Decomposition procedures are used in time series to describe the trend and seasonal factors in a time series. More extensive decompositions might also include long-run cycles, holiday effects and day of week effects and so on. Here, we’ll only consider trend and seasonal decompositions.
Structure used for decomposition procedure is as follows –
- Additive: xt = Trend + Seasonal + Random
- Multiplicative: xt = Trend * Seasonal * Random
Additive model is being used for less seasonal variation overtime while multiplicative model is used for increases in seasonal variation.
- Additive decomposition
Yt = Tt + C t+St+It
Difference of actual to CMA = Yt – Tt- C t
Seasonal index = average of S+I for each quarter
Index for quarter I = 0
= unadjusted seasonal index
Index – mean of all unadjusted index
Realized value = (T+C+I) = Yt – St = Tt + C t+It
Deterred value = (C+I) : Yt – St – Tt = C t+It
Cyclic indices = 3 period MA + CI
- Multiplicative model
Y = TCSI RATIO OF ANNUAL CMA = y /(TC)
Unadjusted seasonal index = Average s + I for each quarter
Seasonal index of quarter I = length of L
Adjusted SI = Unadjusted seasonal index * L/ seasonal Index
Deseasonalised value = (T+C+I) = Y/S= TCI
Trend estimate = T = b+ bt
Deterend value =(C+I) = Y/ (ST)= CI
CYCLIC Indices = 3- period MA of c+I
For cast = TCS
(iii) Exponential smoothing forecast for random variable as the data in consideration has trend and it will not be an acceptable forecast method.
It is similar to weighted moving average more suitable. Value of alpha has been taken from 0.3 to 1 . For alpha = 0 the demand does not changes thee for it is completely insensitive.
For alpha =1 , the forecast has demonstrated a knife forecasting where previous period actual become next period forecast which make alpha highly sensitive.
(iv) Holt method
FT+M = ( L T + MTT)*ST+M-S
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