CS_441_2023_Spring_January_19,_2023.vtt

WEBVTT Kind: captions; Language: en-US

NOTE
Created on 2024-02-07T20:51:27.1993072Z by ClassTranscribe

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Good morning, everybody.

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Morning.

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Alright, so I'm going to get started.

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Just a note.

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So I'll generally start at 9:31

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exactly.

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So I give a minute of slack and.

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At the end of the class, I'll make it

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pretty clear.

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When class is over, just wait till I

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say thank you or something.

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That kind of indicates that class is

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over before you pack up.

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Because otherwise like when students

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start to pack up, like if I get to the

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last slide and then students start to

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pack up, it makes quite a lot of noise

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if like couple 100 people are packing

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up at the same time.

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Right, so by the way these are I forgot

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to mention that brain image is an image

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that's created by Dolly.

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You might have heard of that.

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It's a AI like image generation method

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that can take an image, take a text and

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then generate an image that matches a

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text.

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This is also an image that's created by

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Dolly.

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I forget exactly what the prompt was on

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this one.

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It was it didn't exactly match the

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prompt.

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I think it was like, I think I said

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something like a bunch of animals

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somewhere ring.

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Orange vests and somewhere in green

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vests standing in a line.

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It has some trouble, like associating

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the right words with the right objects,

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but I still think it's pretty fitting

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for nearest neighbor.

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I like how there's like that one guy

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that is like standing out.

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So today I'm going to talk about two

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things really.

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So one is talking a bit more about the

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basic process of supervised machine

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learning, and the other is about the K

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nearest neighbor algorithm, which is

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one of the kind of like fundamental

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algorithms and machine learning.

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And I'll also talk about how we what

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are the sources of error.

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So why is it a?

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What are the different reasons that a

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machine learning algorithm will make

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test error even after it's fit the

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training set?

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And I'll talk about a couple of

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applications, so I'll talk about

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homework, one which has a couple of

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applications in it.

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And I'll also talk about the deep face

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algorithm.

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So a machine learning model is

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something that maps from features to

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prediction.

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So in this notation I've got F of X is

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mapping to YX are the features, F is

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some function that we'll have some

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parameters.

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And why is the prediction?

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So for example you could have a

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classification problem like is this a

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dog or a cat?

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And it might be based on image features

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or image pixels and so then X are the

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image pixels, Y is yes or no, or it

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could be dog or cat depending on how

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you frame it.

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Or if the problem is this e-mail spam

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or not, then the features might be like

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some summary of the words in the

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document and the words in the subject

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and the sender and the output is like

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true or false or one or zero.

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You could also have regression tests,

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for example, what will the stock price

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be of NVIDIA tomorrow?

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And then the features might be the

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historic stock prices, maybe some

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features about what's trending on

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Twitter, I don't know anything you

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want.

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And then the prediction would be the

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numerical value of the stock price

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tomorrow.

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When you're training something like

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that, you've got like a whole bunch of

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historical data.

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So you try to learn a model that can

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predict based, predict the historical

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stock prices given the preceding ones,

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and then you would hope that when you

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apply it to today's data that it would

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be able to predict the price tomorrow.

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Likewise, what will be the high

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temperature tomorrow?

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Features might be other temperatures,

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temperatures in other locations, other

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kinds of barometric data, and the

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output is some temperature.

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Or you could have a structured

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prediction task where you're outputting

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not just one number, but a whole bunch

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of numbers that are somehow related to

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each other.

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For example, what is the pose of this

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person?

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You would output positions of each of

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the key points on the person's body.

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Right.

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All of these though are just mapping a

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set of features to some labeler to some

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set of labels.

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The machine learning has three stages.

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There's a training stage which is when

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you optimize the model parameters.

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There is a validation stage, which is

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when you evaluate some model that's

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been optimized and use the validation

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to select among possible models or to

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select among some parameters that you

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set for those models.

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So the training is purely optimizing

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your some model design that you have on

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the training data.

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The validation is saying whether that

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was a good model design.

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And so you might iterate between the

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training and the validation many times.

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At the end of that, you'll pick what

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you think is the most effective model,

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and then ideally that should be

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evaluated only once on the test data as

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a measure of the final performance.

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So training is fitting the data to

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minimize some loss or maximize some

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objective function.

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So there's kind of a lot to unpack in

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this one little equation.

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So first the Theta here are the

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parameters of the model, so that's what

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would be optimized.

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And here I'm writing it as minimizing

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some loss, which is the most common way

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you would see it.

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Theta star is the Theta that minimizes

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that loss.

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The loss I'll get to it can be

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different different definitions.

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It could be, for example, a 01

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classification loss or a cross entropy

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loss.

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That's evaluating the likelihood of the

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ground truth labels given the data.

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You've got your model F, you've got

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your features X.

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Those errors are slightly off and your

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ground truth prediction.

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So Capital X, capital Y here are the

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training data and they're those are

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pairs of examples or examples, meaning

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that you've got pairs of features and

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then what you're supposed to predict

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from those features.

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So here's one example.

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Let's say that we want to learn to

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predict the next day's temperature

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given the preceding day temperatures.

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So the way that you would commonly

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formulate this is you'd have some

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matrix of features this X.

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So in Python you just have a 2D Numpy

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of A.

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And you would often store it as that

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you have one row per example.

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So each one of these rows.

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Here is a different example, and if you

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have 1000 training examples, you'd have

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1000 rows.

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And then you have one column per

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feature.

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So this might be the temperature of the

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preceding day, the temperature of two

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days ago, three days ago, four days

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ago.

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And this training data would probably

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be based on, like, historical data

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that's available.

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And then Y is what you need to predict.

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So the goal is to predict, for example

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50.5 based on these numbers here, to

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predict 473 from these numbers here,

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and so on South you'll have the same

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number of rows and your Y as you have

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in your X, but X will have a number of

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columns that corresponds to the number

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of features.

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And if Y is just you're just predicting

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a single number, then you will only

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have one column.

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So for this problem, it might be

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natural to use a squared loss, which is

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that we're going to say that the.

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We want to minimize the squared

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difference between each prediction F of

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XI given Theta.

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Is a prediction on the ith training

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features given the parameters Theta.

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And I want to make that as close as

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possible to the correct value Yi and

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I'm going to I'm going to say close as

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possible is defined by a squared

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difference.

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And I might say for this I'm going to

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use a linear model, so we'll talk about

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linear models in more detail next

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Thursday, but it's pretty intuitive.

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You just have a set for each of your

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features.

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You have some coefficient that's

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multiplied by those features, you sum

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them up, and then you have some

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constant term.

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And then if we wanted to optimize this

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model, we could optimize it using

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ordinary least squares regression,

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which again we'll talk about next

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Thursday.

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So the details of this aren't

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important, but the example is just to

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give you a sense of what the training

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process involves.

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You have a feature matrix X.

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You have a prediction vector a matrix

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Y.

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You have to define a loss, define a

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model and figure out how you're going

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to optimize it.

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And then you would actually perform the

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optimization, get the parameters, and

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that's training.

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So often you'll have a bunch of design

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decisions when you're faced with some

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kind of machine learning problem.

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So you might say, well, maybe that

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temperature prediction problem, maybe a

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linear regressor is good enough.

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Maybe I need a neural network.

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Maybe I should use a decision tree.

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So you might have different algorithms

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that you're considering trying.

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And even for each of those algorithms,

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there might be different parameters

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that you're considering, like what's

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the depth of the tree that I should

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use.

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And so you so it's important to have

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some kind of validation set that you

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can use to.

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That you can use to determine how good

00:11:05.022 --> 00:11:07.020
the model is that you chose, or what

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how good the design parameters of that

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model are.

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So for each one of the different kind

00:11:12.766 --> 00:11:13.940
of model designs that you're

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considering, you would train your model

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and then you evaluate it on a

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validation set and then you choose the

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best of those.

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The best of those models as you're like

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final model.

00:11:25.280 --> 00:11:28.296
So in some if you're doing, like if

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you're getting data sets from online,

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sometimes data sets.

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They'll almost always have a train and

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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

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you're ready to do your final

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evaluation.

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They don't always have a trained and

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Val split, so sometimes you need to

00:11:47.900 --> 00:11:49.680
separate out a portion of the training

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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.

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If you evaluate 1000 different models

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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

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test examples because you've optimized

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your model selection for that test set.

00:12:20.220 --> 00:12:22.440
And then the final stages is evaluation

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or testing.

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And here you have some held out test

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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.