1 00:00:11,050 --> 00:00:14,680 In this lecture, we are going to discuss holds linear trend model. 2 00:00:15,430 --> 00:00:21,190 Previously we looked at simple exponential smoothing and we noted that our forecasts were always a straight 3 00:00:21,190 --> 00:00:27,370 horizontal line, holds linear trend model, extends the exponential smoothing model so that we can 4 00:00:27,370 --> 00:00:29,470 capture and forecast trends. 5 00:00:30,130 --> 00:00:33,320 Let's start by considering what linear trend might look like. 6 00:00:33,670 --> 00:00:35,040 Well, linear means line. 7 00:00:35,470 --> 00:00:41,980 We know that the equation for a line in general is Y equals M, X plus B, let's suppose that instead 8 00:00:41,980 --> 00:00:46,240 we say Y, Sebti equals slope times T plus. 9 00:00:46,240 --> 00:00:46,820 Why not? 10 00:00:47,530 --> 00:00:48,730 This doesn't change anything. 11 00:00:48,740 --> 00:00:52,330 All we've done is replaced X with T and be with Y not. 12 00:00:53,350 --> 00:00:59,650 As you know, the coefficient in front of T is called the slope and the constant term by itself is called 13 00:00:59,650 --> 00:01:00,520 the intercept. 14 00:01:01,480 --> 00:01:07,520 Clearly it is the value of Y at time T equals zero, hence the symbol y not or Y zero. 15 00:01:08,500 --> 00:01:13,540 Note that the slope is the amount that y changes when t increases by one. 16 00:01:14,350 --> 00:01:19,900 Understanding this kind of equation is a crucial step in understanding Holtze linear trend model. 17 00:01:25,020 --> 00:01:30,390 The best way to understand holds linear trend model, in my opinion, is to simply look at the model 18 00:01:30,390 --> 00:01:34,320 equations and to try and decipher each component one at a time. 19 00:01:35,070 --> 00:01:39,720 The first thing you'll notice is that it's very similar to the simple exponential smoothing equations 20 00:01:39,720 --> 00:01:40,890 in component form. 21 00:01:41,370 --> 00:01:44,650 That's why it was so important to rearrange them in that way. 22 00:01:45,780 --> 00:01:51,090 You'll notice that, whereas the simple exponential smoothing model had two equations, we now have 23 00:01:51,090 --> 00:01:52,270 three equations. 24 00:01:52,620 --> 00:01:57,580 So we've added one more equation, specifically one more equation for modeling the trend. 25 00:01:58,230 --> 00:02:04,080 So now we have three equations in total, one for the forecast, which we already had, one for the 26 00:02:04,080 --> 00:02:08,070 level which we already had, and one new equation for the trend. 27 00:02:12,970 --> 00:02:18,790 Let's start by studying the forecast equation, as you can see, this is nothing but the equation for 28 00:02:18,790 --> 00:02:19,340 a line. 29 00:02:19,900 --> 00:02:25,210 Of course, we now have different symbols, but I know you're not going to let that intimidate you as 30 00:02:25,210 --> 00:02:25,830 before. 31 00:02:25,840 --> 00:02:29,350 We have the level which represents some sort of average value. 32 00:02:29,890 --> 00:02:36,550 But now we also have a term that increases linearly with H, which is the number of steps in the forecast. 33 00:02:37,660 --> 00:02:44,020 Clearly, this means that B of T is the trend, or in other words, the slope that is elevated, the 34 00:02:44,020 --> 00:02:51,040 level is the starting value of the forecast and then the forecast increases by the amount B of T for 35 00:02:51,040 --> 00:02:56,500 each step into the future that we forecast, of course, but can also be negative. 36 00:02:56,680 --> 00:03:00,490 So if you increase by a negative amount, then you're actually decreasing. 37 00:03:01,270 --> 00:03:07,420 The key point is, thanks to this new linear trend term, our forecast is no longer a horizontal line, 38 00:03:07,630 --> 00:03:09,370 but a line in any direction. 39 00:03:09,910 --> 00:03:13,150 This justifies the name holds linear trend model. 40 00:03:18,130 --> 00:03:23,900 Let's now study the level equation again, this is very similar to the level equation from before. 41 00:03:24,280 --> 00:03:29,100 The only difference is on the far right, which represents this moving average value. 42 00:03:29,830 --> 00:03:35,740 Remember that the prediction is now made up of two components, the level and the trend, which are 43 00:03:35,740 --> 00:03:37,380 represented by LMB. 44 00:03:38,020 --> 00:03:45,130 As you recall, in order to make a forecast, we find L plus H times B, but since this is just one 45 00:03:45,130 --> 00:03:47,080 step ahead, H equals one. 46 00:03:47,230 --> 00:03:54,160 And so the value of our prediction is just L plus B or more specifically L at T minus one, plus B, 47 00:03:54,160 --> 00:03:55,180 A, T minus one. 48 00:03:55,870 --> 00:03:58,050 So that concludes how to update the level. 49 00:03:58,480 --> 00:04:05,080 It is still exponential smoothing on the input signal way T using L of T minus one A plus B of T minus 50 00:04:05,080 --> 00:04:07,840 one as the previous smoothed value. 51 00:04:12,780 --> 00:04:18,990 Finally, we have the trend equation, as you can see, the general form of this equation still follows 52 00:04:18,990 --> 00:04:20,490 that of exponential smoothing. 53 00:04:20,910 --> 00:04:24,670 We have better times, something plus one minus BITA times the old value. 54 00:04:25,230 --> 00:04:29,860 In other words, the trend to B of T is also an exponentially smooth estimate. 55 00:04:30,510 --> 00:04:33,310 But what is it an exponentially smooth estimate of? 56 00:04:33,900 --> 00:04:36,200 Well, that's the thing that goes in front of the beta. 57 00:04:36,390 --> 00:04:37,420 So what goes there? 58 00:04:38,010 --> 00:04:41,460 In fact, it's just elev T minus elev T minus one. 59 00:04:42,090 --> 00:04:43,680 So why does this make sense? 60 00:04:44,400 --> 00:04:48,930 Remember that B of T is trying to estimate the slope or the trend of the signal. 61 00:04:49,470 --> 00:04:55,470 The slope of the signal is the value at one time, minus the value at some other time divided by the 62 00:04:55,470 --> 00:04:56,530 difference in time. 63 00:04:57,150 --> 00:05:02,310 In our case, the time difference is just one, because we're updating B of T on every timestep. 64 00:05:02,550 --> 00:05:05,970 So of T minus elev T minus one makes sense. 65 00:05:07,080 --> 00:05:14,070 You might think y ltt and not Y of T remember that one way of thinking of a time series signal is that 66 00:05:14,070 --> 00:05:15,000 it is noisy. 67 00:05:15,540 --> 00:05:20,450 The level of T represents the smooth version of that signal without the noise. 68 00:05:20,820 --> 00:05:26,910 And so it's making the difference in L.A. is a better estimate of the overall trend because it essentially 69 00:05:26,910 --> 00:05:29,490 removes that noise from the equation. 70 00:05:34,500 --> 00:05:39,780 Recall that previously Alpha was treated as sort of a hyper parameter that could be tuned depending 71 00:05:39,780 --> 00:05:41,040 on what you were trying to do. 72 00:05:41,520 --> 00:05:43,580 In fact, this is often the case. 73 00:05:44,040 --> 00:05:49,770 For example, suppose you're working as an audio engineer and you would like to remove some noise from 74 00:05:49,770 --> 00:05:50,590 a sound file. 75 00:05:51,270 --> 00:05:56,100 Well, if you have experience with sound editing, then you probably know that you often have to choose 76 00:05:56,100 --> 00:05:57,580 some parameters yourself. 77 00:05:58,260 --> 00:06:02,700 You tune these to your tastes depending on what the final output signal sounds like. 78 00:06:03,150 --> 00:06:08,520 You want it to sound subjectively good, which in general is not something that can be quantified. 79 00:06:13,340 --> 00:06:18,320 On the other hand, with Holtze linear trend model, now we're getting closer to the machine learning 80 00:06:18,320 --> 00:06:23,780 picture where what we care about is predictive accuracy, we want our forecast to be good. 81 00:06:23,960 --> 00:06:28,930 And now that our model can exhibit a trend, it actually has a chance of making decent predictions. 82 00:06:29,420 --> 00:06:34,790 Therefore, the parameters that we learned about in this lecture, Alpha and Beta will no longer be 83 00:06:34,790 --> 00:06:39,730 chosen arbitrarily, but rather we can fit them as usual. 84 00:06:39,740 --> 00:06:44,000 If you're familiar with machine learning, then you know, this involves setting up a lost function 85 00:06:44,000 --> 00:06:49,400 like the mean squared error and then minimizing the squarer over the training set using methods such 86 00:06:49,400 --> 00:06:50,420 as gradient descent. 87 00:06:51,050 --> 00:06:55,970 If you are unfamiliar with this process, then you don't need to worry because stats models is going 88 00:06:55,970 --> 00:06:57,610 to do all this work for us. 89 00:07:02,550 --> 00:07:09,090 Finally, let's take a quick look at how the code will look at a high level as before we start by instantiating 90 00:07:09,090 --> 00:07:15,690 the model, unlike cyclosarin and most modern machine learning APIs, this is where we pass in the data, 91 00:07:15,960 --> 00:07:18,150 which is a univariate time series. 92 00:07:18,660 --> 00:07:24,690 Unlike Saikat learn, since the data is UNIVARIATE, it's OK if this data array is one dimensional. 93 00:07:25,710 --> 00:07:31,600 Next, we call the fit function, which in this case doesn't take in any parameters inside yet learn. 94 00:07:31,800 --> 00:07:33,630 This is where the data would usually go. 95 00:07:34,560 --> 00:07:41,060 This returns a result object we can use the result object to obtain the predicted values from the set 96 00:07:41,190 --> 00:07:43,650 by calling the attribute fitted values. 97 00:07:43,890 --> 00:07:50,310 And if we want to forecast, we can call results forecast passing in the number of time steps to forecast.