1 00:00:00,410 --> 00:00:04,740 Welcome to practical time series analysis. 2 00:00:04,740 --> 00:00:07,125 In these introductory lectures, 3 00:00:07,125 --> 00:00:10,230 we're reviewing some basic statistical concepts. 4 00:00:10,230 --> 00:00:12,330 In this particular lecture, 5 00:00:12,330 --> 00:00:19,160 we'll look at linear regression or ordinary least squares as people call it. 6 00:00:19,180 --> 00:00:24,685 In this lecture, we'll learn how to plot time series data 7 00:00:24,685 --> 00:00:29,730 and we'll learn how to fit a linear model to a set of ordered pairs. 8 00:00:29,730 --> 00:00:31,995 If your background in statistics is strong, 9 00:00:31,995 --> 00:00:35,470 you can move very quickly through these lectures. 10 00:00:35,470 --> 00:00:39,835 R comes complete with a variety of datasets. 11 00:00:39,835 --> 00:00:42,800 If you'd like a little narrative summary explaining 12 00:00:42,800 --> 00:00:46,465 your dataset you can use the help command. We've done that here. 13 00:00:46,465 --> 00:00:48,890 We looked at the command help on 14 00:00:48,890 --> 00:00:54,413 the CO2 dataset and I've put some of the output right here for you, 15 00:00:54,413 --> 00:01:01,930 atmospheric concentrations of carbon dioxide over a set of years. 16 00:01:01,930 --> 00:01:03,845 When we plot our data set, 17 00:01:03,845 --> 00:01:07,040 you can see that a linear model is not going to 18 00:01:07,040 --> 00:01:11,645 capture all of the interesting behavior in this data set. 19 00:01:11,645 --> 00:01:16,400 First of all, there is a rising trend to the set of data but it doesn't look like 20 00:01:16,400 --> 00:01:23,110 a straight line is really our best object for this dataset as far as trend goes. 21 00:01:23,110 --> 00:01:24,590 Even worse, there is 22 00:01:24,590 --> 00:01:32,590 this oscillatory piece that a straight line is just not going to capture it all. 23 00:01:32,590 --> 00:01:39,675 The idea behind linear regression is that you have a response variable, 24 00:01:39,675 --> 00:01:42,795 here that will be carbon dioxide concentration, 25 00:01:42,795 --> 00:01:45,660 and you feel that it depends at least 26 00:01:45,660 --> 00:01:48,965 somewhat on the explanatory variable in a linear way. 27 00:01:48,965 --> 00:01:54,930 Our particular dataset shows the deficiencies of this approach and in fact our time 28 00:01:54,930 --> 00:01:57,880 series cores allow us to move beyond 29 00:01:57,880 --> 00:02:02,925 the simple linear regression to more sophisticated techniques. 30 00:02:02,925 --> 00:02:09,825 The response variable is thought to be a linear model plus some noise. 31 00:02:09,825 --> 00:02:12,935 This noise term does a lot of work for us. 32 00:02:12,935 --> 00:02:17,310 If you just want to fit a straight line to a set of data, that's fine. 33 00:02:17,310 --> 00:02:20,630 The ordinary least squares approach will always do that for you. 34 00:02:20,630 --> 00:02:23,483 If you want to start drawing inferences though, 35 00:02:23,483 --> 00:02:26,140 you need to invoke some distributional assumptions 36 00:02:26,140 --> 00:02:29,935 and this error term is our way of doing that. 37 00:02:29,935 --> 00:02:33,550 The error term can come about in a variety of ways. 38 00:02:33,550 --> 00:02:35,430 You might have measurement error, 39 00:02:35,430 --> 00:02:41,360 you might not have all of the important variables in your model. 40 00:02:41,360 --> 00:02:45,845 There are a number of ways that we can produce error in a model. 41 00:02:45,845 --> 00:02:49,745 Now, if you want to do some inference there are some, 42 00:02:49,745 --> 00:02:54,080 what I would think of as, vanilla assumptions. 43 00:02:54,080 --> 00:02:59,885 The errors in the simplest case would be normally distributed in an average zero. 44 00:02:59,885 --> 00:03:02,480 They'd have the same variance. 45 00:03:02,480 --> 00:03:06,680 And when you do regression in a more mature way than we'll do in 46 00:03:06,680 --> 00:03:11,870 this lecture you learn how to critique these assumptions. 47 00:03:11,870 --> 00:03:14,600 Also, we'll assume that the errors in 48 00:03:14,600 --> 00:03:21,190 simple ordinary least squares will assume that the errors are independent. 49 00:03:21,190 --> 00:03:26,280 Now that in a time series course makes the modeling a little bit boring. 50 00:03:26,280 --> 00:03:29,525 But again, we're just starting here. 51 00:03:29,525 --> 00:03:33,910 The basic idea behind ordinary least squares is to get 52 00:03:33,910 --> 00:03:37,780 your observed data point and compare it to 53 00:03:37,780 --> 00:03:42,545 what your choice of slope and intercept would predict. 54 00:03:42,545 --> 00:03:45,170 So I can throw any numbers I like in for 55 00:03:45,170 --> 00:03:49,450 a slope and intercept and that'll give me a prediction. 56 00:03:49,450 --> 00:03:51,620 If I look at what I've observed and compare it to 57 00:03:51,620 --> 00:03:54,220 that prediction then what we're going to 58 00:03:54,220 --> 00:04:00,575 do is square our terms and come up with an aggregate error. 59 00:04:00,575 --> 00:04:03,700 The idea behind ordinary least squares of course, 60 00:04:03,700 --> 00:04:09,125 is that we'll make this aggregate error as small as mathematically possible. 61 00:04:09,125 --> 00:04:13,630 It really only takes a little bit of calculus in order to do this. 62 00:04:13,630 --> 00:04:17,788 Rather than work with the calculations by hand, 63 00:04:17,788 --> 00:04:21,575 we'll let R do the calculation for us. 64 00:04:21,575 --> 00:04:26,114 The LM, command, linear model command, will take CO2, 65 00:04:26,114 --> 00:04:32,473 it knows to come into your time series and extract the variable of interest here, 66 00:04:32,473 --> 00:04:36,855 CO2 concentrations, will take CO2 on. 67 00:04:36,855 --> 00:04:42,000 Now the CO2 time series has a time part to it. 68 00:04:42,000 --> 00:04:46,430 You can think of it as response together with time and in 69 00:04:46,430 --> 00:04:51,780 order to extract the time part we'll use the little command here called time. 70 00:04:51,780 --> 00:04:56,330 I've put parentheses around this line in order to have 71 00:04:56,330 --> 00:05:01,710 the output appear on the screen and I've copied and pasted in this slide for you. 72 00:05:01,710 --> 00:05:03,770 You can see that the intercept, 73 00:05:03,770 --> 00:05:12,925 the best intercept is something like negative 2000 and the best slope is 1.3 or so. 74 00:05:12,925 --> 00:05:16,350 Now take this number here with a little bit of caution. 75 00:05:16,350 --> 00:05:19,475 We're not saying that at time zero, the intercept, 76 00:05:19,475 --> 00:05:24,315 the carbon dioxide concentration would be negative 2000. 77 00:05:24,315 --> 00:05:26,965 That's sort of a meaningless thing to say. 78 00:05:26,965 --> 00:05:29,345 But given our dataset, 79 00:05:29,345 --> 00:05:32,225 the best intercept for that cloud, 80 00:05:32,225 --> 00:05:36,230 that scatterplot, really would be negative 2000. 81 00:05:36,230 --> 00:05:39,245 We will not extrapolate back that far. 82 00:05:39,245 --> 00:05:44,530 Our model utility would have broken down long before then. 83 00:05:44,690 --> 00:05:49,330 If you'd like to plot your line and include 84 00:05:49,330 --> 00:05:52,765 the data I've just reproduced the plot command here. 85 00:05:52,765 --> 00:05:56,485 And I'm going to use the command now a b line. 86 00:05:56,485 --> 00:05:58,660 So this is intercept slope line. 87 00:05:58,660 --> 00:06:02,645 So I'll do that on the model that we developed. 88 00:06:02,645 --> 00:06:10,270 If you do that then you'll see your original data set increasing in time, 89 00:06:10,270 --> 00:06:15,130 though a straight line is not the best model there probably even to capture the trend, 90 00:06:15,130 --> 00:06:20,470 increasing in time but also with an oscillatory part. 91 00:06:20,470 --> 00:06:24,490 In the next lecture we'll look at our errors and 92 00:06:24,490 --> 00:06:28,020 try to say something meaningful about the errors. 93 00:06:28,020 --> 00:06:31,120 But for right now we've been able to fit the best, 94 00:06:31,120 --> 00:06:35,425 arithmetically best, straight line to this dataset. 95 00:06:35,425 --> 00:06:40,330 At this point given a set of x and y values you should be able to 96 00:06:40,330 --> 00:06:45,245 plot your data and fit a linear model to your set of ordered pairs. 97 00:06:45,245 --> 00:06:49,990 We will critique the modeling process in the next lecture.