1 00:00:11,070 --> 00:00:16,050 In this lecture, we are going to bridge the gap between exponential smoothing and the whole winters' 2 00:00:16,050 --> 00:00:16,560 model. 3 00:00:17,130 --> 00:00:19,230 This is going to be done in a series of steps. 4 00:00:19,230 --> 00:00:24,350 And the first step is to understand exponential smoothing from a different perspective. 5 00:00:24,750 --> 00:00:27,960 That is from the perspective of Time series forecasting. 6 00:00:28,890 --> 00:00:34,140 Previously, our perspective was simply what are different ways that we can take the average in a time 7 00:00:34,140 --> 00:00:34,720 series. 8 00:00:35,100 --> 00:00:41,160 Now it's more about how do we fit a model to the Time series and then how do we use that model to forecast 9 00:00:41,160 --> 00:00:42,130 future values? 10 00:00:42,630 --> 00:00:46,880 Mathematically, this is the same as before, but philosophically it's different. 11 00:00:51,920 --> 00:00:55,910 To do this, we're going to have to introduce some new and more precise notation. 12 00:00:56,630 --> 00:01:02,110 Let's begin by reviewing the exponentially weighted moving average once again, but using our new notation. 13 00:01:02,630 --> 00:01:09,200 So now we have we had a time t equal to Alpha times Y of T plus one minus Alpha times. 14 00:01:09,200 --> 00:01:12,620 We had a time T minus one in this form. 15 00:01:12,620 --> 00:01:13,370 Y had a time. 16 00:01:13,370 --> 00:01:21,500 T is our exponentially smooth version of Y at time t y it is the value from our time series at time. 17 00:01:21,500 --> 00:01:24,860 T Alpha is again the smoothing parameter. 18 00:01:29,900 --> 00:01:36,060 The next step is to now phrase this as a forecasting model, in this form, the equation looks as follows. 19 00:01:36,680 --> 00:01:41,720 You'll notice that I've introduced this vertical bar symbol, which, as you know, in probability means 20 00:01:41,720 --> 00:01:48,620 given some on the left hand side, we're saying why hat is the exponentially smooth forecast for a time 21 00:01:48,620 --> 00:01:56,570 T plus one given the known values at time t on the right hand side we have alpha times Y of T as usual, 22 00:01:56,750 --> 00:02:00,710 plus one minus four times the previous exponentially smooth value. 23 00:02:02,350 --> 00:02:08,310 Noticed something interesting about this, the time indices have changed for the current equation, 24 00:02:08,380 --> 00:02:15,640 the Y hat on the left is for the time index T plus one, the Y hat on the right has the time index T. 25 00:02:17,160 --> 00:02:22,290 If you look at the previous slide at the time and next is on the left and at T minus one is on the right. 26 00:02:23,010 --> 00:02:25,650 This is probably extremely confusing at this point. 27 00:02:25,650 --> 00:02:27,480 And I wouldn't blame you for thinking so. 28 00:02:27,930 --> 00:02:32,780 It will make much more sense when we look at the code and you actually observe this behavior for yourself. 29 00:02:33,480 --> 00:02:35,510 Now, realize that I didn't invent this. 30 00:02:35,520 --> 00:02:37,250 This is all part of the whole winters' model. 31 00:02:37,440 --> 00:02:39,420 So don't blame me if you find it confusing. 32 00:02:40,110 --> 00:02:44,820 By the time we reach the full whole Winsor's model and you see it in action, you will see that this 33 00:02:44,820 --> 00:02:45,720 all makes sense. 34 00:02:46,290 --> 00:02:48,840 Again, recognize that this is now a forecast. 35 00:02:49,260 --> 00:02:53,120 Previously we were not forecasting, we were just calculating averages. 36 00:02:53,640 --> 00:03:00,360 The forecast for the next timestep is equal to Alpha Times, the currently observed value plus one minus 37 00:03:00,360 --> 00:03:04,050 alpha times, the current exponentially smooth's fitted value. 38 00:03:09,080 --> 00:03:14,420 The next step is to express the simple exponential smoothing model in what is called component form. 39 00:03:15,050 --> 00:03:19,170 This will make it seem like overkill at this point because there is only a single component. 40 00:03:19,760 --> 00:03:24,870 However, the whole winters' model contains multiple components, which is where this becomes useful. 41 00:03:25,460 --> 00:03:28,490 So in this form, the forecast equation looks as follows. 42 00:03:29,540 --> 00:03:37,700 We say that we had a time at T plus H given T is equal to live T, that is the forecast at an arbitrary 43 00:03:37,700 --> 00:03:41,650 eight steps ahead is equal to Helft, whatever that is. 44 00:03:42,290 --> 00:03:49,370 Then the smoothing equation says this elev T is just the exponentially smooth average of Y of t the 45 00:03:49,370 --> 00:03:50,280 Time series. 46 00:03:50,930 --> 00:03:56,870 Notice that L.A. now becomes the exponentially smooth average and Y hat becomes the forecast. 47 00:03:58,960 --> 00:04:05,770 Also, note that L.A. gets back its original time indices from the previous lecture, that is, we now 48 00:04:05,770 --> 00:04:08,200 have tea on the left and T minus one on the right. 49 00:04:08,950 --> 00:04:15,770 So left is itself not a forecast, but the forecast is based on liberty for this simple model. 50 00:04:16,000 --> 00:04:18,250 The forecast is simply assigned it to be a. 51 00:04:19,000 --> 00:04:21,350 But this will get more complex in later lectures. 52 00:04:22,210 --> 00:04:24,130 So what does Aloft represent? 53 00:04:24,820 --> 00:04:29,760 LTE is called the level, which will become more clear when we put it into context later on. 54 00:04:30,520 --> 00:04:34,890 Basically level can be thought of as the moving average, which kind of makes sense. 55 00:04:35,350 --> 00:04:41,470 The level is the average value of the signal in time, but the actual signal may fluctuate around that 56 00:04:41,470 --> 00:04:42,280 average level. 57 00:04:43,760 --> 00:04:48,170 One important thing to note about this forecasting model is that it's not very expressive. 58 00:04:48,620 --> 00:04:50,940 The forecast is simply a constant value. 59 00:04:50,960 --> 00:04:54,680 It's always left no matter how many steps ahead you want to forecast. 60 00:04:55,310 --> 00:04:59,960 This must be the case because remember, the exponentially weighted moving average is nothing but an 61 00:04:59,960 --> 00:05:05,450 estimate of the mean, since all we're estimating is the mean and that's the only thing we can predict 62 00:05:05,990 --> 00:05:07,750 after we've stopped collecting data. 63 00:05:07,970 --> 00:05:12,150 We have no idea how this will change beyond the last known data point. 64 00:05:12,620 --> 00:05:18,020 And so our forecast simply consists of predicting the same mean value for each timestep. 65 00:05:22,920 --> 00:05:27,110 The last thing we were going to do in this lecture is talk about how this will work in code. 66 00:05:27,870 --> 00:05:32,420 Previously we looked at pandas as a way to calculate the exponentially weighted moving average. 67 00:05:32,730 --> 00:05:38,460 And we've noted that this lecture didn't introduce any new concepts other than some renaming of variables. 68 00:05:39,000 --> 00:05:44,820 However, in the next coding lecture, we're going to use stats models to perform exponential smoothing, 69 00:05:45,060 --> 00:05:48,870 which has a model that works more in line with what we discussed in this lecture. 70 00:05:49,710 --> 00:05:55,170 Furthermore, it's capable of making forecasts so that will allow us to use exponential smoothing as 71 00:05:55,170 --> 00:05:58,800 a predictive model rather than just some way of taking the average. 72 00:06:00,490 --> 00:06:06,220 So how does it work, since this is the first model from stat's models will be looking at in this course, 73 00:06:06,430 --> 00:06:10,110 how it works might be surprising if you come from a cyclone background. 74 00:06:10,810 --> 00:06:14,930 Basically, remember that this is a different library, so it has a different API. 75 00:06:15,790 --> 00:06:22,210 First, we start by importing the class, simple smoothing from the whole winters' module inside stat's 76 00:06:22,210 --> 00:06:22,810 models. 77 00:06:23,620 --> 00:06:29,110 Next, we create a model by instantiating an object of type, simple, smoothy. 78 00:06:30,130 --> 00:06:36,220 Note that when doing this, we pass in the data set, which should be a univariate time series, unlike 79 00:06:36,220 --> 00:06:39,430 Saikat Learn, which expects a two dimensional array, is input. 80 00:06:39,730 --> 00:06:45,000 This model expects a one dimensional array because the only dimension is time inside. 81 00:06:45,070 --> 00:06:48,340 It learn its number of samples by a number of features. 82 00:06:49,250 --> 00:06:55,040 Also, notice that, unlike Saikat, learn where the data is passed into the fit function with stats 83 00:06:55,040 --> 00:06:57,650 models, the data is passed into the constructor. 84 00:06:58,370 --> 00:07:03,560 The stats models object does have a fit function, which is what we will call next its function, except 85 00:07:03,570 --> 00:07:05,220 some parameters such as Alpha. 86 00:07:05,810 --> 00:07:10,340 So it's kind of like the reverse of Saikat, learn in Saikat, learn at the constructor. 87 00:07:10,340 --> 00:07:15,680 It usually takes in the hyper parameters and the fit function it takes in the data in stats models, 88 00:07:15,680 --> 00:07:21,650 the constructor takes in the data and the fit function takes in the hyper parameters for us. 89 00:07:21,650 --> 00:07:27,270 We're going to pass in Alpha into the fit function in addition to setting the argument optimized equal 90 00:07:27,320 --> 00:07:27,920 false. 91 00:07:28,700 --> 00:07:34,010 Note that for all of the models in the whole WINTERS' module, it's possible to fit parameters to the 92 00:07:34,010 --> 00:07:34,460 data. 93 00:07:35,060 --> 00:07:38,780 This makes it more like machine learning than simply calculating a moving average. 94 00:07:38,990 --> 00:07:43,180 But this will make more sense as our model gets more and more complex in later lectures. 95 00:07:44,090 --> 00:07:49,010 In order to make what we're doing equivalent to what we did previously, we will not try to optimize 96 00:07:49,010 --> 00:07:49,970 the value of Alpha. 97 00:07:50,660 --> 00:07:55,040 Basically, Alpha would be optimized in order to minimize the error on the training set. 98 00:07:55,610 --> 00:08:00,350 If you're familiar with machine learning, then you will understand that this involves taking the mean 99 00:08:00,350 --> 00:08:05,270 squared error between the true Time series and the predicted values and then minimizing that with respect 100 00:08:05,270 --> 00:08:05,920 to Alpha. 101 00:08:06,470 --> 00:08:09,830 If you're not comfortable with this idea, it's not necessary to know. 102 00:08:09,830 --> 00:08:12,620 So don't worry about it since we're not going to use it anyway. 103 00:08:15,180 --> 00:08:20,490 All right, so the fit function returns a result object, which is again different from what you're 104 00:08:20,490 --> 00:08:26,190 probably familiar with from Saikat, learn in Saikat, learn you just get a reference to the model object 105 00:08:26,190 --> 00:08:26,820 itself. 106 00:08:27,690 --> 00:08:29,900 In stat's models, we get a result object. 107 00:08:30,390 --> 00:08:36,000 Specifically, we get back a whole winters' results object so you can look up the documentation if you're 108 00:08:36,000 --> 00:08:39,750 interested in seeing what functions can be called on this object. 109 00:08:41,460 --> 00:08:47,490 The first function we are interested in is the predictive function, the way it works is this you pass 110 00:08:47,490 --> 00:08:51,060 in a start date for the start argument and an end date for the end argument. 111 00:08:51,270 --> 00:08:57,380 And then this will return an array containing a prediction of time series values for those dates. 112 00:08:58,790 --> 00:09:03,800 Note that these dates can be in sample or out of sample, meaning that they can be dates from the training 113 00:09:03,800 --> 00:09:09,550 set or after the training set, if the predicted dates are from inside the range of the train set, 114 00:09:09,740 --> 00:09:11,680 we call that an N sample prediction. 115 00:09:12,230 --> 00:09:16,850 If the prediction dates are from outside the range of the train set, we call that an out of sample 116 00:09:16,850 --> 00:09:17,660 forecast. 117 00:09:22,370 --> 00:09:27,770 A simpler way to get predictions is this if you want all of the train predictions, you simply call 118 00:09:27,770 --> 00:09:29,780 the attribute fitted values. 119 00:09:30,320 --> 00:09:35,170 This way you can avoid having to figure out the dates that you need for the start and end arguments. 120 00:09:35,780 --> 00:09:39,760 A simpler way to get a forecast is to simply call a forecast function. 121 00:09:40,430 --> 00:09:43,320 This accepts as input the number of forecasting steps. 122 00:09:43,340 --> 00:09:48,080 So, for example, if you want to forecast ten steps ahead, you just pass in 10. 123 00:09:48,470 --> 00:09:50,600 There's no need to figure out what the data is. 124 00:09:50,600 --> 00:09:51,470 10 steps ahead. 125 00:09:51,650 --> 00:09:53,840 Passing in the Integer 10 is enough. 126 00:09:54,980 --> 00:10:00,350 Note that this assumes that your forecast begins at the timestep after the end of the train set. 127 00:10:00,860 --> 00:10:06,380 So if your train set goes from one up to Big T., then your first forecast prediction will be for the 128 00:10:06,380 --> 00:10:08,270 time index, big T plus one.