# Low-Pass Filtering Method for Poisson Data Time Series
## 3 Results  
### Testing the Method and the Quasi-Gaussian Filter on Model Normalized Poisson Data  
#### 3.1.1 Testing on Model Hourly Data Using Statistical Modeling  
Testing of the proposed method and quasi-Gaussian filter was carried out on model hourly normalized data using statistical modeling [17]. For this purpose, on the basis of the Matlab module exprmd.mat [18], exponentially distributed model random numbers \(\tau_{i}\), \(i=1,2,\ldots\) were generated, with their mean value \(\tau_{M0}\), and the evenly distributed random registration time intervals \(T_{0k},k=1,2,\ldots\) within the range \(T_{0\ min}\leq T_{0k}\leq T_{0\ max}\). The number of Poisson model events \(N_{M}(Tk)\) was counted on the registration time intervals \(T_{0k}\). Finding \(N_{M}(Tk)\) was carried out by solving the conditional maximization problems:  
\[N_{M}(Tk)=\arg\{\max N_{M}\}_{i}>0, \tag{11}\]  
providing that \(T_{0k}-\sum_{i=1}^{N_{M}}\tau\), where for \(T_{0k},\ k\ =\ 1,2,\ldots k_{0}\) the range bounds \(T_{0\ min}\ =\ 57\) s, \(T_{0\ max}\ =\ 59.5\) s were assigned (see the Introduction section). Initial model 1-minute-sampled and normalized Poisson-distributed data were constructed according to (11) and the calculation of relations \(\overline{N}_{M}(Tk)\), similar to (1):  
\[\overline{N}_{M}(Tk)=N_{M}(Tk)/T_{0k},k=1,2,\ldots k_{0}. \tag{12}\]  
The modulation of the average number of Poisson events in order to model decreases (increases) in the mathematical expectation was carried out by specifying the mean value function \(\tau_{M0}(Tk)\) on the intervals \((T(k-1),\ \ Tk)\) for \(k\) from (12). For this, the relative modulation function \(\mu(Tk),\ \ k\ =\ 1,2,\ldots,k_{0}\) was formed and the initial temporal index of the modulation decrease \(k_{a}\), the duration of the decrease \(dk_{a}\) and the depth of the relative decrease \(d\mu\) : The function \(\mu(Tk)\) was represented by the relations \(\mu(Tk)\ =\ 1-d\mu\) for \(k_{a}\leq k\leq k_{a}+dk_{a}\), \(\mu(Tk)\ =\ 1\) for \(1\leq k<k_{a}\), \(k_{a}+dk_{a}+1\leq k\leq k_{0}\). For the calculation example, the average number of Poisson model events per minute was set \(N_{M0}\ =\ 25\), normalized average \(\overline{N}_{M0}\ =\ N_{M0}/T\), modulated normalized mean \(\overline{N}_{M0}(Tk)\ =\ \overline{N}_{M0}\mu(Tk)\ =\ N_{M0}\mu(Tk)/T\), \(k\ =\ 1,2,\ldots,k_{0}\) and the parameter \(\tau_{M0}(Tk)\ =\ 1/(\overline{N}_{M0}(Tk)-1)\) was calculated. Based on [18], random exponentially distributed numbers with \(\tau_{M0}(Tk)\) and random evenly distributed values with \(T_{0\ min}\ =\ 57s\), \(T_{0\ max}\ =\ 59,5s\) were generated, with the use of which by (11), model Poisson data \(N_{M}(Tk)\) and by (12)--normalized Poisson data \(\overline{N}_{M}(Tk)\) were calculated. Further, similarly to (1), a time series of averaged model hourly normalized Poisson data was formed:  
\[Y_{M}(T_{0}n)=\frac{1}{60}\sum_{k=1+60(n-1)}^{k=60n}\overline{N}_{M}(Tk),n=1,2,\ldots,n_{0},n_{0}\ =\ ent(\ k_{0}/60). \tag{13}\]  
For modeling, we assumed \(k_{0}\ =\ 6000\), which corresponded to the model minute data produced during 4.166 days. For the modulation function, the values \(\ k_{a}\ =\ 1920\), \(dk_{a}\ =\ 1440\) and \(d\mu\ =\ 0.02\) were taken. Model hourly averaged data \(Y_{M}(T_{0}n)\) for (13) with \(\ n_{0}\ =\ 100\), \(n_{a1}\ =\ 32,n_{a2}\ =\ n_{1}+dn_{a}\ dn_{a}\ =\ 24\). Figure 3 shows an example of statistical modeling results: the jagged light gray line with index 1 displays the \(Y_{E}(T_{0}n)\) plot; the solid line with index 2 denotes the fragment of \(X_{EG}(T_{0}n)\) which is the result of filtering the model dataset using a quasi-Gaussian filter; for comparison, the dashed line with index 3 denotes the fragment \(X_{EF}(T_{0}n)\) which is the result of filtering the model dataset using the software module fir1.mat [3].