Decomposition of a Time Series
By decomposition of a time series we mean the analysis of the time series by the process of segregation of its four components viz: secular trend, seasonal variations, cyclical fluctuations, and irregular movements. This involves the taking up of the following major steps:
(i) Identification of the various factors whose interaction produces fluctuations in the time series, and
(ii) Measurement of the effects of these factors separately and independently by keeping the effect of the other factors constant.
Mathematical Models
For making proper decomposition of a time series we have three different types of mathematical models before us: They are:
(i) Additive model, (ii) Multiplicative model and (iii) Mixed model. The form and nature of these models are stated as under:
(i) Additive Model
Under this model, the observed value of a time series is given as under:
Y = T + S + C + 1
Where Y = an observed value;
T = trend value;
S = Seasonal variation,
C = Cyclical variation, and
I = Irregular variation
In this model it is assumed that the four components of a time series, T, S, C and I are independent of each other, that none has any effect, whatsoever, on the remaining three components; that the observed value is the some of the four components; that S, C, and I are absolute quantitative deviations from trend values, and that the behaviour of the components is of an additive character.
Under this model, only the absolute values of the other components are added to, or subtracted from the trend value to arrive at the observed value of the time ser5ies. In this model, S, C and I are absolute quantities which can take both positive and negative value so that for any year, and = 0
Example
If T = 500, S = 115, C = 15 and 1 = -50
Then Y = 500 + 115 + 15-50 = 580
However, in actual practice, the assumption of independent character of the various components of the time series does not hold good. A sharply rising, and falling trend may wipe off the effects of seasonal or cyclical variations, and a strong and powerful seasonal variation may intensify the changes in the cyclical variations. Thus, in practice, the additive occurs rarely with a time series.
(ii) Multiplicative
Under this model, it is assumed that the four components, T, S, C and 1, although, occur due to different causes, are not necessarily independent and they can affect one another ; that the behaviour of the components is of multiplicative characters ; and that the geometric mean of all S,C and I would unity or 1.
The above multiplicative model can also be converted into additive model with the help of logarithms as under:
Log Y = log T + log S + log C + log 1
It may be noted that most of the time series relating to business and economics adhere to the multiplicative model in as much as the effect of various factors affecting such time series are not independent of each other.
(iii) Mixed Model
Under this model, an observed value of a time series is obtained by any of the following formulae based on the combination of both the additive and multiplicative models:
Y = TSC + 1, or
Y = TS + CI, or
Y = T + SCI, or
Y = T + S + CI, or
Y = TC + SI.
Note. Theadditive, and multiplicative models explained above can be used to obtain the value of various components of a time series by the process of subtraction, or division shown as under:
Under the Additive Model
Where Y = T +S +C + 1
Y – T = S + C +1
Y-T- S = C + I and
Y – T – S – C = I
When the data are given on annual basis so that there are no seasonal variations it them,
Then y = T + C + 1,
Y – T = C + 1 and
Y-T-C = 1
Under the Multiplicative Model
Where, Y = T × S × C × 1,
SCI = Y/T i.e., Original Value/ Trend Value ;
CI = Y/(T x S) and;
I = Y/T.S.C
When the data are given on an annual basis so that there are no seasonal fluctuations in it,
Then, Y = T × C × 1;
C.I. = Y/T i.e. Original Value/ Trend Value and;
I = Y/T.C
