## 5 Simulation Result
### Fictitious Project Results  
Figure 5 shows the results of the simulator on one randomly selected project from the ProGen dataset. In this diagram, the daily progresses of physical and actual percent completion were compared. The S-curve shape can be seen in these two diagrams; however, this simulated project includes a delay of 56 days. This project has 92 tasks and is scheduled for 87 days, and it is simulated by these input parameters:  
* The project has four disciplines comprising 20%, 10%, 40% and 30% of all of the tasks, and back-loaded, flat, front-loaded and bell are their work contours, respectively. * Eighty percent of task delays are in the range of 10% of scheduled progress. * The project was simulated with a 20% inflation rate for the last six months. Figure 5: Percent completion of the simulated project. Figure 6: Actual cost S-curve. To validate the simulator, another randomly selected project from the ProGen dataset is used, taking the results and comparing them to the S-curve technique's results. The simulator inputs are set, as all tasks are in one discipline with a normal distribution, a 20% inflation rate for last six months and 80% of task delays in the range of 10% of scheduled progress. Figure 6 shows the actual cost S-curve. This actual time series is now used as an example to illustrate the selection of the SSA parameters and to show the reconstruction of the original series in detail. * Selection of the window length L:  
At the decomposition stage, the window length L is a single parameter that has to be chosen and achieving sufficient separation of the components is crucial in selecting L. As stated above, the window length is the dimension of the Euclidean space that the time series is developed. Moreover, in literature considerable attempt and various methods have been presented for choosing the optimal value of L, for example, L = T/4 as a recommended one and L can be larger but not larger than T/2 [19]. More recently, the selection of L was optimized based on the minimization of a loss function [38]. Larger value of L leads to a more detailed decomposition and resolving longer period oscillations so L has been set at a larger value (L = T/2) in these analyses. In this example L = 38. * Selection of r:  
In this method, some auxiliary information and diagrams generated by Caterpillar SSA could be applied to select the parameters L and r. Here, there is a brief explanation of the method which might be beneficial in separating the signal from noise in the time series, which is offered by controlling breaks in the eigenvalue spectra. Furthermore, a pure noise series typically generates a gradually reducing sequence of singular values. It is noteworthy that there are alternate methods for the selection of r, refer to [PERSON] et al. (2016) for instance [20]. With these accomplished results, the principal components are attained from SVD. Choosing L = 38 and performing SVD of the trajectory matrix X, we get 38 eigentriples, organized by the share they contribute to the decomposition. Figure 6 depicts the plan of the logarithms of the 38 singular values. In Figure 7, an enormous decrease in values takes place close to Element 9, which can be taken as the beginning of the noise floor. Grouping the major elements can be done by another technique in which we analyze the w-correlations of the matrix of the absolute values. Figure 8 represents the w-correlations for  
Figure 7: Logarithms of the 38 eigenvalues. Figure 9 shows the first main elements plotted as time series. Practically, the singular values of the two eigentriples of a harmonic series are usually very near to one another, and this makes the visual identification of the harmonic components easy. A close look at pairwise scatterplots of the singular vectors enables us to visually distinguish those eigentriples that match the harmonic elements of the series, given that these components can be separated from the residual component.