## 4 Results and Discussion
### Discussion and General Recommendations for the Application of the Proposed Methodology  
The results obtained from the three case studies allow us to make general recommendations for the application of the proposed methodology. The generation of realistic synthetic data about the power consumption of a consumer depends on the chosen number of states. On the one hand, it is important to make sure the following requirements are met:  
* The Frobenius distances, describing the difference between the transition matrices of the original dataset and of the generated data, are as close as possible to 0.
* The \(R^{2}\) coefficients, describing how well the generated transition matrices describe the original ones, are as close as possible to 1.  
Figure 13: Seven days of sample data from the original time series for a printing house (**a**), and from the generated synthetic data for 11 states (**b**), 16 states (**c**), and 22 states (**d**).  
* The statistical measures of the generated data, such as its variance and standard deviation, are as close as possible to the statistical measures of the original dataset. However, the obtained results also showed that
* The Frobenius distances and \(R^{2}\) coefficients were usually closer to their optimal values for a lower number of states.
* The difference between the statistical measures of the generated and original datasets could vary in a wide range, depending on the number of states of the transition matrices.
* The generated datasets with a lower number of states were usually more scattered than the training ones.  
In other words, it is not always possible to make sure all of the above requirements are met at the same time. In this case, it is necessary to choose appropriate criteria for optimality. Based on the obtained results, we suggest the following methodology for choosing the optimal number of states:  
1. Local minimums of the average Frobenius distances and/or local maximums of the average \(R^{2}\) coefficients should be looked for at a higher number of states of the transition matrices. It is recommended that the number of states be more than 20 or at least 15.
2. Local minimums of the statistical measures of variance and/or standard deviation could be looked for.
3. The optimal number of states can be chosen for the situation where the measures from p. 1 and p. 2 have their local peak values or values that are very close to them.  
There is another opportunity to mitigate the earlier-mentioned problem, which is expressed in greater scattering of the generated data when fewer states of the transition matrices are used. According to the presented methodology, once the new state of power consumption is chosen, the specific value within the boundary of this state is generated with a uniform distribution. This part of the methodology could be altered to generate random numbers with an appropriate frequency distribution. To do this, an individual frequency distribution should be obtained for each state of each matrix (i.e., for each hour of the day or a total of 24 \(\times\) N number of distributions), based on the training data. In theory, this could allow generating synthetic data that are very similar to the original one, using fewer states. This approach could be useful when the training data are limited; however, this would also require splitting each state into substates for the frequency distribution, which additionally complicates the algorithm.