## 5 Results and Discussion
### Numerical Efficiency
#### 5.3.4 Simulation Effectiveness  
Last but not least, simulation models can be compared in terms of how well they are able to represent the variability present in the empirical evidence, in this case, in the MISR-HR FAPAR records. This is done by contrasting the \(\chi^{2}\) values at the end of the inversion.  
Table 7 summarizes the results and shows that the PDSF actually best matches the available remote sensing data, when they are available. However, as was shown above, it is also the model that achieved the smallest number of successful inversions.  
As noted above, the PDHyTgF and PDLF models yield very similar results: This was expected because they can be made to match each other exactly by manipulating the model coefficients. However, these functions are not identical: both are defined on the domain \(\mathbb{R}\) of real numbers, or in the context of this application, on the subdomain \(\mathbb{R}_{l}\) of real numbers suitable to represent Julian days, but their co-domains are the intervals \([-1.0,1.0]\) and \([0.0,1.0]\), respectively. More importantly, they do differ from each other in terms of their numerical efficiency: the PDLF model achieves, on average, a slightly better fit than the PDHyTgF, though the latter did manage to achieve a successful inversion in more cases than the former.  
Again, these last results depend on the availability of a floating-point coprocessor, or on the implementation of the algorithms to compute transcendental functions, such as exponential and trigonometric functions. However, since all results were obtained on the same computer, the results  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Iterations** & \multicolumn{2}{c}{**PDGF**} & \multicolumn{2}{c}{**PDHyTgF**} & \multicolumn{2}{c}{**PDLF**} & \multicolumn{2}{c}{**PDSF**} \\ \hline
**Cases** & **s** & **u** & **s** & **u** & **s** & **u** & **s** & **u** \\ \hline Minimum & 1.00008 & 1.00008 & 1.00008 & 1.33341 & 1.00000 & 1.00008 & 1.00008 & 1.00008 \\
25 th percentile & 1.66675 & 1.66675 & 1.66683 & 1.66675 & 1.66675 & 1.66675 & 1.66683 & 1.33341 \\ Median & 2.00008 & 2.00008 & 2.00016 & 2.00016 & 2.00016 & 1.66683 & 2.00016 & 1.66683 \\ Mean & 2.26680 & 2.53349 & 2.66686 & 2.13346 & 2.26680 & 2.06678 & 2.46685 & 1.93348 \\
75 th percentile & 2.00016 & 2.33349 & 2.33357 & 2.33349 & 2.33349 & 2.00016 & 2.33349 & 2.00016 \\ Maximum & 4.66708 & 5.66708 & 6.33383 & 3.33357 & 4.33365 & 4.00023 & 5.33372 & 3.66700 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Execution time, relative to the fastest case (successful PDLF model inversion in 0.00299978 second), for successful (s) and unsuccessful (u) model inversions.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**PDGF** & **PDHyTgF** & **PDLF** & **PDSF** \\ \hline
0.022 & 0.020 & 0.018 & 0.015 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Average \(\chi^{2}\) value at the end of the model inversion step.  
exhibited here may be indicative of the relative performance of these models. These findings may need to be confirmed over a larger number of cases to provide general guidance about the selection of such models.