Estimation Results Rel err Test function SNR EL MSS EL MSS (A) 100 observations, Δt=0.5, Θ(0) (1,0.1) Θ1 1.00 ± 0.19 1.56 ± 0.92 15% 71% SNR: 0.21 Θ2 0.101 ± 0.02 0.156 ± 0.09 16% 72% (B) 2000 observations, Δt=0.5, Θ(0) =(1,0.1) Θ1 1.00 ± 0.04 1.00 ± 0.15 2% 13% SNR: 0.17 Θ2 0.101 ± 0.00 0.101 ± 0.015 2% 13% (C) 500 observations, Δt=5, Θ(0) =(1,0.1) Θ1 1.01 ± 0.09 1.02 ± 0.11 7% 9% SNR: 0.50 Θ2 0.101 ± 0.01 0.102 ± 0.01 7% 9% (D)100 observations, Δt=10, Θ(0) =(0.6,0.06) Θ1 0.60 ± 0.12 0.65 ± 0.17 15% 21% SNR: 0.55 Θ2 0.061 ± 0.01 0.065 ± 0.018 16% 22%
Table 1: Statistics of the estimation results for Immigration-Death model. The table shows that the MSS method works well in this example for three out of four experimental designs and the test functions identify the problematic design.
50 data sets are simulated using the Gillespie algorithm with initial value v0=10. For each of the 50 data sets an estimation is performed with an exact method (EL) and the MSS method. The table shows a statistic over the 50 estimates for both the exact and the MSS method: Parameter name (column 1), averages for exact method (column 2), standard deviation of exact method (column 3), averages of MSS method (column 4), standard deviations of MSS method (column 5), relative errors of exact method (column 6) and MSS method (column 7) and the signal to noise (SNR) test function (column 7).
The SNR test function indicates a weak SNR in situation (A) and therefore identifies the case in which the MSS approximation is problematic. If this is the case many more observations are needed to resolve the problem despite still low SNR, situation (B), or another experimental design (C).