Mathematical Statement of the Composition of Time Series
A time series may not be affected by all type of variations. Some of these type of variations may affect a few time series, while the other series may be effected by all of them. Hence, in analysing time series, these effects are isolated. In classical time series analysis it is assumed that any given observation is made up of trend, seasonal, cyclical and irregular movements and these four components have multiplicative relationship.
O = T × S × C × I
where O refers to original data,
T refers to trend.
S refers to seasonal variations,
C refers to cyclical variations and
I refers lo irregular variations.
This is the most commonly used model in the decomposition of time series.
There is another model called Additive model in which a particular observation in a time series is the sum of these four components.
O = T + S + C + I
To prevent confusion between the two models, it should be made clear that in Multiplicative model S, C, and I are indices expressed as decimal percents whereas in Additive model S, C and I are quantitative deviations about trend that can be expressed as seasonal, cyclical and irregular in nature. If in a multiplicative model. T = 500, S = 1.4, C = 1.20 and I = 0.7 then
O = T × S × C × I
By substituting the values we get
O = 500 × 1.4 × 1.20 × 0.7 = 608
In additive model, T = 500, S = 100, C = 25, I = –50
O = 500 + 100 + 25 – 50 = 575
The assumption underlying the two schemes of analysis is that whereas there is no interaction among the different constituents or components under the additive scheme, such interaction is very much present in the multiplicative scheme. Time series analysis, generally, proceed on the assumption of multiplicative formulation.