Finding Seasonal Variations

Once a Trend has been established, by whatever method, Seasonal Variation can be calculated.

Procedure for Finding Seasonal Component using the:

1. Additive Model

The additive model assumes that the components of the series are independent of each other, an increase trend not affecting the seasonal variations for example.

• The Additive Model for Time Series Analysis is Y = T + S + I

• If Trend is deducted from the model, we get Y  - T = S + I

• If we assume that I, the random, or irregular component of the time series is relatively small and therefore negligible, then S = Y – T

Therefore, the seasonal component, S = Y – T (The De-Trended Series)

Example

Use the data on the ‘Factory Output’ to calculate the seasonal variation for each of the 15 days, and adjust the seasonal variations for each day of the week.

Solution

		Output (Y)	Five day Moving Total	Trend (T)	Seasonal Variation (Y-T)
Week 1	Monday	80	-	-	-
	Tuesday	104	-	-
	Wednesday	94	460	92.00	2.00
	Thursday	120	462	92.40	27.60
	Friday	62	468	93.60	-31.60
Week 2	Monday	82	471	94.20	-12.20
	Tuesday	110	476	95.20	14.80
	Wednesday	97	478	95.60	1.40
	Thursday	125	480	96.00	29.00
	Friday	64	486	97.20	-33.20
Week 3	Monday	84	489	97.80	-13.80
	Tuesday	116	494	98.80	17.20
	Wednesday	100	496	99.20	0.80
	Thursday	130	-	-	-
	Friday	66	-	-	-

It will be noticed that the variation between the actual result on any one particular day and the Trend line average is not the same from week to week. This is because Y-T contains not only seasonal variations but also random variations.

Adjusting Seasonal Variation

An average of the seasonal variation (Y-T) is made, and if the total is not equal to zero,  adjust one or more of them so that their total is zero.

 Monday Tuesday Wednesday Thursday Friday
Week 1   2.00 27.60 -31.60
Week 2 -12.20 14.80 1.40 29.00 -33.20
Week 3 -13.80 17.20 0.80  
Total -26.00 32.00 4.20 56.60 -64.80
Average -13.00 16.00 1.40 28.30 -32.40

The Total of the Seasonal Variation Averages = 0.30.

• Spread the Total of Daily variation (0.30) across the five days (0.3 5) so that final total of the daily variation goes to zero.

• Subtract 0.06 (i.e.0.3 5) from each seasonal variation Average.

Monday Tuesday Wednesday Thursday Friday  Total
      
-13.00 16.00 1.40 28.30 -32.40  0.30
-0.06 -0.06 -0.06 -0.06 -0.06  -0.30
-13.06 15.94 1.34 28.24 -32.46  0.00

Therefore, the Adjusted Seasonal Variations for the five days are;

Monday -13.06
Tuesday 15.94
Wednesday 1.34
Thursday 28.24
Friday -32.46

2. Multiplicative Model

In multiplicative model each actual figure is expressed as a proportion of the Trend. Sometimes this method is called the proportional mode.

Time Series = Trend x Seasonal x Random
     Y  = T x S x I
Assuming the value of Random Variable (I) as negligible, then the Seasonal variation becomes S = Y/T for the multiplicative model.

Example;

Consider the ‘Factory output’ example covered in additive model, use it to calculate the Seasonal Variations using multiplicative model.

  Output Five day Moving Total Trend (T) Seasonal Variation (Y/T)
  (Y)   
Week 1 Monday 80   
 Tuesday 104   
 Wednesday 94 460 92.00 1.022
 Thursday 120 462 92.40 1.299
 Friday 62 468 93.60 0.662
Week 2 Monday 82 471 94.20 0.870
 Tuesday 110 476 95.20 1.155
 Wednesday 97 478 95.60 1.015
 Thursday 125 480 96.00 1.302
 Friday 64 486 97.20 0.658
Week 3 Monday 84 489 97.80 0.859
 Tuesday 116 494 98.80 1.174
 Wednesday 100 496 99.20 1.008
 Thursday 130   
 Friday 66   

The summery of the seasonal variations is as follows;

 Monday Tuesday Wednesday Thursday Friday  
Week 1   1.022 1.299 0.662  
Week 2 0.870 1.155 1.015 1.302 0.658  
Week 3 0.859 1.174 1.008    
Total 1.7290 2.3290 3.0450 2.6010 1.3200  
Average 0.8645 1.1645 1.0150 1.3005 0.6600  5.0045

Instead of summing to Zero as with Additive Model, the Multiplicative model should sum (in this case) to 5 (an average of 1).

The averages summed up to 5.0045 so 0.0009 has to be deducted from each one. This is two small to make a difference to the figures above, so we should deduct 0.002 and 0.0025 from Monday and Tuesday respectively.

Therefore, the Adjusted Seasonal Variations for the five days for multiplicative modal are;

Monday 0.8625
Tuesday 1.162
Wednesday 1.015
Thursday 1.3005
Friday 0.66

Note; The multiplicative model is better than the additive model for forecasting when the trend is increasing or decreasing over time. In such circumstances, seasonal variations are likely to be increasing or decreasing too. The additive model simply adds absolute and unchanging seasonal variations to the trend figures where as the multiplicative model, by multiplying increasing or decreasing trend values by a constant seasonal variation factor, takes account of changing seasonal variations.