.220 variance. 01:03:45.220 --> 01:03:47.178 If you have a really simple model, it's 01:03:47.178 --> 01:03:50.216 going to have high bias but low 01:03:50.216 --> 01:03:50.650 variance. 01:03:52.150 --> 01:03:53.400 The variance means that you have 01:03:53.400 --> 01:03:55.200 trouble fitting your model given a 01:03:55.200 --> 01:03:56.510 limited amount of training data. 01:03:57.880 --> 01:03:59.030 You can also have things like 01:03:59.030 --> 01:04:00.850 distribution shift that some things are 01:04:00.850 --> 01:04:03.150 more common and some samples are more 01:04:03.150 --> 01:04:04.220 common in the test set than the 01:04:04.220 --> 01:04:06.450 training set if they're not IID, which 01:04:06.450 --> 01:04:07.610 I discussed before. 01:04:08.710 --> 01:04:10.350 Or you could have in the worst case of 01:04:10.350 --> 01:04:12.360 function shift, which means that the. 01:04:13.490 --> 01:04:16.375 That the answer and the test data, the 01:04:16.375 --> 01:04:17.691 probability of a particular answer 01:04:17.691 --> 01:04:20.023 given the data given the features is 01:04:20.023 --> 01:04:21.560 different in testing than training. 01:04:21.560 --> 01:04:24.305 So one example is if you're trying if 01:04:24.305 --> 01:04:26.065 you're doing like language prediction 01:04:26.065 --> 01:04:28.070 and somebody says what is your favorite 01:04:28.070 --> 01:04:31.250 TV show and you trained based on data 01:04:31.250 --> 01:04:36.197 from 2010 to 2020, then probably the 01:04:36.197 --> 01:04:38.192 answer in that time range, the 01:04:38.192 --> 01:04:40.047 probability of different answers then 01:04:40.047 --> 01:04:41.560 is different than it is today. 01:04:41.560 --> 01:04:42.980 So you actually have. 01:04:43.030 --> 01:04:44.470 Changed your. 01:04:44.470 --> 01:04:48.510 If you're test set is from 2022, then 01:04:48.510 --> 01:04:50.980 the probability of Y the answer to that 01:04:50.980 --> 01:04:53.910 question is different in the test set 01:04:53.910 --> 01:04:55.550 than it is in a training set that came 01:04:55.550 --> 01:04:57.130 from 2000 to 2020. 01:05:00.450 --> 01:05:03.714 Then there's other things that are that 01:05:03.714 --> 01:05:06.760 are that are also can be issues if 01:05:06.760 --> 01:05:08.210 you're imperfectly optimized on the 01:05:08.210 --> 01:05:08.880 training set. 01:05:09.660 --> 01:05:12.550 Or if you are not able to optimize. 01:05:13.420 --> 01:05:16.050 For the same, if you're a training loss 01:05:16.050 --> 01:05:17.480 is different than your final 01:05:17.480 --> 01:05:18.190 evaluation. 01:05:18.980 --> 01:05:20.450 That's actually happens all the time 01:05:20.450 --> 01:05:22.310 because it's really hard to optimize 01:05:22.310 --> 01:05:23.040 for training error. 01:05:26.620 --> 01:05:27.040 So. 01:05:28.040 --> 01:05:28.830 Here's a question. 01:05:28.830 --> 01:05:31.540 So what happens if so? 01:05:31.540 --> 01:05:34.222 Suppose that you train a model and then 01:05:34.222 --> 01:05:35.879 you increase the number of training 01:05:35.879 --> 01:05:38.200 samples, and then you train it again. 01:05:38.200 --> 01:05:40.170 As you increase the number of training 01:05:40.170 --> 01:05:41.880 samples, do you expect the test error 01:05:41.880 --> 01:05:43.850 to go up or down or stay the same? 01:05:48.170 --> 01:05:49.710 So you'd expect it. 01:05:49.710 --> 01:05:51.260 Some people are saying down as you get 01:05:51.260 --> 01:05:54.052 more training data you should fit a 01:05:54.052 --> 01:05:54.305 better. 01:05:54.305 --> 01:05:55.710 You should have like a better 01:05:55.710 --> 01:05:57.540 understanding of your true parameters. 01:05:57.540 --> 01:05:59.110 So the test error should go down. 01:05:59.870 --> 01:06:01.510 So it might look something like this. 01:06:03.130 --> 01:06:07.910 If I get more training data and then I 01:06:07.910 --> 01:06:09.170 measure the training error. 01:06:10.070 --> 01:06:12.510 Do you expect the training error to go 01:06:12.510 --> 01:06:14.080 up or down or stay the same? 01:06:16.740 --> 01:06:17.890 There are how many people think it 01:06:17.890 --> 01:06:18.560 would go up? 01:06:21.510 --> 01:06:23.280 How many people think the training area 01:06:23.280 --> 01:06:25.080 would go down as they get more training 01:06:25.080 --> 01:06:25.400 data? 01:06:27.750 --> 01:06:29.760 OK, so there's a lot of uncertainty. 01:06:29.760 --> 01:06:32.593 So what I would expect is that the 01:06:32.593 --> 01:06:35.410 training error will go up because as 01:06:35.410 --> 01:06:37.170 you get more training data, it becomes 01:06:37.170 --> 01:06:38.670 harder to fit that data. 01:06:38.670 --> 01:06:40.720 Given the same model, it becomes harder 01:06:40.720 --> 01:06:42.660 and harder to fit an increasing size 01:06:42.660 --> 01:06:43.250 training set. 01:06:44.120 --> 01:06:46.920 And if you get infinite examples and 01:06:46.920 --> 01:06:49.230 you don't have any things like a 01:06:49.230 --> 01:06:51.000 function shift, then these two will 01:06:51.000 --> 01:06:51.340 meet. 01:06:51.340 --> 01:06:54.122 If you get infinite examples, then you 01:06:54.122 --> 01:06:54.520 will. 01:06:54.520 --> 01:06:56.030 You're training and tests are basically 01:06:56.030 --> 01:06:56.520 the same. 01:06:57.140 --> 01:06:58.690 And then you will have the same error, 01:06:58.690 --> 01:07:00.030 so they start to converge. 01:07:02.070 --> 01:07:03.490 And this is important concept 01:07:03.490 --> 01:07:04.350 generalization error. 01:07:04.350 --> 01:07:06.530 Generalization error is the difference 01:07:06.530 --> 01:07:08.240 between your test error and your 01:07:08.240 --> 01:07:08.810 training error. 01:07:08.810 --> 01:07:10.805 So your test error is your training 01:07:10.805 --> 01:07:12.479 error plus your generalization error. 01:07:12.479 --> 01:07:15.250 Generalization error is due to the 01:07:15.250 --> 01:07:19.370 ability of your or the failure of your 01:07:19.370 --> 01:07:21.520 model to make predictions on the data 01:07:21.520 --> 01:07:22.500 hasn't seen yet. 01:07:22.500 --> 01:07:24.670 So you could have something that has 01:07:24.670 --> 01:07:26.480 absolutely perfect training error but 01:07:26.480 --> 01:07:28.370 has enormous generalization error and 01:07:28.370 --> 01:07:29.140 that's no good. 01:07:29.140 --> 01:07:30.780 Or you could have something that has a 01:07:30.780 --> 01:07:32.170 lot of trouble fitting the training. 01:07:32.230 --> 01:07:33.750 Data, but its generalization error is 01:07:33.750 --> 01:07:34.320 very small. 01:07:39.000 --> 01:07:39.460 So. 01:07:41.680 --> 01:07:43.470 If you train so suppose you have 01:07:43.470 --> 01:07:45.820 infinite training examples, then 01:07:45.820 --> 01:07:48.508 eventually you're training error will 01:07:48.508 --> 01:07:51.175 reach some plateau, and your test error 01:07:51.175 --> 01:07:53.239 will also reach some plateau. 01:07:54.150 --> 01:07:56.773 This these will reach the same point if 01:07:56.773 --> 01:07:58.640 you don't have any function shift. 01:07:58.640 --> 01:08:01.795 So if you have some difference, if you 01:08:01.795 --> 01:08:03.820 have some gap to where they're 01:08:03.820 --> 01:08:06.130 converging, it either means that you 01:08:06.130 --> 01:08:07.640 have that you're not able to fully 01:08:07.640 --> 01:08:10.056 optimize your function, or that the 01:08:10.056 --> 01:08:11.890 that you have a function shift that the 01:08:11.890 --> 01:08:13.160 probability of the true label is 01:08:13.160 --> 01:08:14.760 changing between training and test. 01:08:16.520 --> 01:08:19.117 Now, this gap between the test area 01:08:19.117 --> 01:08:20.750 that you would get from infinite 01:08:20.750 --> 01:08:22.733 training examples and the actual test 01:08:22.733 --> 01:08:24.310 area that you're getting given finite 01:08:24.310 --> 01:08:28.020 training examples is due to the model 01:08:28.020 --> 01:08:28.670 variants. 01:08:28.670 --> 01:08:30.615 It's due to the model complexity and 01:08:30.615 --> 01:08:32.500 the inability to perfectly solve for 01:08:32.500 --> 01:08:34.200 the best parameters given your limited 01:08:34.200 --> 01:08:34.770 training data. 01:08:35.900 --> 01:08:38.800 And it can also be exacerbated by 01:08:38.800 --> 01:08:40.840 distribution shift if you like, your 01:08:40.840 --> 01:08:42.710 training data is more likely to sample 01:08:42.710 --> 01:08:44.410 some areas of the feature space than 01:08:44.410 --> 01:08:45.110 your test data. 01:08:46.970 --> 01:08:49.990 And this gap the training error. 01:08:50.830 --> 01:08:52.876 Is due to the limited power of your 01:08:52.876 --> 01:08:55.360 model to fit whatever whatever you give 01:08:55.360 --> 01:08:55.590 it. 01:08:55.590 --> 01:08:58.200 So it's due to the model bias, and it's 01:08:58.200 --> 01:09:00.120 also due to the unavoidable intrinsic 01:09:00.120 --> 01:09:02.580 error that even if you have infinite 01:09:02.580 --> 01:09:04.180 examples, there's some error that's 01:09:04.180 --> 01:09:04.950 unavoidable. 01:09:05.780 --> 01:09:07.420 Either because it's intrinsic to the 01:09:07.420 --> 01:09:09.320 problem or because your model has 01:09:09.320 --> 01:09:10.250 limited capacity. 01:09:16.100 --> 01:09:16.590 All right. 01:09:16.590 --> 01:09:18.230 So I'm bringing up a point that I 01:09:18.230 --> 01:09:19.590 raised earlier. 01:09:20.930 --> 01:09:24.070 And I want to see if you can still 01:09:24.070 --> 01:09:25.350 explain the answer. 01:09:25.350 --> 01:09:27.510 So why is it important to have a 01:09:27.510 --> 01:09:28.570 validation set? 01:09:30.680 --> 01:09:32.180 If I've got a bunch of models that I 01:09:32.180 --> 01:09:35.400 want to evaluate, why don't I just take 01:09:35.400 --> 01:09:37.060 do a train set and test set? 01:09:37.710 --> 01:09:39.110 Train them all in the training set, 01:09:39.110 --> 01:09:40.760 evaluate them all in the test set and 01:09:40.760 --> 01:09:42.650 then report the best performance. 01:09:42.650 --> 01:09:43.970 What's the issue with that? 01:09:43.970 --> 01:09:46.120 Why is that not a good procedure? 01:09:47.970 --> 01:09:49.590 I guess back with the orange shirt, 01:09:49.590 --> 01:09:50.370 easier in first. 01:09:52.350 --> 01:09:54.756 So your risk overfitting the model, so 01:09:54.756 --> 01:09:56.190 that the problem is that. 01:09:56.980 --> 01:09:59.915 You're the problem is that your test 01:09:59.915 --> 01:10:02.840 error measure will be biased, which 01:10:02.840 --> 01:10:05.170 means that it won't be the expected 01:10:05.170 --> 01:10:07.620 value is not the true value. 01:10:07.620 --> 01:10:08.980 In other words, you're going to tend to 01:10:08.980 --> 01:10:11.400 underestimate the error if you do this 01:10:11.400 --> 01:10:13.800 procedure because you're choosing the 01:10:13.800 --> 01:10:15.529 best model based on the test 01:10:15.530 --> 01:10:16.430 performance. 01:10:16.430 --> 01:10:18.370 But this test sample is just one random 01:10:18.370 --> 01:10:19.880 sample from the general test 01:10:19.880 --> 01:10:21.250 distribution, so if you're to take 01:10:21.250 --> 01:10:22.530 another sample, it might have a 01:10:22.530 --> 01:10:23.200 different answer. 01:10:24.770 --> 01:10:28.290 And there's been cases where one time 01:10:28.290 --> 01:10:30.840 somebody had some agency had some big 01:10:30.840 --> 01:10:34.819 challenge they had, they had, they 01:10:34.820 --> 01:10:35.840 thought they were doing the right 01:10:35.840 --> 01:10:36.045 thing. 01:10:36.045 --> 01:10:37.898 They had a test set, they had a train 01:10:37.898 --> 01:10:38.104 set. 01:10:38.104 --> 01:10:40.827 They said you can only evaluate on the 01:10:40.827 --> 01:10:43.176 train set and only test on the test 01:10:43.176 --> 01:10:43.469 set. 01:10:43.470 --> 01:10:45.135 But they provided both the train set 01:10:45.135 --> 01:10:46.960 and the test set to the researchers. 01:10:47.600 --> 01:10:50.780 And one group like iterated through a 01:10:50.780 --> 01:10:53.400 million different models and found a 01:10:53.400 --> 01:10:55.451 model that got that you could train on 01:10:55.451 --> 01:10:57.080 the train set and achieved perfect 01:10:57.080 --> 01:10:58.400 error on the test set. 01:10:58.400 --> 01:11:00.182 But then when they applied a held out 01:11:00.182 --> 01:11:02.459 test set, it did like really really 01:11:02.460 --> 01:11:04.180 badly, like almost chance performance. 01:11:05.170 --> 01:11:08.930 So the so training on your, even doing 01:11:08.930 --> 01:11:10.319 model selection on your. 01:11:11.920 --> 01:11:13.850 On your test set, it's called like meta 01:11:13.850 --> 01:11:16.405 overfitting that you're kind of still 01:11:16.405 --> 01:11:17.920 like an overfit to that test set. 01:11:21.020 --> 01:11:21.330 Right. 01:11:21.330 --> 01:11:24.730 So I have just a little more time. 01:11:26.140 --> 01:11:28.790 And I'm going to show you two things. 01:11:28.790 --> 01:11:30.660 So one is homework #1. 01:11:31.810 --> 01:11:33.840 So, homework one you have. 01:11:35.670 --> 01:11:37.000 2 problems. 01:11:37.000 --> 01:11:38.580 One is digit classification. 01:11:38.580 --> 01:11:40.140 You have to try to assign each of these 01:11:40.140 --> 01:11:42.960 digits into a particular category. 01:11:43.900 --> 01:11:47.060 And so the digit numbers are zero to 01:11:47.060 --> 01:11:47.440 10. 01:11:48.430 --> 01:11:52.110 And these are small images 28 by 28. 01:11:52.110 --> 01:11:53.910 The code is there to just reshape it 01:11:53.910 --> 01:11:56.150 into a 784 dimensional vector. 01:11:57.270 --> 01:11:59.500 And I've split it into multiple 01:11:59.500 --> 01:12:02.650 different training and test sets, so I 01:12:02.650 --> 01:12:03.940 provide starter code. 01:12:05.220 --> 01:12:07.720 But the starter code is really just to 01:12:07.720 --> 01:12:09.025 get the data there for you. 01:12:09.025 --> 01:12:11.550 I don't do the actual like K&N or 01:12:11.550 --> 01:12:13.100 anything like that yourself. 01:12:13.100 --> 01:12:14.422 So this is starter code. 01:12:14.422 --> 01:12:15.660 You can look at it to get an 01:12:15.660 --> 01:12:17.120 understanding of the syntax if you're 01:12:17.120 --> 01:12:19.140 not too familiar with Python, but it's 01:12:19.140 --> 01:12:20.735 just creating train, Val, test splits 01:12:20.735 --> 01:12:22.460 and I also create train splits at 01:12:22.460 --> 01:12:23.310 different sizes. 01:12:24.090 --> 01:12:25.210 So you can see that here. 01:12:26.210 --> 01:12:27.980 And darn it. 01:12:29.460 --> 01:12:30.040 OK, good. 01:12:33.290 --> 01:12:34.290 Sorry about that. 01:12:36.090 --> 01:12:38.060 Alright, so here's the starter code. 01:12:39.120 --> 01:12:42.110 So you fill in like the K&N function, 01:12:42.110 --> 01:12:43.740 you can change the function definition 01:12:43.740 --> 01:12:45.540 if you want, and then you'll also do 01:12:45.540 --> 01:12:47.232 Naive Bayes and logistic regression, 01:12:47.232 --> 01:12:49.000 and then you can have some code for 01:12:49.000 --> 01:12:51.550 experiments, and then there's a 01:12:51.550 --> 01:12:52.850 temperature regression problem. 01:12:54.950 --> 01:12:57.770 So there's a couple things that I want 01:12:57.770 --> 01:12:59.640 to say about all this. 01:12:59.640 --> 01:13:02.930 So one is that there's two challenges. 01:13:02.930 --> 01:13:05.830 One is digit classification. 01:13:06.810 --> 01:13:08.400 And one is temperature regression. 01:13:08.400 --> 01:13:10.210 For temperature regression, you get the 01:13:10.210 --> 01:13:11.750 previous temperatures of a bunch of 01:13:11.750 --> 01:13:11.960 U.S. 01:13:11.960 --> 01:13:13.397 cities, and you have to predict the 01:13:13.397 --> 01:13:14.400 temperature for the next day in 01:13:14.400 --> 01:13:14.930 Cleveland. 01:13:16.170 --> 01:13:17.881 And you're going to use. 01:13:17.881 --> 01:13:18.907 You're going to. 01:13:18.907 --> 01:13:20.960 For both of these you'll use Canon 01:13:20.960 --> 01:13:22.720 Naive Bayes, and for one you'll use 01:13:22.720 --> 01:13:24.190 logistic regression, the other linear 01:13:24.190 --> 01:13:24.690 regression. 01:13:25.510 --> 01:13:26.900 At the end of today you should be able 01:13:26.900 --> 01:13:28.440 to do the key and part of these. 01:13:29.520 --> 01:13:30.940 And then for. 01:13:32.620 --> 01:13:34.880 For the digits, you'll look at the 01:13:34.880 --> 01:13:37.830 error versus training size and also do 01:13:37.830 --> 01:13:39.300 some parameter selection. 01:13:40.350 --> 01:13:43.790 Using a validation set and then for 01:13:43.790 --> 01:13:46.280 temperature, you'll identify the most 01:13:46.280 --> 01:13:47.270 important features. 01:13:47.270 --> 01:13:49.450 I'll explain how you do that next 01:13:49.450 --> 01:13:51.070 Thursday, so that's not something you 01:13:51.070 --> 01:13:52.270 can implement based on the lecture 01:13:52.270 --> 01:13:52.670 today yet. 01:13:53.370 --> 01:13:55.070 And then there's also a stretch goals 01:13:55.070 --> 01:13:56.890 if you want to earn additional points. 01:13:57.490 --> 01:13:59.230 So these are just trying to improve the 01:13:59.230 --> 01:14:00.540 classification or regression 01:14:00.540 --> 01:14:03.430 performance, or to design a data set. 01:14:03.430 --> 01:14:05.160 We're naive's outperforms the other 01:14:05.160 --> 01:14:05.390 two. 01:14:07.080 --> 01:14:09.400 When you do these homeworks you have 01:14:09.400 --> 01:14:11.145 this is linked from the website, so 01:14:11.145 --> 01:14:12.280 this gives you like the main 01:14:12.280 --> 01:14:12.840 assignment. 01:14:14.200 --> 01:14:16.920 There's a starter code the data. 01:14:17.620 --> 01:14:19.290 You can look at the tips and tricks. 01:14:19.290 --> 01:14:25.780 So this has different examples of 01:14:25.780 --> 01:14:28.510 Python usage in this case that might be 01:14:28.510 --> 01:14:30.740 handy, and also talks about Google 01:14:30.740 --> 01:14:32.820 Colab which you can use to do the 01:14:32.820 --> 01:14:33.230 assignment. 01:14:33.230 --> 01:14:34.900 And then there's some more general tips 01:14:34.900 --> 01:14:35.710 on the assignment. 01:14:38.340 --> 01:14:42.380 And then for when you report things, 01:14:42.380 --> 01:14:44.990 you'll report you'll do like a PDF or 01:14:44.990 --> 01:14:46.810 HTML of your Jupiter notebook. 01:14:47.470 --> 01:14:50.540 But you will also mainly just fill out 01:14:50.540 --> 01:14:53.700 these numbers which are the like kind 01:14:53.700 --> 01:14:56.120 of the answers to the experiments, and 01:14:56.120 --> 01:14:57.655 this is the main thing that we'll look 01:14:57.655 --> 01:14:58.660 at to grade. 01:14:58.660 --> 01:15:00.340 And then they'll only they may only 01:15:00.340 --> 01:15:01.955 look at the code if they're not sure if 01:15:01.955 --> 01:15:03.490 you did it right given your answers 01:15:03.490 --> 01:15:03.710 here. 01:15:04.620 --> 01:15:05.970 So you need to fill this out. 01:15:07.150 --> 01:15:09.060 And you say, how many points do you 01:15:09.060 --> 01:15:10.115 think you should get for that? 01:15:10.115 --> 01:15:12.190 And so then the TAS will say, the 01:15:12.190 --> 01:15:14.148 graders will say the difference between 01:15:14.148 --> 01:15:15.790 the points that you get and what you 01:15:15.790 --> 01:15:16.460 thought you should get. 01:15:20.560 --> 01:15:22.590 So I think that's all I want to say 01:15:22.590 --> 01:15:23.740 about homework one. 01:15:26.900 --> 01:15:27.590 Let me see. 01:15:27.590 --> 01:15:28.155 All right. 01:15:28.155 --> 01:15:29.480 So we're out of time. 01:15:29.480 --> 01:15:31.130 So I'm going to talk about this at the 01:15:31.130 --> 01:15:33.470 start of the next class and I'll do a 01:15:33.470 --> 01:15:35.390 recap of KNN. 01:15:37.160 --> 01:15:40.330 And so next week I'll talk about Naive 01:15:40.330 --> 01:15:43.010 Bayes and linear logistic regression. 01:15:44.260 --> 01:15:44.810 Thanks.