1 00:00:00,810 --> 00:00:04,690 Welcome back to practical time series analysis. 2 00:00:04,690 --> 00:00:07,440 In these lectures we're looking at forecasting. 3 00:00:07,440 --> 00:00:09,500 We're trying to say something interesting and 4 00:00:09,500 --> 00:00:15,720 intelligent about a future state of the system based upon present and past states. 5 00:00:15,720 --> 00:00:18,730 We've already seen how to simple exponential smoothing. 6 00:00:18,730 --> 00:00:21,480 We will determine a smooth level. 7 00:00:21,480 --> 00:00:25,720 We've done double exponential smoothing where we come up with a smooth level and 8 00:00:25,720 --> 00:00:28,020 a smooth trend. 9 00:00:28,020 --> 00:00:33,980 And now we're going to bring in a seasonal effect with triple exponential smoothing. 10 00:00:36,950 --> 00:00:40,300 Our goals in this lecture are somewhat modest. 11 00:00:40,300 --> 00:00:42,006 We'll be engaging software and 12 00:00:42,006 --> 00:00:45,618 datasets more fully in the next lecture in the follow on to this one. 13 00:00:45,618 --> 00:00:49,780 But for this lecture, we're going to try to work you through the basic steps 14 00:00:49,780 --> 00:00:53,675 of triple exponential smoothing in such a way that you could state and 15 00:00:53,675 --> 00:00:55,840 explain it to a friend or a colleague. 16 00:00:58,926 --> 00:01:03,557 There's a really terrific website from Nikolaos Kourentzes to try to 17 00:01:03,557 --> 00:01:08,678 capture the difference between additive and multiplicative seasonality. 18 00:01:08,678 --> 00:01:12,900 If you want to explain to somebody what a cat is, 19 00:01:12,900 --> 00:01:14,710 I think you're going to have a hard time. 20 00:01:14,710 --> 00:01:17,600 But if you show them 20 different cats, I think they'll get the idea. 21 00:01:17,600 --> 00:01:20,837 In the same spirit, in the readings and 22 00:01:20,837 --> 00:01:25,850 in past lectures, we've already discussed additive or multiplicative seasonality. 23 00:01:25,850 --> 00:01:30,450 The virtue of this website and the exercises that he has there for 24 00:01:30,450 --> 00:01:35,580 you is that you can look at additive and multiplicative seasonality, and 25 00:01:35,580 --> 00:01:40,340 try to understand in a very intuitive way, the differences between the two. 26 00:01:40,340 --> 00:01:45,380 I mean, as I look here on the additive seasonal series, 27 00:01:45,380 --> 00:01:49,710 you can see that there is certainly a repeating pattern. 28 00:01:49,710 --> 00:01:53,315 There is a seasonality to this dataset. 29 00:01:54,620 --> 00:01:56,792 When I look at the variability, 30 00:01:56,792 --> 00:02:00,980 the variability appears to be somewhat constant in time. 31 00:02:00,980 --> 00:02:02,670 Okay, perhaps with a little bit of noise. 32 00:02:02,670 --> 00:02:04,140 But basically, 33 00:02:04,140 --> 00:02:08,820 I don't see any systematic changes in the variability as I move along. 34 00:02:08,820 --> 00:02:13,040 Multiplicative seasonality gives you more characters to fanning structure. 35 00:02:13,040 --> 00:02:16,760 And the variability really does seem to be in this case growing in time. 36 00:02:18,020 --> 00:02:21,330 So there's a difference between additive seasonality, 37 00:02:21,330 --> 00:02:25,320 where to look into the future we just add a constant amount. 38 00:02:25,320 --> 00:02:29,790 And multiplicative seasonality where we'll multiply by a constant amount. 39 00:02:32,350 --> 00:02:36,980 Let's see which one Holt Winters exhibits in their famous example here, 40 00:02:36,980 --> 00:02:38,980 Air Passenger Miles. 41 00:02:38,980 --> 00:02:43,031 So we have an original source for the data set, but 42 00:02:43,031 --> 00:02:47,757 what these guys were looking at was during the 1950s, 43 00:02:47,757 --> 00:02:52,881 air passenger miles and essentially the volume of air travel. 44 00:02:52,881 --> 00:02:57,903 So as I go from the late 40s to the very early 60s, I certainly 45 00:02:57,903 --> 00:03:03,855 would expect the volume of air passenger miles to be increasing, and I think 46 00:03:03,855 --> 00:03:10,010 I can talk myself into believing that there'd be a seasonality there as well. 47 00:03:13,173 --> 00:03:15,780 And the data set does not disappoint. 48 00:03:15,780 --> 00:03:17,900 As I move from the left to the right, 49 00:03:17,900 --> 00:03:22,980 I certainly see that there is a season involved here. 50 00:03:22,980 --> 00:03:27,230 So I get this repetitive structure every 12 months. 51 00:03:27,230 --> 00:03:30,430 And also, as I move from left to right, 52 00:03:30,430 --> 00:03:35,630 there's a trend upwards and the variability also seems to be growing. 53 00:03:35,630 --> 00:03:40,110 So I see the same essential structure in the data set as I look here. 54 00:03:40,110 --> 00:03:43,650 But as I move from left to right, the magnitude of the structure, 55 00:03:43,650 --> 00:03:45,660 the size of the structure seems to be increasing. 56 00:03:46,790 --> 00:03:48,380 If we take logs, 57 00:03:48,380 --> 00:03:53,720 we can look at something that seems to have more apparent additive seasonality. 58 00:03:53,720 --> 00:03:54,920 This is an old trick. 59 00:03:54,920 --> 00:04:00,810 You know that the log whatever base E, 10, 2 are very common bases 60 00:04:00,810 --> 00:04:06,240 that the logarithm is going to take multiplication and turn it into addition. 61 00:04:06,240 --> 00:04:09,920 Log of a times b is log a plus log b. 62 00:04:09,920 --> 00:04:13,060 So we're not surprised that we can get from multiplicative seasonality to 63 00:04:13,060 --> 00:04:15,800 additive based upon taking logs. 64 00:04:17,810 --> 00:04:22,460 Now our current methods are pretty disappointing on this data set. 65 00:04:22,460 --> 00:04:29,810 If I look on the graph on the left, then I can see the actual time series in black. 66 00:04:29,810 --> 00:04:34,440 The forecasts are in red and 67 00:04:34,440 --> 00:04:38,250 the amount of error here is I guess I would say unacceptable. 68 00:04:38,250 --> 00:04:41,930 There's really, the forecast is not very good using 69 00:04:41,930 --> 00:04:44,440 this first kind of simple exponential smoothing. 70 00:04:45,520 --> 00:04:49,610 Perhaps a little surprisingly as you move to double exponential smoothing, 71 00:04:49,610 --> 00:04:52,160 we still have a pretty poor forecast. 72 00:04:52,160 --> 00:04:56,540 So it's really important to include this seasonal effect in our modeling, 73 00:04:56,540 --> 00:04:58,830 if we want to have a good forecast. 74 00:05:00,380 --> 00:05:04,630 Just as a reminder, the types of exponential smoothing we've been doing, 75 00:05:04,630 --> 00:05:08,370 of course, have a geometric series lurking in the background. 76 00:05:08,370 --> 00:05:12,240 But there's some simplifications that you can make, as we discussed in the readings. 77 00:05:12,240 --> 00:05:15,690 And our terms tend to look like this. 78 00:05:15,690 --> 00:05:20,690 We'll take some Greek letter that controls how much of the current value of 79 00:05:20,690 --> 00:05:23,140 the system you want to see compared to history. 80 00:05:23,140 --> 00:05:27,864 We'll take Greek letter times some sort of fresh new value + 81 00:05:27,864 --> 00:05:32,890 1- Greek letter times the history of the system. 82 00:05:32,890 --> 00:05:35,030 So this is the common pattern here. 83 00:05:36,070 --> 00:05:39,483 If we want to weight the new information higher, 84 00:05:39,483 --> 00:05:43,906 that's going to come at the expense of the history of the system. 85 00:05:47,245 --> 00:05:52,408 Triple exponential smoothing, we can look at sums of squares of errors or 86 00:05:52,408 --> 00:05:55,930 whatever you like for your criterion of quality. 87 00:05:55,930 --> 00:06:02,800 Triple exponential smoothing visually in fact is quite strikingly better. 88 00:06:02,800 --> 00:06:06,810 So we seem to be onto something good here with this triple exponential smoothing. 89 00:06:06,810 --> 00:06:10,990 We want to be able to include level, trend, and seasonality. 90 00:06:13,110 --> 00:06:17,480 So we're going to look at, first, smoothing the level. 91 00:06:18,750 --> 00:06:25,140 So x sub n is a data point and this is our seasonal bump. 92 00:06:25,140 --> 00:06:27,840 So either adding or subtracting a certain amount. 93 00:06:29,190 --> 00:06:32,622 And the way we'll smooth the level is by 94 00:06:32,622 --> 00:06:37,519 looking at our smooth level as alpha times fresh value. 95 00:06:37,519 --> 00:06:43,060 There's your current data point minus the current seasonal smooth piece. 96 00:06:43,060 --> 00:06:47,944 And in fact, we're assuming that there are m data points as 97 00:06:47,944 --> 00:06:52,948 we get through a cycle or a season, I think people would say. 98 00:06:52,948 --> 00:06:56,838 So seasonal n- m is sitting right here. 99 00:06:56,838 --> 00:07:02,560 We've got 1- alpha over on this side times level n plus trend. 100 00:07:02,560 --> 00:07:08,440 So the smooth level essentially looks like what we did before, only now instead 101 00:07:08,440 --> 00:07:14,830 of just having alpha times x sub n, we're doing a seasonally adjusted level. 102 00:07:14,830 --> 00:07:18,100 And we'll try to pick up the seasonal bump differently in a moment. 103 00:07:20,640 --> 00:07:24,257 We're going to smooth the trend so the trend, 104 00:07:24,257 --> 00:07:30,029 the current smooth trend is going to look like beta times level minus level. 105 00:07:30,029 --> 00:07:34,761 Again we're using these smooth levels to try to get rid of the effects of noise, 106 00:07:34,761 --> 00:07:38,153 and look at the more persistent, or systematic pieces, 107 00:07:38,153 --> 00:07:41,210 plus 1- beta times the history here trend n-1. 108 00:07:41,210 --> 00:07:43,750 This is just what we did in the last lecture. 109 00:07:45,318 --> 00:07:50,118 Finally, we want to smooth the season, and so we'll bring in a new parameter gamma. 110 00:07:50,118 --> 00:07:55,459 We'll take a gamma times and now here we're going to bring in new information 111 00:07:55,459 --> 00:08:00,905 as current data point minus the current smooth level and we'll add to that. 112 00:08:00,905 --> 00:08:02,625 Now look at this piece, 113 00:08:02,625 --> 00:08:08,100 we're going to take 1- gamma times the past smooth seasonal piece. 114 00:08:08,100 --> 00:08:13,080 So seasonal at time n is going to look at seasonal at time n- m. 115 00:08:13,080 --> 00:08:15,768 So we're going to reach back along 116 00:08:15,768 --> 00:08:20,650 the time series to the start of that little cycle going on. 117 00:08:20,650 --> 00:08:25,032 And we'll take our old best guess at the seasonal bump and 118 00:08:25,032 --> 00:08:29,240 augment it with gamma times this new piece right here. 119 00:08:32,205 --> 00:08:36,006 In order to update the forecast for additive seasonality, 120 00:08:36,006 --> 00:08:39,740 it's really quite simple and intuitive. 121 00:08:39,740 --> 00:08:44,790 We'll get our prediction h into the future as our current smooth level, 122 00:08:45,870 --> 00:08:47,170 and now remember what we did. 123 00:08:47,170 --> 00:08:52,290 We'll take our smooth trend, that's essentially the step size, and 124 00:08:52,290 --> 00:08:56,850 we'll walk up or down depending upon the sign, h steps. 125 00:08:56,850 --> 00:08:59,350 So this first piece is accounting for 126 00:08:59,350 --> 00:09:04,190 the smooth level then we'll bring in our trend contribution. 127 00:09:04,190 --> 00:09:06,660 And we're also going to bring in a seasonal contribution. 128 00:09:06,660 --> 00:09:11,582 So we're going to get our seasonal bump as we're looking for 129 00:09:11,582 --> 00:09:15,508 the forecast that n + h, so h into the future. 130 00:09:15,508 --> 00:09:21,135 So we'll take our seasonal piece here, n + h, and then- m, 131 00:09:21,135 --> 00:09:28,188 and that's going to give us the updated forecast using additive seasonality. 132 00:09:32,080 --> 00:09:35,271 If you want multiplicative seasonality, 133 00:09:35,271 --> 00:09:38,911 then instead of adding you're going to multiply. 134 00:09:38,911 --> 00:09:43,590 So instead of adding a fixed amount you multiply by 120%, 150%, 135 00:09:43,590 --> 00:09:46,240 whatever's appropriate in your model. 136 00:09:47,880 --> 00:09:51,710 Now people enjoy programming will go out and write their own code to find alpha, 137 00:09:51,710 --> 00:09:53,500 beta, and gamma, and that's fine. 138 00:09:53,500 --> 00:09:58,440 I solute you, but I think for our approach here, we'll let the software 139 00:09:58,440 --> 00:10:01,600 find the best values for us, and also do the predictions for us. 140 00:10:04,770 --> 00:10:09,357 So at this point, you should be able to turn around to somebody next to you and 141 00:10:09,357 --> 00:10:13,440 explain the whole triple exponential smoothing approach. 142 00:10:13,440 --> 00:10:15,170 That we're going to smooth level, 143 00:10:15,170 --> 00:10:20,170 we're going to smooth trend, we're going to smooth the seasonal contribution. 144 00:10:20,170 --> 00:10:24,390 And then for additive seasonality, we'll add the three of them together. 145 00:10:24,390 --> 00:10:26,820 And in the multiplicative seasonality, 146 00:10:26,820 --> 00:10:32,170 we'll get the seasonal effect as level plus H trend. 147 00:10:32,170 --> 00:10:35,841 And then, instead of adding we'll be multiplying by the seasonal bump.