In this video, learn how smoothing uses errors in prior forecasts to improve subsequent forecasts.
- [Narrator] Many approaches to quantitative forecasting begin with this sort of data. Particularly the date, or time, in column A, and the quantity that you're forecasting in column B. Together, those two columns make up what's called the baseline. You apply the formulas to that baseline in order to generate your forecasts. Formulas that you use in exponential smoothing take on one of two basic forms: error correction and smoothing. Those two forms are mathematically equivalent, and return precisely the same results.
We'll look at both in this course. Let's start here by looking at the error correction form. That form uses four components. The letter Y usually represents your actual observations. The letter T usually represents the date or time on which the observation took place. So for time one, which is shown in row two, the value of Y is 2,008. You had $2,008 worth of sales at time one. The letter Y with a caret over it is usually pronounced "Y hat." Y hat has a subscript, and on this screen, the subscript is either T+1 or just T.
If you have an observation for each day, then each value of T represents a different day. So if the value of T is three, Y hat T might represent your forecast for Wednesday, and Y hat T+1 would represent your forecast for Thursday. There's also the smoothing constant, often symbolized as the Greek letter Alpha. The smoothing constant is under your control. You can set its value to whatever you want, but it is usually the same value throughout the entire forecast.
I'll show you how to optimize the smoothing constant in later videos. Finally, we have the forecast error, represented by the Greek symbol Epsilon. It is simply the difference between the observation at time T, and the forecast for time T. So for example at time two on the screen, we observed 1,857 sales dollars, but we had forecast 1,954 sales dollars. So we have an error of minus 97 dollars, the result of subtracting the forecast 1954 from the actual 1857.
We symbolize the next forecast value as Y hat sub T+1. To calculate the next forecast, we start with Y hat sub T, which is the forecast for time T. Then we take the smoothing constant, or Alpha, and multiply it by the forecast error at time T. We add the forecast value, Y hat sub T, to the product of Alpha times the error at time T. The result of the addition is the forecast for time T+1.
In the context of the worksheet shown at the bottom of the screen, our forecast for time two appears in cell C3, a forecast of 1954. That's the result of adding the value in C2, the forecast for time T, to the product of the smoothing constant in cell G1 times the error at time one. The forecast formula is shown as the formula for time two in cell E3. You can verify that the formula in cell E3 adds the prior forecast of 1941 to the product of the smoothing constant in cell G1 times the $67 error in the prior forecast.
The error correction form of the smoothing equation emphasizes the fact that each forecast is self-correcting. That is, if the prior forecast was too low, the next forecast is adjusted up a bit, and vice versa. Here's how that works in the example shown in the worksheet. At time one in row two, our actual observation was 2,008. But we forecast 1941. Our forecast was too low by $67, as shown in cell D2.
For the forecast at time two, in row three, we add the prior forecast of 1941 to the smoothing constant of 0.2 times the prior error of 67. That results in 1941 plus 13, or 1954, as shown in cell C3. So we've adjusted our forecast for time two by an amount that depends in part on the amount of forecast error at time one. And in a direction that would have improved that time one forecast.
Similarly, our forecast of 1954 for time two turns out to be an overestimate. It is $97 too high. So for time three, our forecast will be the prior forecast of 1954, plus the smoothing constant, 0.2, times the prior error, which is -97. That is, for time three, we had 1954 plus -20, which is approximately .2 times -97.
Again, our forecast for time two was too high, so we wind up subtracting a portion of the forecast error in creating our next forecast for time three. Now that you know how self-corrections work, let's take a look at the equation in its smoothing form.
- Using correlograms to identify the nature of a baseline
- Assembling the forecast equation
- Methods of identifying the first forecast
- Getting a measure of overall forecast accuracy
- Optimizing a smoothing constant by minimizing RMSE