## 2 Materials and Methods
### Fundamental Differences between Classical and Non-Classical Methods of Mathematical Modeling
Thus, in [20], a method of a posteriori control over the stability of observation conditions identified by modern absolute ballistic gravimeters of free fall acceleration based on the methods of the non-classical error theory was developed alongside a NETM diagnosis of the probabilistic form of measurement distributions to improve the methodology of these high-precision determinations. Algorithm for Estimating the Accuracy of the Results of GNSS Measurements Identified by the Non-Classical Error Theory of Measurements  
The essence of the theory of the mathematical diagnostics of models in GNSS time series is reduced to the statistical analysis of the differences in \"observation-calculation\" (O-C) based on the following weight function:  
\[p(x_{i})=\frac{f^{\prime}(x_{i})}{x_{i}\cdot f(x_{i})}=\left[\left(\frac{m-0.5} {m}\right)^{3}\cdot\sigma+\frac{x_{i}^{2}}{2m}\right]^{-1} \tag{5}\]  
which allows for a fairly rigorous, simple, and visual assessment of the modeling accuracy. Usually, the differences (O-C) are obtained as residuals of the time series. The influence of weak, non-excludable correlated systematic errors can be neglected only if the weighting function of measurement errors (5) is non-singular. It should be noted that the weighting function is non-singular only if it is sufficiently proven that the actual distribution of errors corresponds to Formula (2). Weight function (5) is singular when the error distribution has significant asymmetry. Let us consider the algorithm of actions for evaluating the results of high-precision GNSS observations, namely, modeling diagnostics using the NETM. The software package for diagnosing GNSS stations uses data of high-precision determinations of spatial topocentric rectangular coordinates (\(N,E,U\)) at permanent observation stations as part of international, regional, or national networks. It should be noted that their reliable accuracy should be at the level of several mm, and the errors in the rate of coordinate changes should not exceed 0.1-0.3 mm/year. Therefore, the following algorithm for diagnosing time series modeling using mathematical methods of statistics and NETM is proposed below:  
1. Finding the arithmetic mean of sample n > 500;
2. Calculating errors and central sampling points;
3. Calculating the skewness (\(A\)) and kurtosis (\(\varepsilon\)) of the sample using unbiased central moments \(m_{2}-m_{8}\)[20]: \[m_{r}=\frac{\sum(x_{i}-\overline{x})^{r}}{n}\] (6) where \(r=2,\ 3,\ 4,\ 5,\ 6,\ 8;\overline{x}=\frac{\sum x_{i}}{n}\);
4. Finding the standard of skewness (\(\sigma_{A}\)) and kurtosis (\(\sigma_{\varepsilon}\));
5. Building confidence intervals for skewness and kurtosis. To diagnose the modeling, it is enough to find 90% confidence intervals, i.e., using the quantile \(t_{\alpha}=1.645,\alpha=10\%\), and the following formulas: \(A\pm 1.645\cdot\sigma_{A}\), \(\varepsilon\pm 1.645\cdot\sigma_{\varepsilon}\). Provided that the confidence intervals for \(A\) and \(\varepsilon\) cover zero, processing the GNSS observations can be limited to classical estimation methods;
6. Performing diagnostics of mathematical modeling through <<observation-calculation>> differences based on the constructed confidence intervals for the skewness and kurtosis. NETM methods should be used when the confidence interval for (\(H_{0}\): \(A=0\)) covers zero and when the entire interval for \(\varepsilon\) is in the positive region (\(H_{0}\): \(\varepsilon>0\)) or the confidence interval is in the negative region (\(H_{0}\): \(\varepsilon<0\)) without covering zero. All other cases indicate various pathologies in the operation of GNSS equipment, the processing, or unacceptable observation conditions (the state of the antenna installation center, the presence of a constant source of multipath signals, specific local geophysical parameters, etc.). All the diagnostic parameters based on the constructed confidence intervals for skewness and kurtosis can be found in Table 1;7.