Examples of MSE

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

$$s_i=\frac {Z_{i-1)} }{ Z_{p(i-1)}},$$ (140)

where $p(i-1)$ is the index of the day preceding the present one. In this example we regard a multiple SE as two single SE: Monday with marketing and Tuesday without the one.

Another way is to regard the impacts of those two SE by defining the scale $s_{i-1}$ as a product of two scales .

$$s_{i-1}=s^1_{i-1}\ s^2_{i-1}$$ (141)

Here

$$s^1_{i-1}=\frac {z_{p^1(i-1)} }{ z_{p(p^1(i-1))}} .$$ (142)

where $p^1(i-1)$ is the index of the day preceding Monday without marketing and $p(p^1(i-1))$ is the index of the day preceding $p^1(i-1)$ and

$$s^2_i=\frac {z_{p^2(i-1)} }{ z_{p(p^2(i-1))}} .$$ (143)

where $p^2(i-1)$ is the index of the day preceding the day with marketing and $(p(p^2(i-1))$ is the index of the day preceding $p^2(i-1)$.

The expression (2.80) is an example of empirical approach when multiple SE are regarded as different special events. The expression (2.81) needs less data but is based on the assumption that scales are multiplicative, what is not true strictly speaking. The expression (2.81) involves expert opinion indirectly by assuming that the scales are multiplicative.