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And the following I want to explain to you two methods and how we can fit the data that we have just

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created.

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The first method will be quite mathematical.

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So not really related to a physical model, but just like the mathematically ideal approach here, I

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will consider Splice.

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So maybe you have heard of spleens already.

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They are typically used in graphics programs, for example.

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But yeah, they're very, very useful, but not really related to any physical model.

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So then the second option will, of course, be then in related motor physics so that we will use some

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model function that could be motivated by some theory.

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And then we have coefficients in this model functions that we are trying to optimize to reproduce our

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data.

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So first of all, let us use the unperturbed data and let's try to use Blind's to interpolate the data.

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And yeah, I wrote down that explain interpolation always fits the data perfectly and even have it has

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a continuous derivative.

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If a cubic spined blind is used and therefore the spleen is simply defined piece wise.

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So basically, you take neighboring points like this and that, and then you use either a linear function

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between those.

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So this would be really the most easy way of interpolating.

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These just take linear functions connecting the dots or you use third order polynomials.

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And so what is really, really nice about the third order polynomial is that then the derivative were

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two of these political formulas.

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Meat is still continuous, so it looks really, really smooth.

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And that is very useful also for some algorithms that you can do then later on with this data.

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But of course, it is a nice thing on one hand side that every data point is fitted perfectly, but

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also it's a bit unrealistic.

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For example, if we think about this in terms of physical data that we have some noise, so then we

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do not actually want that.

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We have some very complicated function connecting all of these dots, but we want to have a rough function

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going roughly through these dots.

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So let's see how this works out in detail.

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And that's yeah, let's use this spline interpolation.

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So we begin by looking once again at our data.

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So for this, we can just load this one just so that we have it here.

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And then the next step we will do is we will load the molecule so pi.

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And from there, we will load just to interpolate commands.

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So you see inside Pi, which is a module for Python.

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There is already a method implements.

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It's called Interpolate, and we will use this in the following we will not construct these lines by

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ourselves.

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So we will start with the linear splice basically just connecting the dots.

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This would actually be pretty easy to do ourselves.

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We would just have to define a Piece Y's function.

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And then we would just have to define the lines connecting the dots.

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But here, since we have loaded this interpolate command, it's even more easy so we can just write

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interpolate dots and then the command is into p 1d and then we must give the X values.

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So this would be theta zero zero and then the Y values theta zero one, and then we must define the

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type of the interpolation.

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And here the argument is then kind and the string with the type of interpolation.

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And here, first of all, we want to use linear and we want to store this function that will be generated

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in the variable or in the function.

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We give it a name called spline linear zero.

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And now we have our spleen function.

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We can, of course, plot this.

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So I will take this one, copy it here and then use a new line p t dot plot and here right data zero

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zero for the X values and then for the Y values, I will use spline linear zero acting on data zero

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zero.

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And so you see the lines here, they are really explained supplying function and the function connects

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all of the dots.

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So maybe now it's a good idea to zoom in.

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So I have to zoom here and.

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To zoom in, we will just redefine the X range into why rates, so I would write X plot Dot X line going

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from two to four.

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And then we have to adapt also the Y limits.

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And I've found that it looks quite good when we use minus five to 15.

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So then this will be our plot and you see that we have now points that are our data points and they

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are connected just by straight lines.

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And of course, they are just connected by straight lines because we say that we want to create a plot

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where we have pairs of X and Y values and they can just be connected by dots.

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So just to make sure that these are really straight lines, we should increase our resolution for this

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plot here.

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So I write X list, which will be a list for the plot here.

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So just for the plot, not for the fit and for this arrow right and p dot line space from two to four.

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And now I will not just use a very few points, so here I have five points.

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Instead, I will use 201 points and you see the plot doesn't change at all.

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So, for example, if it is your setup, then oh sorry my, that I have to change this, of course x

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list.

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And you're also x list.

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Then you can see we have no many, many points.

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So let's go back to plots and you see it's still the straight line.

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So all the points that are just correct by this line, linear zero are these piece wise linear functions.

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And this is, of course, OK.

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In a sense, if you have a lot of data points and you don't really care about the derivatives because

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on this plot range here looks quite good.

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But when you zoom in, you see the derivative is discontinuous and this is, of course, very on physical.

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If you relate this data to some real world problem, then this can be very, very, very bad.

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And to solve this problem, we can use a cubic spine and we can do this in the following way.

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Let me copy the plot here, and then let me copy also the command for creating this plane.

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And then we could call this here blind cubic.

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And here we just have to change the optional argument to cubic.

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And then here are right.

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Also quick, and I think that should be it already.

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And you see, now we have cubic spines.

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So a third order polynomial connecting these dots and you see the derivative is now continuous and it

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works very, very well.