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Welcome to the section on error sanctioning.

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In the previous section we saw how to build a simple linear regression model.

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In this section, we'll see how to calculate the model's error in order to permit it make or update

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its parameters such that it makes, uh, such that this error is reduced.

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So let's get back here.

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We had this y equals mx plus c and then we had this line.

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But um, again this line could be just any other line depending on how we initialize um m um and c.

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So we could have this line.

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It could be this line, it could be this line.

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It could be this line, or it could be maybe say even this line.

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Now, supposing that it is this line right here, then for each and every point, the way we are going

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to sanction, um, the error or the model's error is by, first of all, measuring it.

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So we would, um, get let's, let's start with this point.

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For example, if we have this point here, uh, what the model predicts is that the price if you use

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the model's prediction and then you link this up, this point up with the model's prediction, that

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is a straight line.

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Then you would find that, uh, model model predicts is actually this value.

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So the model predicts uh, the, let's say a value of um, the well this this prices go from 8 to 10.

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So let's suppose that this is a value of about five, whereas it's supposed to predict ten, because

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here we have the true value.

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Let's take this, let's copy this and um just drag it right here.

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So as we're saying here we have the true value from this point.

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The true value is actually this ten.

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But what the model predicts is five.

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And so now this difference of five is actually what we would call the error.

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So this this difference that's the error.

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But it happens that this error is not always this large for all other values.

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If you let's zoom in let's zoom in.

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You will find that for this value.

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For example, for this, um, specific value we have here, what the model predicts and its actual value

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is the same.

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Let's get back, copy this and paste out.

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So we have that here.

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So let's say the model here predicts um 5.5 or 5 point um two.

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So here we have 5.2.

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Well this is five here.

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Let's drag this down.

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So here we have this five.

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And then just right beside it we have 5.2.

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So this is 5.2.

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And then this is five.

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So what we're saying for this point what the the model predicts.

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That's when you link the point with the with a line with is our function y equals mx plus c.

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You will find that what it predicts is um 5.2.

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And the actual value itself is 5.2.

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So what this tells us now is that our error our error is equal to zero.

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So that's it.

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We have an error of zero.

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Um, well, just above here at this, uh, at the level of this point, we have an error which is,

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um, quite large of about five.

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Let's let's see there we have an error of, um, about five.

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Okay.

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So that's it.

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We see that depending on the depending on what the point, we could have different, um, error values.

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Now generally what we want to do is we want to get an average of all these values.

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And then, um, that will be our overall error.

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So our overall error here will be for each and every point we have here we will get the errors.

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So let's say uh we will take this point.

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You see when you when you link this up with a, with a line, you find that what the model predicts

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here is this value, whereas the actual value is right here.

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And so the error is going to be this for this point.

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And you could repeat the same process for all other points.

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But what's important to note is the fact that, um, the error of the model is going to be um, the,

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the, the mean or the sum total, um, averaged of all these different points.

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Let's suppose that we had, um, this line something like this.

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So let's suppose that this line was like this.

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In that case the, the error for all these values, you see that the error for all these values when

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you compare, well, let's get back, you would find that when you compare this model, this model here

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with this other model, the error here will be much larger than the error for a.

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So let's call this model A and let's call this model B.

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Um our error.

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Our error.

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For model B will be much larger or let's say greater than the error for model A.

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And this makes sense because for most of the points we have here, they are individual errors.

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When we're using this method, B is going to be much larger than the individual errors.

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When we're using, um model A diving into some code, we get to the documentation.

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You have TensorFlow losses.

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Let's reduce this.

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We have TensorFlow losses Keras.

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And then we have losses.

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Here we have this mean squared error.

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Let's click on this mean squared error.

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And it's going to be defined your computer mean of squares of errors between labels and predictions.

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So that's it.

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Simple definition.

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And then you have some examples.

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So just right here we have um y true.

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And so what that means is if you consider in this example here, if you consider this example it's y

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true is ten.

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Whereas what the model predicts is five.

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You see.

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So here is y true it's going to be ten while the y pred by the model is going to be five.

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And then with this um mean squared error loss we'll be able to compute the the loss very easily.

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So let's copy this.

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Get back to our notebook, space it out.

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And you see here we have y true zero one.

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Let's just let's rearrange this zero one uh and zero zero and one one and one zero.

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So let's run this and then see what we get.

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Well let's print this out.

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So let's have print print that out and see what we get.

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So you see we have 0.5.

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And the way we could get this is simple.

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We will have for each and every position you compare the values zero and one.

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That gives you zero minus one.

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And the mean square error as the name goes is squared.

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So we have zero minus one all of that squared.

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So we have that squared.

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Um plus it's a mean.

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So it's an average.

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So we have plus um one here and one.

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So let's just copy this and paste out here.

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Take that off.

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We have one and 1 or 1 minus one all that squared.

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Plus um for the next zero minus one that is zero minus one all that squared.

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And then here we have zero -0 or that squared.

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So zero -0 other square.

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Take that off and we have zero.

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So all this is going to give us um one this is one square one plus zero plus one plus zero.

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That's two.

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And if you divide this by the total number of um elements we have four.

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So this gives us 0.5.

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Now let's modify this slightly and say we will have one.

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So if we have one then we will have one divided by four which will give us 0.25.

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Here one divided by four should give us 0.25.

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Let's change this from 0 to 1.

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Now we have one minus.

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We have zero minus one.

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So here is zero minus one.

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And then um this should be one.

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This is one plus zero plus one one plus one.

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So that's three divided by four.

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Now we should have 0.75.

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So let's comment that and see what we get.

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There we go.

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You see we have 0.75.

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So essentially we have this error which is for each and every point here what the model predicts, which

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in this case is um, five actually.

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Because when you, when you make use of the line you see is five what the model predicts that, uh,

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five minus what the model predicts, minus what um, is actually like minus the actual value.

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So that's why a y actual.

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And you take all this and you square, then what you do is you repeat this for all the different points,

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and then you look for the average.

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So simply, um, the sum of all this y minus y is um, all of that square.

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So you sum all those up, and then you divide by the total number of points or total number of samples

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in, in this case is, um, say thousand.

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Now, the next problem we have with this mean square error is that it doesn't do well when it comes

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to, um, dealing with outliers.

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A simple outlier we could detect, um, by looking at this plot.

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Is this one right here?

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Why is this an outlier?

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The answer is simple.

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We have high horsepower, but we have low price.

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We would expect that the the higher the horsepower.

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Like most others here, we should have, um, a higher price, just like, um, as we've seen with all

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these others.

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So with this outliers, given that the deviate a lot from the usual pattern of the whole, um, data

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set of, uh, different samples using y minus y, that's y minus y, let's say y prime square using

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all this, this kind of formula where we have, uh, y minus y prime square, um, isn't the best,

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uh.

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Here.

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The different the the the error is much larger as compared to all the others.

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So here we would have an error.

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Um, if we take this and paste it out here, let's try to link this up with our with our plot.

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So let's, let's, let's keep dragging this.

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See we link this up and then we drag this to up.

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There we go.

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You see that this error is much larger as compared to all the other errors.

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So um, compared to this is much larger than this, much larger than maybe the error would have here,

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much larger than this error and much larger than this error.

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And so because of that, um, the model will pay more attention to this outlier as compared to all these

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other, um, points.

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And this is for the simple reason that the loss is going to be much higher, especially as we're going

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to be squared.

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So remember, if you have, um, a square, um, you would have something like this plot like this.

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So if you have um say an error of three.

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An era of three.

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Well, you would have a square of nine, but if you have an error of ten, an error of ten, you have

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a square of about 100.

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So the, the, the gap you have here will be much larger than as compared to the gap you have here.

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So this kind of outliers tend to trick the model into updating its parameters M and C based on um,

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its own particular loss when um, it's actually an outlier.

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So what we use in this kind of cases is we make use of the mean absolute error.

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So instead of squaring we just simply say y minus y prime and then we take the absolute value.

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So instead of this kind of plot we would have a plot like um this instead.

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And the difference now is that although the we still have that error margin.

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So although the error or the difference between the y predicted and the Y um, actual is going to still

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be large, we we don't have to square it so it doesn't, um, aggravate the situation.

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So that's it.

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We have this mean absolute error again with TensorFlow.

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You could, um, use that without writing any extra line of code.

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Now, if you want the best of both worlds, that's if you want to use the mean squared error and the

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mean absolute error.

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Then you could go in for the Uber loss.

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Now with the Uber loss spurred Uber loss with the Uber loss is actually a mixture of the two.

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So what we have here is if we have um if the error is too large, then we'll use the mean absolute error.

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So we just we just compare if we have um, error, if our, our error, um, error is too large.

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So if error is large then we use our mean absolute error.

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And then if, if we have an error that is two that is small or that is usual like um, in the case of

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all these other points, if you have usual errors then our usual error margins, then you would go in

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for the mean square error.

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So in that case if your error, if your error, um, error is small, then you will go in for the mean

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square error.

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Diving back to the documentation, you have the mean absolute error just right here.

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This is the mean squared error square of y minus y prime square or rather the square y minus y prime.

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And then for the mean absolute error computes the mean of absolute difference between the labels and

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the predictions.

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Now um, remember that the absolute value, the absolute value actually is um x.

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That's let's, let's, let's write that out.

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In case you don't know, when you talk about absolute function, the absolute value is simply x if x

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is greater than zero.

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So if x is already positive then it remains the same.

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But if x is negative, then you add a negative there so that it turns into a positive value.

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So in that case, the absolute value of minus seven is seven and the absolute value of seven is seven.

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So that's it.

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We get back here.

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That's that's it for the mean absolute error.

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So let's get here.

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We have Uber.

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There we go.

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We have this Uber loss.

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Let's scroll up.

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Compute the Uber loss between y true and y pred.

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So we have this parameter delta.

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And the role of this parameter delta is simple.

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Um the way we know that the error is large is by comparing it with our parameter delta.

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So if the error is greater than the parameter delta, then we consider that it's large and we use the

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mean absolute error.

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If the error is less than the parameter delta, we consider that the error is small and we use the mean

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squared error.

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So getting back here we have that.

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We should have an example showing that okay this is it.

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You see we have um x squared.

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That's y minus y prime square.

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If x is less than that parameter delta.

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Yeah denoted as d.

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And um if it's not then we're going to use um the absolute value.

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Well looking at this here, this definition, it turns out that we made an error, uh, on the board.

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Um, and it's actually not just absolute value.

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So we have uh, with the Uber, if it's less than the delta, that is, if the error is small, then

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we use the mean squared error y minus y prime um square.

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But for the Uber it's still the absolute value.

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But now we subtract half of the sigma and then multiply by sigma.

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So or delta.

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So this delta um comes in again in the formula.

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So it's not just y minus y prime.

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And then one uh times one over n.

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Anyways the good thing with TensorFlow is you don't really need to dive into all these details before

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making use of, um, the Uber loss in practice.

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So here you see how to make use of it.

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Um, usage with the compile API.

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So after you've created your model, you now compile that model.

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You specify the loss and you see you specify the optimizer, which we shall look at in the next section.

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So let's just copy this and then um get back to the code.

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Take this off.

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We have now model compile.

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We have the optimizer SGD and the loss which is the over loss.