Time-Series Forecasting
Notes abridged from Operations Management by K.N. Dervitsiotis, Mc Graw Hill, 1981
Time-Series Components
Trend (T)
Business-cycle (C)
refers to long-term growth/decline in average level of demand
refers to large deviation of actual demand values from expected (trend line) due
to complex environmental influences
Seasonal component (S) refers to annual repetitive demand fluctuations
Random component (R)
is the irregular residual in the demand due to many complex random forces in
the environment
Additive Model:
Y=T+C+S+R
Multiplicative Model
Y = TCSR
Illustration of Multiplicative Model:
Demand Data for Hermes Co., thousands of gallons of paint (Table 11-3)
Quarter
Q1
Q2
Q3
Q4
Year
Winter
Spring
Summer
Fall
1974
289
410
301
213
1975
212
371
374
333
1976
293
441
411
363
1977
324
462
379
301
1978
347
520
540
521
1979
381
594
573
504
1980
444
592
571
507
Annual
total
1213
1290
1508
1466
1928
2052
2114
Step 1: Calculation of trend-line parameters
The trend component in the time series can be specified by a straight line of the form Y' = a + bX. For 1974
Q1 as base quarter, by least-squares method we get Y' = 271.78 + 10.48X, where X = 0 for 1974 Q1, = 1 for
1974 Q2, = 4 for 1975 Q1, and so on.
Step 2: Seasonal-Component Analysis
Moving average based on as many periods as required to cover a full year is calculated. When quarterly demand
data is provided, moving average is based on 4 periods. When monthly demand data is provided, moving
average is based on 12 periods.
Computation of specific seasonal-index values (Table 11-6)
Year
1974
1975
1976
1977
Quarter
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
Actual
4-quarter Centered Specific
demand
moving
moving
seasonal
Y = TCSR average average TC index SR
289
410
301
303.25
293.63
1.025
213
284.00
279.13
0.763
212
274.25
283.38
0.748
371
292.50
307.50
1.207
374
322.50
332.63
1.124
333
342.75
351.50
0.947
293
360.25
364.88
0.803
441
369.50
373.25
1.182
411
377.00
380.88
1.079
363
384.75
387.38
0.937
324
390.00
386.00
0.839
462
382.00
374.25
1.234
379
366.50
369.38
1.026
301
372.25
379.50
0.793
Year
1978
1979
1980
Quarter
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
Actual
4-quarter Centered Specific
demand
moving
moving
seasonal
Y = TCSR average average TC index SR
347
386.75
406.88
0.853
520
427.00
454.50
1.144
540
482.00
486.25
1.111
521
490.50
499.75
1.043
381
509.00
513.13
0.743
594
517.25
515.13
1.153
573
513.00
520.88
1.100
504
528.75
528.50
0.954
444
528.25
528.00
0.841
592
527.75
528.13
1.121
571
528.50
507
Note that four-quarter average for Q1, Q2, Q3 and Q4 of 1974 is MA1 = (289 + 410 + 301 + 213)  4 = 303.25.
Similarly, four-quarter average for Q2, Q3, Q4 of 1974 and Q1 of 1975, MA2 = (410 + 301 + 213 + 212)  4
= 284.00. The values of MA1 and MA2 are defined at the midpoint of the time interval covered. Thus, MA1 is
between Q2 and Q3 of 1974 and MA2 is between Q3 and Q4 of 1974. Therefore, neither one is representative
of any of these quarters. In order to associate a moving average with a particular quarter, a centered moving
average is calculated by adding two simple moving averages at a time. Or, CMA1 = (MA1 + MA2)  2 = (303.25
+ 284.00)  2 = 293.63.
The centered moving average is also a deseasonalized measure of quarterly demand. As such, it includes the
effect on quarterly demand of the long-term T and the business cycle C. To isolate the seasonal effect for Q3
of 1974, the actual demand of Q3 of 1974 is divided by CMA1. Or, (SR)Q3, 1974 = (TCSR)Q3, 1974 / (TC)Q3, 1974 =
301  293.63 = 1.025
By averaging the values of specific seasonal index for the same period in successive years, the random
influence is smoothened out.
SQ3 = [(SR)Q3, 1974 + (SR)Q3, 1975 +      + (SR)Q3, 1979]  6
= (1.025 + 1.124 + 1.057 + 1.079 + 1.111 + 1.100)  6
= 1.083
Similarly, SQ1, SQ2 and SQ4 are equal to 0.8045, 1.173 and 0.906, respectively.
SQ1 + SQ2 + SQ3 + SQ4 = 3.967
However, this sum should be equal to 4 (number of periods per year). Hence, seasonal value of each quarter is
adjusted by multiplying by 4/3.967.
Thus, SQ1 = 0.812 (winter)
SQ2 = 1.186 (spring)
SQ3 = 1.087 (summer)
SQ4 = 0.915 (fall)
4.000
Step 3: Cyclical-Component Analysis
The cyclical component is isolated by dividing TC by T. The quarterly (or monthly) original data is
deseasonalized by first dividing by the typical seasonal index values S determined in previous step. This is
equal to TCR. Using regression, the trend line for the quarterly (or monthly) original data is constructed. This
is equal to T. In this example the trend line is described by 271.78 + 10.48X, where X = 0 for Q1 of 1974, = 1
for Q2 of 1974, and so on. By dividing the deseasonalized values by T, the CR component of each quarter is
obtained. The cyclic component, C, is obtained by smoothing the CR values using a three-quarter moving
average.
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