1 00:00:00,480 --> 00:00:03,600 So let's continue discussing other type of plots. 2 00:00:04,620 --> 00:00:10,050 So here you have seen how two plots, two dimensional data, so an X coordinate and the Y coordinate, 3 00:00:10,050 --> 00:00:10,800 for example. 4 00:00:11,580 --> 00:00:16,500 And we have also encountered here our example where we had X, Y and Z coordinates. 5 00:00:17,010 --> 00:00:21,570 But of course, this was a bit of a problem because we could only make two separate plots, which we 6 00:00:21,570 --> 00:00:26,940 have then combined into one plots, but it's not really a three dimensional representation. 7 00:00:28,020 --> 00:00:31,200 What we can do instead is to use a so-called density plot. 8 00:00:31,890 --> 00:00:38,640 So, for example, if you think of when you go hiking and you have some map of a mountain, then you 9 00:00:38,640 --> 00:00:42,180 often see these type of density or contour plots. 10 00:00:42,810 --> 00:00:48,360 And here you have the X and Y coordinates, which are really spatial coordinates. 11 00:00:48,720 --> 00:00:53,490 And then the Z coordinate is the height of the profile of the mountain. 12 00:00:54,300 --> 00:01:01,470 And we will do here something very similar, and we will use density or control plots to plot three 13 00:01:01,470 --> 00:01:02,400 dimensional data. 14 00:01:03,540 --> 00:01:09,900 So the first thing that we are going to do is we are going to have to discuss a bit more about arrays. 15 00:01:10,560 --> 00:01:19,770 So you see plotting, especially in Python, is often related to handling arrays and even multi-dimensional 16 00:01:19,770 --> 00:01:20,220 arrays. 17 00:01:20,850 --> 00:01:27,330 And this is really the most difficult thing about plotting to really have to syntax correct and to manipulate 18 00:01:27,330 --> 00:01:30,030 these arrays so that they fit the correct syntax. 19 00:01:31,020 --> 00:01:34,020 And so, for example, what we had previously was something like this. 20 00:01:35,790 --> 00:01:38,000 I don't know why there is this mistake here. 21 00:01:38,010 --> 00:01:39,930 Let's try it again. 22 00:01:40,470 --> 00:01:41,190 OK, now it's gone. 23 00:01:41,550 --> 00:01:42,150 So we right? 24 00:01:42,150 --> 00:01:46,880 And he got a little space for linear space. 25 00:01:47,340 --> 00:01:50,880 Basically, a list or an array starting from zero to five. 26 00:01:51,210 --> 00:01:54,810 And we want to have six values here, which gives us something like this. 27 00:01:56,190 --> 00:01:58,650 Now let's use two of these. 28 00:02:01,070 --> 00:02:08,270 And, for example, for the other one, I think other values, let's just use five to 10 and also six 29 00:02:08,270 --> 00:02:08,720 steps. 30 00:02:09,470 --> 00:02:14,720 And now we are going to use a non-paid method called and p dot Magritte. 31 00:02:16,310 --> 00:02:22,520 So let's fix the syntax here, and I would say, Yeah, let's do it like this. 32 00:02:23,420 --> 00:02:29,330 So now we have used the command and mesh grid on these two grids or on these two arrays. 33 00:02:30,020 --> 00:02:37,190 And what it does is it basically spans a whole space for the coordinates. 34 00:02:38,690 --> 00:02:46,940 So if we would now transpose this by writing dot t, sorry, that's not working because it's the wrong 35 00:02:46,940 --> 00:02:47,510 syntax. 36 00:02:48,530 --> 00:02:54,530 Let me show you what I mean here, if we would right and p dot transpose. 37 00:02:57,680 --> 00:03:00,620 Which basically means reorder the whole thing. 38 00:03:02,910 --> 00:03:05,410 And then what we get is this. 39 00:03:05,680 --> 00:03:10,420 So then we combine this value and this value and so on and so on. 40 00:03:10,990 --> 00:03:21,070 So it means now we have 25 data points which contain basically the numbers that are spent by these two 41 00:03:21,070 --> 00:03:21,760 lists here. 42 00:03:22,540 --> 00:03:30,200 So, you know, we have this list, we have another list which was this one here, and this will be 43 00:03:30,220 --> 00:03:33,450 our X-axis who are the coordinates go from zero to five. 44 00:03:33,460 --> 00:03:36,890 And this will be all why access where the coordinates go from five to 10. 45 00:03:37,690 --> 00:03:45,760 So we have used mass grids, which we have then transposed to end up with 25 pairs of coordinates where 46 00:03:45,760 --> 00:03:50,380 the X value goes from zero to five and the Y value goes from five to 10. 47 00:03:51,340 --> 00:04:00,700 So now we have really a two dimensional arrow array, which, yeah, which basically describes our range 48 00:04:00,700 --> 00:04:03,190 for which we want to plot the height profile. 49 00:04:03,370 --> 00:04:05,650 So we have the X range and the Y range. 50 00:04:05,950 --> 00:04:07,840 So let's get to the actual example. 51 00:04:08,560 --> 00:04:15,130 So what we're going to do next is we are going to take take this command here and increase basically 52 00:04:15,130 --> 00:04:16,269 the number of points. 53 00:04:17,290 --> 00:04:25,270 And one thing that I forgot to mention is that this year may be, for me, the more reasonable way to 54 00:04:25,270 --> 00:04:27,520 write down the coordinates just pairs of numbers. 55 00:04:27,970 --> 00:04:32,290 But we have seen that actually what we have to provide in these plots. 56 00:04:32,860 --> 00:04:39,670 So just for the scatterplot and also for the upcoming plot is first only the X coordinates and then 57 00:04:39,670 --> 00:04:40,900 only the Y coordinates. 58 00:04:41,620 --> 00:04:48,550 So it turns out, let me scroll down again that this output here has already to correct syntax and we 59 00:04:48,550 --> 00:04:49,570 do not need this one. 60 00:04:50,320 --> 00:04:53,230 So we do not need to transpose one with this one, actually. 61 00:04:54,520 --> 00:04:57,580 So let's delete this and use this one instead. 62 00:04:58,750 --> 00:05:05,340 So basically what we have is we basically have such a thing here. 63 00:05:05,350 --> 00:05:08,470 We can also write it down like this to save a bit of space. 64 00:05:09,010 --> 00:05:14,860 And I would say we go from minus 10 to 10 and 201 points. 65 00:05:15,550 --> 00:05:19,000 And here the same thing, 201 hundred. 66 00:05:20,260 --> 00:05:24,490 And this one will be our coordinates. 67 00:05:24,880 --> 00:05:33,820 And so I would say we just saved them as x, we could say. 68 00:05:33,850 --> 00:05:35,950 I don't know, x two and Y two. 69 00:05:37,240 --> 00:05:42,010 And so if I run this, we have now our lists for the coordinates. 70 00:05:42,010 --> 00:05:47,080 For example, x one or three x two would be all of these coordinates. 71 00:05:48,370 --> 00:05:50,770 So you will see in a second that this really works. 72 00:05:50,950 --> 00:05:57,460 So basically, we have stored now these two individual arrays for the X coordinates Y coordinates in 73 00:05:57,640 --> 00:06:01,180 X two and Y two, and we have increased the number of points. 74 00:06:02,560 --> 00:06:07,060 So now what we're going to do is we are going to generate the actual values for the height. 75 00:06:07,900 --> 00:06:10,000 For this, I will call this Z. 76 00:06:11,110 --> 00:06:13,990 Yeah, we'll call it a Z two and Z two. 77 00:06:14,350 --> 00:06:16,750 Here we just have to provide some kind of function. 78 00:06:17,380 --> 00:06:18,280 It doesn't really matter. 79 00:06:18,280 --> 00:06:24,190 I just take a function x plus y squared. 80 00:06:25,570 --> 00:06:31,900 So you see, this is supposed to be in the raid itself with where every combination of X and Y coordinate 81 00:06:31,900 --> 00:06:33,640 must have, of course, the Z value. 82 00:06:34,060 --> 00:06:36,430 So we have to use here these arrays. 83 00:06:38,390 --> 00:06:45,890 So now when I run this, I can call the Z two, and you see it's a very large array. 84 00:06:47,060 --> 00:06:54,430 It's basically 200 one times 200 one array with all the values of two. 85 00:06:55,670 --> 00:06:57,500 And now we can finally do the plot. 86 00:06:58,430 --> 00:07:02,230 And this you can do by writing down Pulte thoughts control. 87 00:07:03,230 --> 00:07:08,180 And then first at the X coordinates the Y coordinates and then the Z coordinates. 88 00:07:09,260 --> 00:07:11,450 And this will be our results. 89 00:07:12,740 --> 00:07:17,810 So you see, like on a height map, you have these lines where on the line the height is always the 90 00:07:17,810 --> 00:07:18,290 same. 91 00:07:18,650 --> 00:07:23,270 So you see here will probably be the maximum, or it could also be the minimum hard to see. 92 00:07:23,630 --> 00:07:28,670 And then here on these lines, you climb up or down the mountain. 93 00:07:29,570 --> 00:07:32,420 So to find out what is the maximum and the minimum? 94 00:07:32,720 --> 00:07:35,330 It would be a good idea to add a legend. 95 00:07:35,900 --> 00:07:39,050 And here I think we can do even some fancy stuff. 96 00:07:40,070 --> 00:07:45,900 What we can do is we first see this plot and this will give it a name. 97 00:07:45,920 --> 00:07:49,520 Basically, we called it, I don't know, control plot. 98 00:07:51,830 --> 00:07:54,680 And now what we are going to do is we are going to add a legend. 99 00:07:55,040 --> 00:07:59,390 And for this, I write Pulte Dot sea level. 100 00:07:59,780 --> 00:08:03,200 So see label is a special type of label or legend. 101 00:08:03,400 --> 00:08:08,360 And now we first have to provide the type of plots that we want to apply this legend to. 102 00:08:09,230 --> 00:08:11,210 And then just some options. 103 00:08:11,240 --> 00:08:14,250 So you will see in a second look, this does in line equal one. 104 00:08:14,660 --> 00:08:18,890 And then we can also provide some font size for you guys. 105 00:08:18,980 --> 00:08:21,350 Just try like this and let's see. 106 00:08:22,070 --> 00:08:22,620 All right. 107 00:08:22,640 --> 00:08:28,730 So now you see, we have added to every one of those lines the value of the function. 108 00:08:28,970 --> 00:08:30,470 So basically the value of Z. 109 00:08:31,340 --> 00:08:36,620 So this means on this line, all the Z values are the same and they're equal to zero. 110 00:08:37,039 --> 00:08:42,140 And then here in this line, they are 15, which means in this area, the value is between zero and 111 00:08:42,140 --> 00:08:42,590 15. 112 00:08:43,280 --> 00:08:48,620 So actually, this means this is not the hail, this is the bottom, and we have to climb up in this 113 00:08:48,620 --> 00:08:49,160 direction. 114 00:08:50,400 --> 00:08:53,730 All right, so now you see how we can do such a control plot. 115 00:08:54,750 --> 00:09:02,340 We can do also something very similar by using a so-called Contour F plot, which basically means we 116 00:09:02,340 --> 00:09:03,510 fill the area here. 117 00:09:04,590 --> 00:09:05,820 So let me show this to you. 118 00:09:07,620 --> 00:09:08,760 So we will. 119 00:09:08,920 --> 00:09:09,750 Sorry, not here. 120 00:09:10,170 --> 00:09:21,270 We will add a new cell, and here I will name these X three, y three and Z three and function. 121 00:09:21,270 --> 00:09:25,500 Maybe we use a different function here or I have tested before that. 122 00:09:25,500 --> 00:09:26,730 This looks pretty cool. 123 00:09:26,730 --> 00:09:36,210 So let's use cosine function of the X coordinate plus the Y coordinate, and then let's add a linear 124 00:09:36,210 --> 00:09:37,290 function to it. 125 00:09:37,800 --> 00:09:46,470 Zero point zero five Maybe some spaces here zero point zero five times and then X coordinates minus 126 00:09:47,430 --> 00:09:48,240 y coordinate. 127 00:09:49,560 --> 00:09:55,980 So yeah, this is how we generate our data, the X and Y values and the corresponding Z values. 128 00:09:56,580 --> 00:09:58,260 And now we must plot its. 129 00:09:58,410 --> 00:10:06,750 And for this, I write Potti Dot Contour F for Contour plot, which is filled. 130 00:10:07,530 --> 00:10:12,450 And of course, I must provide now that lists X three y three z three. 131 00:10:13,140 --> 00:10:16,200 And we run this and we have this nice plot. 132 00:10:17,100 --> 00:10:17,880 So what does it mean? 133 00:10:18,090 --> 00:10:20,730 Well, we can just add a legend is a what color bar. 134 00:10:21,120 --> 00:10:24,420 And yeah, apparently here it's really easy to do this. 135 00:10:24,690 --> 00:10:27,540 So we just tried to collaborate with no arguments. 136 00:10:28,170 --> 00:10:29,820 And we have to call a bar to the right. 137 00:10:30,690 --> 00:10:38,580 So you see, now you can read the value of Z, which could be the height of a mountain, for example, 138 00:10:39,150 --> 00:10:39,990 from this legend. 139 00:10:40,050 --> 00:10:44,130 So here for the yellow color, it's pretty high and here it's low for the blue color. 140 00:10:45,450 --> 00:10:52,680 So these are two nice methods to visualize three dimensional data like you would do for a height map 141 00:10:52,680 --> 00:10:55,620 or some topographical data map. 142 00:10:56,190 --> 00:11:01,080 And these types of plots are called density plots or contour plots.