Hi. In this video, we're going to talk a little bit about Holt-Winters Forecasting model, and it builds upon the moving average that we discussed in the previous video. In the previous video, we talked about simple exponential smoothing, and one of the limitations of simple exponential smoothing is that it does not perform well with data that has some trend or some seasonality. By trend, I mean something that looks like, maybe the data goes up and there's an upward trend like this, or there's a seasonal cycles. So you might see something an irregular period maybe around the holiday season, there's an increase in sales, and then an increase in sales like at Christmas time, or bathing suits in the springtime as people are getting ready for the summer, or ski equipment before the winter, things like that. So those are seasonal cycles that we might see. So Holt's exponential smoothing is used for data with trends. So we're going to look at that in a minute. Then the next type of modeling is Holt-Winters Exponential Smoothing, which takes the trend and then adds another component to the model for seasonality. So before we begin, let's look at some data. This is international airline passengers in thousands from the years '49-1960. You can see going down the left-hand side of the screen is the year, and then months going across. If this were a time series data, you would look at this row first of 1949, and then wrap around to 1950 etc. Here's the graph of the data, and one of the first things you want to do when you do get some data is to graph it, make some visualizations. We can see here some interesting observations. Just by eyeballing it, there does look like there is some trend, it's an upward trend, so as we go into the future, more and more people are riding the airplanes, the international passengers. The other thing I notice is that there seems to be some cycle here. You can see it is up and down, and it seems to be at regular intervals. So if we look at here, it seems to be just before at the end of the year. So in the end of the year 1950, in the third or fourth quarter, there's an increase, a spike maybe around the holiday season, and we seem to see that toward the third quarter as we go through time. There does seem to be some seasonal cycle. So let's look at the whole Winters model. But before we do, let's take a look at the equations. These are the equations that I introduced in the moving average section for simple exponential smoothing, and recall our prediction for the next time period, T plus 1 is some combination of today's value, y_T and our previous estimate of T minus 1. Recall that Alpha is between zero and one. So if Alpha is 60 percent of your weight from y_T, and 1 minus 60, or 0.4 of y_T minus 1. Notice the subscripts go from one to the capital T. So this first equation is for the immediate future one period forward, and this is the prediction given the past data of any value of T. Now, before I go into the equations for the Holt-Winters Model, I want to re-phrase the simple exponential smoothing into component forms. Basically, here's your y_T plus h, and I'm just going to call it l_T. Well, l_T is this, the level equation, and that's the equation we saw before. So if you compare the last slide, you see y_T plus h equals Alpha y_T plus one minus Alpha like this. In this case, we're just breaking it up into two pieces. L is for the level occasion or the smoothed value, and h is the lag. So if h equals one, it gives the fitted value for one period difference, but we can look multiple periods in the future. You might ask why we're bothering to put the equation in this form and separating it into two equations, and right now quite frankly, it's not very useful because there's a simple substitution here, and I could just say y_T plus h, this is equal to this, and not bother with that l_T notation. But as we go forward, it become very, very useful. So let's leave it at that for the moment, and here's where we see why. So this is the linear trend model. Recall in this model, we're looking for some trend, and we're going to add some trend in there. So here is the forecast T plus h periods ahead, and there's the level equation. We see that there really is no difference in the level equation, it's the exact same level equation that we saw before. The difference here that I want you to focus on, is this term here hb_t. So if h is one, that means look one period into the future. But if there's multiple periods in the future, you want to look at one period, two period into the future, three periods in the future. So b_t is some trend equation. So am now looking at one period on the trend, two periods on the trend, etc. So h is the number of pairs looking forward and I multiply it by this b_t, which helps describe what the trend is. This trend is nothing more than a weighted average of the prior trends. So if you look at it, here's the trend equation, and here again, we see that pattern of Beta and 1 minus Beta. So again, Beta has to be between zero and one. What are we weighting here? Well, on this side of the equation, we're going to weight our past estimate of our trend, and here we're taking the difference between the level of equation of today and the level equation of yesterday. So if it goes up, we can see a trend. If it goes down, this will be negative and we can see a trend, and that will be our weighted average of the trend.