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So you see, we are almost at the end of our notebook here, because the only thing left to do is we

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have to loop over the error of its gradient function over and over again and then update our values

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to do the actual fitting of our model function.

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So let's do it.

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I will define here the number of iterations to be maybe 1000.

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I think we need quite a few of those several thousands, but let's start with one thousand.

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And then we must define some update rate and to be honest, looking at these arrows here.

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So I know already that these are the correct set of parameters.

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So even if it's correct, the gradient will never be zero, which is due to the random numbers that

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we have added.

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So the fit is never really perfect, so we will always have some error on some gradient.

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So these will be the smallest gradients that we will encounter.

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So if these are the smallest gradients and such values shouldn't really affect the A0 at all.

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So we need a value of age that is much smaller than the reciprocal of these values, so much smaller

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than this one.

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So I don't know.

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Maybe I will add just here to more zero, so we will have to test and see if this works or not.

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And then we will start with some values for our eight.

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And as I said previously, we will of course not use a zero because, yeah, that would be stupid,

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because then we would have to perfect fit right away.

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We have to start from some random values.

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We could, of course, guess something just first of all, fit this function by I and then improve the

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fit by using this algorithm.

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But here I want to do them in a very brute force way and just say, OK, I used two times and p dot

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random dot around four minus one.

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So this means a is now a vector.

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It has four components for the four coefficients, and all the values are in the range of minus one

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and one.

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And they are randomly selected.

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For example, we can check this are now the starting points for our eight coefficients.

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And when I run this once again, of course, then the starting parameters are different.

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All right.

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So now the only thing that we have to do is you have to write a loop for I in range iterations.

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Now, the only thing that we have to do is we have to update the values for eight eight is the old eight.

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And then I told you that we will go along the opposite direction of the gradient and we will multiply

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by the step with.

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And then we will just update this with this error.

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And now, of course, we have to use our parameters and not the A0.

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And I think this is already it.

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So when we run this, then we should end up with some parameters.

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And so we could, for example, say here at the starting point, print A and then at the end print a

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again.

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So you see the A values have been updated.

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All right.

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So we do not really know if these are good values yet, but we could check this by comparing a with

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a zero because we know that these zeros are the good values.

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And it doesn't look good yet.

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So probably we have to increase the numbers of iterations.

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Let's go with 10000.

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So you see, it takes a moment now,

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and this time at least the last two coefficients fit quite well.

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And maybe this is a bit surprising for you now.

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But the last two coefficients are the most important ones because they determine the highest order polynomial

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terms.

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So this would be the coefficient for X to the power of three.

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This one four x squared.

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And of course, these will determine the shape of the function, especially for large X values.

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So it's really, really good that we see that these values are quite close to the actual values.

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So maybe just go to 100K.

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So you see here it runs for a few seconds now.

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And maybe if your PC is slower than mine, this could take quite a few seconds.

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Let's see what's happening here.

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Actually, I could leave this here because we print the values anyway, so this time, let's see.

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Um.

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Yeah, I didn't really get much better, but I would say, let's just look at the plots because it's

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hard to say just by comparing the parameters.

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So even though these are the parameters that we have used for creating the data points, they do not

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necessarily have to be the perfect parameters for the fit because we have added this random noise.

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So I would say, let's just go ahead and plot the function.

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So let me copy this one.

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And what we will do is we will use here a plot, so we will use people, we will plot the actual line

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and we will plot through, of course, the function with our fitted coefficients.

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And then in the background, we will use a scatter point as scatterplot and sorry, using the data points.

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Will use data zero.

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Common data one.

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And you see, it looks really good.

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So even though, as I said, the parameters do not agree perfectly.

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If it looks good and this could be, for example, for the following reason here this is the zero with

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all the terms, and this is just a shift in my direction.

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So actually, our fit is shifted by 18 units to the top compared to the function that we have used to

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create the data.

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But the linear term is also quite different.

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So maybe this compensates the whole thing.

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So what we could do is we could also add here the other fit or not the other fit, but the function

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that we have used to create the data and.

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Now, I'm a bit surprised.

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I just stopped the recording because I was surprised that the orange function that we have just plotted

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doesn't really reproduce our data points, so something was off here.

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I went back to the very start of this section.

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So where we have started with the interpolation and generated the data, I have seen that here.

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These are actually the coefficients and the first coefficient is actually 15.

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And for some reason, I redefined here, let me search for it.

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Here in three point 2.3.1, a redefined a zero to be minus two for some reason, so that was, of course,

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wrong.

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So let's updated to 15, and then I was, let's say let's rerun the whole notebook, our restyled run

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all.

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And now, of course, it takes a few seconds, few seconds.

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Actually, it's very fast, except for the very last thing, because here we will have to loop 100000

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times.

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This takes a while.

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And then we can compare the plots and we can compare to coefficients this time in an accurate manner.

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So sorry for having made this mistake.

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I hope this wasn't too confusing, but this is sometimes how it goes.

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You make mistakes, and the good thing or good methods to finding these mistakes is always plotting.

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I think because I have seen the orange function was way too low, so I noticed there had to be something

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wrong here and now everything looks good.

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You see, the orange function was the one that we have used for creating this data, and then we have

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added the noise on top of it.

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And the black function is the one that we have received from the fitting.

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So it could really be that due to the random data that we have added that the black function is actually

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better than the orange one, and we could even verify this by calculating the error.

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So this time we can use our Arrow Fit function so that it isn't useless.

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We can just go ahead and calculate here.

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Print error fits polynomial model, a zero data and then a data.

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So, first of all, will be our fit.

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Second run will be a zero.

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The data are the function we have used initially.

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And do you really see our fit data has a lower error.

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So really, at least for the error that we have to find our fit is better than the function that we

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have started with.

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So this is maybe a bit counterintuitive and a bit unexpected.

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So for me, it's also unexpected, but it makes sense because due to the randomly added points here,

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it can happen that the ideal function for fitting this data is different than the function that we have

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started out with.

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And so now if we compare two coefficients, they are a bit different.

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But you see, it makes sense because our fit is actually the better fit to the data.

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I think this is a very good closing remark and really the ultimate proof that our fit works very, very

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well and that this method is very useful.

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And so comparing this result with these blind methods, we have now really fitted our data with a model

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function.

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And this is really what you want to do in many physical problems where you have some mechanism that

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follows some physical law and then you measure some experimental data and then you do not always want

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to have the perfect fit, which would be a splain function.

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It would be the mathematically ideal fit.

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But do you want to have the physically optimal fit so you want to fit it to the physical model?

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And therefore, for example, you disregard these thermal fluctuation effects.

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So this is at least for many scientific problems, the better way to go to really think first about

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it, what would be our model function and then fit this model of function to your data?

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So in our case, we have used here a polynomial, and that's for our model function.

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We have made it really general, so you could now go ahead and test with a higher order polynomial.

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But please be warned, this can often leads to problems because the higher order you go into polynomial,

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the higher these terms will get and the higher the error will get, and you will soon run to some memory

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issues.

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But in general, it should work to you to consider also higher order polynomials.

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And of course, also you could use a completely different model function.

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You could use an exponential function sine function, anything that you want, basically.

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And this will be the only thing that you have to update.

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Of course, you have to be careful here a bit with the parameters of the coefficient so that you still

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call them and define them as an array.

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But if you do this, then you have to, of course, also take care a bit of the error function.

151
00:12:00,270 --> 00:12:06,750
But the general idea in the finding the error function and then the finding the error of its gradient

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function and then looping over and over again so that we can update the value of a this will all remain

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the same no matter which function you use for the fit.