Moving average and exponential smoothing models
This example teaches you how to apply exponential smoothing to a time series in Excel. Exponential Click in the Damping factor box and type Literature. Tutorial on how to do simple exponential smoothing in Excel. similar to that shown in Figure 2 of Simple Moving Average, except that a Damping Factor field is. Exponential smoothing is a rule of thumb technique for smoothing time series data using the . The time constant of an exponential moving average is the amount of time for the .. IBM SPSS includes Simple, Simple Seasonal, Holt's Linear Trend, Brown's Linear Trend, Damped Trend, Winters' Additive, External links.
Simple Exponential Smoothing
Unfortunately, there is no underlying statistical theory that tells us how the confidence intervals ought to widen for this model. For example, you could set up a spreadsheet in which the SMA model would be used to forecast 2 steps ahead, 3 steps ahead, etc.
You could then compute the sample standard deviations of the errors at each forecast horizon, and then construct confidence intervals for longer-term forecasts by adding and subtracting multiples of the appropriate standard deviation.
If we try a 9-term simple moving average, we get even smoother forecasts and more of a lagging effect: If we take a term moving average, the average age increases to Notice that, indeed, the forecasts are now lagging behind turning points by about 10 periods. Which amount of smoothing is best for this series? Here is a table that compares their error statistics, also including a 3-term average: Model C, the 5-term moving average, yields the lowest value of RMSE by a small margin over the 3-term and 9-term averages, and their other stats are nearly identical.
So, among models with very similar error statistics, we can choose whether we would prefer a little more responsiveness or a little more smoothness in the forecasts.
Exponential Smoothing Explained.
Return to top of page. Brown's Simple Exponential Smoothing exponentially weighted moving average The simple moving average model described above has the undesirable property that it treats the last k observations equally and completely ignores all preceding observations. Intuitively, past data should be discounted in a more gradual fashion--for example, the most recent observation should get a little more weight than 2nd most recent, and the 2nd most recent should get a little more weight than the 3rd most recent, and so on.
The simple exponential smoothing SES model accomplishes this. One way to write the model is to define a series L that represents the current level i. The value of L at time t is computed recursively from its own previous value like this: The forecast for the next period is simply the current smoothed value: Equivalently, we can express the next forecast directly in terms of previous forecasts and previous observations, in any of the following equivalent versions.
In the first version, the forecast is an interpolation between previous forecast and previous observation: The interpolation version of the forecasting formula is the simplest to use if you are implementing the model on a spreadsheet: This is not supposed to be obvious, but it can easily be shown by evaluating an infinite series. For a given average age i.
Another important advantage of the SES model over the SMA model is that the SES model uses a smoothing parameter which is continuously variable, so it can easily optimized by using a "solver" algorithm to minimize the mean squared error. The long-term forecasts from the SES model are a horizontal straight line, as in the SMA model and the random walk model without growth.
However, note that the confidence intervals computed by Statgraphics now diverge in a reasonable-looking fashion, and that they are substantially narrower than the confidence intervals for the random walk model.
The SES model assumes that the series is somewhat "more predictable" than does the random walk model. For example, if you fit an ARIMA 0,1,1 model without constant to the series analyzed here, the estimated MA 1 coefficient turns out to be 0.
It is possible to add the assumption of a non-zero constant linear trend to an SES model. The long-term forecasts will then have a trend which is equal to the average trend observed over the entire estimation period. You cannot do this in conjunction with seasonal adjustment, because the seasonal adjustment options are disabled when the model type is set to ARIMA.
However, you can add a constant long-term exponential trend to a simple exponential smoothing model with or without seasonal adjustment by using the inflation adjustment option in the Forecasting procedure.
When we start an exponential smoothing calculation, we need to manually plug the value for the 1st forecast.
So in Cell B4, rather than a formula, we just typed in the demand from that same period as the forecast. So each subsequent exponential smoothing calculation inherits the output of the previous exponential smoothing calculation.
It will draw connector lines to the 1st level of precedents, but if you keep clicking Trace Precedents it will draw connector lines to all previous periods to show you the inherited relationships.
Figure 1B Figure 1B shows a line chart of our demand and forecast. You case see how the exponentially smoothed forecast removes most of the jaggedness the jumping around from the weekly demand, but still manages to follow what appears to be an upward trend in demand.
Any time you use smoothing when a trend is present; your forecast will lag behind the trend. This is true for any smoothing technique. In fact, if we were to continue this spreadsheet and start inputting lower demand numbers making a downward trend you would see the demand line drop, and the trend line move above it before starting to follow the downward trend.
Exponential Smoothing Explained
There is a lot more to forecasting than just smoothing out the bumps in demand. We need to make additional adjustments for things like trend lag, seasonality, known events that may effect demand, etc. But all that is beyond the scope of this article. You will likely also run into terms like double-exponential smoothing and triple-exponential smoothing. These terms represent using exponential smoothing on additional elements of the forecast. So with simple exponential smoothing, you are smoothing the base demand, but with double-exponential smoothing you are smoothing the base demand plus the trend, and with triple-exponential smoothing you are smoothing the base demand plus the trend plus the seasonality.