1 00:00:10,030 --> 00:00:17,620 In this lecture, we are going to check our understanding of the Akef and encode this lecture is going 2 00:00:17,620 --> 00:00:23,320 to walk you through a prepared CoLab notebook, although a very good exercise, which I always recommend, 3 00:00:23,590 --> 00:00:29,240 is once you know how this is done, to try and recreate it yourself with as few references as possible. 4 00:00:29,770 --> 00:00:35,440 As always, you can check the lectures, how to code by yourself and how to practice for a more in-depth 5 00:00:35,440 --> 00:00:36,220 discussion. 6 00:00:36,700 --> 00:00:42,040 If there's anything in this lecture you didn't understand or you think I missed a step or didn't explain 7 00:00:42,040 --> 00:00:45,270 why we were doing something, please use the Q&A to inquire. 8 00:00:45,880 --> 00:00:50,890 As usual, you can look at the title of the notebook to determine what notebook we are currently looking 9 00:00:50,890 --> 00:00:51,200 at. 10 00:00:53,180 --> 00:00:59,900 So the first thing we're going to do is import plot and a plot akef from stat's models, these are more 11 00:00:59,900 --> 00:01:06,440 useful than just generic HCF functions that only give you back an array, since these will make a plot 12 00:01:06,560 --> 00:01:08,660 and draw the confidence bounce automatically. 13 00:01:12,790 --> 00:01:15,880 Next, we're going to import numpty and matplotlib. 14 00:01:19,040 --> 00:01:22,850 Next, we're going to generate IDE noise from the standard normal. 15 00:01:26,260 --> 00:01:31,420 Just to build a point of comparison for what we will see later in this lecture, let's plot what this 16 00:01:31,420 --> 00:01:35,650 noise looks like, OK, so I'm pretty sure this is not too surprising. 17 00:01:41,900 --> 00:01:47,930 Next, let's call the function of plot, we'll be looking at auto regressive models first. 18 00:01:48,110 --> 00:01:54,110 And as you know, the relevant plot for that is the notice that I'm using. 19 00:01:54,110 --> 00:01:56,970 The subplots function to manage the size of a plot. 20 00:01:57,680 --> 00:02:02,160 This returns an access object, which I then pass into the plot function. 21 00:02:03,290 --> 00:02:03,680 All right. 22 00:02:03,680 --> 00:02:04,820 So what do we see? 23 00:02:05,720 --> 00:02:10,880 Well, as expected, most of the lagged values are very near zero. 24 00:02:11,450 --> 00:02:16,430 The first value is one, since that's just the autocorrelation of each point with itself. 25 00:02:17,600 --> 00:02:23,160 Notice how there are multiple values in which the pickoff goes just outside the confidence bounds. 26 00:02:23,690 --> 00:02:27,590 However, you have to remember the definition of the confidence interval. 27 00:02:28,250 --> 00:02:33,470 Basically, this is allowed to happen randomly five percent of the time, even when there is no true 28 00:02:33,470 --> 00:02:34,330 correlation. 29 00:02:35,030 --> 00:02:41,240 Since these values are so close to the threshold, we can intuit that it's probably OK to ignore them. 30 00:02:46,620 --> 00:02:53,260 Next, let's create an air one process and test that again to generate this process. 31 00:02:53,280 --> 00:02:57,520 I'm going to create a list called X1 within initial value of zero. 32 00:02:58,200 --> 00:03:04,040 Then I'm going to enter a loop that goes for 1000 iterations inside this loop. 33 00:03:04,050 --> 00:03:10,950 I'm going to create the next value by taking zero point five times the previous value, plus some Gaussian 34 00:03:10,950 --> 00:03:12,960 noise with standard deviation. 35 00:03:12,960 --> 00:03:16,500 At zero point one, we'll call this variable X. 36 00:03:17,900 --> 00:03:20,180 Next, we append X to X1. 37 00:03:21,860 --> 00:03:28,730 When we're outside the loop, we can't explain it to an umpire in the next block, we plot one. 38 00:03:29,330 --> 00:03:35,300 As you can see, it looks pretty stationary and there's no real observable difference between this and 39 00:03:35,300 --> 00:03:36,640 ID noise. 40 00:03:37,460 --> 00:03:41,420 It's not obvious that each value depends on the previous value. 41 00:03:46,610 --> 00:03:55,580 Next, we plot the pickoff for X1, as you can see, we have a value at like one which is far outside 42 00:03:55,580 --> 00:03:56,990 the confidence threshold. 43 00:03:57,590 --> 00:04:04,690 Therefore, if we were to look just at this plot, we would conclude that our TIME series is an A1 process. 44 00:04:05,240 --> 00:04:09,350 And of course, we know that this is true because that is what we just created. 45 00:04:14,250 --> 00:04:18,880 Next, we're going to do another experiment again for an R-1 process. 46 00:04:19,380 --> 00:04:24,120 I'm not going to run through the details of this code again, since it's very similar to last time. 47 00:04:24,690 --> 00:04:30,330 The main difference with this code is that the coefficient for the previous X term is now negative zero 48 00:04:30,330 --> 00:04:33,310 point five instead of positive zero point five. 49 00:04:33,840 --> 00:04:39,810 We'll see what effect this has on the Earth if we look at the plot of the Time series. 50 00:04:40,080 --> 00:04:45,260 Again, it's not at all obvious that this is any different from just plain Idy noise. 51 00:04:51,230 --> 00:04:57,890 All right, so next we look at the picture, this time we see that there is, again, a non-zero value 52 00:04:57,890 --> 00:05:05,080 at like one, but this time it's negative corresponding to the negative coefficient of our R-1 process. 53 00:05:12,620 --> 00:05:19,550 Next, we're going to generate an R2 process, so this time the Time series will depend linearly on 54 00:05:19,550 --> 00:05:23,690 to pass values instead of just one to initialize. 55 00:05:23,690 --> 00:05:27,750 Our list will set the first two values in X two to be zero. 56 00:05:28,670 --> 00:05:31,490 Inside the loop, we use the following equation. 57 00:05:32,180 --> 00:05:35,570 The coefficient for the one value is zero point five. 58 00:05:36,080 --> 00:05:40,290 The second coefficient for the two value is minus zero point three. 59 00:05:40,910 --> 00:05:45,260 And again, we have Gaussian noise with a standard deviation, a zero point one. 60 00:05:50,130 --> 00:05:55,800 From our Time series plot, it is again the case that we probably cannot distinguish this from idee 61 00:05:55,800 --> 00:06:00,780 noise, although the variants in the later parts of the Times series seems to be quite high. 62 00:06:04,700 --> 00:06:10,850 If we look at the pickoff, we can see that there are now two non-zero lag's corresponding to the two 63 00:06:10,850 --> 00:06:18,980 coefficients of our R2 process, the first non-zero value is positive and the second at non-zero value 64 00:06:18,980 --> 00:06:19,760 is negative. 65 00:06:20,330 --> 00:06:26,390 If we were to use this plot to choose the value of P for fitting in auto regressive model, we would 66 00:06:26,390 --> 00:06:27,640 definitely choose to. 67 00:06:27,950 --> 00:06:30,400 So this confirms that this method works. 68 00:06:35,740 --> 00:06:44,350 Next, we're going to generate an five process to initialize X5 or create an array of all zeroes inside 69 00:06:44,350 --> 00:06:51,190 the loop to generate the next value of X. We will make X depend on three different lagged values rather 70 00:06:51,190 --> 00:06:53,820 than all five in particular. 71 00:06:53,830 --> 00:07:00,070 These will be the last value at T minus one, the second class value at T minus two and the fifth less 72 00:07:00,070 --> 00:07:01,830 value at T minus five. 73 00:07:02,350 --> 00:07:08,830 The weights for these will be zero point five minus zero point three and minus zero point six respectively. 74 00:07:09,460 --> 00:07:14,680 Again, we'll add Gaussian noise with mean zero and standard deviation at zero point one. 75 00:07:18,420 --> 00:07:23,400 Notice that when we plot the process is a time series, it definitely looks a little different from 76 00:07:23,400 --> 00:07:30,120 Idy noise and all of the previous plots, there seems to be more of a pattern in a more wavelike structure 77 00:07:30,210 --> 00:07:31,560 to the Time series. 78 00:07:37,140 --> 00:07:44,970 Next, we noticed something interesting with this, even though our Time series does not depend on the 79 00:07:44,990 --> 00:07:50,740 lag, three and like four values, the corresponding values in the pickoff are still non-zero. 80 00:07:51,510 --> 00:07:56,910 So although we generated the process ourselves and we know that it doesn't depend on the three and like 81 00:07:56,910 --> 00:08:02,070 four values, when we fit in ERP model, we will always include those terms. 82 00:08:02,610 --> 00:08:08,400 So this is a sign that just because you have some in between values that don't affect the next value 83 00:08:08,400 --> 00:08:12,780 in a time series, this does not mean that their values will be zero. 84 00:08:13,650 --> 00:08:14,040 All right. 85 00:08:14,050 --> 00:08:19,200 So since this lecture has been pretty long already, I will see you in the next lecture to look at the 86 00:08:19,200 --> 00:08:21,480 akef and moving average models.