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So in the previous lecture, we have seen that a cube explained function that's that is defined piece

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wise works very well for describing these data points.

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So the general trend of the data points is nicely reproduced by this function.

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And the good thing is that now we have the function that describes the trend of the data between the

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data points.

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So for example, if we want to calculate now some derivative at this point, we do not have to use these

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and these two points as the reference.

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But we can just take some small area near the point to calculate the derivative so it can be very,

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very helpful and it makes the life a lot easier.

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So these cubic lines, they work well and they have a continuous derivative at the data points.

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But so far, the data was very smooth, so it wasn't so difficult to generate this interpolation.

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So let's see what happens if we used to perturb data from here.

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So I will scroll down where we have left our notebook and now we come to the perturb data.

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And this time I will just basically copy what we had written down here, and then I will just paste

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it at the bottom where we have left.

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And this time I will write spline Kubik without the zero, because this time we will fit not the zero

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data, but the actual data.

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And of course, we will use a cubic methods.

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So now we use here once again such an ex list that we have to find previously, so we don't have to

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redefine it again.

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We can just write here ex list.

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And

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no, sorry, that was not correct.

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I wanted to change the other one.

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So this one or two data points, we leave them as they are, and then we just changed this plot.

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Exactly.

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So we use X list and then Blind Cubic X list.

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And then we can run it, and we see, OK, Rick's list is defined in an odd way, so we have to change

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it.

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So let me change it.

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Maybe let's go from minus four point five to four point five with nine hundred and one points so that

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it looks like this.

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So now you may wonder what is going on here.

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It looks a bit odd.

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Um, so I think, yeah, this is because we plot here the the unperturbed data and not the perturbed

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ones.

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So maybe let me do it like this.

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I will write your plot.

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So then now blue is the actual data.

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Orange is the blind fit of the new data, and I will know the plot, the new data as well.

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So let me do it like this and I will do it in a scatterplot.

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Yes, exactly.

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Now it looks good.

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So we have in blue the unperturbed data we have in the blue dots, the perturb data, and we have an

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orange, the cubic blind fitting the perturbed data.

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And do you see, as I mentioned earlier, every individual data point is really fitted, which can be

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good, but actually we call this in numeric that it is over fitted.

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So of course, if you think of it in terms of physics, then the physical dependence on a physical model

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behind it will, of course, never be so difficult here.

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It will not have all of these fluctuations.

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It's pretty, pretty clear that this is just noise.

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So what we would want here is we want to smooth these blind and we don't want every point to be fitted

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perfectly.

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And to do this, we will come to new methods in the next lecture, which is called We are fitting a

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model function.

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But I want to show you here that is.

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It is also possible with slides.

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However, we must use a different command this time.

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It's called interpolate and then it's called uni variance on line, and we must fit data zero and data

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one.

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And this time we don't use an optional argument.

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It works just as it is.

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And I recall this function supplying smooth.

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And let's see, maybe I have a typo.

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Yeah, here of a typo.

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Um, you need very splain.

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Maybe still a typo.

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You need to vary it with an A.

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Yeah, like this.

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Sorry about the type of A. works we have to command interplay the univariate splicing data zero and

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data one and restore this function and name it spline smooth.

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So now we can run this and we can plot it.

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So once again, let me copy this here without the first line.

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So now we plot the data.

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So actually, I would say we don't have to plot the initial data this time.

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We can just do it like this.

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Just plot the scatter data and then the updated function here, which is called splice moved.

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And if you compare this to the other plots, maybe it looks already a bit more smooth.

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For example, here you have this large peak, which is totally unreasonable.

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And here it is, almost gone, so it looks much better already.

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And to improve this, we can tune the smoothness factor so we can write splain smoothed dot set on a

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school smoothing on the score factor.

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Now we must specify some number, for example.

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50.

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See, it's even smoother if we go 500.

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And you see, it's very, very smooth.

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And this is exactly what we wanted.

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So you see, by controlling the smoothing factor, we you yeah, we basically disregards that every

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point must be fitted perfectly and we must just roughly fit all the points by, such as flying here.

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And you see, this is a much better function that describes the physics much better.

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But of course, it is not any more.

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A piece wise defines blind, so it is a bit of a different thing.

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And this is because we have called here to command univariate line, which is something different.

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But I want I don't want to go into more details here.

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I want to go straight to the next lecture where we will define a model function.

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And then we try to optimize the parameters so that this model of function describes what data best because

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this is typically the way physicists or I would say other scientists also fit some experimental data

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and disciplines are, of course, a very, very nice because they are a nice mathematical tool for fitting

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or for interpolating the data.

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But often they don't reproduce the correct mechanism that is behind the data.

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So I hope you stay tuned and let's go together into the next lecture.