# Low-Pass Filtering Method for Poisson Data Time Series
## 2 Method
### Quasi-Gaussian Filter Synthesis Procedure  
We can synthesize the quasi-Gaussian filter based on the technology of stochastic global minimization of the (7) functional with the constraints (8) and (9) using the optimization algorithm for annealing simulation [14; 15]. To implement it, we will use the simulannealbnd.mat software module from the Matlab Global Optimization Toolbox [16]. Let us form a parallelepiped of constraints \(\overline{A}_{0}\) of dimension \(r_{0}\) with boundaries \(\overline{a}_{r}\), \(r=1,\ldots,r_{0}\)--\(a\in\overline{A}_{0}\), \(\overline{A}_{0}=\{a:(0\leq a_{r}\leq\overline{a}_{r},\ r=1,\ldots,r_{0})\}\) and a new--with respect to (7)--functional \(\overline{S}(H_{0g},a)\) with a penalty term taking into account the constraint equality (8). Let us implement the global minimization of \(\overline{S}(H_{0g},a)\) taking into account \(\overline{A}_{0}\) using [16]. Let us set the initial vectors for the first iteration \(a_{1}(I)\in\overline{A}_{0}\), uniformly distributed in \(\overline{A}_{0}\), \(I\)--a single descent procedure, \(I=1,2,\ldots,I_{0}\), \(I_{0}\)--a total number of descent procedures. Let us assume that each descent procedure consists of \(m_{0}\)--a total number of iterations, \(m\)--a single iteration, \(m=1,2,\ldots,m_{0}\). During descent, we assume that the initial value of the vector of parameters for \((m+1)\)-st iteration is equal to the calculated optimal value for the vector of parameters for \(m\)-th iteration--\(a_{m+1}(I)=a_{m}^{\circ}(I)\). In each iteration, we perform \(n_{0}\) descent steps, \(n\) is a descent step number, \(n=1,2,\ldots,n_{0}\). Next, we will calculate the sequence of the functional \(S(m_{0},I)\)= \(\overline{S}(H_{0g},a^{\circ})\) values and the corresponding optimal vectors \(a^{\circ}(m_{0},I)\), \(I=1,2,\ldots,I_{0}\). For global minimization, we search for the optimal index \(I^{\circ}\) corresponding to the minimum of the \(S(m_{0},I)\) functional, and the optimal vector \(a^{\circ}\) using brute force:  
\[I^{\circ}=\arg(\min_{1\leq I\leq I_{0}}S(m_{0},I)\},a^{\circ}=a^{\circ}(m_{0},I^{\circ}).\]  
On Figure 1, the example plots of the minimized \(S(m,I)\) functionals are displayed, depending on iteration number \(m\) and the descent procedure number \(I\). Functionals are shown starting with \(m=2\), since for \(m=1\) their values are very large. Here, \(m_{0}=20\); as the iteration number increases, the values of the functionals decrease. During the optimization process, a movement is made in a \(r_{0}\)-dimensional space from one local minimum to another. Let us consider an example of quasi-Gaussian FIR filter synthesis. Based on the analysis of hourly experimental MH data from [12], it was found that the durations of possible fluctuations of the mathematical expectation in them were, on average, \(\approx\)10 \(\div\) 20 h and more. The dimension value \(r_{0}\), that could possibly allow the recognition of such fluctuations in mathematical expectations, was equal to 8. For a prototype filter FR (6), the parameter \(w_{c0}\) was related to the assigned cutoff frequency \(w_{c}\) based on (6)  
\[(0.5)^{1/2}=\exp(-(w_{c}/w_{c0})^{2}),w_{c0}=w_{c}/(0.5\cdot\ln 2)^{1/2}.\]  
We assign the cutoff frequency \(w_{c}=0.1\), find \(w_{c0}\) and define \(H_{0g}(w,w_{c})\)--the prototype filter FR. By defining \(L\) we set the number of discrete normalized frequencies \(w_{l}\) of calculations of the functional (7) for \(0\leq w_{l}\leq 1.0\), let us assume that \(L=100\) in our calculations. The polyharmonic FR function \(|H(jw)|\) (5) is formed from components performing 1, 2,..., \(s_{0}\) fluctuations in this interval. For the accepted values \(L\) and \(r_{0}\), one period of the polyharmonic component with the maximum frequency corresponding to the number \(s_{0}\) in (5), accounted for \(\approx\)15 sampling points of normalized frequencies \(w_{l}\), \(l=1,\ldots,L\), which fully provided a fairly accurate calculation of the functional (7) necessary for direct search.