# Low-Pass Filtering Method for Poisson Data Time Series
## 4 Discussion  
The comparison between the model data filtering result obtained using the proposed filter and the one obtained using the fir1 (plots on Figure 3) shows that the resulting time series are close to each other; however, the \(X_{MG}(T_{0}i)\) seems to be closer to the initial model. The main quantitative result of testing the method and the quasi-Gaussian filter on model normalized Poisson datasets included the calculations for (14) for a set of realizations/ The resulting errors \(\sigma_{MG}\) for \(X_{MG}(T_{0}i)\), on average, by 15-30% less errors \(\sigma_{MF}\) for \(X_{MF}(T_{0}i)\). This means that the proposed filtering method provided better filtering (noise reduction) than the standard FIR filter. Consideration of the results of statistical modeling made it possible to draw a conclusion about the efficiency of the method and the quasi-Gaussian filter.  
Further tests of the new method on model data, aimed at estimating the mathematical expectation and its RMS errors with respect to the durations and magnitudes of model decreases, showed the method capabilities in disturbance recognitions. It can be seen on Figure 4 that the coefficients \(\epsilon^{\circ}_{gm}\) turned out to be less than the values of the coefficients \(\epsilon^{\circ}_{fm}\), on average, by about 10-30%. The nature of the plots of coefficients \(\epsilon^{\circ}_{ge}\) and \(\epsilon^{\circ}_{for}\) for the RMS for the same parameters \(d\mu_{a},d\mu\) is almost the same--the coefficients \(\epsilon^{\circ}_{gv}\) are less than the values of the coefficients \(\epsilon^{\circ}_{for}\), on average, also by \(\approx\)10-30%. From the point of view of recognizing decreases in duration and magnitude, the use of a quasi-Gaussian filter is more preferable than a common FIR filter.  
Finally, tests made on real experimental datasets from a muon hodoscope display the method application to data processing and recognition of intervals of decreasing and recovering muon flux intensity. Due to the noise reduction in \(X_{EG}(T_{0}n)\), it became possible to clearly see the intervals of quiet data ( Figure 5), intervals with decreases and recoveries and intervals with declines in mathematical expectation; all these recognized intervals were denoted by a line \(X_{ES}(T_{0}n)\) (index 3 on Figure 5). On the intervals with the boundary points 5900-5954, 6057-6121, 6197-6276 there were quiet data, on the time intervals 5969-6043, 6127-6189 a decrease in mathematical expectation was observed, the time intervals 5955-5970, 6044-6056, 6122-6126, 6190-6196 corresponded to data with decreases and recoveries. On the intervals 5969-6043, 6127-6189, it is quite possible to recognize relative reductions in mathematical expectation. The errors of the mathematical expectations estimates were \(\Delta m^{\circ}\) = 0.0010-0.0015, which is 10-30% from the mathematical expectation values obtained, and this led to successful recognition of decreases with the relative decrease rates of 0.02-0.03 and an average duration of \(~{}\)_mathrm approx_ 10 h.  
Testing the proposed method and quasi-Gaussian filter for data variants with indices \(j_{0}~{}=~{}31\), \(i_{0}~{}=~{}30\), allowed to obtain results that are almost similar to those depicted on Figure 5); the errors in the estimation of the boundary points of the sections during  
Figure 5: Results of quasi-Gaussian filtering and identification of regions with Forbush decreases: original data (line 1), filtered data (line 2), recognized intervals of various muon flux intensity (line 3).  
recognition with depressions amounted to \(\delta n\approx\) 2-5 h. Thus, testing on experimental MH data allowed us to make a conclusion about the efficiency of the method and the quasi-Gaussian filter and their satisfactory capabilities for recognizing mathematical expectation small in duration and magnitude.