1 00:00:00,430 --> 00:00:04,310 In this lecture, we'll talk about autocovariance coefficients. 2 00:00:04,310 --> 00:00:06,000 Objectives are the following. 3 00:00:06,000 --> 00:00:09,710 We'll recall the covariance coefficient for a bivariate data set. 4 00:00:10,740 --> 00:00:15,520 We will define autocovariance coefficients for a time series. 5 00:00:15,520 --> 00:00:19,774 And we'll estimate autocovariance coefficients of a time series at 6 00:00:19,774 --> 00:00:20,884 different lags. 7 00:00:22,581 --> 00:00:26,370 Remember if you have two random variables x and y, covariance is 8 00:00:26,370 --> 00:00:31,350 basically measuring the linear dependence in those two random variables. 9 00:00:31,350 --> 00:00:35,670 And we have seen the same definition of covariance in the previous video lecture. 10 00:00:35,670 --> 00:00:40,702 Covariance of x, y is expectation of x, 11 00:00:40,702 --> 00:00:43,960 y minus expectation of y. 12 00:00:45,550 --> 00:00:52,720 Now, usually we do not get the random variables in real life. 13 00:00:52,720 --> 00:00:55,400 We get, let's say, datasets. 14 00:00:55,400 --> 00:00:59,160 We have this paired dataset (x1, y1) (x2, y2) (xN, y1). 15 00:00:59,160 --> 00:01:03,510 And we would like to estimate the covariance between these two sets, 16 00:01:03,510 --> 00:01:07,540 this is basically sample covariance. 17 00:01:07,540 --> 00:01:12,142 We would like to look at the set x1, x2, xN, and y1, y2, yN, and 18 00:01:12,142 --> 00:01:16,363 somehow measure the linear dependence between these two sets. 19 00:01:16,363 --> 00:01:20,200 And the estimation formula is basically Sxy. 20 00:01:20,200 --> 00:01:26,099 We sum xt- xy yt- y bar. 21 00:01:26,099 --> 00:01:30,140 X bar and y bar here are the sample average of each data set. 22 00:01:30,140 --> 00:01:33,540 And we divide by not n, but n-1. 23 00:01:33,540 --> 00:01:40,223 Now in R, we don't have to calculate this by hand or with any loop. 24 00:01:40,223 --> 00:01:45,480 We can just use the covariance routine in R with cov. 25 00:01:45,480 --> 00:01:49,029 And we can put the dataset one and dataset two, 26 00:01:49,029 --> 00:01:52,311 it will calculate this covariance for us. 27 00:01:55,133 --> 00:01:57,910 Now we going to talk about autocovariance coefficient. 28 00:01:57,910 --> 00:02:02,662 So autocovariance coefficients at different lags defined to be lambda, 29 00:02:02,662 --> 00:02:03,971 I'm sorry gamma k. 30 00:02:03,971 --> 00:02:06,350 And this is what we defined the last lecture. 31 00:02:06,350 --> 00:02:10,530 Gamma k's the covariance between xt and xt plus k. 32 00:02:10,530 --> 00:02:15,720 And since we assume the weak stationarity, it doesn't matter what t is. 33 00:02:15,720 --> 00:02:20,373 We would have same gamma k as long as the distance between 34 00:02:20,373 --> 00:02:23,147 these two random variables is k. 35 00:02:23,147 --> 00:02:26,780 And ck is going to be an estimation for gamma k. 36 00:02:28,470 --> 00:02:31,430 And this is how we're going to define our estimation. 37 00:02:31,430 --> 00:02:35,680 It's very much like what we defined to be covariance. 38 00:02:35,680 --> 00:02:40,010 In this case, we don't have x's and y's, we just have x's. 39 00:02:40,010 --> 00:02:44,174 So we look at xt values from 1 to N-k, 40 00:02:44,174 --> 00:02:48,346 x values starting from k+1 to the N. 41 00:02:48,346 --> 00:02:52,551 And we calculate their difference from the x bar, 42 00:02:52,551 --> 00:02:58,870 x bar being the sample average, and we calculate whole sum and divide by N. 43 00:02:58,870 --> 00:03:04,105 This is going to be our estimation for our autocovariance coefficients. 44 00:03:05,428 --> 00:03:08,540 Now, again in R, we will not do it by hand. 45 00:03:08,540 --> 00:03:12,290 Although we can write just one simple loop to calculate this 46 00:03:12,290 --> 00:03:16,850 autovariance coefficients, we will use what's called acf routine. 47 00:03:16,850 --> 00:03:19,810 Now, acf stands for autocorrelation function, 48 00:03:19,810 --> 00:03:22,636 which I'm going to talk about in my next lecture. 49 00:03:22,636 --> 00:03:27,820 But for now, we'll use acf routine in the following way. 50 00:03:27,820 --> 00:03:31,130 We're going to acf, the time series, and 51 00:03:31,130 --> 00:03:34,380 the type, we're going to type in covariance. 52 00:03:34,380 --> 00:03:37,130 If we type in covariance, it will give us all 53 00:03:38,530 --> 00:03:44,800 autocovariance coefficients. 54 00:03:44,800 --> 00:03:48,220 Now we're going to simulate a purely random process. 55 00:03:48,220 --> 00:03:53,080 It's a purely random process the time series with no special pattern. 56 00:03:53,080 --> 00:03:55,230 And we're going to use rnorm routine. 57 00:03:55,230 --> 00:03:59,397 We will call our time series purely_random_process, and 58 00:03:59,397 --> 00:04:04,543 we will use the ts routine, which will take the dataset that we generate and 59 00:04:04,543 --> 00:04:07,440 put time series structure on it. 60 00:04:07,440 --> 00:04:11,190 And inside that ts routine, I have rnorm routine. 61 00:04:11,190 --> 00:04:14,780 R stands for random, norm stands for normal random variables, so 62 00:04:14,780 --> 00:04:21,360 we will generate, let's say, 100 data points from normal distribution. 63 00:04:21,360 --> 00:04:26,940 In fact, it's going to generate 100 data points from standard normal distribution 64 00:04:26,940 --> 00:04:29,840 with a mean 0 and standard deviation 1. 65 00:04:29,840 --> 00:04:36,400 When I do that, now we have purely random process. 66 00:04:36,400 --> 00:04:42,628 Let's just print, purely_random_process. 67 00:04:45,365 --> 00:04:51,023 And if I print it, I see that it's a time series object that starts at time 1, 68 00:04:51,023 --> 00:04:52,775 ends up at time 100. 69 00:04:52,775 --> 00:05:00,030 And the frequency is 1, and we have our 100th data points. 70 00:05:00,030 --> 00:05:02,290 So we will be using acf routine. 71 00:05:02,290 --> 00:05:04,380 Acf routine usually gives us a plot. 72 00:05:04,380 --> 00:05:07,020 We're going to change the type to the covariance 73 00:05:07,020 --> 00:05:09,270 because we would like to get autocovariance. 74 00:05:09,270 --> 00:05:12,330 And I'm going to put parenthesis around it so that it will print out 75 00:05:12,330 --> 00:05:17,090 the data that it produces, and you obtain the plot. 76 00:05:17,090 --> 00:05:22,400 Along with the plot it produces autocovariance coefficients for 77 00:05:22,400 --> 00:05:23,938 every single lag. 78 00:05:23,938 --> 00:05:28,830 So when we type in this command, it gives us a plot, 79 00:05:28,830 --> 00:05:30,800 which I will talk about in the next lecture. 80 00:05:30,800 --> 00:05:35,100 But what we are concentrating on would be these numbers here. 81 00:05:35,100 --> 00:05:38,327 Basically, this is autocorrelation coefficient estimation for 82 00:05:38,327 --> 00:05:40,840 our autocorrelation coefficient at lag 0. 83 00:05:40,840 --> 00:05:44,270 This is at lag 1, this is at lag 2, and this is at lag 3, and so forth. 84 00:05:46,008 --> 00:05:50,451 And this is how we're going to calculate our autocovariance coefficients in R. 85 00:05:50,451 --> 00:05:52,316 So, what have learned in this lecture? 86 00:05:52,316 --> 00:05:55,629 You have learned the definition of autocovariance coefficients 87 00:05:55,629 --> 00:05:56,658 at different lags. 88 00:05:56,658 --> 00:06:00,922 And you have learned how to estimate the autocovariance coefficient of a time 89 00:06:00,922 --> 00:06:02,440 series using acf routine.