## 5 Simulation Result
### Fictitious Project Results
Figure 10 presents scatterplots, accompanied by lines joining consecutive points, of the first nine paired major elements. The trend is a gradual change in the element of a time series that does not include oscillatory components. Thus, to distinguish a trend in the series, one must search for gradually-changing eigenvectors. Figure 11 depicts the trend, which is extracted from the eigentriples 1 to 9. This curve explicitly shows the general tendency in the series. Figure 12 represents the selected harmonic components (4 to 9) and clearly shows the harmonic behavior of the original series. Therefore, we can categorize the remaining components of the eigentriples (10 to 38) as noise. Figure 13 displays the residuals extracted from these eigentriples. Figure 11: Reconstructed trend from the eigentriples 1 to 9 compared with ACWP and S-Curve. Figure 10: Scatterplots corresponding to the first nine paired principal components. Finally, as is described in Section 3.3, with chosen values \(B_{\sfrac{1}{2}}\) and \(r_{0.67}\), the parameter \(\gamma\) can be obtained from Equation (1), and Equation (2) can plot the desired evolution S-curve. To fit this curve with an actually simulated curve (the generated time series from the selected fictitious project), we set required parameters as \(Y_{0}=100\), \(Y_{\infty}=160,000\) and project duration = 77 days. To fulfill the other desired conditions, we set \(B_{\sfrac{1}{2}}=0.49\), \(\gamma=29.02\) and \(\delta=0.09\). Figure 14: Principal components related to the first three eigentriples. Figure 12: Harmonic from the eigentriples 4 to 9. Figure 13: The residuals from the eigentriples 10 to 38. In the next step, the analysis of the S-curve technique time series was implemented similarly to the previously mentioned selected project by SSA method. To show the similarity of these two time series, principal components of two time series, simulator output and S-curve technique output are extracted and compared. The most important components concerning the first three eigentiples, for both the simulator and S-curve technique series, are shown in Figure 14. As shown, these two series have quite similar components. Accordingly, we can conclude that the simulator and S-curve technique time series have the same structure, and this verifies the accuracy of the simulator result. The deviations between the trend of simulated results (SSA trend) and the S-curve technique with the first generated time series are shown in Figure 15. Clearly, these two deviations have the same behavior over time, with the exception of the start and end points. However, SSA results show a lower change range. Generally, these two deviations are near to one another, which results in the fact that the simulated fictitious results have a high accuracy and that this accuracy can be confirmed by similar tests on empirical data. To validate deviations between SSA and the S-curve method, two measures, RMSE and R, are employed that assess the quality of the generated time series. These two measures, R and RMSE, are used to determine the extent to which the model fits the data and are calculated as below:  
Figure 6:0  
Table 1 presents the results of the two measures for SSA trend and the S-curve method with the generated time series. These two measures prove that the generated time series fit the actual trend better than the standard S-curve. Both RMSE values are less than 5% of the maximum member of each series that prove the validity of estimated time series. RMSE of the SSA trend is less than half of the S-curve one. Although the R measure is perfect in both, SSA is judged slightly better.