Examples of MSE

Suppose that the impact of the SE "marketing messages send" will start at the next day $i$, which is Tuesday. The present day $i-1$ is Monday with no additional special events. Then the scale

$$s_i=\frac {Z_{p(i)}}{Z_{p(i-1)}}.$$ (130)

Here $p(i)$ is the subscript of the most recent previous Tuesday with the same impact SE. $p(i-1)$ is the subscript of the most recent previous Monday with no SE. In this example we regarded a multiple SE as two single SE, Monday without marketing and Tuesday with marketing.

Another way is to regard the impacts of those two SE by defining the scale $s_i$ as a product of two scales .

$$s_i=s^1_i \ s^2_i$$ (131)

Here

$$s^1_i=\frac {Z_{p^1(i)} }{ Z_{p^1(i-1)}} .$$ (132)

where $p^1(i)$ is the index of Monday and $p^1(i-1)$ is the index of Tuesday of the previous week, and

$$s^2_i=\frac {Z_{p^2(i)} }{Zz_{p^2(i-1)}} .$$ (133)

where $p^2(i)$ is the index of the most recent day with "marketing" and $p^1(i-1)$ the most recent day with no "marketing".

The expression (2.70) is an example of empirical approach when multiple SE are regarded as different special events. The expression (2.71) needs less data but is based on an expert assumption that scales are multiplicative.