History-based User Delay Model

The psychological model is content-based. It predicts user delay length knowing what users will see. In some cases, it may be difficult to know beforehand the user interface content. Therefore, we devise a user delay model based on history, $i.e.$, multiple observations from the past. It works in a fashion similar to the median filter in order to filter out randomness in user behavior. Let $D^i$ denote the $i$th last observed delay for an STD state and $M$ the total number of user delays recorded for that state.

The delay record is sorted to generate a new series so that

$$D^1_s<D^2_s<...<D^M_s$$

Then we define the habitual set $\Phi(p)$ as

$$\Phi(p) = \{D^{{\lfloor}M{\cdot}p\rfloor+1]}_s,D^{{\lfloor}M{\cdot}p\rfloor+2}_s,...,D^{{\lfloor}M{\cdot}(1-p)\rfloor}_s\}$$

where $0{\leq}p{\leq}0.5$ with a typical value of $0.25$. $\Phi(p)$ contains the central $M{\cdot}(1-2{\cdot}p)$ items of the sorted series. It is a set extension of the concept of a median. Let $m_\Phi$ denote the mean of $\Phi(p)$. Then the next user delay for the state is predicted as

$$D = \alpha{\cdot}m_\Phi\vspace*{-1mm}$$

where $0<\alpha< 1$. $\alpha$ is called the $pessimism$ factor.

When $p=0.5$, the above approach reduces to using the median of the delays in the recorded history. When $p=0$, it reduces to using the mean instead.