Posts Tagged ‘double exponential smoothing’

Forecast Friday Topic: Double Exponential Smoothing

May 20, 2010

(Fifth in a series)

We pick up on our discussion of exponential smoothing methods, focusing today on double exponential smoothing. Single exponential smoothing, which we discussed in detail last week, is ideal when your time series is free of seasonal or trend components, which create patterns that your smoothing equation would miss due to lags. Single exponential smoothing produces forecasts that exceed actual results when the time series exhibits a decreasing linear trend, and forecasts that trail actual results when the time series exhibits an increasing trend. Double exponential smoothing takes care of this problem.

Two Smoothing Constants, Three Equations

Recall the equation for single exponential smoothing:

Ŷt+1 = αYt + (1-α) Ŷt

Where: Ŷt+1 represents the forecast value for period t + 1

Yt is the actual value of the current period, t

Ŷt is the forecast value for the current period, t

and α is the smoothing constant, or alpha, 0≤ α≤ 1

To account for a trend component in the time series, double exponential smoothing incorporates a second smoothing constant, beta, or β. Now, three equations must be used to create a forecast: one to smooth the time series, one to smooth the trend, and one to combine the two equations to arrive at the forecast:

Ct = αYt + (1-α)(Ct-1 + T t-1)

Tt = β(Ct - Ct-1) + (1 – β)T t-1

Ŷt+1 = Ct + Tt

All symbols appearing in the single exponential smoothing equation represent the same in the double exponential smoothing equation, but now β is the trend-smoothing constant (whereas α is the smoothing constant for a stationary – constant – process) also between 0 and 1; Ct is the smoothed constant process value for period t; and Tt is the smoothed trend value for period t.

As with single exponential smoothing, you must select starting values for Ct and Tt, as well as values for α and β. Recall that these processes are judgmental, and constants closer to a value of 1.0 are chosen when less smoothing is desired (and more weight placed on recent values) and constants closer to 0.0 when more smoothing is desired (and less weight placed on recent values).

An Example

Let’s assume you’ve got 12 months of sales data, shown in the table below:

Month t

Sales Yt

1

152

2

176

3

160

4

192

5

220

6

272

7

256

8

280

9

300

10

280

11

312

12

328

You want to see if there is any discernable trend, so you plot your sales on the chart below:

The time series exhibits an increasing trend. Hence, you must use double exponential smoothing. You must first select your initial values for C and T. One way to do that is to again assume that the first value is equal to its forecast. Using that as the starting point, you set C2 = Y1, or 152. Then you subtract Y1 from Y2 to get T2: T2 = Y2 – Y1 = 24. Hence, at the end of period 2, your forecast for period 3 is 176 (Ŷ3 = 152 + 24).

Now you need to choose α and β. For the purposes of this example, we will choose an α of 0.20 and a β of 0.30. Actual sales in period 3 were 160, and our constant-smoothing equation is:

C3 = 0.20(160) + (1 – 0.20)(152 + 24)

= 32 + 0.80(176)

= 32 + 140.8

= 172.8

Next, we compute the trend value with our trend-smoothing equation:

T3 = 0.30(172.8 – 152) + (1 – 0.30)(24)

= 0.30(20.8) + 0.70(24)

= 6.24 + 16.8

=23.04

Hence, our forecast for period 4 is:

Ŷ4 = 172.8 + 23.04

= 195.84

Then, carrying out your forecasts for the 12-month period, you get the following table:

     

Alpha=

0.2

Beta=

0.3

Month t

Sales Yt

Ct

Tt

Ŷt

Absolute Deviation

1

152

       

2

176

152.00

24.00

152.00

 

3

160

172.80

23.04

176.00

16.00

4

192

195.07

22.81

195.84

3.84

5

220

218.31

22.94

217.88

2.12

6

272

247.39

24.78

241.24

30.76

7

256

268.94

23.81

272.18

16.18

8

280

290.20

23.05

292.75

12.75

9

300

310.60

22.25

313.25

13.25

10

280

322.28

19.08

332.85

52.85

11

312

335.49

17.32

341.36

29.36

12

328

347.85

15.83

352.81

24.81

       

MAD=

20.19

 

Notice a couple of things: the absolute deviation is the absolute value of the difference between Yt (shown in lavender) and Ŷt (shown in light blue). Note also that beginning with period 3, Ŷ3 is really the sum of C and T computed in period 2. That’s because period 3’s constant and trend forecasts were generated at the end of period 2 – and onward until period 12. Mean Absolute Deviation has been computed for you. As with our explanation of single exponential smoothing, you need to experiment with the smoothing constants to find a balance that most accurate forecast at the lowest possible MAD.

Now, we need to forecast for period 13. That’s easy. Add C12 and T12:

Ŷ13 = 347.85 + 15.83

= 363.68

And, your chart comparing actual vs. forecasted sales is:

As with single exponential smoothing, you see that your forecasted curve is smoother than your actual curve. Notice also how small the gaps are between the actual and forecasted curves. The fit’s not bad.

Exponential Smoothing Recap

Now let’s recap our discussion on exponential smoothing:

  1. Exponential smoothing methods are recursive, that is, they rely on all observations in the time series. The weight on each observation diminishes exponentially the more distant in the past it is.
  2. Smoothing constants are used to assign weights – between 0 and 1 – to the most recent observations. The closer the constant is to 0, the more smoothing that occurs and the lighter the weight assigned to the most recent observation; the closer the constant is to 1, the less smoothing that occurs and the heavier the weight assigned to the most recent observation.
  3. When no discernable trend is exhibited in the data, single exponential smoothing is appropriate; when a trend is present in the time series, double exponential smoothing is necessary.
  4. Exponential smoothing methods require you to generate starting forecasts for the first period in the time series. Deciding on those initial forecasts, as well as on the values of your smoothing constants – alpha and beta – are arbitrary. You need to base your judgments on your experience in the business, as well as some experimentation.
  5. Exponential smoothing models do not forecast well when the time series pattern (e.g., level of sales) is suddenly, drastically, and permanently altered by some event or change of course or action. In these instances, a new model will be necessary.
  6. Exponential smoothing methods are best used for short-term forecasting.

Next Week’s Forecast Friday Topic: Regression Analysis (Our Series within the Series!)

Next week, we begin a multi-week discussion of regression analysis. We will be setting up the next few weeks with a discussion of the principles of ordinary least squares regression (OLS), and then discussions of its use as a time-series forecasting approach, and later as a causal/econometric approach. During the course of the next few Forecast Fridays, we will discuss the issues that occur with regression: specification bias, autocorrelation, heteroscedasticity, and multicollinearity, to name a few. There will be some discussions on how to detect – and correct – these violations. Once the regression analysis miniseries is complete, we will be set up to discuss ARMA and ARIMA models, which will be written by guest bloggers who are well-experienced in those approaches. We know you’ll be very pleased with the weeks ahead!

Still don’t know why our Forecast Friday posts appear on Thursday? Find out at: http://tinyurl.com/26cm6ma

Forecast Friday Topic: Exponential Smoothing Methods

May 13, 2010

(Fourth in a series)

In last week’s Forecast Friday post, we discussed moving average forecasting methods, both simple and weighted. When a time series is stationary, that is, exhibits no discernable trend or seasonality and is subject only to the randomness of everyday existence, then moving average methods – or even a simple average of the entire series – are useful for forecasting the next few periods. However, most time series are anything but stationary: retail sales have trend, seasonal, and cyclical elements, while public utilities have trend and seasonal components that impact the usage of electricity and heat. Hence, moving average forecasting approaches may provide less than desirable results. Moreover, the most recent sales figures typically are more indicative of future sales, so there is often a need to have a forecasting system that places greater weight on more recent observations. Enter exponential smoothing.

Unlike moving average models, which use a fixed number of the most recent values in the time series for smoothing and forecasting, exponential smoothing incorporates all values time series, placing the heaviest weight on the current data, and weights on older observations that diminish exponentially over time. Because of the emphasis on all previous periods in the data set, the exponential smoothing model is recursive. When a time series exhibits no strong or discernable seasonality or trend, the simplest form of exponential smoothing – single exponential smoothing – can be applied. The formula for single exponential smoothing is:

Ŷt+1 = αYt + (1-α) Ŷt

In this equation, Ŷt+1 represents the forecast value for period t + 1; Yt is the actual value of the current period, t; Ŷt is the forecast value for the current period, t; and α is the smoothing constant, or alpha, a number between 0 and 1. Alpha is the weight you assign to the most recent observation in your time series. Essentially, you are basing your forecast for the next period on the actual value for this period, and the value you forecasted for this period, which in turn was based on forecasts for periods before that.

Let’s assume you’ve been in business for 10 weeks and want to forecast sales for the 11th week. Sales for those first 10 weeks are:

Week (t)

Sales (Yt)

1

200

2

215

3

210

4

220

5

230

6

220

7

235

8

215

9

220

10

210

From the equation above, you know that in order to come up with a forecast for week 11, you need forecasted values for weeks 10, 9, and all the way down to week 1. You also know that week 1 does not have any preceding period, so it cannot be forecasting. And, you need to determine the smoothing constant, or alpha, to use for your forecasts.

Determining the Initial Forecast

The first step in constructing your exponential smoothing model is to generate a forecast value for the first period in your time series. The most common practice is to set the forecasted value of week 1 equal to the actual value, 200, which we will do in our example. Another approach would be that if you have prior sales data to this, but are not using it in your construction of the model, you might take an average of a couple of immediately prior periods and use that as the forecast. How you determine your initial forecast is subjective.

How Big Should Alpha Be?

This too is a judgment call, and finding the appropriate alpha is subject to trial and error. Generally, if your time series is very stable, a small α is appropriate. Visual inspection of your sales on a graph is also useful in trying to pinpoint an alpha to start with. Why is the size of α important? Because the closer α is to 1, the more weight that is assigned to the most recent value in determining your forecast, the more rapidly your forecast adjusts to patterns in your time series and the less smoothing that occurs. Likewise, the closer α is to 0, the more weight that is placed on earlier observations in determining the forecast, the more slowly your forecast adjusts to patterns in the time series, and the more smoothing that occurs. Let’s visually inspect the 10 weeks of sales:

The Exponential Smoothing Process

The sales appear somewhat jagged, oscillating between 200 and 235. Let’s start with an alpha of 0.5. That gives us the following table:

Week (t)

Sales (Yt)

Forecast for This Period (Ŷt)

1

200

200.0

2

215

200.0

3

210

207.5

4

220

208.8

5

230

214.4

6

220

222.2

7

235

221.1

8

215

228.0

9

220

221.5

10

210

220.8

Notice how, even though your forecasts aren’t precise, when your actual value for a particular week is higher than what you forecasted (weeks 2 through 5, for example), your forecasts for each of the subsequent weeks (weeks 3 through 6) adjust upward; when your actual values are lower than your forecast (e.g., weeks 6, 8, 9, and 10), your forecasts for the following week adjusts downward. Also notice that, as you move to later periods, your earlier forecasts play less and less of a role in your later forecasts, as their weight diminishes exponentially. Just by looking at the table above, you know that the forecast for week 11 will be lower than 220.8, your forecast for week 10:

Ŷ11 = 0.5Y10 + (1-0.5) Ŷ10

= 0.5(210) + 0.5(220.8)

= 105 + 110.4

=215.4

So, based on our alpha and our past sales, our best guess is that sales in week 11 will be 215.4. Take a look at the graph of actual vs. forecasted sales for weeks 1-10:

Notice that the forecasted sales are smoother than actual, and you can see how the forecasted sales line adjusts to spikes and dips in the actual sales time series.

What if we Had Used a Smaller or Larger Alpha?

We’ll demonstrate by using both an alpha of .30 and one of .70. That gives us the following table and graph:

Week (t)

Sales (Yt)

Forecast α=0.50

Forecast α=0.30

Forecast α=0.70

1

200

200.0

200.0

200.0

2

215

200.0

200.0

200.0

3

210

207.5

204.5

210.5

4

220

208.8

206.2

210.2

5

230

214.4

210.3

217.0

6

220

222.2

216.2

226.1

7

235

221.1

217.3

221.8

8

215

228.0

222.6

231.1

9

220

221.5

220.4

219.8

10

210

220.8

220.2

219.9

 

As you can see, the smaller the α, the smoother the curve for forecasted sales; the larger the α, the bumpier the curve, as you can see as you move from .30 to .50 to .70. Notice how much faster an α of .70 adjusts to the actual sales than the smaller α’s. The forecasts for week 11 would be 217.2 with an α=.30 and 213 with an α=.70.

Which α is best?

As with moving average models, the Mean Absolute Deviation (MAD) can be used to determining which alpha best fits the data. The MADs for each alpha are computed below:

Week

Absolute Deviations

α=.30

α=.50

α=.70

1

-

-

-

2

15.0

15.0

15.0

3

5.5

2.5

0.5

4

13.9

11.3

9.8

5

19.7

15.6

13.0

6

3.8

2.2

6.1

7

17.7

13.9

13.2

8

7.6

13.0

16.1

9

0.4

1.5

0.2

10

10.2

10.8

9.9

MAD=

9.4

8.6

8.4

 

Using an alpha of 0.70, we end up with the lowest MAD of the three constants. Keep in mind that judging the dependability of forecasts isn’t always about minimizing MAD. MAD, after all, is an average of deviations. Notice how dramatically the absolute deviations for each of the alphas change from week to week. Forecasts might be more reliable using an alpha that produces a higher MAD, but has less variance among its individual deviations.

Limits on Exponential Smoothing

Exponential smoothing is not intended for long-term forecasting. Usually it is used to predict one or two, but rarely more than three periods ahead. Also, if there is a sudden drastic change in the level of sales or values, and the time series continues at that new level, then the algorithm will be slow to catch up with the sudden change. Hence, there will be greater forecasting error. In situations like that, it would be best to ignore the previous periods before the change, and begin the exponential smoothing process with the new level. Finally, this post discussed single exponential smoothing, which is used when there is no noticeable seasonality or trend in the data. When there is a noticeable trend or seasonal pattern in the data, single exponential smoothing will yield significant forecast error. Double exponential smoothing is needed here to adjust for those patterns. We will cover double exponential smoothing in next week’s Forecast Friday post.

Still don’t know why our Forecast Friday posts appear on Thursday? Find out at: http://tinyurl.com/26cm6ma

 

 

 


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