Exponential smoothing

Exponential smoothing is a forecasting technique that uses a weighted average of past time series values to compute a smoothed value that serves as a forecast for the next period in the time series. The simple exponential smoothing model that we consider here is based on the assumption that there is no significant trend or seasonal effect in the data.

Suppose that a time series is given. Our aim is to predict the unknown future values .The simple exponential smoothing method of forecasting can be used to predict future values of

To apply this method we accept

.

Then use ;

;

where is a smoothing constant whose value is fixed between 0

and 1. And standing at time , we obtain forecasts of future values, of the series by

;

Example:

The following table shows the price of a share of common stock for a well-known computer firm over the past 8 weeks. The price shown is the closing price on the same day of the week for 8 consecutive weeks.

Week	Stock price

Use the method of simple exponential smoothing to obtain forecasts of stock price over the next three weeks. Use a smoothing constant of . Graph the observed time series and the forecasts.

Solution:

As mentioned above, we let the smoothed value of the time series for the first period equal the actual first value of the time series. So,

To illustrate the nature of the computation, we will use a smoothing constant of . Thus the smoothed value for period two becomes

The smoothed value for period 3 becomes

Continuing this process in the end we obtain the following complete set of smooth values shown in the table:

Week	Stock price	Smoothed time series value
		50.00 51.80 50.12 50.05 45.22 52.29 52.12 55.05

Now, let us use the results of exponential smoothing to develop a forecast of the stock price for the and weeks. The assumption of the simple exponential smoothing is that the smoothed value of the time series at one period provides the best estimate of the time series for the next periods. Thus, the simple exponential smoothing model forecast of the stock price for the following 3 weeks is $55.05.

; ;

Figure 7.2 shows the plot of smoothed values for the time series.

Remark:

To use MINITAB menu follow the following instructions

1. Select Stat>Time series>Single exponential smoothing

2. Enter time series variable (for example, C1)

3. Select

4. Insert

5. Select “generate forecasts”

6. Number of forecasts: Insert an integer to indicate number of forecasts you want.

7. Starting from origin: Enter a positive integer to specify a starting point for the forecasts. For example, if you specify 4 forecasts and 10 for the origin, Minitab computes forecasts for periods 11, 12, 13, and 14, based on the level and trend components at period 10. If you leave this space blank, Minitab generates forecasts from the end of the data.

8. Select Options

9. Select graphics, outputs

10. Enter 1 to the window “Use average of __ observations”

11. Click OK.

Дата добавления: 2015-08-05