1 00:00:00,005 --> 00:00:05,476 In this lecture, we will be talking about Simulating MA (2), 2 00:00:05,476 --> 00:00:08,662 moving average of order two process. 3 00:00:11,246 --> 00:00:14,057 Objectives are to simulate a moving average process and 4 00:00:14,057 --> 00:00:16,760 interpret correlogram of a Moving average process. 5 00:00:18,500 --> 00:00:23,039 So let's remember, moving average process of order two for 6 00:00:23,039 --> 00:00:27,845 example, Xt depends on the noise two days back or two step back. 7 00:00:27,845 --> 00:00:31,030 It depends on Zt, Zt minus 1, zt minus 2. 8 00:00:31,030 --> 00:00:34,540 All of these are identical distributed independent, 9 00:00:34,540 --> 00:00:39,140 normal random variables with some mean and variance. 10 00:00:39,140 --> 00:00:44,720 And Zt minus 1 and Zt minus 2 has some weight on it, theta1 and theta2. 11 00:00:45,770 --> 00:00:47,700 So in this simulation, 12 00:00:47,700 --> 00:00:53,009 we're going to use standard normal distribution for our noise. 13 00:00:53,009 --> 00:00:57,377 So Xt = Zt + 0.7 Zt- 1 + 0.2 Zt- 2 and 14 00:00:57,377 --> 00:01:02,653 all of these noises are standard normal distribution. 15 00:01:06,941 --> 00:01:12,718 So have here are up and running, we can use a shell but for this simulation, 16 00:01:12,718 --> 00:01:18,150 I would like to open a document editor and write the code on the editor. 17 00:01:18,150 --> 00:01:23,340 So you can do it by just going to the file and say a new document and 18 00:01:23,340 --> 00:01:26,610 once you push new document, there's a new document opened up. 19 00:01:26,610 --> 00:01:31,500 You can write our code here and we can WAN the code but I have already 20 00:01:31,500 --> 00:01:35,790 opened up a document and write some piece of code here. 21 00:01:35,790 --> 00:01:37,180 So let's go step by step here. 22 00:01:37,180 --> 00:01:39,920 At the beginning these are my comments. 23 00:01:39,920 --> 00:01:42,410 Comments always starts with a number sign. 24 00:01:43,570 --> 00:01:47,970 So I'm generate noise, remember, noise are coming from normal distribution. 25 00:01:47,970 --> 00:01:53,380 I'm going to generate 10,000 dat point from standard normal distribution. 26 00:01:53,380 --> 00:01:55,470 I'm going to call MA2. 27 00:01:55,470 --> 00:01:57,300 This is my variable. 28 00:01:57,300 --> 00:02:01,740 I'm going to introduce that variable into the RM, I'm going to call it null and 29 00:02:01,740 --> 00:02:06,280 here with this loop, I am actually getting a rating MA2 process. 30 00:02:06,280 --> 00:02:13,270 So, syntax for loop in R is that 4I, I being the index in the range. 31 00:02:13,270 --> 00:02:16,611 My range will start from three to 10,000. 32 00:02:16,611 --> 00:02:19,960 Remember, my MA and MA2 process. 33 00:02:19,960 --> 00:02:25,190 My Xt values will depend on Ztn, Zt minus one and 34 00:02:25,190 --> 00:02:30,090 Zt minus two, so I'm going to use Z1 and Z2. 35 00:02:30,090 --> 00:02:32,750 That's why I need to start from X3. 36 00:02:32,750 --> 00:02:37,500 So I am starting from index three. 37 00:02:37,500 --> 00:02:43,360 This is my noise from right now, noise from yesterday, noise from two days ago. 38 00:02:43,360 --> 00:02:46,650 And I have appropriate weights on them, 0.7 and 0.2, 39 00:02:46,650 --> 00:02:52,350 that would generate a data set where the first two points are naught. 40 00:02:52,350 --> 00:02:55,010 So the first two points are missing. 41 00:02:55,010 --> 00:02:59,760 So I'm going to shift it, so I'm going to shift all of it by two units to the left. 42 00:02:59,760 --> 00:03:05,550 I'm going to call MA2 from 3 to the n, all of these basically shifted. 43 00:03:05,550 --> 00:03:10,710 Once I shift it I'm going to call a new process, a moving average process. 44 00:03:10,710 --> 00:03:14,210 Remember, this moving average process still is just a data point. 45 00:03:14,210 --> 00:03:17,830 It does not have time series structure on it. 46 00:03:17,830 --> 00:03:20,170 So let's put some time structure on this vanilla data. 47 00:03:20,170 --> 00:03:27,250 I have this data, I'm using ts operator, which will make it a time series. 48 00:03:27,250 --> 00:03:30,180 And I'm going to put it exactly back into the moving average process. 49 00:03:30,180 --> 00:03:34,000 Now I truly have a moving average process. 50 00:03:34,000 --> 00:03:40,120 Now it's time to plot this moving average process and at its column graph. 51 00:03:41,170 --> 00:03:44,510 So before I do that, I'm going to do this partition. 52 00:03:44,510 --> 00:03:47,840 I'm going to partition my multiple graphics. 53 00:03:47,840 --> 00:03:51,460 My graphs was going to be basically one frame. 54 00:03:51,460 --> 00:03:54,790 My frame's going to have a multi-frame. 55 00:03:54,790 --> 00:03:57,010 Basically I'm going to have two rows. 56 00:03:57,010 --> 00:03:59,850 And each row is going to be the plot and 57 00:03:59,850 --> 00:04:03,540 the ACF and to do this, I'm going to use this par routine. 58 00:04:03,540 --> 00:04:09,370 So par(mfrow), this stands for multi frame row. 59 00:04:09,370 --> 00:04:12,715 And if I put 2, 1, this means two rows and one columns. 60 00:04:13,750 --> 00:04:14,520 Now I can plot. 61 00:04:14,520 --> 00:04:18,810 I'm going to plot my moving average process and I'm going to plot my ACF. 62 00:04:18,810 --> 00:04:21,970 I put a title for both of them and 63 00:04:21,970 --> 00:04:26,690 I am going to color my plot to blue in the first row. 64 00:04:28,300 --> 00:04:33,740 So I do have it, I can save it using dot our extension. 65 00:04:33,740 --> 00:04:37,750 And all I have to do, if you are Mac, you just do command E. 66 00:04:37,750 --> 00:04:41,630 That will execute the source code. 67 00:04:41,630 --> 00:04:44,630 If you are on Windows, I think you will do CTRL + E. 68 00:04:44,630 --> 00:04:49,538 So if I did Command E, basically it just go to the shelve, 69 00:04:49,538 --> 00:04:54,230 runs it on the shelve, and gives you the output. 70 00:04:54,230 --> 00:04:59,760 Here I have a moving average process of order 2. 71 00:04:59,760 --> 00:05:00,980 Without analyzing it, 72 00:05:00,980 --> 00:05:04,970 it would be impossible to tell this is actually moving average process. 73 00:05:04,970 --> 00:05:06,610 But here is the thing. 74 00:05:06,610 --> 00:05:12,380 I do have my ACF and ACF has some particular structure. 75 00:05:12,380 --> 00:05:15,440 I realize that I have a correlation at lag 0. 76 00:05:15,440 --> 00:05:16,950 This is always the case. 77 00:05:16,950 --> 00:05:19,733 It always starts ACF one but 78 00:05:19,733 --> 00:05:24,582 then there's a high correlation with lag one. 79 00:05:24,582 --> 00:05:29,190 This makes sense because we are still getting noises from the previous step. 80 00:05:29,190 --> 00:05:33,795 And there's also noise coming from the two steps back, two days back and 81 00:05:33,795 --> 00:05:35,847 then boom, there's nothing. 82 00:05:35,847 --> 00:05:37,149 So this is very, 83 00:05:37,149 --> 00:05:43,710 very much a characteristic of MAQ process if you are looking at MAQ process. 84 00:05:43,710 --> 00:05:48,696 If you take ACF off MA q process, it lags or 85 00:05:48,696 --> 00:05:53,268 the correlation will cut off at LAGQ. 86 00:05:53,268 --> 00:05:59,538 In this case, we have MA2 process and then ACF has to cut from lag 2. 87 00:05:59,538 --> 00:06:02,910 And this is lag 1, this is lag 2, boom, I have nothing. 88 00:06:02,910 --> 00:06:08,886 So this is one of the ways we're going to model our time series later on. 89 00:06:08,886 --> 00:06:14,374 You're going to look at ACF and if you see an ACF cuts off after some lag, 90 00:06:14,374 --> 00:06:19,880 that gives us a reason to model our data using a moving average process. 91 00:06:21,740 --> 00:06:22,950 What have you learned in this lecture? 92 00:06:22,950 --> 00:06:27,600 You have learned how to simulate the moving average processes in R. 93 00:06:27,600 --> 00:06:30,650 And you have learned that ACF or 94 00:06:30,650 --> 00:06:35,120 moving average process of all the q cuts off at lag q.