You thought Holt's Exponential Smoothing was, wow, wait until you see what we've got in store for you. This week. and next, we will look at business forecasting methods for when our time series data exhibits a seasonal component. The previous models were appropriate for time series that were horizontal, or had a trend, but these will not be appropriate for a time series that has a seasonal component. Time series may exhibit a seasonal component due to whether holiday periods, weekends, or even accounting reporting periods. Seasonal components typically lead to fluctuations of the time series. The fluctuation pattern is typically repeated for every seasonal cycle. Here is a chart of monthly sales data over a few years. There is an overall long-term trend in the data from start to finish, you can see the data increasing in the long run, and this trend could be due to population growth, or market size. Apart from the overall long-run trend, we can see a regular seasonal fluctuation where the data spikes, known as peaks, occur in December each year. If you examine the chart even closer, you will see that the low points in the data, known as the troughs, also occur on the same month, in this case, February each year.That is, the periodicity is constant. Seasonality can be classified into two broad categories, additive and multiplicative. Additive is when the seasonal fluctuations of the time series can be modeled by addition of a defined seasonal component. The seasonal component size is absolute, you can draw two parallel lines to bind the time series. The seasonal changes are a fixed value over the time series, and the magnitude of seasonal fluctuation does not vary with the level of the series. Multiplicative is when the seasonal fluctuations of the time series can be modeled by multiplication of a defined seasonal component. The seasonal component is relative to the level of the time series. You cannot draw two parallel lines, but you can draw two diverging lines to bind the time series. The seasonal changes are a fixed percentage of the time series, and the magnitude of seasonal fluctuation varies with the level of the time series. If the time series has seasonal components, none of the previous models studied so far will be adequate. We will need to include a seasonal component in our models, or adjust for seasonality when forecasting. An extension of Holt's method can accommodate seasonal effects. This extension of Holt's model for business forecasting is the winters exponential smoothing model. Now, in winters exponential smoothing, we have a third equation to take the seasonality into account, and a third parameter, gamma, which is the weight between zero and one. In the Excel screen flow videos, we will focus on Multiplicative winters exponential smoothing, which works all the time, even when the seasonality is additive as the multiplicative factor is then equal to one. If you want to explore additive winters, you can find resources for this in the toolbox for this week. The level equation is almost exactly the same as for Holt's, except the actual data Y t, is scaled by a seasonal factor, capital S. Lowercase s here represents the number of periods in a year. For monthly data, lowercase s is 12, and for Courtly data, lowercase s is four. Capital S subscript t minus lowercase S is the seasonal factor one year ago. The trend equation is exactly the same as for Holt's exponential smoothing, so that's good news.The third and newest equation, capital S, for the seasonal component, follows the same pattern as the rest. We have gamma times the actual data divided by the level, plus one minus gamma times the seasonal factor from one year ago. To forecast, we bring these three equations together. In f t plus p, p is the number of periods we want to forward forecast. P is like m in Holt's. The first part of the equation is the same as Holt's, then we multiply that entire term by the relevant seasonal factor. Those subscripts for the seasonal factor, capital S, look rather tricky. Don't worry. All that means is, if you are forecasting for December, you need the seasonal factor for December from a year ago. That's it. It's pretty intuitive. The model also needs initialization seed values, for the first value of the level, the first value of the trend, and the first year of seasonal factors. Twelve seasonal seeds, if the data is monthly, four seasonal seeds, if the data is quarterly. As you would have guessed, the values for alpha, beta, and gamma are arbitrary. Typically, you would begin with low values of these parameters no bigger than 0.5. Once the model is properly setup as before, changes to the smoothing parameters may reduce error levels and improved accuracy, usually based on an error criterion like the Mean Squared Error. As with Holt's and simple exponential, the Solver tool can be used to derive the optimum combination of the smoothing parameters. Just as last week, over to the Excel Screen Flow videos. As always, download the relevant Excel workbook, and work alongside me as you watch the video, attempt the quiz for practice and feedback. You also have a practice exercise for this week where you will create your own Winters model, and when you see the model fits so neatly and give you those important forecasts, everyone say wow. Over to Excel.
