[MUSIC] Welcome back. This week, we'll be looking at business forecasting for when our Time Series Data exhibits a trend component. We will look at two forecasting methods, Trend Fitting and Holt's Exponential Smoothing. Let's look at trend fitting first. Where the time series has a trend, a trend fitting and extrapolation approach may be used for prediction. Trend fitting is a method where the time series is linked to some function of a time index. The data,Y, is the function of Time, T. Prediction is based on extrapolation by substitution of the appropriate value of the time index by converting our dates to t equals 123 and so on. While a linear trend equation is what we typically assume, there are other options for trends such as exponential or logarithmic. Assuming a linear trend, the equation is Yt equals A plus B times T. Does this equation look familiar? Yes, your high school mathematics teacher was preparing you for this very moment. It's a simple linear equation. Your t equals a time index, and A and B are constants. The values of A and B typically are estimated by regression of the time series, Y, against the time Index, t, in a statistical software package. Excel has numerous alternative ways of estimating these. We will look at one of these methods this week, and we will explore a more rigorous approach in the next course, which is specifically on regression models for business forecasting. Now let's look at Holt's Exponential Smoothing. The two general methods already studied, moving average, MA, and simple exponential smoothing, SCS, are useful when the time series is predominantly horizontal, but will not be good predictors when the Time series has other systematic components. If the Time series has a trend, then MA and SCS will be poor predictors. If you were to try and SCS on data that was trending upwards, you would notice that the forecast consistently underestimates the actual data. This is because the forecast does not take the increasing nature of the data, as SCS cannot account for a trend, but rather assumes that the data is level. A simple extension of the SCS model is the Holt's model. Holt's Exponential Smoothing incorporates a trend component that can be used for better prediction of trending Time series. Like simple exponential smoothing, Holt's Exponential Smoothing uses a smoothing algorithm to remove random influences from the time series, revealing the underlying systematic components. Holt's Exponential Smoothing is characterized by three equations. The first equation looks similar to the simple exponential smoothing equation, except in Holt's Exponential Smoothing that last term is the sum of Lt minus one and capital Tt minus one instead of just Ft minus one like in the SCS method. The second equation follows the same pattern as the first equation. It is a weighted average using a parameter beta whose value is between zero and one. The trend value at time period t is equal to Beta times the difference between the levels, this periods level minus last period's level, plus one minus beta times the previous trend. If we want to forecast one period forward, then we have Ft plus one equals the level in period T plus the trend in period T. If we want to forecast two periods forward, then we have Ft plus two equals the levelling period T plus two times the trend in period T. And so on for three or more periods forward. Just like in SCS, the values of alpha and beta are arbitrarily determined, and just as with SCS, we can try different alphas and betas to determine the optimum combination as assessed by an error criteria such as the mean squared error. Again, the Solver tool in Excel can be used to find the optimum alpha and beta by minimizing the chosen error criteria. When you see the Solver find those optimum parameters, and your beautiful smooth graph and those amazing forecasts, you guessed it, you will say, everyone say Wow. One last thing to note with Holt's Exponential Smoothing. As there are two equations, we will need two seeds, one for each equation. The seed for L is usually the initial observed time series value, the seed for the trend T is the difference between the second observation and the first observation. Although at your discretion and expert judgment, if you have a large time series, you could take the average of this difference and the next difference Taking three observations into account, or the average of three, or even four differences. The seed values actually don't end up making much of a difference in either simple exponential or Holt's forecasting when we have a large sample. So don't worry. The way exponential smoothing works, the weight given to the very first few values is very, very small, and thus the impact of the seed is almost negligible. Okay, you know the drill. Next comes the Excel screen flow videos. After these, you've got quizzes to practice and discussion boards to have a Q&A session with your peers in the course. You've also got an assessment do this week to see how you are going. With practice you will do well. Now over to the Excel screen flow videos. [MUSIC]
