1 00:00:00,060 --> 00:00:07,890 So now we will go away from the mathematical blinds and instead we will try to choose the ideal parameters 2 00:00:08,340 --> 00:00:13,790 of a physically motivated model functions such that some area is minimized. 3 00:00:14,320 --> 00:00:17,310 That sounds a bit cryptic, so let's go to an example. 4 00:00:18,120 --> 00:00:24,270 So we have previously defined the detail, for example, this one that we tried to fit in some way, 5 00:00:24,840 --> 00:00:31,200 and we know already that we have created the data with some polynomial function and some thermal fluctuations, 6 00:00:31,200 --> 00:00:32,159 some random numbers. 7 00:00:32,610 --> 00:00:37,920 And so by just looking at the data, I think it is reasonable that you would assume that this could 8 00:00:37,920 --> 00:00:41,700 be a polynomial function that describes the data. 9 00:00:42,300 --> 00:00:47,490 Also, since it's an odd function, it must be an odd power for the highest power. 10 00:00:48,090 --> 00:00:51,780 And yeah, it looks not so difficult. 11 00:00:51,780 --> 00:00:55,500 So I think it's reasonable to assume that this is a third order polynomial. 12 00:00:55,950 --> 00:01:01,470 But in general, we could say, let's use some polynomial function to fit the data. 13 00:01:02,220 --> 00:01:09,540 So I wrote down that our model function is at the polynomial f of X is a zero plus a one times X to 14 00:01:09,540 --> 00:01:14,430 the power of one plus a two times x two and then more and more terms. 15 00:01:14,760 --> 00:01:21,240 So we can say this is a sum where the index case starts from zero goes to sum end, so maximum value. 16 00:01:21,570 --> 00:01:28,380 And then the terms are just these evolution coefficients a k and then we have these polynomial x to 17 00:01:28,380 --> 00:01:29,280 the power of K. 18 00:01:30,330 --> 00:01:34,710 So for two stars, let us define such a model function in general. 19 00:01:35,520 --> 00:01:44,820 So I will write definition of a polynomial model and we need X and we need support coefficients. 20 00:01:45,570 --> 00:01:51,380 So of course, we could say we defined the coefficients and then just define our polynomials. 21 00:01:51,390 --> 00:01:58,470 We don't put it as an argument here, but we want what we want to try to do is we want to fit these 22 00:01:58,470 --> 00:01:59,130 parameters. 23 00:01:59,160 --> 00:02:00,930 This is the whole purpose of the methods. 24 00:02:01,470 --> 00:02:06,510 So these are the things that we try to change and that we try to optimize. 25 00:02:07,020 --> 00:02:13,140 And therefore it's a very good idea to define the function with the h-he's as the arguments. 26 00:02:14,250 --> 00:02:18,930 And we don't have to define every of these coefficients individually. 27 00:02:19,290 --> 00:02:25,230 We can just define this as an array and this array can be of variable size. 28 00:02:25,830 --> 00:02:31,380 So you see, we can just go ahead now and define or program this sum here. 29 00:02:31,380 --> 00:02:40,680 So as previously, we just say T is zero that we make a loop for K and range O. 30 00:02:40,680 --> 00:02:43,560 We could then say, for example, length of P. 31 00:02:44,310 --> 00:02:53,310 So this means if I were a A-rated, we give for the polynomials has four terms, then this loop will 32 00:02:53,310 --> 00:03:00,540 be run four times and we will then say T will be updated old T plus. 33 00:03:01,110 --> 00:03:08,070 And then we just say use the case elements of the array of eight and multiply it by X to the power of 34 00:03:08,070 --> 00:03:08,400 K. 35 00:03:08,530 --> 00:03:10,200 So this is exactly what's written here. 36 00:03:11,070 --> 00:03:14,460 And then, of course, return the value of T. 37 00:03:16,050 --> 00:03:24,990 And if we now say our starting point for these is will be a zero, which is and three and tier I use 38 00:03:24,990 --> 00:03:28,980 known to the coefficients that we have used for our function from the beginning. 39 00:03:29,400 --> 00:03:39,420 So like minus two two point four minus 0.5 and minus zero point three five, then we can go ahead and 40 00:03:39,420 --> 00:03:55,590 plot this so we can just say T Dot X Label X to Y Label Y and then X list. 41 00:03:56,100 --> 00:03:57,720 So let's redefine the X list. 42 00:03:57,720 --> 00:04:05,050 I think we have overwritten it at some point, so let's just redefine it in space minus five to five. 43 00:04:05,070 --> 00:04:09,630 This is what we have chosen in the beginning and the number of points we called end points. 44 00:04:10,830 --> 00:04:23,580 And then we can finally plot it's plotted or plots x list come out and then purely nominal model of 45 00:04:23,580 --> 00:04:26,940 X list, and then the coefficients will be a zero. 46 00:04:30,780 --> 00:04:35,190 And so then we have plotted here our polynomial. 47 00:04:35,430 --> 00:04:37,620 So we did not plot our data this time. 48 00:04:37,950 --> 00:04:39,660 We just plotted the polynomial. 49 00:04:40,380 --> 00:04:47,310 So ideally, we will start from just some random values of zero because we shouldn't know what these 50 00:04:47,310 --> 00:04:48,150 values are yet. 51 00:04:48,630 --> 00:04:54,030 And then we use our method that we will implement in the following lectures, and then we will update 52 00:04:54,030 --> 00:04:55,380 the values. 53 00:04:55,410 --> 00:05:00,330 At some point, we should hopefully end up with values that are quite similar to these. 54 00:05:00,750 --> 00:05:07,260 And then our function will look very similar to this one so that our data points are described very 55 00:05:07,260 --> 00:05:07,530 well. 56 00:05:08,760 --> 00:05:12,990 And for this, we will continue with the finding an error of function. 57 00:05:13,560 --> 00:05:19,890 So assuming that we do not have the correct values of a yet, then there will be some error. 58 00:05:20,160 --> 00:05:22,560 And first of all, we need a measure for this error.