ISEN 315
Spring 2011
Dr. Gary Gaukler
Forecasting for Stationary Series
A stationary time series has the form:
Dt = m + e t where m is a constant and e t
is a random variable with mean 0 and
var s2 .
Two common methods for forecasting
stationary series are moving averages
and exponential smoothing.
Moving Averages
In words: the arithmetic average of the n
most recent observations. For a onestep-ahead forecast:
Ft = (1/n) (Dt - 1 + Dt - 2 + . . . + Dt - n )
(Go to Example.)
Exponential Smoothing Method
A type of weighted moving average that applies
declining weights to past data.
1. New Forecast = a (most recent observation)
+ (1 - a) (last forecast)
or
2. New Forecast = last forecast
a (last forecast error)
where 0 < a < 1 and generally is small for
stability of forecasts ( around .1 to .2)
Comparison of ES and MA
• Similarities
– Both methods are appropriate for stationary series
– Both methods depend on a single parameter
– Both methods lag behind a trend
• Differences
–
–
Two-equation Smoothing Model
Add linear trend:
Assume Dt = m + t G + et
St = a Dt + (1-a ) [St-1 + 1 Gt-1],
where Gt -1 = 1-period trend estimate
Two-equation Smoothing Model:
Update G by exponential smoothing:
Gt = b (St - St-1) + (1 - b) Gt-1
Then forecast is:
Ft, t+t = St + t Gt
Example
Demand: 200, 250, 175
Estimates: S0=200, G0=10
Parameters: a= b=0.1
Estimate demand in weeks 4 - 6
Using Regression for Forecasting
(Linear) regression methods can be used when trend
is present
–
Model: Dt = a + bt, or y = a + bx
How do we find the a and b?
Deriving the Regression Parameters
Deriving the Regression Parameters
Deriving the Regression Parameters
Deriving the Regression Parameters
Deriving the Regression Parameters
Using Regression for Forecasting
Least squares estimates for a and b are computed as
follows:
1)
Set Sxx = n2 (n+1)(2n+1)/6 - [n(n+1)/2]2
2)
Set Sxy = n Σ (i Di)- [n(n + 1)/2] Σ Di
3)
Let b = Sxy / Sxx
and a = Davg - b (n+1)/2
Example
Assume demand for periods 1 through 5 is as
follows:
200, 250, 175, 186, 235
What is the regression forecast for period 7?
The Difficulty with Long-Term Forecasts