1 00:00:00,420 --> 00:00:02,480 This lecture is about random walk. 2 00:00:02,480 --> 00:00:04,370 Objectives are the following. 3 00:00:04,370 --> 00:00:05,787 We will get familiar with random walk model. 4 00:00:05,787 --> 00:00:10,848 We will simulate a random walk model in R. 5 00:00:10,848 --> 00:00:14,237 We will obtain the correlogram of a random walk, and 6 00:00:14,237 --> 00:00:17,170 we will see a difference operator in action. 7 00:00:19,440 --> 00:00:20,440 The model is the following. 8 00:00:21,990 --> 00:00:25,769 Xt is equal to Xt minus 1 plus Zt. 9 00:00:25,769 --> 00:00:28,338 So here's how you can interpret this. 10 00:00:28,338 --> 00:00:34,010 Xt can be location of your particle at this moment. 11 00:00:34,010 --> 00:00:37,788 And Xt minus 1 will be location of that particle one step before. 12 00:00:37,788 --> 00:00:43,879 In other words, it can be location of a particle one minute ago or one day ago. 13 00:00:43,879 --> 00:00:48,124 And Zt is just sound residual, sound white noise. 14 00:00:48,124 --> 00:00:51,090 The random walk works as following. 15 00:00:51,090 --> 00:00:53,840 Wherever you are, we just add a little bit of noise to it, 16 00:00:53,840 --> 00:00:55,190 now you're in the next step. 17 00:00:55,190 --> 00:00:57,489 You add a little bit of noise to it, now you're in the next step. 18 00:00:58,510 --> 00:01:03,064 There's another interpretation where you can think Xt is the price of 19 00:01:03,064 --> 00:01:04,298 a stock today, and 20 00:01:04,298 --> 00:01:09,026 Xt minus y was the price of a stock yesterday, and Zt is the random noise. 21 00:01:09,026 --> 00:01:14,280 So stock is changing from the yesterday's price by adding some random noise into it. 22 00:01:16,160 --> 00:01:20,215 And this random noise, which is white noise or residual, 23 00:01:20,215 --> 00:01:25,866 that's just a normal random variable with some expectation and some variance. 24 00:01:29,964 --> 00:01:33,920 We can assume that maybe we are starting at point 0. 25 00:01:33,920 --> 00:01:35,936 At times 0, we are at 0. 26 00:01:35,936 --> 00:01:40,404 Which means that time 1 X1 would be X0, which is 0, 27 00:01:40,404 --> 00:01:43,524 plus the Z1, so X1 is actually Z1. 28 00:01:43,524 --> 00:01:49,050 And the next times step we are X2, which has to be X1 plus the Z2. 29 00:01:49,050 --> 00:01:53,105 But X1 is Z1 so it becomes Z1 plus the Z2. 30 00:01:53,105 --> 00:01:54,120 That's how you go. 31 00:01:54,120 --> 00:02:00,820 So X3 will become X2, which is Z1 plus Z2 plus additional noise, which is Z3. 32 00:02:00,820 --> 00:02:04,410 So as you go in this random work, you accumulate the noises. 33 00:02:04,410 --> 00:02:11,229 So at set T, XT, you basically have the sum of all noises until time T. 34 00:02:14,354 --> 00:02:19,161 If you look at expectation of Xt, well it is expectation of sum. 35 00:02:19,161 --> 00:02:23,360 And expectation of sum is the same thing as the sum of expectations. 36 00:02:23,360 --> 00:02:30,101 And since all of Zis are have the same mean mu, you will get mu t. 37 00:02:30,101 --> 00:02:32,470 So expectation is mu t. 38 00:02:32,470 --> 00:02:36,806 Expectation of this stochastic process is changing by the location. 39 00:02:36,806 --> 00:02:40,860 It is definitely not a stationary process. 40 00:02:40,860 --> 00:02:47,180 And variants of Xt is variants of the sum, which I wrote as the sum of the variants. 41 00:02:47,180 --> 00:02:51,973 This is only true if the random variables, the are independent. 42 00:02:51,973 --> 00:02:56,408 So we assume that in our model, that noises are independent from each other, 43 00:02:56,408 --> 00:03:00,366 which would mean that variance of the sum is the sum of the variances, 44 00:03:00,366 --> 00:03:02,428 which will give us signal square t. 45 00:03:02,428 --> 00:03:06,140 So there's systematic change in mean. 46 00:03:06,140 --> 00:03:07,890 There is systematic change in variance. 47 00:03:07,890 --> 00:03:10,835 This is definitely a non-stationary time series, or 48 00:03:10,835 --> 00:03:13,081 a non-stationary stochastic process. 49 00:03:14,450 --> 00:03:16,270 So let's do a simulation, in this simulation, 50 00:03:16,270 --> 00:03:17,910 we're going to start from X1 not X0. 51 00:03:19,350 --> 00:03:23,222 X1 is 0 and our random variable is standard, 52 00:03:23,222 --> 00:03:27,111 our noise is a standard normal distribution. 53 00:03:27,111 --> 00:03:32,976 And we're going to simulate it using a for loop and we're going to plot it and 54 00:03:32,976 --> 00:03:38,962 we're going to look at the correlation function of it for the simulation. 55 00:03:38,962 --> 00:03:42,977 So I'm going to say x=NULL, okay. 56 00:03:42,977 --> 00:03:47,838 And that x1, my starting point, is actually 0. 57 00:03:47,838 --> 00:03:52,474 Then for later on, I have to start from my previous step add some noise to it. 58 00:03:52,474 --> 00:03:54,863 So I'm going to use for loop. 59 00:03:54,863 --> 00:03:57,160 The syntax is for parentheses. 60 00:03:57,160 --> 00:04:04,370 We do the index i which is n, starting from 2 until let's say 1,000. 61 00:04:04,370 --> 00:04:06,801 We going to generate 1,000 data points. 62 00:04:06,801 --> 00:04:09,910 I'm going to do the brackets. 63 00:04:09,910 --> 00:04:11,970 Open bracket and I'm going to close the bracket. 64 00:04:11,970 --> 00:04:17,972 Everything in the bracket will have a loop, will be inside the loop basically. 65 00:04:17,972 --> 00:04:23,254 And I'm going to say x[i]=x[i-1]+ some noise, and 66 00:04:23,254 --> 00:04:28,440 the noise we assume to be standard normal distribution. 67 00:04:28,440 --> 00:04:31,110 So I'm going to say rnorm, just 1 data point, 68 00:04:31,110 --> 00:04:34,610 because I want to add one noise to it. 69 00:04:34,610 --> 00:04:38,080 And then we generated our beta set. 70 00:04:38,080 --> 00:04:43,483 If I say print(x), we see that we have thousand, 71 00:04:43,483 --> 00:04:46,256 thousand, data points. 72 00:04:46,256 --> 00:04:49,060 But it does not have a time series structure on it. 73 00:04:49,060 --> 00:04:52,190 So let me just go ahead and clear the console. 74 00:04:53,330 --> 00:04:58,610 And x is a data set, but it doesn't have a time state structure on it. 75 00:04:58,610 --> 00:05:02,987 So I'm going to define a random walk and I'm going to say ts. 76 00:05:02,987 --> 00:05:07,850 And ts is going to basically transform the data set to a time series. 77 00:05:07,850 --> 00:05:09,420 I'm going to put x in it. 78 00:05:09,420 --> 00:05:14,370 And then I have a random walk, which is basically, time series. 79 00:05:14,370 --> 00:05:15,620 Now let's plot this. 80 00:05:15,620 --> 00:05:20,100 Let's plot random_walk. 81 00:05:20,100 --> 00:05:21,730 Let's out some title to it. 82 00:05:21,730 --> 00:05:26,670 Title would be a random walk. 83 00:05:27,960 --> 00:05:32,070 Into the y label, let's put nothing. 84 00:05:32,070 --> 00:05:37,406 And the x label, let's say this is basically days. 85 00:05:37,406 --> 00:05:38,870 And let's put some color into it. 86 00:05:38,870 --> 00:05:46,240 We can put a blue color to it, and we can increase the width of our line by 2. 87 00:05:46,240 --> 00:05:50,303 And once I do that, we obtain the following. 88 00:05:50,303 --> 00:05:51,949 A random walk. 89 00:05:51,949 --> 00:05:57,804 This is a very, very typical time plot for a random walk. 90 00:05:57,804 --> 00:06:02,040 Now, random walk we just said, is not a stationary time series. 91 00:06:02,040 --> 00:06:06,710 It would not make sense to actually find acf of it, 92 00:06:06,710 --> 00:06:12,061 because acf, we define acf for stationary time series. 93 00:06:12,061 --> 00:06:15,920 But let's just do it because we can just do it. 94 00:06:15,920 --> 00:06:18,620 Let's just try to find the acf in r. 95 00:06:20,880 --> 00:06:27,710 If I say acf(random_walk), we obtain the following plot. 96 00:06:27,710 --> 00:06:32,620 As you see, there's a high correlation, 97 00:06:32,620 --> 00:06:37,180 even 30 laps back, which just again shows that there is 98 00:06:37,180 --> 00:06:41,080 a high correlation in this data set and there is no stationality. 99 00:06:42,240 --> 00:06:45,710 Now let's deviate from the topic, random walk, and 100 00:06:45,710 --> 00:06:50,040 say there is a trend, definitely trending here. 101 00:06:50,040 --> 00:06:51,050 Goes up and down. 102 00:06:52,150 --> 00:06:53,550 Can you remove that trend? 103 00:06:53,550 --> 00:06:54,930 It turns out that yes we can. 104 00:06:54,930 --> 00:06:56,406 Look at here, look what we have here. 105 00:06:56,406 --> 00:07:02,180 Xt is Xt-1 + Zt, I'm going to take this Xt-1 to the left hand side. 106 00:07:02,180 --> 00:07:06,045 So basically we have Xt- Xt-1 = Zt. 107 00:07:06,045 --> 00:07:10,523 Let's define Xt- Xt-1 as delta. 108 00:07:13,719 --> 00:07:17,482 Well, this is not exactly delta, this is a difference operator. 109 00:07:17,482 --> 00:07:18,497 So let's call it delta Xt. 110 00:07:18,497 --> 00:07:20,619 So this is our difference operator. 111 00:07:20,619 --> 00:07:24,932 So difference operator applied to the Xt, DXt. 112 00:07:24,932 --> 00:07:28,778 This is a new time series, which is equal to Zt. 113 00:07:28,778 --> 00:07:30,830 I remember Zt is a random noise. 114 00:07:30,830 --> 00:07:33,460 Zt is a purely random process. 115 00:07:33,460 --> 00:07:38,990 Which means that my difference data delta Xt is purely random processed, 116 00:07:38,990 --> 00:07:44,280 which is a stationary time series, which is stationary statistic process. 117 00:07:44,280 --> 00:07:48,160 So it means that if we have a random walk, simulation for 118 00:07:48,160 --> 00:07:53,140 a random walk, if we can take difference and 119 00:07:53,140 --> 00:07:56,030 look at the difference, the difference is going to be stationary. 120 00:07:56,030 --> 00:07:59,470 Let's confirm that using R. 121 00:07:59,470 --> 00:08:03,559 What we begin to do, we going to use difference operators. 122 00:08:03,559 --> 00:08:07,360 I diff and I write the random walk. 123 00:08:07,360 --> 00:08:11,002 That will give me difference of like 1. 124 00:08:11,002 --> 00:08:14,509 So it's going to give me x2 minus x1, x3 minus x2, 125 00:08:14,509 --> 00:08:18,195 x4 minus x3 dot dot dot dot x1000 minus x900. 126 00:08:18,195 --> 00:08:22,143 So this will actually give me 999 data points. 127 00:08:22,143 --> 00:08:27,170 We are missing one point at the beginning, but that's all right. 128 00:08:27,170 --> 00:08:31,280 Once I do that, I have another time series, which is just differences. 129 00:08:31,280 --> 00:08:33,743 For example, I can try to just plot it. 130 00:08:35,701 --> 00:08:39,850 Let me not just put any title on it. 131 00:08:39,850 --> 00:08:44,880 By plotting this, I should get a purely random process because where you saw 132 00:08:44,880 --> 00:08:49,689 that by just taking xt minus 1 to the left hand side the difference is Zt. 133 00:08:49,689 --> 00:08:51,510 It is purely [INAUDIBLE] process. 134 00:08:51,510 --> 00:08:55,343 Let's plot this, And 135 00:08:55,343 --> 00:08:59,720 you get what looks like white noise. 136 00:09:02,550 --> 00:09:06,620 Now we can also look at a acf of the difference. 137 00:09:06,620 --> 00:09:10,190 So, I write acf of the difference of the random walk. 138 00:09:10,190 --> 00:09:16,599 And I look at the difference, I get an acf, which I have seen before. 139 00:09:16,599 --> 00:09:21,480 This is acf of the purely random process we generated a few lectures back. 140 00:09:23,490 --> 00:09:26,410 So we just apply difference operator to remove the trend. 141 00:09:26,410 --> 00:09:29,997 There was some kind of trend going on, we removed the trend. 142 00:09:29,997 --> 00:09:32,614 By just applying the difference operator, 143 00:09:32,614 --> 00:09:36,020 it'll get the pot of ACF of the differenced time series. 144 00:09:36,020 --> 00:09:37,770 So what have we learned? 145 00:09:37,770 --> 00:09:40,450 We have learned a random walk model. 146 00:09:40,450 --> 00:09:43,400 We learned how to simulate a random walk in R. 147 00:09:43,400 --> 00:09:46,000 And we learned how to get a stationary time series. 148 00:09:46,000 --> 00:09:51,160 In fact, the purely random process [INAUDIBLE] Random Walk using 149 00:09:51,160 --> 00:09:52,130 difference operator.