Techniques for Seasonality
• Seasonal variations
– Regularly repeating movements in series values
that can be tied to recurring events.
• Seasonal relative
– Percentage of average or trend
• Centered moving average
– A moving average positioned at the center of the
data that were used to compute it.
1
Seasonality
• Short-term cyclical, not
random, change in
demand
• Measurement:
– Relative to overall
amount
– Seasonal Index
– Estimated using
Centered Moving
Average (CMA)
ActualDemand
SI 
UnderlyingDemand
Question: What does SI tell us
about the season if SI = 1.2? 0.8?
3-2
Seasonality
Month
1
2
3
4
5
6
Season
1
2
3
Sum
Sales
40
46
42
41
49
44
CMA(3) Relatives
42.67
43.00
44.00
44.67
Normalized
SI
SI
0.93
0.93
1.09
1.09
0.98
0.98
3.00
3.00
1.08
0.98
0.93
1.10
Season#
2
3
1
2
Average of
months 2 & 5
3-3
Seasonality
Even number of periods per cycle:
1. Average periods 1-4;
2. Average periods 2-5;
3. Average the 2 averages.
Avg of 2
averages
centered at 3
Avg of 1-4
centered
between 2-3
1
Avg of 2-5
centered
between 3-4
2
3
4
5
6
3-4
Seasonality
Quarter
Quarter
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
Sales
Sales
40
40
46
46
42
42
38
38
44
44
49
49
47
47
42
42
47
47
54
54
51
51
45
45
CMA(4)
CMA(4)
41.50
41.5
42.50
42.5
43.25
43.25
44.50
44.5
45.50
45.5
46.25
46.25
47.50
47.5
48.50
48.5
49.25
49.25
CMA(2)
CMA(2)
42.00
42
42.88
42.875
43.88
43.875
45.00
45
45.88
45.875
46.88
46.875
48.00
48
48.88
48.875
Relatives Season#
Relatives Season#
1.00
1
0.89
0.886297
1.00
1.002849
1.09
1.088889
1.02
1.024523
0.90
0.896
0.98
0.979167
1.10
1.104859
1
1
2
2
3
3
4
4
1
1
2
2
3
3
4
4
1
1
2
2
3
3
4
4
3-5
Seasonality: Estimate SI’s
Season
1
2
3
4
Sum
SI
0.99
1.10
1.01
0.89
3.99
Normalized
SI
0.99
1.10
1.01
0.89
4.00
3-6
Seasonality: Deseasonalize
demand
Quarter
Sales
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
40
46
42
38
44
49
47
42
47
54
51
45
Deseason
Normalized alized
Season# SI
Demand
Trend
Forecast
1
0.99
40.28
40.16
39.89
2
1.10
41.85
41.13
45.21
3
1.01
0.00
4
0.89
0.00
1
0.00
2
0.00
3
0.00
4
0.00
1
0.00
2
0.00
3
0.00
4
0.00
1
0.00
2
0.00
3
0.00
3-7
Seasonality: Deseasonalize
demand
Quarter
Sales
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Season#
40
46
42
38
44
49
47
42
47
54
51
45
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
Normalized
SI
0.99
1.10
1.01
0.89
0.99
1.10
1.01
0.89
0.99
1.10
1.01
0.89
0.99
1.10
1.01
Deseason
alized
Demand
Trend
Forecast
40.28
40.16
39.89
41.85
41.13
45.21
41.40
42.09
42.70
42.55
43.05
38.45
44.30
44.01
43.71
44.58
44.97
49.43
46.33
45.93
46.60
47.03
46.89
41.88
47.32
47.85
47.53
49.12
48.82
53.66
50.27
49.78
50.50
50.39
50.74
45.31
51.70
51.35
52.66
57.89
53.62
54.40
3-8
Seasonality: Estimate trend
SUMMARY OUTPUT Deseasonalized Data
Regression Statistics
Multiple R 0.990982
R Square 0.982045
0.980249
Adjusted R Square
0.491529
Standard Error
12
Observations
ANOVA
df
Regression
Residual
Total
Significance F
F
MS
SS
1 132.1412 132.1412 546.9401 4.63E-10
10 2.416009 0.241601
11 134.5572
Standard Error t Stat
Coefficients
129.589
39.20265 0.302515
Intercept
X Variable 10.961283 0.041104 23.38675
P-value Lower 95% Upper 95%
39.8767
1.84E-17 38.52861
4.63E-10 0.869698 1.052868
3-9
Seasonality: Forecast
Quarter
Sales
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
40
46
42
38
44
49
47
42
47
54
51
45
Deseason
Normalized alized
Season# SI
Demand
Trend
Forecast
1
0.99
40.28
40.16
39.89
2
1.10
41.85
41.13
45.21
3
1.01
0.00
4
0.89
0.00
1
0.00
2
0.00
3
0.00
4
0.00
1
0.00
2
0.00
3
0.00
4
0.00
1
0.00
2
0.00
3
0.00
4
0.00
3-10
Seasonality: Forecast
Quarter
Sales
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Season#
40
46
42
38
44
49
47
42
47
54
51
45
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Normalized
SI
0.99
1.10
1.01
0.89
0.99
1.10
1.01
0.89
0.99
1.10
1.01
0.89
0.99
1.10
1.01
0.89
Deseason
alized
Demand
Trend
Forecast
40.28
40.16
39.89
41.85
41.13
45.21
41.40
42.09
42.70
42.55
43.05
38.45
44.30
44.01
43.71
44.58
44.97
49.43
46.33
45.93
46.60
47.03
46.89
41.88
47.32
47.85
47.53
49.12
48.82
53.66
50.27
49.78
50.50
50.39
50.74
45.31
51.70
51.35
52.66
57.89
53.62
54.40
54.58
48.75
3-11
Seasonality: Forecast
60
50
40
Actual
30
D.Actual
20
10
0
1 2
3 4 5 6
7 8 9 10 11 12
3-12
Seasonality: Forecast
60
55
Actual
50
D.Actual
45
Trend
40
Forecast
35
15
13
11
9
7
5
3
1
30
3-13
Dealing with Trend & Seasonality
• Estimate seasonal indexes using CMA
• Deseasonalize demand data by dividing
demand with seasonal indexes
• Estimate trend based on deseasonalized data
using either TAF or Linear Trend Equation
• Project the trend to the future
• Forecast by multiplying trend values with
seasonal indexes
3-14