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So the previous lecture, we have defined our model function, which is supposed to be a polynomial.

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And then we have tested that if we use the parameters that we used for the beginning, that indeed we

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get the correct function.

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So this is, of course, not the solution that we are going to use because, yeah, we are trying to

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figure out these parameters and we don't just want to use the correct solution right away.

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This was just for testing if our model function works and we see it works quite well.

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So say, we don't know these values yet and say we use here instead some other values, for example,

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a starting point could be just ones.

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So like these values, then you see the function looks totally different and there is a deviation from

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the actual data points.

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So how would we now say how good is a fit for this?

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We need an error function.

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So let me just for the notebook.

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Restore this and let's come to the error function.

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We will now define the error, which basically also defines the quality of our fit.

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As I wrote down, there are actually many reasonable definitions of an error function, but a very common

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choice, especially in the beginning, is such a square into differences methods.

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So basically, you sum up of all your data points and you just take the difference in the Y values and

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you square all of the values and then you add them up.

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And this gives you, of course, always a positive number.

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And so since you square it up when there is a single point with a large difference, this will give

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you a very large error.

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So it's such an error of function tries to basically minimize the distance between fit and data for

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all points at the same time, and mainly it tries to avoid some very large distance and one of these

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terms.

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So the function F is the fit function that is determined by the coefficients.

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And we have several coefficients a i so in our case, we have a zero, a one, a two and a three because

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we have a third order polynomial.

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So for parameters.

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But the way that we have programmed everything is very generous.

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We could also use a ninth order polynomial.

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It would work.

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And the points XIII and y i.

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These are the data points that we tried to fit.

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And I think there we had 21 data points between minus five and plus five.

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So now let's go ahead and just take this definition of the error and define it.

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So I will call this error fit.

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The name isn't really that important.

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You could give it another name and the arguments will be if the function that we have, the fish shoes

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and the data points, of course.

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So if it will be the fit function or the model, then the coefficients will be the values that we try

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to optimize.

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So the MRI that we try to optimize and then we have the data, which will be our data.

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We try to fit.

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And once again, we just add things up here.

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So you see, whenever we have these sons, we can just use a loop.

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And as I said, there are other methods, but I think the loop is just fine, even though it's a bit

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clumsy, maybe, but doesn't matter here because all of these calculations are very quick anyway.

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So it doesn't really matter.

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If we try to optimize things here, we can just go ahead.

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So here is maybe a thing where you could make a mistake if you're not careful.

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So we calculate here some and we have an index AI that ranges from.

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And then here I wrote, down from one to end.

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But in our case, we have programmed it to start at zero and it goes to end.

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So this, I would say, is totally fine.

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So since we don't include and we go from zero to and minus one.

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So it's the same thing.

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Basically here and here, you have to be careful because if you write in range length of data, then

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this value would just be too because we have X and Y, so the length would then be just too.

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But if we now just look at the X coordinates, for example, then we have the actual numbers of our

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points.

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So make sure you don't forget this one here.

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Otherwise, the court will still work, but it will give you an incorrect error because it will only

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take into account the first two data points, which is, of course, not good.

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But anyway, since we have it correct now, we can have data on error, so we will be old error plus

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and then we just write the square here in the brackets, we have data one which will be the Y coordinates

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and then the five point minus f of.

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Date Zero Comma II, which the X coordinates, and then we have the coefficients, which are just two

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coefficients coefficients.

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And then we could say just for testing purposes, we want to print the arrow just to see if it works,

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because this will, we will delete later on and then we can return the error after the loop.

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So we run this and now we can write a fit of our colleague in non-lethal model.

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Then we use a zero for the starting point for the parameters, and then we use the data points for the

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data.

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And when I run this.

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What's going on?

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I think for some reason, this is a markdown cell, so this is just my fault.

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They have somehow missed something else.

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Let's copy it in an actual cell that is a code cell, and now it works.

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Yes.

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So you see here the error will be added up and we start from a very small value.

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And then you see it just keeps on increasing, which is clear since we are adding here square numbers.

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So these can only be positive and the error will increase.

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And then at some point the loop will stop and it will ultimately give us the error.

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All right.

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So it seems to me reasonable.

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So that's common.

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This one, we don't need it anymore.

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And so now the error just gives us a single value.