## 3 Materials and Methods
### Time Series Decomposition
Although the processing procedure was performed for signals recorded by all SRDN-3 probes, it is exemplified here only for the signal from probe no. 2. The least squares method (LSM) is a classical statistical tool used to fit a function, most often a linear function, to a set of measurement points. In the context of time series analysis, it enables the estimation of a linear trend in the data, which is useful for identifying long-term changes in environmental radon levels. Let us assume a series of measurement data in the form: (\(t_{1}\), \(x_{1}\)), (\(t_{2}\), \(x_{2}\)),..., (\(t_{n}\), \(x_{n}\)), where t\({}_{i}\) denotes the measurement time, and x\({}_{i}\) the signal value. The aim of the method is to find a straight line (2),  
\[x(t)=at+b \tag{2}\]  
The sum of the squared deviations of the actual values x\({}_{i}\) from the values predicted by the model is minimized (3). \[S=\sum_{i=1}^{n}(x_{i}-at_{i}-b)^{2} \tag{3}\]  
To find the coefficients a (slope) and b (intercept), one must solve the system of normal Equations (4) and (5). \[a=\frac{n\sum t_{i}x_{i}-\sum t_{i}\sum x_{i}}{n\sum t_{i}^{2}-\left(\sum t_{i }\right)^{2}} \tag{4}\]  
\[b=\frac{\sum x_{i}-a\sum t_{i}}{n} \tag{5}\]  
Once the coefficients are determined, the fitted line can be used to predict values at future time points, assess long-term trends, or remove the trend component for further analysis of signal residuals. The least squares method assumes linearity of the model, as well as independence and constant variance of the measurement errors. The stationarity of all data in the time series (for probes no. 2-6) was proven using the Dickey-Fuller test [58]. The Augmented Dickey-Fuller (ADF) test is a statistical test used to determine whether a time series is stationary, i.e., whether its statistical properties (mean, variance, autocorrelation) remain constant over time. It is an extension of the Dickey-Fuller test that accounts for  
Figure 5: Trend line for radon probe SRDN-3 No. 2.  
higher-order autoregressive processes by including the lagged differences of the dependent variable. The general regression equation for the ADF test is (6),  
\[\Delta y_{t}=\alpha+\beta t+\gamma y_{t}^{-1}+\sum\left(\delta_{\mathrm{i} \Delta y_{t-1}}\right)+\epsilon_{t,\ \ for\ i=1\ to\ p} \tag{6}\]  
where:  
\(y_{t}\)--value of the time series at time t,  
\(\Delta y_{t}=y_{t}-y_{t-1}\)--first difference of the series,  
\(\alpha\)--constant term (drift),  
\(\beta t\)--deterministic time trend (optional),  
\(\gamma\)--coefficient on the lagged level of the series (used to test for a unit root),  
\(\delta_{t}\)--coefficients on lagged differences (adjust for autocorrelation),  
\(\epsilon_{t}\)--white noise error term,  
\(p\)--number of lagged difference terms included. In all cases, the H0 null hypothesis about the unit root was rejected Table 2.  
The removal of the linear trend allowed the residuals to be represented with a clearly seasonal character (Figure 6). After the linear components were reduced, a periodicity analysis was performed on the radon time series. As mentioned above, the total study period was four months, and it was further divided into approx. 1-month sub-periods. Thus, seasonality had to be removed from the series so that it did not disturb the short-term variability (the function inflection points could disturb the correlations). The most commonly used method for determining periodicity is the Fourier Transform (FT), which allows the identification of repetitive components contained in the signal. The transform provides global information on the frequencies present in the signal. The Fourier Transform decomposes the signal x(t) into orthogonal trigonometric functions (7),  
\[X_{FT}(f)=\int_{-\infty}^{\infty}x(t)e^{-\beta\pi ft}dt \tag{7}\]