1 00:00:00,120 --> 00:00:06,890 So far, we have discovered the Taylor expansion, we have discussed the properties of polynomials, 2 00:00:06,900 --> 00:00:13,770 and we have found that basically any continuous function can be expanded in terms of a Taylor expansion. 3 00:00:14,490 --> 00:00:20,040 So of course, in this total expansion, we can never take into account all terms because there is an 4 00:00:20,040 --> 00:00:23,880 infinite number of these terms, but we have to stop at some point. 5 00:00:24,750 --> 00:00:31,470 But we've also seen that even for difficult functions, the approximate form of the Taylor expansion 6 00:00:31,470 --> 00:00:38,670 works quite well in the vicinity of the value zero that we have chosen for the expansion of the series. 7 00:00:39,660 --> 00:00:47,190 So what we do next is we will use polynomials to interpolate functions or better to say, to interpolate 8 00:00:47,190 --> 00:00:51,060 data, to get the function that describes this data. 9 00:00:51,690 --> 00:00:57,300 So this is a very, very common phenomenon in science and in physics, for example, where you do some 10 00:00:57,300 --> 00:01:04,530 measurement and you know, the mechanism that is behind the data that gave rise to the data. 11 00:01:04,980 --> 00:01:10,770 And then you have a model function, for example, and then you try to fit some parameters so that you 12 00:01:10,770 --> 00:01:18,600 find out what exactly are the parameters that allow you to describe the measured data in terms of the 13 00:01:18,600 --> 00:01:19,500 model function? 14 00:01:20,250 --> 00:01:27,840 And of course, I could have just started this lecture here by giving you some data points, but I think 15 00:01:27,840 --> 00:01:34,380 it's also a good practice and a good exercise to to find these data points and to generate the data. 16 00:01:34,860 --> 00:01:40,680 And also then I can be very honest with you what the data is and how it has been generated. 17 00:01:41,850 --> 00:01:48,480 So first of all, I will define here a function called correct data. 18 00:01:49,320 --> 00:01:51,210 Sorry, correct function. 19 00:01:53,180 --> 00:01:59,570 And the argument would just be so this will be a one dimensional problem and the future will be some 20 00:01:59,570 --> 00:02:02,900 polynomial because we have seen that polynomials are quite nice. 21 00:02:03,350 --> 00:02:07,940 So let's use a polynomial here, but it's in general not really necessary to do. 22 00:02:08,509 --> 00:02:14,630 But I will use it because then we know also the coefficients. 23 00:02:15,140 --> 00:02:17,810 So I will talk about this in a second. 24 00:02:18,350 --> 00:02:19,760 So this will be our function. 25 00:02:22,310 --> 00:02:28,400 And you see, it's a third order polynomial where we have four coefficients 15 two point four minus 26 00:02:28,400 --> 00:02:32,690 0.5 and minus zero point three five. 27 00:02:33,680 --> 00:02:40,390 And what we will do later on this, we will establish a model function, which will be a polynomial 28 00:02:40,400 --> 00:02:41,990 and a third order polynomial. 29 00:02:42,380 --> 00:02:50,240 And then we will try to fit the coefficients here to fit the data that we are going to create using 30 00:02:50,240 --> 00:02:51,290 this correct function. 31 00:02:51,290 --> 00:02:56,810 And then we can compare if the parameters that we have determined by this fit are correct because we 32 00:02:56,810 --> 00:02:58,280 know what are the correct values. 33 00:02:58,280 --> 00:02:59,690 These are these coefficients you. 34 00:03:01,230 --> 00:03:09,570 So let me go ahead and define some some constants here, called number of points. 35 00:03:10,200 --> 00:03:16,650 And let's use 21 and then we have an ex list that is given violin space. 36 00:03:18,820 --> 00:03:24,730 And it goes from minus five to five, and the number of points will be and points. 37 00:03:26,020 --> 00:03:34,330 And our data with which we will start will be data zero will be an array and the coordinates will of 38 00:03:34,330 --> 00:03:43,330 course be given by the list at the Y, coordinates will be given by a correct function acting on the 39 00:03:43,330 --> 00:03:44,020 ex list. 40 00:03:45,460 --> 00:03:54,310 So I run both of these cells and now we should have our data so we can we can look at it by writing. 41 00:03:57,070 --> 00:03:59,200 But clotting experts is why. 42 00:04:00,040 --> 00:04:01,990 So we can do it like this, of course. 43 00:04:02,470 --> 00:04:10,720 And then we just write penalty dot plot data, zero comma data one. 44 00:04:11,620 --> 00:04:15,610 And of course, we want to plot here data zero and data zero. 45 00:04:16,750 --> 00:04:19,329 So this will be our third order polynomial. 46 00:04:19,720 --> 00:04:21,640 And if I use your scatter. 47 00:04:24,250 --> 00:04:30,070 With Tuti, of course, then you see, this is our data that we have generated, and these are the points 48 00:04:30,070 --> 00:04:31,480 that we try to fit. 49 00:04:33,040 --> 00:04:39,670 However, in reality, when you do measurement and experiment, the measured data is not always that 50 00:04:39,670 --> 00:04:40,360 perfect. 51 00:04:40,390 --> 00:04:44,230 So there is some noise in the data and there is some error in the data. 52 00:04:44,830 --> 00:04:48,190 And here, for example, you could definitely see easily, OK? 53 00:04:48,220 --> 00:04:51,310 We just take a 3.0 polynomial and then it should work. 54 00:04:51,310 --> 00:04:56,650 But actually, this data will be blurred and it will be distorted in reality. 55 00:04:57,520 --> 00:05:03,520 So what we will do next is we will modify the X and Y values by adding some random numbers, and I will 56 00:05:03,520 --> 00:05:08,860 do this in the following way we will right and p dot random dot rant. 57 00:05:10,630 --> 00:05:12,910 And then we can just say, for example, end points. 58 00:05:13,900 --> 00:05:21,220 And this will give us an array of 21 points in this or 21 random values in this case, and the random 59 00:05:21,220 --> 00:05:23,800 values will be in the range of zero and one. 60 00:05:24,790 --> 00:05:32,680 So if we want to have it in the range of minus one and one, we just have to subtract one and then we 61 00:05:32,680 --> 00:05:33,880 have to multiply 62 00:05:36,670 --> 00:05:37,140 two. 63 00:05:37,150 --> 00:05:40,270 But only two the first yeah, to to randomize itself. 64 00:05:41,080 --> 00:05:44,890 So then the range will be from minus one to plus one. 65 00:05:46,150 --> 00:05:51,100 OK, and then we can, of course, scale it with any value that we want, for example, zero point one 66 00:05:51,100 --> 00:05:57,340 and then the range will be in the range of minus 0.1 + + 0.1. 67 00:05:58,510 --> 00:06:05,770 So what I will do next is I will create P file or an R equal data and it will be an array. 68 00:06:08,440 --> 00:06:18,340 And this array will be based on our previous data as a data zero and data one, data zero zero and data 69 00:06:18,340 --> 00:06:18,940 zero one. 70 00:06:19,360 --> 00:06:25,270 And so I can basically it could just copy them here. 71 00:06:25,270 --> 00:06:35,230 But what I want to do instead is I will write X List, which is data zero zero plus, and then I tested 72 00:06:35,230 --> 00:06:36,040 what looks good. 73 00:06:38,290 --> 00:06:46,000 Basically, this random random command here, this array of random numbers scaled by a 0.25. 74 00:06:46,480 --> 00:06:47,980 So this will be the will you. 75 00:06:48,550 --> 00:06:50,920 And then for the Y value, we take 76 00:06:53,860 --> 00:06:56,390 our basically or. 77 00:06:57,790 --> 00:07:01,420 So so we have we could also see just data zero zero. 78 00:07:02,020 --> 00:07:04,570 And then for the Y value, we take data zero one. 79 00:07:05,740 --> 00:07:08,650 And here we add a similar thing. 80 00:07:09,550 --> 00:07:15,580 But I tested that's 0.25 is maybe a bit too low. 81 00:07:15,850 --> 00:07:18,910 Let's use a higher value of five point zero. 82 00:07:19,990 --> 00:07:23,010 So I think, or I hope that I don't have a typo here. 83 00:07:23,020 --> 00:07:30,320 Let's test let's copy this code and use data instead of data zero. 84 00:07:32,290 --> 00:07:36,490 And OK, yeah, we have some blurred values here. 85 00:07:38,160 --> 00:07:41,400 So what we can do now also is we can 86 00:07:44,220 --> 00:07:50,790 plot the actual function to see what is here, the difference so we can write X list. 87 00:07:51,300 --> 00:07:55,530 Well, we could also write, you know, later zero 88 00:07:57,900 --> 00:08:05,910 zero and date zero one and the color will be black. 89 00:08:08,040 --> 00:08:12,960 So this will be the correct function that we have used for creating the data. 90 00:08:13,320 --> 00:08:20,880 And then you see, due to the random fluctuations we have that our data on this one will be a bit different. 91 00:08:21,300 --> 00:08:29,490 So this could be quite real realistic physical situation from a measurement where you have some noise 92 00:08:29,730 --> 00:08:30,810 on top of your data. 93 00:08:32,130 --> 00:08:38,100 And so in the following, we will discuss which options we have to fit this function. 94 00:08:38,370 --> 00:08:43,799 For example, I mean, we know what is the correct function to fit this because we have created the 95 00:08:43,799 --> 00:08:44,250 data. 96 00:08:44,790 --> 00:08:47,370 But let's say we have just a measurement with these points. 97 00:08:47,760 --> 00:08:53,520 How would we find out what would be a good fit or a good description of this data? 98 00:08:54,180 --> 00:08:59,580 And then the following lectures we will talk about spleens and fitting model functions.