Hello. In this model, I like to go over again, the model components of the ARIMA model and the three parts. The first part is the integrated portion of the model is parameterized with the d. If it's integrated with order zero, that means you take the raw data and use it as is, you don't have to do any differencing. If it's integrated of order one, that means you did some first differencing. If you see an integrated data of two, that means you also did a second differencing. Recall that this is done using the ADF, the Augmented Dickey-Fuller test, and you just keep doing that until the data is stationary, and that's the first thing you want to do. Next, you want to look at the autoregressive components of the model p. P is the number of lags that you decide to use for your time series. If you just used an ARIMA(1,0,0), that means p has a value of one, and this is essentially an autoregressive model with one lag. So the P tells us how many lags we're going to use in the equation. A moving average queue, the MA part, q stands for the number of lags and the moving average to the error terms. If you used just ARIMA(0,0,1) that has a value one, that just tells you use the one period moving average for your forecasting. So now, we know or understand that three parameters that are essential for an ARIMA model. But there are some special cases of the ARIMA model that I want to show you. The first case is ARIMA(0,0,0) which means that there is no autoregressive component, there is no differencing, and there is no moving average lags. So I want to show you what that looks like in R. So I'm in the RStudio environment and I want to first show you what white noise looks like, and I'm going to simulate white noise. I use this command, arima.sim, that's the command. Here's my parameters, it's order of (0,0,0). I'm going to take 100 data points in this time series with a mean of zero. So my time series will be 100 points long, so let's just see what that looks like. Line three will show you the data. Here, we can see it just looks like a bunch of numbers. Let's plot it, and that will really give us a feel for the data. You can see that it has in this case a mean of zero and there's some standard variance there. If you increase n to maybe 1,000 or 5,000, 10,000 even, you would see a darker line but it would look pretty filled in and pretty constant. So that's what white noise looks like. You can also do this just by generating random numbers from a normal distribution. An ARIMA model (0,1,0) data that just has first differences is also known as a random walk model. So the way to think about a random walk model is if here at some position at time t and then you have a random draw, for this example, just say you can go up one or down one, you're here, and you flip a coin, and you go up one, you flip another coin, you go up on again. You flip a coin, this time you go down. If it's tails, up, down, up, down, and it's cumulative. It's always based on this random flipping of a coin. More concretely though, it's the random draw from a normal distribution, let's say a mean of zero, standard deviation of one. Then you're just bouncing around so that each of your bounces are the same, but you're not necessarily in that same position. So that's a random walk. So I'm simulating random walk data here in lines six through nine, and I'm going to plot it. Again, the parameters here pdq are (0,1,0). Here in this case, you can see this random walk. We start at zero, I went down a little bit, and then it happened again, a sequence of positive numbers and you can see them here, 32, 48, 147, 235. It just seems to be going up, and up, and down, and then it trends downwards. If I run this again, I'm not going to get the same random walk. Because it's using a random number generator and there was another random walk. The next type of data that I want to talk about is a random walk with a constant drift. So you might be going in a random walk but you might be continually going upwards or downwards, and that's what this +c component is. On the slide, you can see that the first part ARIMA(0,1,0) is a random walk plus some constant. So you're always have some drift in your model or trend. I'm going to show you now the random walk with drift in the ARIMA model using this simulating some data. When we plot it, you can see that there's a positive trend. Even though it's bouncing around, it constantly rises upwards. Next, we have a first-order autoregressive model (1,0,0). That's where you just have an autoregressive component in your model. I mean, R now and I'm going to show you some simulated data for a first-order autoregressive model. Just going to run it all at the same time, now that you understand what's going on. There it is. Today's value is based on some proportion of yesterday's model. Finally, I want to show you a moving average model ARIMA(0,0,1), so that uses only one lag component for your moving average. Here's the code, here again, arima.sim simulates the data, there's the (0,0,1) for the moving average, term of 0.6. We're going to run that. There you have it, and that's what the data looks like. So hopefully, by now, you have seen a random walk, a random walk with drift, white noise, an autoregressive one, a moving average one. By playing around with this R code, I encourage you to play with these order parameters, the pdq parameters, to look at different graphs. That'll give you some intuition hopefully about what different types of data will look like.