# 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 the Method and the Quasi-Gaussian Filter on Experimental Normalized Poisson Data from the URAGAN Hodoscope  
Testing in this section consisted of determining the performance and capabilities of the proposed method and the quasi-Gaussian filter for recognizing small in duration and magnitude decreases in time intervals for the experimental hourly normalized Poisson data registered by the URAGAN hodoscope, taken from [12]. For analysis, a time interval was selected from 09/02/2017, 20:00 UTC to 09/18/2017, 15:00 UTC, with a total duration of 15.6 days. During this interval, the heliosphere was turbulent due to strong solar coronal mass ejections (CMEs) The CMEs that occurred on that period, caused intense geomagnetic storms that were discussed, for example, in [19; 20]. The emerging CMEs caused modulations of muon fluxes recorded in MH and led to lower mathematical expectations (including the ones due to Forbush decreases) in Poisson MH data. MH data were the matrix series of distribution functions of the intensities of muon fluxes \(Y_{E}(i,j,T_{0}n)\), defined in a rectangular region \(i=1,\ldots,N_{1},j=1,\ldots,N_{2},N_{1}=90\), \(N_{2}=76\), \(n=1,2,\ldots\) Solid angles correspond to azimuth and zenith indices \(i\), \(j\), \(\vartheta_{i}=\Delta\vartheta(i-1)\), \(\vartheta_{j}=\Delta\vartheta(j-1)\), \(\Delta\varphi=1^{\circ}\), \(\Delta\theta=4^{\circ}\) in which the registered particles were counted. MH data \(Y_{E}(j,i_{0},T_{0}n)\) were a time series with indices \(j_{0}\), \(i_{0}\); the considered interval was determined for \(n_{E\min}\leq n\leq n_{E\max}\), \(n_{E\min}=5900\), \(n_{E\max}=6275\) (counting hours for [12] began from the first hour of 2017). Figure 5 shows the results of quasi-Gaussian filtering and interval recognition with reductions in mathematical expectation. The original data \(Y_{E}(T_{0}n)\) for \(j_{0}=30\), \(i_{0}=31\) were denoted by light gray jagged lines (index 1). Fluctuations in data with a period of \(\approx\)24 h and an amplitude of \(\approx\)0.0037-0.0040 are due to the daily rotation of the MH with the Earth. Line with index 2 depicts the data \(X_{EG}(T_{0}n)\) filtered based on quasi-Gaussian filter. The recognized intervals of intensity decrease, intensity recovery and intensity mathematical expectation decrease were denoted by a piecewise linear spline-like dashed line \(X_{ES}(T_{0}n)\) (index 3). Analysis of intervals 5969-6043, 6127-6189 based on \(X_{ES}(T_{0}n)\) leads to a conclusion that the mathematical expectation values of decreases on them were \(\Delta m_{1}=0.01\), \(\Delta m_{2}=0.005\) for the relative decrease rates \(d\mu_{1}=0.027\), \(d\mu_{2}=0.020\). For \(Y_{E}(T_{0}n)\) and \(X_{EG}(T_{0}n)\), the mathematical expectations on these intervals were \(m^{\circ}_{E1}=0.3505\), \(m^{\circ}_{E1}=0.3510\), and \(m^{\circ}_{EG1}=0.490\), \(m^{\circ}_{EG2}=0.3520\), respectively, on average. 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. Testing on experimental MH data made it possible to draw a conclusion about the efficiency of the quasi-Gaussian filtering method and its satisfactory capabilities for recognizing small fluctuations of the mathematical expectations. Figure 4: Results of calculating the coefficients of relative errors \(\epsilon^{\circ}_{gm}\), \(\epsilon^{\circ}_{fm}\).