# Detecting and Responding to Concept Drift in Business Processes
## 5 Evaluation on Artificial Data
### Experimental Results  
The HMDP approach was validated under different penalties. That is, the penalty for action'retrain' is increased from zero to one with an increment of 0.1, associated with a 50% discount for 'adapt'. For example, under a penalty of 0.6 for'retrain', the penalty is 0.3 for 'adapt'. Tables 2 and 3 demonstrate the average reward of HMDP and other action selection policies under all the above-mentioned datasets. Column 'penalty' represents the penalty for action'retrain', and the best policy is highlighted in bold type separately for sudden and gradual drift. Without action penalty, the average reward ranges from zero to two because the \(p\)-value of [PERSON]'s method for both transitions and duration ranges from 0 to 1. Under action penalty one, there can be a negative penalty ranging from \(-2\) ('retrain_retrain') to 0 ('keep_keep') and, therefore, the average reward can be negative but within the period between \(-1\) (action 'adapt_adapt' with both [PERSON]'s \(p\)-value equal to 0) and 2 (action 'keep_keep' with [PERSON]'s \(p\)-value as 1). As we expected, without a penalty, action'retrain' can achieve 100% accuracy in modelling new data's transitions and durations because the bottom layer representation is updated using new data and then used to estimate the new data's goodness of fit. Consequently, the HMDP approach becomes invalid for detecting concept drift and manage model updating. On the contrary, using penalty one, [PERSON]'s method can have 100%  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Penalty**} & \multicolumn{6}{c}{**Sudden Drifts**} & \multicolumn{6}{c}{**Gradual Drifts**} \\ \cline{2-10}  & **Keep** & **Adapt** & **Retrain** & **Random** & **HMDP** & **Keep** & **Adapt** & **Retrain** & **Random** & **HMDP** \\ \hline
0 & 1.042 & 1.196 & **2.000** & 1.339 & 1.741 & 1.051 & 1.244 & **2.000** & 1.384 & 1.873 \\
0.1 & 1.041  
accuracy with the \(p\)-value as one, but there is a negative penalty of \(-1\), leading to zero reward at the end. There is no penalty for action 'keep', so it remains stable, with the reward fluctuating around one, indicating its ineffectiveness. Because all datasets solely contain one drift type, either in transitions or durations, the action 'keep' can achieve a \(p\)-value close to one for the unchanged part, i.e., it has a bottom line around one. In other words, the action 'keep' has an average reward of around one, indicating that it can not represent new data under concept drift scenarios. Action 'adapt' performs unsatisfactorily under all scenarios because it combines both the old and new data, and the mixed data can not precisely represent the environment. The HMDP approach can outperform the baseline policy 'random' and other policies under appropriate penalties, indicating its effectiveness. Basically, a penalty between 0.4 and 0.7 is suitable. In conclusion, the action'retrain' concentrates on the goodness of fit of the process environment, requiring updating the bottom layer customer representation every time. Action 'keep' saves computational resources since it does nothing during the environment evolution and, consequently, is disabled to represent the environment. The HMDP approach benefits both the accuracy and efficiency in environment modelling by automatically retraining the model under concept drift and maintaining the model otherwise. Using penalty 0.5 as an example, we illustrate the average reward for transition and duration drift in Figure 7. The first two subfigures refer to the average reward of the five action selection policies under sudden and gradual transition drift, respectively. Figure 7c,d relate to the average reward under duration and concept drift. The horizontal axis of Figure 7a,b is the 18 control-flow change datasets. In Figure 7c,d, the horizontal axis refers to the 16 duration drift patterns with the first eight elements as changes in mixing proportions and the latter eight elements as drift in the gamma shape parameter.