1 00:00:11,050 --> 00:00:16,750 In this lecture, we are going to discuss another integral part of the ARIMA model, the model. 2 00:00:17,380 --> 00:00:20,410 As you know, M-A stands for Moving Average. 3 00:00:20,770 --> 00:00:26,020 However, it's important not to conflate this with the simple moving average and the exponentially weighted 4 00:00:26,020 --> 00:00:26,770 moving average. 5 00:00:26,920 --> 00:00:33,160 We discussed earlier in this course, as you will see shortly, this model is very different from those. 6 00:00:33,550 --> 00:00:36,640 So let's get right to what the moving average model looks like. 7 00:00:41,610 --> 00:00:47,160 The moving average model is similar to the auto regressive model in that it is a linear function of 8 00:00:47,160 --> 00:00:47,700 something. 9 00:00:48,360 --> 00:00:50,160 But what is it a linear function? 10 00:00:51,360 --> 00:00:55,050 In fact, it is a linear function of past error terms. 11 00:00:55,500 --> 00:00:59,940 This should definitely strike you as odd in pretty much all of machine learning. 12 00:01:00,210 --> 00:01:03,150 We typically create models which depend on input data. 13 00:01:03,570 --> 00:01:07,050 In this moving average model, there is no input data to be seen. 14 00:01:08,010 --> 00:01:11,340 Instead, at the time, series depends only on errors. 15 00:01:11,880 --> 00:01:17,190 Of course, in order to know these errors, we must have compared at the previous predictions with the 16 00:01:17,190 --> 00:01:19,260 previous values of the Time series. 17 00:01:20,760 --> 00:01:26,790 Note that our abbreviated form for a moving average model with order Q is may Q. 18 00:01:27,240 --> 00:01:34,200 This means that the output Y depends on Q passed error terms in addition to the latest error term epsilon 19 00:01:34,200 --> 00:01:34,940 sub t. 20 00:01:39,820 --> 00:01:44,350 Here's one way to think of the moving average model that might help you rationalize its name. 21 00:01:45,340 --> 00:01:48,580 Consider what the expected value of Y of T should be. 22 00:01:49,750 --> 00:01:54,460 As you know, we treat errors as normals with mean zero and variance Sigma Square. 23 00:01:55,030 --> 00:02:01,210 Well, if we take the expected value of y of t, we simply get C, since the expected value of each 24 00:02:01,210 --> 00:02:02,680 of the errors is zero. 25 00:02:03,370 --> 00:02:10,210 Hence we can think of the biased term C as the average value and then each of the errors as fluctuations 26 00:02:10,210 --> 00:02:13,210 that make y of t go up or down around c. 27 00:02:18,260 --> 00:02:22,460 Another way to think about the moving average model is in terms of simulation. 28 00:02:23,490 --> 00:02:28,830 In fact, simulating a moving average process is somewhat easier than an auto regressive process. 29 00:02:29,400 --> 00:02:32,430 Let's suppose that we want to simulate an may to process. 30 00:02:32,640 --> 00:02:38,010 So the output y depends on two past error terms in addition to the current error term. 31 00:02:39,360 --> 00:02:42,360 First, we need to generate Epsilon one and Epsilon two. 32 00:02:42,630 --> 00:02:47,040 These are just samples from some normal with mean zero and variance sigma squared. 33 00:02:47,580 --> 00:02:53,730 Then we generate epsilon three, then we calculate y three according to our formula, which depends 34 00:02:53,730 --> 00:02:58,620 on Epsilon one, two and three, as well as the model weights, which we assume have been given. 35 00:02:59,700 --> 00:03:05,310 Next we generate Epsilon four, then we calculate y four, which depends on Epsilon two, three and 36 00:03:05,310 --> 00:03:05,850 four. 37 00:03:06,420 --> 00:03:11,070 Then we generate Epsilon five and we five, Epsilon six and Y six and so on. 38 00:03:11,760 --> 00:03:17,520 So as you see, this process really amounts to nothing but generating samples from the normal and adding 39 00:03:17,520 --> 00:03:18,210 them together. 40 00:03:18,900 --> 00:03:24,780 Therefore, you would use this model if you think that nature actually behaves this way to generate 41 00:03:24,780 --> 00:03:26,460 the data which you are observing.