1 00:00:11,130 --> 00:00:17,190 All right, so the main reason I wanted to discuss the Akef and the pickoff was because in a Time series 2 00:00:17,190 --> 00:00:23,880 analysis courses, instructors often talk about how to use the Akef and the pickoff as if you're following 3 00:00:23,880 --> 00:00:25,560 a set of arbitrary rules. 4 00:00:26,130 --> 00:00:31,770 But they never go on to show you that these rules actually do what they say they do by comparing the 5 00:00:31,770 --> 00:00:38,950 Akef and the pickoff to known data by looking at the acronym for known data. 6 00:00:39,210 --> 00:00:41,940 We can confirm that those rules actually work. 7 00:00:42,450 --> 00:00:44,650 Following them thus becomes natural. 8 00:00:45,240 --> 00:00:50,700 It's also important to recognize that these are standard topics and time series analysis, so it wouldn't 9 00:00:50,700 --> 00:00:51,730 be right to leave them out. 10 00:00:52,650 --> 00:00:55,020 Even so, there are important caveats. 11 00:00:55,590 --> 00:01:00,580 The caveats are that even if you do follow these rules, they might not lead to the best model. 12 00:01:01,110 --> 00:01:05,140 Of course, we haven't yet defined what best model actually means, but we will. 13 00:01:05,910 --> 00:01:11,880 The real question is why do all that work when we can get a computer to do it for us, especially when 14 00:01:11,880 --> 00:01:14,280 a computer will do a better job than we can? 15 00:01:14,820 --> 00:01:16,720 That is what this lecture is all about. 16 00:01:21,770 --> 00:01:26,900 So it turns out that the computers of today are fast enough that you don't need to go through all that 17 00:01:26,900 --> 00:01:30,620 work, especially since you might end up with a suboptimal answer. 18 00:01:31,370 --> 00:01:38,120 Instead, popular time series packages usually contain a function called auto arima that automatically 19 00:01:38,120 --> 00:01:40,150 finds the best model for you. 20 00:01:40,730 --> 00:01:46,070 That is to say, it tries a bunch of different settings and returns the best settings according to some 21 00:01:46,070 --> 00:01:51,110 criteria on what these criteria might be will be discussed in the next lecture. 22 00:01:51,800 --> 00:01:57,470 In the other language, for example, you can simply call a function called Auto Arima and pass in your 23 00:01:57,470 --> 00:01:59,180 Time series, which is very nice. 24 00:01:59,660 --> 00:02:02,450 In stat's models, there exists no such function. 25 00:02:03,260 --> 00:02:09,560 However, there is a package called PMed Yarema, which uses stat's models under the hood that does 26 00:02:09,560 --> 00:02:11,530 implement the auto arima function. 27 00:02:12,260 --> 00:02:15,920 So basically we have the power of auto arima at our fingertips. 28 00:02:16,070 --> 00:02:21,140 We just need to install a different library, which luckily still uses Stass models under the hood. 29 00:02:26,000 --> 00:02:31,430 Now, the PMed Arima API is a little different from stat's models, so that would be worth mentioning 30 00:02:31,430 --> 00:02:38,400 briefly assuming that we have our data as a series, the first step will be to call the auto Arima function. 31 00:02:39,050 --> 00:02:43,520 This allows us to pass in an argument called seasonal as well as the seasonal period. 32 00:02:43,530 --> 00:02:50,360 M We will get to what this means later on, but it's basically a seasonal extension of the vanilla Arima 33 00:02:50,360 --> 00:02:51,890 we've been discussing so far. 34 00:02:52,820 --> 00:02:57,410 This will return to us a model object from which we can make predictions. 35 00:02:57,950 --> 00:03:02,090 Note that this is not a static model object, so the API is a bit different. 36 00:03:02,840 --> 00:03:06,380 When we want to forecast, we call the model predict function. 37 00:03:07,070 --> 00:03:14,030 This takes into arguments which are relevant to us, which are in periods and return content and periods. 38 00:03:14,030 --> 00:03:20,210 Is the forecast horizon or the number of time steps to forecast and return content should be set to 39 00:03:20,210 --> 00:03:23,850 true if you want the confidence bounce along with your prediction. 40 00:03:24,530 --> 00:03:26,480 So that's for the out of sample data. 41 00:03:27,080 --> 00:03:31,700 For the sample data you want to call the function model predict in sample. 42 00:03:32,270 --> 00:03:37,450 This takes into arguments start and end along with some others, which we won't worry about. 43 00:03:38,270 --> 00:03:42,980 Technically, you can return confidence intervals for the sample predictions as well, although that 44 00:03:42,980 --> 00:03:44,830 is less common than for forecasts. 45 00:03:45,800 --> 00:03:51,530 Unlike stats, models start and end should be specified as integers which correspond to the indices 46 00:03:51,530 --> 00:03:53,810 of the train set in stats. 47 00:03:53,810 --> 00:03:59,510 Models start and end can be integers, but they can also be daytime objects or strings corresponding 48 00:03:59,510 --> 00:04:02,630 to the index of the the series you passed in for training. 49 00:04:07,600 --> 00:04:13,120 In order to better understand what Auto Arima is doing, we'll have to discuss seasonal Arima. 50 00:04:13,690 --> 00:04:18,300 This will be brief since it's not as important to know compared to the vanilla Arima. 51 00:04:18,790 --> 00:04:22,010 Just understand the basic high level points and you will be fine. 52 00:04:22,960 --> 00:04:28,870 Again, I want to mention that seasonality is not usually observed in stock prices, but stock prices 53 00:04:28,870 --> 00:04:32,650 do not comprise all of what can be considered financial data. 54 00:04:33,070 --> 00:04:37,450 And so these concepts do show up in financial literature as they should. 55 00:04:38,290 --> 00:04:43,210 One example of this is that the weather or in other words, actual seasons like summer, winter and 56 00:04:43,210 --> 00:04:47,040 so forth, affect when and where people purchase real estate. 57 00:04:47,770 --> 00:04:51,430 So that's one example of where seasonality affects finance. 58 00:04:52,150 --> 00:04:59,080 OK, so seasonal Arima, abbreviated as SCREAMO, gives us three new hyper parameters called Capital 59 00:04:59,080 --> 00:05:05,030 P, Capital D and Capital Q, which are analogous with the lowercase versions of the Eurema. 60 00:05:05,710 --> 00:05:08,340 So now we have six hyper parameters in total. 61 00:05:08,980 --> 00:05:17,890 We usually write this as a Arima PD to cross capital P Capital, the capital Q Subscript M where M is 62 00:05:17,890 --> 00:05:19,020 the seasonal period. 63 00:05:20,170 --> 00:05:25,780 The reason why we add the cross is because it turns out when you write the model using operators, you 64 00:05:25,780 --> 00:05:30,310 end up multiplying the non seasonal parts by the seasonal parts to get your full model. 65 00:05:30,940 --> 00:05:35,560 This is outside the scope of this course, since the notation is very messy and doesn't really add any 66 00:05:35,560 --> 00:05:38,020 benefit on top of what we are already doing. 67 00:05:38,770 --> 00:05:43,840 However, if you want some simple intuition, we can discuss what the seasonal part might look like 68 00:05:43,840 --> 00:05:44,980 in isolation. 69 00:05:49,970 --> 00:05:56,330 So let's suppose we want to consider only the seasonal part of the seasonal Arima, if we have capital, 70 00:05:56,340 --> 00:06:01,310 the one that means we difference YFC once using the seasonal period. 71 00:06:02,730 --> 00:06:10,180 Our notation for this is Delta sub M y of T, which is equal to Y of T, minus Y of T minus M. 72 00:06:10,980 --> 00:06:16,440 So this is just like regular differences, except that regular different saying subtracts from one timestep 73 00:06:16,440 --> 00:06:19,080 behind and that measures the slope of the trend. 74 00:06:20,650 --> 00:06:27,070 With the seasonal part of seasonal Arima, we subtract from one period ago, so if we have monthly data 75 00:06:27,070 --> 00:06:30,880 and T is equal to March, then we subtract from the previous March. 76 00:06:31,750 --> 00:06:37,270 Note that when we discussed stationery earlier, we discussed certain characteristics of Time series 77 00:06:37,450 --> 00:06:43,670 that indicate that it is not stationary examples of this where trend and changing variance. 78 00:06:44,080 --> 00:06:49,060 If either the level or the variance changes over time, then the series is not stationary. 79 00:06:49,990 --> 00:06:57,050 Seasonality is the other key component of this stationary time series should also not exhibit seasonality. 80 00:06:57,640 --> 00:07:02,710 So if you find a time series that does exhibit seasonality, then you would say is not stationary. 81 00:07:07,710 --> 00:07:13,530 Interestingly, what we found in our previous, quote, examples was that it is possible for a non seasonal 82 00:07:13,530 --> 00:07:19,410 Arima to model the airline passengers data set quite well, even though it has no seasonal component. 83 00:07:20,130 --> 00:07:26,750 In fact, it's possible for a purely auto regressive H2 model to perfectly represent a sine wave. 84 00:07:27,300 --> 00:07:34,560 So it's not that Arima cannot model seasonality, but including seasonality explicitly by using SCREAMO 85 00:07:34,740 --> 00:07:36,200 can lead to a better model. 86 00:07:41,060 --> 00:07:45,740 So without loss of generality, let's suppose that we have not difference so that we can keep using 87 00:07:45,740 --> 00:07:51,740 the variable y t or you can pretend y t now represents the new difference, the time series doesn't 88 00:07:51,740 --> 00:07:52,150 matter. 89 00:07:52,790 --> 00:07:56,420 The seasonal auto regressive part is, again, what you might expect. 90 00:07:57,050 --> 00:08:05,240 Y t is regressed on past data points in the Time series at multiples of M away from T, so for example, 91 00:08:05,390 --> 00:08:10,280 by of T depends on Y of T minus M Y of T minus two M and so on. 92 00:08:10,850 --> 00:08:13,760 For the moving average part, we again have the same pattern. 93 00:08:14,270 --> 00:08:17,700 Y of T depends on past errors at multiples of M. 94 00:08:18,080 --> 00:08:22,190 So that would be Epsilon at T minus M Epsilon at T, minus two M and so on. 95 00:08:23,410 --> 00:08:29,170 Finally, once we have this season apart, we multiply it with the non seasonal part, and that gives 96 00:08:29,170 --> 00:08:30,700 us the false Arima model. 97 00:08:35,670 --> 00:08:42,210 In code, there is actually one extra component which gives us the sorry max model, the X part is actually 98 00:08:42,210 --> 00:08:47,310 pretty nice and I think it will make you feel like this model can actually be applied to so-called real 99 00:08:47,310 --> 00:08:48,000 world data. 100 00:08:48,780 --> 00:08:51,620 The ex parte refers to exogenous variables. 101 00:08:52,080 --> 00:08:58,290 So imagine that you have a time series of length Big T at each point in the Time series, you have some 102 00:08:58,290 --> 00:09:04,740 feature vector of exogenous data, such as maybe the sentiment of tweets by Elon Musk or the sentiment 103 00:09:04,740 --> 00:09:06,920 from various financial news sites. 104 00:09:07,410 --> 00:09:13,740 You can include this in your model by passing in an array of feature vectors of size T bidi into the 105 00:09:13,740 --> 00:09:15,060 auto arima function. 106 00:09:15,870 --> 00:09:20,610 One caveat to this is that when you want to make predictions, you have to know what these features 107 00:09:20,610 --> 00:09:21,020 are. 108 00:09:21,330 --> 00:09:24,920 So you want to consider whether or not this is actually practical for you. 109 00:09:25,470 --> 00:09:29,090 In any case, we won't be making use of this, but we will see on our output. 110 00:09:29,250 --> 00:09:30,110 Sorry, Max. 111 00:09:30,300 --> 00:09:32,900 So this is just in case you wanted to know what that means. 112 00:09:33,420 --> 00:09:36,770 Otherwise, consider this topic outside the scope of this cause.