Examples of Partial Special Events (PSE)

The only difference from the previous example is that impact of "marketing" starts at the 20-th minute of the next hour $i$. Let us regard PSE as a special sort of SE and define its scale as

$$s_i=\frac {z_{p(i)}}{z_{p(i-1)}}.$$ (124)

Here $p(i)$ is the subscript of the most recent previous day when the "marketing" starts at the 20-th minute of second working hour. $p(i-1)$ is the subscript of the most recent previous first hour of a working day with no marketing.

Another way is to regard the impact of PSE by decreasing the marketing impact scale $s_i$ in proportion to the duration of the impact

$$s^{20}_i=2/3\ s_i .$$ (125)

Here

$$s_i=\frac {z_{p(i)} }{ z_{p(i-1)}},$$ (126)

where $p(i)$ is the subscript of the most recent previous day when the "marketing" starts at the second working hour and $p(i-1)$ is the subscript of the most recent previous first hour of a working day with no marketing.

Expression (2.65) needs less data but is based on the expert assumption that scales are proportional to the impact duration. Expression (2.64) is an example of empirical approach when a PSE is regarded as a different special event. Expression (2.65) involves expert opinion by assuming the proportionality of scales.