# Low-Pass Filtering Method for Poisson Data Time Series
## 3 Results
### Testing the Method and the Quasi-Gaussian Filter on Model Normalized Poisson Data
#### 3.1.2 Estimation of Mathematical Expectation and Its Root Mean Square Errors  
Testing of the method and quasi-Gaussian filter for estimating the mathematical expectation and the RMS of its errors depending on \(dn_{a}\)--the duration of decreases and \(d\mu\)--the relative decrease value were carried out using statistical tests [17]. Random datasets \(Y_{M}(s,T_{0}i)\), \(X_{MG}(s,T_{0}i)\), \(X_{MF}(s,T_{0}i)\), \(s\ =\ 1,2,\ldots,M\), where \(s\) is the number of the dataset, \(M\) is the total quantity of datasets. The estimates of mathematical expectation \(m_{g}^{\circ}(dn_{a},d\mu)\) and RMS values \(\sigma_{g}^{\circ}(dn_{a},d\mu)\) for \(X_{MG}(s,T_{0}i)\) for a set of values \(dn_{a}\) and \(d\mu\)  
\[m_{g}^{\circ}(s,dn_{a},d\mu)\ =\ \frac{1}{n_{a}}\sum_{n=n_{a}}^{n_{a}+dn_{a}} X_{MG}(s,T_{0}n),m_{g}^{\circ}(dn_{a},d\mu)\ =\ \frac{1}{M}\sum_{s=1}^{M}m_{g}^{\circ}(s,dn_{a},d\mu),\] \[\sigma_{g}^{\circ}(s,dn_{a},d\mu)\ =\ \frac{1}{n_{a}-1}\sum_{n=n_{a}}^{n_{a}+dn_{a}} (X_{MG}(s,T_{0}n)-m_{g}^{\circ}(s,dn_{a},d\mu))^{2}, \tag{15}\] \[\sigma_{g}^{\circ}(dn_{a},d\mu)\ =\ \frac{1}{M}\sum_{s=1}^{M} \sigma_{g}^{\circ}(s,dn_{a},d\mu).\]  
The coefficients of relative errors \(\varepsilon_{gm}^{\circ}(dn_{a},d\mu)\), \(\epsilon_{gg}^{\circ}(dn_{a},d\mu)\) of the quasi-Gaussian filter as ratios of errors \(m_{g}^{\circ}(dn_{a},d\mu)-m_{0}\) and RMS \(\sigma_{g}^{\circ}(s,dn_{a},d\mu)\) to the values of \(dm\) reductions are the following:  
\[\varepsilon_{gm}^{\circ}(dn_{a},d\mu)\ =\ (m_{g}^{\circ}(dn_{a},d\mu)-m_{0})/dm, \varepsilon_{gg}^{\circ}(dn_{a},d\mu)\ =\ (\sigma_{g}^{\circ}(dn_{a},d\mu))/dm\ldots \tag{16}\]  
The coefficients \(\varepsilon_{gm}^{\circ}\), \(\varepsilon_{gg}^{\circ}\), calculated for \(dn_{a}\), \(d\mu\), characterized the recognition capabilities of quasi-Gaussian filtering model decreases. Similarly, using (15) and (16) \(m_{f}^{\circ}(dn_{a},d\mu)\) and \(\sigma_{f}^{\circ}(dn_{a},d\mu)\) for \(X_{MF}(s,T_{0}n)\) and the coefficients \(\varepsilon_{fm}^{\circ}(dn_{a},d\mu)\), \(\varepsilon_{f\circ}^{\circ}(dn_{a},d\mu)\). On Figure 4, the results of statistical tests are displayed, where \(M=500\). The \(\varepsilon_{gm}^{\circ}(dn_{a},d\mu)\) coefficients plots are the solid lines with indices 1, 2, and the \(\varepsilon_{fm}^{\circ}(dn_{a},d\mu)\) plots are the dashed lines with indices 3, 4. The coefficients \(\varepsilon_{gm}^{\circ}\), \(\varepsilon_{fm}^{\circ}\) are given depending on the duration with the values \(dn_{a}\ =\ 12,24,48,72\) h and relative decreases in \(d\mu\), taking the values of 0.01 (indices 1, 3) and 0.03 (indices 2, 4). The effect of quasi-Gaussian filtering was determined based on the calculation of \(\delta\varepsilon_{fg,m}^{\circ}\)--the rates of errors with respect to the mathematical expectations:  
\[\delta\varepsilon_{fg,m}^{\circ}(dn_{a},d\mu)\ =\ (\varepsilon_{fm}^{\circ}(dn_{a},d\mu)- \varepsilon_{gm}^{\circ}(dn_{a},d\mu))/\varepsilon_{gm}^{\circ}(dn_{a},d\mu) \tag{17}\]  
The results of the \(\delta\varepsilon_{fg,m}^{\circ}\) calculations according to (17) for some \(d\mu\) and \(dn_{a}\) values are:  
1. \(\delta\varepsilon_{fg,m}^{\circ}=0.115\) (11.5%) for \(dn_{a}=24\) and \(d\mu=0.01\);
2. \(\delta\varepsilon_{fg,m}^{\circ}=0.196\) (19,6%) for \(dn_{a}=24\) and \(d\mu=0.03\). Analysis of the error values showed that the \(\varepsilon_{gm}^{\circ}\) rate values appeared to be about 10-30% lower than the \(\varepsilon_{fm}^{\circ}\) values. The nature of the dependencies of the estimates of the error coefficients for the \(\varepsilon_{gg}^{\circ}\) and \(\varepsilon_{f\sigma}^{\circ}\) root mean square values for the same \(dn_{a}\) and \(d\mu\) parameters is almost the same: the \(\varepsilon_{g\sigma}^{\circ}\) are also \(\approx\)10-30% lower than the \(\varepsilon_{f\sigma}^{\circ}\). This means that, for the recognition of decreases small in duration and magnitude, the use of a quasi-Gaussian filter is more preferable than the use of a common FIR filter.