Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis
6.4.3. What is Exponential Smoothing?
|Exponential smoothing weights past observations with exponentially decreasing weights to forecast future values||
This smoothing scheme begins by setting S2 to
y1, where Si stands for smoothed
observation or EWMA, and y stands for the original observation.
The subscripts refer to the time periods, 1, 2, ..., n. For
the third period, S3 =
S2; and so on. There is no S1;
the smoothed series starts with the smoothed version of the second
For any time period t, the smoothed value St is found by computing
Note: There is an alternative approach to exponential smoothing that replaces yt-1 in the basic equation with yt, the current observation. That formulation, due to Roberts (1959), is described in the section on EWMA control charts. The formulation here follows Hunter (1986).
|Setting the first EWMA|
|The first forecast is very important||
The initial EWMA plays an important role in computing all the
subsequent EWMA's. Setting S2 to
y1 is one method of initialization. Another way
is to set it to the target of the process.
Still another possibility would be to average the first four or five observations.
It can also be shown that the smaller the value of , the more important is the selection of the initial EWMA. The user would be wise to try a few methods, (assuming that the software has them available) before finalizing the settings.
|Why is it called "Exponential"?|
|Expand basic equation||
Let us expand the basic equation by first substituting for
St-1 in the basic equation to obtain
= yt-1 + (1-) yt-2 + (1-)2 St-2
|Summation formula for basic equation||
By substituting for St-2, then for
St-3, and so forth, until we reach
S2 (which is just y1), it can be
shown that the expanding equation can be written as:
|Expanded equation for S5||
For example, the expanded equation for the smoothed value
|Illustrates exponential behavior||
This illustrates the exponential behavior. The weights,
t decrease geometrically, and their sum is unity
as shown below, using a property of geometric series:
|Example for = .3||
Let = .3.
Observe that the weights
t decrease exponentially (geometrically)
|What is the "best" value for ?|
|How do you choose the weight parameter?||
The speed at which the older responses are dampened (smoothed) is a
function of the value of
is close to 1, dampening is quick and when
is close to 0, dampening is slow. This is illustrated in the
We choose the best value for so the value which results in the smallest MSE.
Let us illustrate this principle with an example. Consider the
following data set consisting of 12 observations taken over time:
The sum of the squared errors (SSE) = 208.94. The mean of the squared errors (MSE) is the SSE /11 = 19.0.
|Calculate for different values of||The MSE was again calculated for = .5 and turned out to be 16.29, so in this case we would prefer an of .5. Can we do better? We could apply the proven trial-and-error method. This is an iterative procedure beginning with a range of between .1 and .9. We determine the best initial choice for and then search between - and + . We could repeat this perhaps one more time to find the best to 3 decimal places.|
|Nonlinear optimizers can be used||But there are better search methods, such as the Marquardt procedure. This is a nonlinear optimizer that minimizes the sum of squares of residuals. In general, most well designed statistical software programs should be able to find the value of that minimizes the MSE.|
|Sample plot showing smoothed data for 2 values of|