1. DSCI 346 Yamasaki Lecture 7
Forecasting

2. Quantitative Forecasting Models based on Time-Series
Time-Series data is data collected over time where each data point reporesnts a specific point in time.
All time-series exhibit one or more of the following:
Trend component: Long term increase or decrease
Seasonal component: Shorter term repeating pattern
Cyclical component: Longer term upward or downward swings, not as regular are seasonal components
Random component: Unpredictable
DSCI 346 Lecture 7 (19 pages)

3. Example Monthly Revenue of an Office Supply Company
Notice : Overall upward trend
DSCI 346 Lecture 7 (19 pages)

4. Example
Notice : Seasonal trend
DSCI 346 Lecture 7 (19 pages)

5. Forecasting using Smoothing Methods
Smoothing Methods are used to “smooth” out the random component in a time series.
We will discuss 3 methods. These methods work best when there is no significant trend, seasonal, or cyclical component.
Simple Moving Average
Weighted Moving Average
Exponential Smoothing
DSCI 346 Lecture 7 (19 pages)

6. Forecasting using Smoothing Methods
Example: A company looks at the past 10 weeks of potential incoming customer sales calls
Week
Sales Calls 1
400 2
430 3
420 4
440 5
460 6 7
470 8 9 10
DSCI 346 Lecture 7 (19 pages)

7. Simple Moving Average
Here we simply average the prior p values. For our example, let’s use p=3
So our forecast for week 4 would be ( )/3 = 416.7
Week
Sales Calls
Forecast (SMA p=3) 1
400 2
430 3
420 4
440
416.7 5
460
430.0 6
440.0 7
470
446.7 8
456.7 9 10
Forecast for 11
DSCI 346 Lecture 7 (19 pages)

8. Mean Absolute Deviation
One way to measure overall fit is called the Mean Absolute Deviation (MAD)
Where yt is actual value at time t and Ft is forecasted value at time t
Week
Sales Calls
Forecast (SMA p=3)
Absolute Error 1
400 2
430 3
420 4
440
416.7
23.33 5
460
430.0
30.00 6
440.0
0.00 7
470
446.7 8
456.7
26.67 9
6.67 10
MAD
19.52
DSCI 346 Lecture 7 (19 pages)

9. Comparing models using MAD
Week
Sales Calls
Forecast (SMA p=3)
Absolute Error
Forecast (SMA p=4) 1
400 2
430 3
420 4
440
416.7
23.33 5
460
430.0
30.00
422.5
37.50 6
440.0
0.00
437.5
2.50 7
470
446.7 8
456.7
26.67
452.5
22.50 9
6.67
450.0
10.00 10
445.0
25.00
MAD
19.52
21.25
DSCI 346 Lecture 7 (19 pages)

10. Weighted Moving Average
Here we weight more recent values. For our example, let’s use p=3 with weights 3 for most recent, 2 for next and 1 for furthest
So our forecast for week 4 would be ((1)*400+(2)*430+(3)*420)/(1+2+3) = 420
Week
Sales Calls
Forecast (WMA p3)
Absolute Error 1
400 2
430 3
420 4
440
420.00
20.00 5
460
431.67
28.33 6
446.67
6.67 7
470
23.33 8
458.33 9
445.00
5.00 10
441.67
21.67
MAD
19.05
DSCI 346 Lecture 7 (19 pages)

11. Exponential Smoothing
Another way to weight more recent events is called exponential smoothing. In this methodology, the present forecast uses the prior forecast plus some portion of the prior forecast’s forecasting error:
Ft = F t-1 + a(At-1 – Ft-1)
Where Ft is the forecast for time t, At is the actual for time t , a is the smoothing factor
F1 = A1
DSCI 346 Lecture 7 (19 pages)

12. Back to our sales calls example
Ft = F t-1 + a(At-1 – Ft-1)
Let a = 0.4
F1 = A1 = 400
F2 = F 1 + (.4)(A1 – F1) = (.4)*( ) = 400
F3 = F 2 + (.4)(A2 – F2) = (.4)*( ) = = 412
F4 = F 3 + (.4)(A3 – F3) = (.4)*( ) = = 415.2
DSCI 346 Lecture 7 (19 pages)

13. Week Sales Calls Forecast (ES a=.4) Absoute Error 1 400 400.00 2 4302
430
30.00 3
420
412.00
8.00 4
440
415.20
24.80 5
460
425.12
34.88 6
439.07
0.93 7
470
439.44
30.56 8
451.67
21.67 9
443.00
3.00 10
441.80
21.80
Forecast for 11
433.08
MAD
19.51
DSCI 346 Lecture 7 (19 pages)

14. Smoothing when trend exists
Exponential Smoothing with Trend Adjustment Forecast (also called double exponential smoothing)
FITt = Ft + At
Ft = FITt-1 + a(At-1 – FITt-1)
Tt = b(Ft – Ft-1) + (1-b)Tt-1
Where FITt is the forecast including trend for time t
Ft is the exponentially smoothed forecast for time t
Tt is the exponentially smoothed trend for time t
At-1 is the actual value for time t-1
a is the smoothing factor for the forecast
b is the smoothing factor for the trend
F1 =A1
T1 = 0
DSCI 346 Lecture 7 (19 pages)

15. Forecasting trend with Regression Analysis
Week
Sales Calls
regression
Absoute Error 1
400
24.91 2
430
2.85 3
420
9.40 4
440
8.36 5
460
433.88
26.12 6
3.88 7
470
31.64 8
10.61 9 10
445.09
25.09
Forecast for 11
MAD
14.57
DSCI 346 Lecture 7 (19 pages)

16. Autocorrelation
Recall one of the assumptions of regression independence of residuals. When residuals are not independent, the condition is called autocorrelation
DSCI 346 Lecture 7 (19 pages)

17. Durbin-Watson statistic
Where et is the residual for time t
Week Calls Regression et et (et – et-1)2 1
400 2
430
2.846 3
420
-9.396 4
440
8.362 5
460
433.88
26.12 6
3.878 7
470
31.636 8 9
-2.848 10
445.09
-25.09 S
-0.01
DSCI 346 Lecture 7 (19 pages)

18. K = number of independent variables (1 in our example)
Table 9 Appendix A
K = number of independent variables (1 in our example)
If d < dl conclude positive autocorrelation
If dl < d< du test is inconclusive
If d > du conclude no positive autocorrelation
For our example n = 10 , a = So n is too small to run test
DSCI 346 Lecture 7 (19 pages)

19. Dealing with seasonality
Methods exist, e.g., Multiplicative Time Series Model
Beyond scope of class
DSCI 346 Lecture 7 (19 pages)