A study of the radon seasonality with temporal dummy variables

Abstract

Radon, a naturally occurring radioactive gas, has garnered significant attention due to its health risks and its potential role as a seismic indicator. Variations in radon levels have been observed in correlation with seismic activity, suggesting that radon could serve as an early warning signal for earthquakes. However, accurately forecasting radon concentrations remains challenging due to the influence of various factors, including meteorological conditions and seasonal fluctuations. Considerable effort has been dedicated to investigating the use of regression models to predict radon levels by incorporating meteorological parameters such as temperature, humidity, and atmospheric pressure. The aim of this study is to improve the modeling of baseline radon concentrations by removing periodic sources of variability (primarily environmental and seasonal effects), rather than directly focusing on radon anomaly detection. Accurate background modeling is a prerequisite for reliable anomaly detection, as it enables a clearer distinction between normal fluctuations and potential anomalies arising from endogenous factors, including seismic or structural phenomena. In particular, we show that the impact of meteorological parameters on radon prediction can be effectively replaced by a seasonal regression model based on temporal dummy variables, without significant loss in predictive accuracy. This method offers a promising alternative for radon modeling, enabling early estimation of average radon levels and facilitating the timely identification of anomalous behavior. Our findings suggest that seasonal regression models based on dummy variables provide a robust and accurate framework for forecasting radon, with potential implications for improved seismic monitoring. Importantly, any future application to anomaly detection must also account for structural changes at the measurement site, which may independently affect radon emissions.

Introduction

This study focuses on ²²²Rn, a naturally occurring radioactive isotope of radon. Its presence in the environment results from the decay of uranium and thorium in the Earth’s crust, and the subsequent migration of radon from the lithosphere towards the Earth’s surface and atmosphere. Over the past decades, radon emanation, exhalation, and diffusion mechanisms have been extensively studied, due to its significance as both a health hazard and a geophysical tracer^1,2. Prolonged exposure to high concentrations of radon has been linked to lung cancer³, highlighting the importance of measuring and monitoring radon levels in indoor environments. In addition, radon is acknowledged to be an efficient marker for dynamic geological processes⁴, providing valuable insights into seismic activity, groundwater monitoring, volcanic eruptions, and tectonic movements.

Radon has a half-life of 3.824 days and it is detectable primarily through advection, with key carriers including carbon dioxide (), methylene (), and nitrogen (). The study of radon is complicated by the stochastic nature of its emissions and the wide range of factors that can affect its exhalation and diffusion (Fig. 1). It is widely acknowledged that meteorological parameters, including temperature, rainfall, barometric pressure, relative humidity, and wind variations, play a significant role in shaping the dynamics of radon release. Indeed, radon measurements are known to exhibit seasonal variability, driven by climatic factors that collectively cause dynamic fluctuations in radon levels: temperature affects radon diffusion, with warmer conditions enhancing gas transport; heavy rainfall reduces soil permeability, limiting radon’s ability to escape from the ground into the air; low atmospheric pressure increases the gradient between soil and atmosphere, driving radon release, while moist air tends to trap radon near the ground, limiting its dispersion; strong winds dilute and disperse radon, reducing concentrations near the ground by mixing it with the surrounding air. All of these factors result in radon seasonal patterns, and numerous efforts have been made to isolate the impact of weather-related parameters, with the aim of improving the accuracy of radon monitoring and its potential applications in geophysical studies and risk assessments. Table 1 summarizes some contributions proposed in the literature from the 2000s to the present. Indoor, outdoor, and soil radon concentrations have been monitored over various time intervals and regions around the world. Models capable of replicating the observed seasonality have been proposed by selecting subsets of meteorological parameters and applying different analysis techniques of increasing complexity - from correlation analysis^5–11 to Multiple Linear Regression (MLR) models^12–16 and Multilayer Perceptron (MLP) neural networks^17–21. The wide range of solutions proposed, both in number and type of meteorological parameters identified as the most influential on radon concentrations, highlights the challenge of accurately quantifying the impact of weather on radon. While most proposed methods rely on linear, constant-parameter models, the general consensus is that radon concentrations show significant temporal and spatial variability, primarily influenced by variations in geological structures and climatic conditions. This variability points to potential cross-correlation among the selected predictors, and the influence of additional, unaccounted-for factors, like tidal forces¹³ and tectonic activity. The proposed additional mechanisms involve both the direct influence of tidal forces on the water table and the mixture of gases within the rocks (e.g. , water vapor, radon), as well as the indirect deformations caused by Earth tides, which promote rock relaxation and create pathways for gas transport. These processes are highly dependent on site-specific subsurface geological and hydro-geological conditions, adding further complexity to the assessment of the impact of weather on radon.

Fig. 1.

Key factors influencing radon emanation, exhalation, and diffusion processes²². After radon is produced from the decay of radium within solid grains, it migrates towards lower concentration regions near the Earth’s surface, eventually entering the atmosphere. Radon flux is governed by geophysical, geological, and atmospheric factors. The intricate relationships among these variables entangle the understanding and prediction of radon concentration levels over time.

Table 1.

Summary of some contributions from the literature (references are specified in the first column) in which the seasonality of radon was modelled by selecting a set of meteorological parameters (T, temperature (soil and/or air); P, barometric pressure; H, relative humidity; R, rainfall; W, wind speed and/or direction; S, solar radiation) and a specific technique (see “Analysis” column: MLR, multiple linear regression” PCR, principal component regression; BDT, boosted decision trees; MLP, multilayer perceptron). The table also shows the region in which the radon measurement was made, the time interval during which it was monitored, and the results of the analysis (see “Location”, “Time”, and “Outcome” columns, respectively).

References	Radon	Location	Time	Weather						Analysis	Outcome
				T	P	H	R	W	S
20015	Indoor	UK	<1y							Correlation analysis	Radon primarily dependent on barometric pressure, vapor pressure*, and wind variations
200212	Indoor	UK	2y							MLR	Most of the radon variation explained by temperature, less by wind speed, rainfall, and pressure
20036	Indoor, outdoor	Brazil	1y							Correlation analysis	Radon inverse correlation with temperature and rainfall, after correction for a lag time
200323	Outdoor (borehole)	Slovenia	1.5y							Regression trees	Regression trees outperform other regression models, radon anomalies before or during earthquakes
200524	Indoor, outdoor	USA	1y							Comparative analysis	Radon heavily influenced by indoor and outdoor temperature differentials and rainfall
200613	Indoor	UK	<1y							Correlation analysis, MLR	Weak radon correlations with rainfall and mean daily temperature
20067	Outdoor	Italy	3y							Correlation analysis	Radon negative correlation with temperature, positive correlation with pressure and relative humidity
200825	Soil, indoor	Norway	<1y							Comparative analysis	Radon variations predominantly caused by changes in air temperature, less by air pressure
201017	Outdoor (borehole)	Slovenia	2y							MLP	Radon highest sensitivity to soil and air temperature, lowest to rainfall. Prediction of seismic events
20108	Indoor	Italy	1y							Correlation analysis	Radon strong correlation with rainfall and humidity, difference between building floors
201414	Soil	India	2y							MLR	Radon primarily dependent on relative humidity, correlation with seismic activity
201418	Indoor	Serbia	<1y							BDT, MLP	BDT method is better than MLP, soil temperature is the most important parameter
201510	Indoor	Italy	4y							Correlation analysis	Radon significant positive correlation with temperature, less with pressure; negative with rainfall
20159	Indoor	USA	1y							Correlation analysis	Radon negative correlation with indoor humidity, outdoor temperature and wind speed
201615	Outdoor	Italy	4y							MLR, PCR	Radon primarily dependent on temperature and humidity, no clear correlation with volcanic activity
201916	Outdoor	Germany	2y							MLR	Soil wetness strongly affects radon, different influencing-mechanisms of soil and air temperature on radon
202019	Soil	Italy	7y							MLP	Radon forecasting and anomaly detection; volcanic unrest affects radon emissions**
202120	Soil	Pakistan	<1y							MLP, MLR, Decision Trees	MLP outperforms other techniques, radon anomalies around earthquakes’ occurrences
202321	Soil	India	3y							Wavelets, MLR, MLP	MLP outperforms other techniques, radon anomalies correlated with seismic events

*Computed from relative humidity and temperature.

**Also fumarolic tremor, background seismicity and gas concentration used to train the neural network.

In this paper, we analyze long-term and continuous radon emissions recorded by the IRON (Italian Radon mOnitoring Network), which consists of multiple radon monitoring stations distributed across the Italian peninsula (see “Radon monitoring stations” section for details on the IRON installation sites and instrumentation used for this study). The radon time series are examined alongside hourly meteorological data (refer to “Meteorological stations” section for information about the meteorological station locations and data). Leveraging these extensive datasets, we demonstrate that long-term radon levels can be accurately estimated using linear regression models that incorporate temporal dummy functions designed to capture seasonal patterns, achieving performance comparable to models that rely on meteorological parameters. It should be emphasized that our approach is strictly statistical in nature and does not incorporate any modeling of the underlying physical mechanisms governing radon variability.

Results

The IRON has been monitoring radon emissions across Italy since 2009. This study analyzed 18 radon time series, selecting those with at least two and a half years of data and corresponding meteorological records. As detailed in “Regression analysis” section, we applied multiple regression analyses to the radon time series, a summary of which is presented in Table 2. First, we used temperature, pressure, rainfall, relative humidity, wind strength, and solar radiation meteorological variables as predictors to model variations in the radon time series based on meteorological conditions. As alternative approaches, we used dummy variables representing monthly, fortnightly, weekly, and half-weekly intervals (with a separate regression for each interval) to capture temporal patterns in radon fluctuations. More details about the dummy-based regression models are described in “Regression analysis” section. The predictive accuracy of each model was assessed using the Mean Absolute Percentage Error (MAPE) evaluated over extended test periods, similar to the approach used by Ambrosino¹⁹. The results of regression analyses applied to each of the 18 radon time series are shown in Tables 3 and 4. As an example, the regressions applied to the ISAR time series are shown in Fig. 2. Finally, Figs. 3 and 4 show the probability density functions and the cumulative distributions of MAPE values for the different cases.

Table 2.

Summary of the regression models used for analysis of radon time series.

Regression model	Predictors	Dependent variable
Meteorological model	6 meteorological variables	Radon
Half-weekly model	104 half-weekly dummy variables	Radon
Weekly model	52 weekly dummy variables	Radon
Fortnightly model	26 fortnightly dummy variables	Radon
Monthly model	12 monthly dummy variables	Radon

Table 3.

Summary of the train Mean Absolute Percentage Error (MAPE) values resulting from the five regression models applied to the radon time series. The mean and the median values for each column (model) are also reported.

Station	Meteo model	Half-weekly model	Weekly model	Fortnightly model	Monthly model
AQU	27.7820	17.9739	18.4055	19.1432	21.6786
BADI	7.8455	6.1138	6.1216	6.1795	6.4609
BATI	31.4061	22.3017	22.4146	22.6624	23.5177
BAT2	25.3646	18.2463	18.5067	19.1186	21.0716
CDCA	31.0440	21.1901	21.2111	21.5050	21.6961
CMDO	14.9466	14.3038	14.3766	14.6659	14.7001
CMPL	13.1382	11.3898	11.5913	12.0082	12.8327
CTTR	11.2221	8.7434	8.7961	8.8331	8.9027
DUR	10.9860	7.5774	7.5914	7.5724	7.6837
FRME_E146	36.7184	22.6046	22.8946	25.3004	24.7512
FRME_H146	13.1882	9.7227	9.7959	10.0501	9.7003
GUAR2	25.1024	19.9661	20.0449	20.0886	21.1411
ISAR	11.8780	9.0291	9.2322	9.7397	11.0554
MMN	48.3965	39.1922	40.1925	40.5053	41.6651
MMNG	30.9141	19.5807	19.9170	20.2785	20.2163
MTR1	30.8632	21.7546	22.1629	22.6970	23.7447
NRCA	22.3031	14.5267	14.5600	14.6166	14.6356
SSFR	72.3525	51.0414	51.9348	53.7294	59.8330
Median	25.2335	18.1101	18.4561	19.1309	20.8941
Mean	26.0383	18.9150	19.1254	19.5422	20.6712

Table 4.

Summary of the test Mean Absolute Percentage Error (MAPE) values resulting from the five regression models applied to the radon time series. The mean and the median values for each column (model) are also reported.

Station	Meteo model	Half-weekly model	Weekly model	Fortnightly model	Monthly model
AQU	36.0843	24.2457	24.3190	24.2303	26.6852
BADI	17.3287	16.4086	16.3971	16.4291	16.6514
BAT1	24.7033	38.7609	38.5701	38.2332	37.1515
BAT2	90.0601	96.0027	95.7498	95.6324	92.4582
CDCA	26.6328	19.2363	19.1746	17.9338	18.0511
CMDO	13.4836	13.1286	13.1059	13.0816	12.7315
CMPL	22.7445	17.4978	17.3654	16.3228	16.1692
CTTR	17.1341	15.5653	15.5348	15.5279	15.5077
DUR	6.2596	10.1388	10.1361	9.7264	8.5610
FRME_E146	30.1782	13.0719	12.3976	11.5404	11.6641
FRME_H146	16.0860	13.1607	13.0251	12.9233	12.6491
GUAR2	20.2106	17.3486	17.1175	17.0071	17.9196
ISAR	12.5604	11.2503	10.8161	11.4446	9.7241
MMN	47.8595	36.0497	35.9958	36.5308	37.4817
MMNG	32.6649	18.0612	17.3993	17.8657	18.0950
MTR1	24.0605	22.7551	22.7559	22.3802	20.6080
NRCA	19.1912	19.6213	19.4909	19.3692	19.7476
SSFR	133.5172	125.4396	125.8500	126.6282	127.7041
Median	23.4025	17.7795	17.6481	17.4364	17.9267
Mean	32.8200	29.3191	29.2074	29.0851	28.8579

Fig. 2.

Comparison of regression models for the ISAR station. The black line represents the observed data, the red line shows the model predictions during the training period, and the blue line shows the predictions during the test period. Subfigure a presents the regression model that uses meteorological parameters as predictors. Subfigure b shows the model incorporating half-weekly dummy variables, while subfigure c uses weekly dummies. Subfigure d displays the model based on fortnightly dummy variables, and subfigure e illustrates the regression using monthly dummy variables.

Fig. 3.

Kernel smoothing for Probability Density Function (PDF) estimate of test MAPE values. To control the smoothness of the resulting density curve, it was used a bandwidth value calculated by the normal approximation method (or Silverman’s rule of thumb). As one can see, there is a shift between the meteorological model MAPE peak and the dummy functions density ones. On the other hand, the overall distributions of the two approaches exhibit a very similar pattern.

Fig. 4.

Empirical cumulative distribution function (CDF) of test MAPE values. As observed, the elbow point in the meteorological approach shows a displacement compared to the ones in the dummy functions cases. However, the overall distributions (particularly in the tails) appear very similar across all approaches.

To compare the accuracies of the meteorological regression model and the four dummy-based models in a statistical meaningful way, we applied two different non-parametric statistical tests: the Smirnov test and the Wilcoxon signed rank test (see “Statistical tests” section). The obtained p-values are summarized in Table 5.

Table 5.

Resulting p-values from statistical tests comparing the meteorological regression model with dummy-based models.

	Half-weekly model	Weekly model	Fortnightly model	Monthly model
Smirnov	0.4255	0.4255	0.4255	0.4255
Wilcoxon right-tailed	0.0203	0.0183	0.0164	0.0083
Wilcoxon two-tailed	0.0386	0.0347	0.0311	0.0156

Discussion

In this study, we compared the performance of a meteorological regression model with four alternative models incorporating dummy variables. It is important to emphasize that the primary objective of this analysis was to obtain statistical evidence supporting the hypothesis that the use of dummy variables does not compromise predictive accuracy relative to models based on meteorological predictors. A preliminary qualitative analysis of the MAPE values reported in Table 5 further supports this hypothesis. Indeed, models employing meteorological variables tend to exhibit higher prediction errors compared to those using dummy variables.

Although a model comparison was not the primary focus of our work, we evaluated the differences between various dummy variable configurations. As shown in the results, these differences are relatively small for both mean and median MAPEs (below ). Therefore, no single configuration can be universally recommended, and the choice largely depends on the characteristics of the data and the specific objectives of the analysis. In general, the selection of the temporal resolution for dummy variables should reflect a balance between the degree of variability observed in the time series (based on prior exploratory analysis) and the risk of introducing multicollinearity. When intra-month variability is limited, a monthly approach may be sufficient and preferable due to its simplicity and robustness. Conversely, in the presence of rapid and high-amplitude fluctuations, a finer temporal resolution (such as half-week intervals) may provide a more accurate representation of the seasonal pattern.

A series of statistical tests was conducted to determine whether the dummy-based models could be considered statistically equivalent to the meteorological model in predicting radon concentrations. All p-values obtained from the Smirnov test exceed 0.42, indicating the absence of statistically significant differences between the cumulative MAPE distributions of the meteorological model and those of the various dummy-based models. According to the Smirnov statistical meaning, this suggests that the overall error distributions of the meteorological model and each dummy-based model are similar, with no strong evidence of disparity.

Notably, we obtain identical p-values for the different Smirnov tests. This can be due to the following considerations. First, the Smirnov statistic is defined as the maximum vertical distance between the Empirical Cumulative Distribution Functions (ECDFs) of two samples (see “Statistical tests” section). Because ECDFs are step functions, small perturbations in the input data may not alter the maximum distance between them. As a result, the test statistic remains unchanged across slightly different datasets. In addition, the p-value associated with the Smirnov statistic is typically computed using an asymptotic approximation derived under the null hypothesis that the two samples originate from the same continuous distribution. As discussed in²⁶, this approximation becomes more accurate as sample size increases. For small samples, it may produce identical p-values for a range of statistic values due to limited resolution and rounding in the asymptotic formula.

Further insights were gained from the two-tailed and right-tailed Wilcoxon signed-rank tests, with the right-tailed test performed under the assumption that the dummy-based models’ errors exceed that of the meteorological model. The p-values from both sets of tests are all below 0.05, indicating that the dummy-based models may even outperform the meteorological model. However, this consideration extends well beyond the scope of our analysis. In our view, the mere equivalence of the two models is already a highly significant scientific finding.

The apparent discrepancy between the “no difference” outcome of the Smirnov test and the difference detected by the Wilcoxon test—a situation that can occasionally arise in statistical analysis—can be attributed to the differing sensitivities of the two methods. The Smirnov test is more responsive to differences in the overall geometry of the MAPE distribution, whereas the Wilcoxon test is primarily sensitive to shifts in central tendency, particularly median values. To clarify this point, we plotted both the probability density functions and the cumulative distribution functions of the MAPE values for the different models (Figs. 3 and 4), which visually support this interpretation. Nevertheless, the results of all statistical tests align with our main objective, reinforcing the conclusion that the dummy-based regression models do not suffer any loss in predictive accuracy compared to the model employing meteorological variables as predictors.

While it is widely acknowledged that the relationship between radon and meteorological parameters is nonlinear, suggesting a shift towards machine learning approaches (such as neural networks) for radon forecasting, it is important to note that the success of neural network models is highly dependent on their architecture and data processing procedures. In particular, when exogenous variables such as meteorological parameters are involved, the performance of neural networks becomes highly sensitive to the quality, availability, and predictive reliability of the inputs. Furthermore, deep learning methods are computationally intensive and time-consuming, making the selection of relevant variables a critical step before implementing such approaches. We plan to apply neural network approach using dummy functions in the future. In this study, we demonstrated that long-term radon forecasting can be effectively achieved using convenient seasonal dummy variables, offering an efficient and practical alternative to more complex machine learning techniques. However, before engaging in the design of more complex architectures, it was essential to assess whether the use of simple, interpretable seasonal dummy variables alone could provide a comparable level of accuracy. This preliminary step was necessary to justify the added complexity of future models and to verify if meteorological variables are indispensable for long-term radon forecasting. Future work will explore whether neural network architectures, potentially hybridized with seasonal dummy inputs, can further improve forecasting performance. The goal is to determine if such hybrid models can enhance predictive accuracy without compromising the operational feasibility and computational efficiency required for practical applications.

While the results support the effectiveness of our approach, some limitations should be acknowledged. First, the method is purely statistical and does not account for the physical processes underlying radon production, migration, and emission. This simplification may reduce its applicability in contexts where such processes are dominant or where site-specific geological or hydrological conditions introduce complex, non-seasonal dynamics. Moreover, the use of temporal dummy variables assumes the stability of seasonal patterns over time, an assumption that could be violated in the presence of structural changes at the monitoring site or long-term environmental shifts. Although our statistical tests on independent test sets do not suggest overfitting, models with higher-resolution dummy variables involve a greater number of parameters. Despite these limitations, the proposed dummy-based regression approach provides a simple, transparent, and computationally efficient method to model long-term radon variability. Its ability to reproduce seasonal patterns without relying on external meteorological inputs makes it particularly valuable in contexts where such data are unavailable or unreliable, and it offers a solid foundation for subsequent anomaly detection analyses.

Methods

The starting dataset

This section provides information about the dataset used for the analysis, consisting of radon and meteorological time series.

Radon monitoring stations

Since 2009, the IRON network²⁷ has been providing near real-time measurements of radon emissions from various stations located throughout Italy, mainly concentrated in the Central-Southern Apennines (see Fig. 5). The stations included in this study differ in both their installation types and the radon detectors employed. Specifically, radon has been measured passively using proprietary INGV instruments based on Lucas cell detectors, from here named just Lucas, and AER-C Algade©(http://www.algade.com/) detectors. For Lucas, radon gas diffuses into the detector’s flask, whose inner wall is coated with silver-activated zinc sulfide (ZnS), serving as the scintillating material that detects radon progeny. Lucas configured acquisition window is about 2 hours long (115 minutes of data acquisition followed by 5 minutes of standby time). The minimum detectable concentration ranges from 3 to , depending on the electronics used and variations in the deposition of the ZnS scintillating layer within the Lucas cell. AER-C is a small sized commercial solid-state radon detector. The sensitivity of this instrument ranges between 15 and per pulse per hour. The acquisition window has been configured to 4 hours and measurements have been adjusted for local absolute humidity²⁸.

Fig. 5.

Locations of the IRON stations used in this study (green dots), distributed across central and southern Italy (refer to Table 6 for latitude/longitude coordinates of each station). The figure has been realized using MATLAB^® version 2024a).

As detailed in Table 6, the majority of detectors’ installations are indoor () and shelter (). Indoor detectors are located in the basement of a building, typically in a room with the smallest aeration system possible and with restricted access, in order to reduce the influence of any anthropogenic activities. In the case of shelter installation, the radon instrument is housed in a small shelter alongside other seismic and/or geodetic monitoring equipment. In borehole () and cavity () installations, the radon detector is placed in boreholes less than 2 meters deep and in underground cavities, such as aqueducts, tunnels or mines, respectively.

Table 6.

List of IRON stations from which radon time series data were collected. For each station, the table provides the station name, the installation type, the radon detector used (Lucas cells, AER-C), start and end dates of correspondent time series, and the total number of effective acquisition days (after excluding periods when the detector was off).

Station	Site	Instrument	Start date	End date	# days
AQU	Shelter	AERC	2019-06-29	2021-12-31	917
BADI	Shelter	LUCAS	2014-08-28	2020-06-01	1784
BAT1	Borehole	AERC	2018-01-13	2021-06-29	1038
BAT2	Borehole	AERC	2018-01-12	2021-07-06	1041
CDCA	Shelter	LUCAS	2014-01-23	2020-06-03	2145
CMDO	Indoor	AERC	2018-02-23	2021-12-31	1388
CMPL	Indoor	LUCAS	2015-08-01	2019-06-25	1253
CTTR	Indoor	LUCAS	2014-01-01	2021-12-02	2593
DUR	Shelter	AERC	2019-04-16	2021-12-31	991
FRME_E146	Indoor	LUCAS	2014-01-01	2017-01-14	978
FRME_H146	Indoor	LUCAS	2017-08-05	2021-12-31	1532
GUAR2	Shelter	AERC	2018-02-03	2021-12-31	1397
ISAR	Indoor	AERC	2018-12-11	2021-12-31	1109
MMN	Indoor	LUCAS	2014-01-01	2021-09-03	2741
MMNG	Indoor	LUCAS	2014-01-01	2021-11-26	2308
MTR1	Cavity	AERC	2018-07-04	2021-12-31	1214
NRCA	Shelter	LUCAS	2016-08-26	2021-12-31	1807
SSFR	Borehole	LUCAS	2014-01-01	2021-12-31	1976

Starting from the available IRON dataset^29,30, this study focuses on radon data corrected for internal humidity dependency. Among these, radon time series with corresponding meteorological data were retrieved from a PostgreSQL relational database (specifically designed and implemented to support IRON^31,32). Since the installation of different instruments at the same station can generate inconsistent time series, we treated as separate series those defined by different station-instrument pairs. We then selected radon time series spanning at least two and a half years. Finally, we excluded the following ones, from stations

• RDP, RDPT, RPD1, RDP2: These series are highly discontinuous due to significant human activity and ongoing construction work in the area.

• MURB: This serie exhibits an excessively high trend toward the end, making it completely unpredictable.

As a result, the final dataset consists of 18 time series: two from the FRME station (denoted as FRME_E146 and FRME_H146) and the remaining from different stations, which are referred to simply only by their station acronyms.

Table 6 provides the start and the end dates for each downloaded radon time series. Some data gaps exist, corresponding to periods when no data were acquired. The last column of Table 6 indicates the number of effective days, excluding the intervals when the radon detector was inactive. Despite the data gaps, the selected time series are sufficiently long to capture the seasonal variations of radon, with all time series providing a sufficient coverage for analyzing radon seasonal trends. Figure 6 highlights the consistency of data collection across the stations, showing the number of days per month in the respective time series.

Fig. 6.

(Top) Heat map displaying the total number of days per month during which each station collected data throughout its entire deployment period. The intensity of the color indicates the frequency of data collection, with brighter shades representing more measurements. For example, the “CTTR” station shows 248 days of data collection in October over a span of 8 years, indicating that it operated every day of October each year. (Bottom) A heat map similar to the top one, but highlighting the number of times the station recorded at least one measurement in a given month throughout its entire deployment. For example, the “CDCA” station shows the number 6 for July, meaning that over its 7-year deployment, there was at least one data point recorded in July for 6 of those years.

The radon levels used in this study were smoothed using a 15-day moving average, a procedure previously adopted for managing IRON acquisitions^10,11. Each data point was smoothed by a mean calculated over a sliding window of length 15 days across neighboring elements, centered about the current and previous data points.

Meteorological stations

Radon measurements have been analyzed together with the time series of the following meteorological variables: temperature (), pressure (mb), rainfall (mm/h), relative humidity (), solar radiation (), and wind strength (kn). Temperature was measured along with radon at each IRON station, while other meteorological data were collected from 3B Meteo meteorological stations located near the IRON stations (see Table 7). On average, the meteorological stations are approximately away from the corresponding IRON stations, with the farthest distance being 22 km. Dedicated procedures were developed to automatically retrieve data from the 3B Meteo stations on a daily basis. Weather data were collected hourly, resampled to match the time intervals of the radon measurements, and then smoothed applying a 15-day moving average, as for radon data.

Table 7.

Location names and latitude/longitude coordinates (in degrees) of the IRON stations (left columns) and 3B-Meteo meteorological stations (right columns) used in the analysis. The very last column specifies the distance R (in km) between the IRON station and the correspondent meteorological station. For AQU and CTTR time series, meteorological data have been downloaded from two 3B-Meteo stations.

Station	IRON station location	Lat [°]	Lon [°]	Meteo station location	Lat [°]	Lon [°]	R [km]
AQU	Preturo (AQ)	42.383	13.316	Rocca di Cambio (AQ)	42.214	13.462	22.29
				Aquila Periferia Ovest (AQ)	42.394	13.265	4.36
BADI	Badiali (PG)	43.509	12.244	Città di Castello (PG)	43.457	12.231	5.85
BAT1	Gubbio (PG)	43.381	12.435	Pietralunga (PG)	43.465	12.426	9.30
BAT2	Pietralunga (PG)	43.370	12.409	Pietralunga (PG)	43.465	12.426	10.61
CDCA	Città Di Castello (PG)	43.458	12.233	Città Di Castello (PG)	43.457	12.231	0.15
CMDO	Montedoro (CL)	37.463	13.822	Montedoro (CL)	37.454	13.815	1.23
CMPL	Campoli Appennino (FR)	41.737	13.678	Campoli Appennino (FR)	41.735	13.682	0.40
CTTR	Cittareale (RI)	42.617	13.159	Cittareale (RI)	42.589	13.090	6.48
				Norcia (PG)	42.790	13.151	19.19
DUR	Duronia (CB)	41.650	14.466	Duronia (CB)	41.659	14.458	1.29
FRME _E146	Forme (AQ)	42.111	13.442	Massa d’albe (AQ)	42.107	13.394	3.98
FRME _H146	Forme (AQ)	42.111	13.442	Massa d’albe (AQ)	42.107	13.394	3.98
GUAR2	Guarcino (FR)	41.794	13.312	Guarcino (FR)	41.800	13.311	0.71
ISAR	Ischia Castello (NA)	40.731	13.964	Barano d’Ischia (NA)	40.709	13.914	4.90
MMN	Mormanno Faro (CS)	39.899	15.990	Mormanno (CS)	39.880	15.980	2.35
MMNG	Mormanno Ghiro (CS)	39.885	16.025	Mormanno (CS)	39.880	15.980	3.90
MTR1	Monte Trocchio (FR)	41.467	13.863	Cassino (FR)	41.486	13.833	3.26
NRCA	Pie La Rocca (PG)	42.833	13.114	Norcia (PG)	42.790	13.090	5.23
SSFR	Sassoferrato (AN)	43.436	12.782	Sassoferrato (AN)	43.430	12.857	6.12

Regression analysis

Regression analysis is a statistical technique used to quantify and model the relationship between a dependent variable and one or more independent variables. The simplest form of regression is a linear model, where the dependent variable is expressed as a linear combination of the input variables:

1

In a supervised learning framework, the goal is to estimate the parameters that best describe this relationship, ensuring that the predicted values of y are as close as possible to the observed data. The most common method is least squares estimation, which minimizes the sum of squared residuals across all data points.

In this study, we used two distinct regression approaches to analyze radon time series, focusing specifically on capturing periodic variations on an annual timescale, as it is well known that radon levels exhibit just diurnal and yearly periodicity¹¹. In the first approach, we used the meteorological variables listed in “Meteorological stations” section as predictors. In the second approach, we introduced dummy variables to better account for radon temporal patterns. A temporal dummy variable is a binary categorical variable that segments a time series into distinct seasonal components, allowing the regression model to account for periodic variations of the dependent variable³⁵. We introduced four types of dummy functions: monthly, fortnightly, weekly, and half-weekly. Each function takes the value 1 during its corresponding time period and 0 otherwise. For example, the monthly dummy function for January is set to 1 for all January observations, regardless of the year, and 0 for all other months.

The two approaches just presented result in five distinct regression models:

1. Meteorological model: capturing temporal variations of radon as a response of meteorological condition,

   2

   where , , , , , and represent the regression coefficients associated with temperature , pressure , rainfall , relative humidity , solar radiation , and wind strength respectively.
2. Monthly model: capturing seasonal variations on a monthly scale,

   3

   where represents the dummy variable for month m, which takes the value 1 if the observation belongs to month m and 0 otherwise.
3. Fortnightly model: accounting for biweekly cycles in radon concentrations,

   4

   where represents the fortnightly dummy variable, dividing the year into 26 two-week periods.
4. Weekly model: identifying weekly fluctuations in radon levels,

   5

   where denotes the weekly dummy variable, which takes the value 1 for observations in week w and 0 otherwise.
5. Half-weekly model: capturing finer temporal variations,

   6

   where represents a half-weekly dummy variable, dividing the year into 104 half-week intervals.

These regression models have been applied to each of the 18 of radon time series (Table 6), and results have been compared to assess the effectiveness of dummy variables in reconstructing the average radon trend. Each regression was trained using the first of the data and tested on the remaining last to ensure robust evaluation. It is important to highlight that the test periods were specifically chosen to occur at the end of each time series, allowing us to assess the models’ forecasting performance for future, unseen data. Table 8 specifies training and test time intervals for each station. The table also specifies which seasons are covered by the test time interval for at least one month, confirming that the models were tested across different seasonal conditions. The Mean Absolute Percentage Error (MAPE) was used to assess model performance. MAPE provides a standardized measure of predictive accuracy and is defined as follows:

7

where is the actual value, is the forecast value and N is the total number of points.

Table 8.

Regression training and test time intervals for each IRON station. Train and test time intervals correspond to 80 and 20 of total coverage respectively. The test seasons column specifies which seasons (check marks) are covered by the test time interval for at least one month.

Station	Train data interval		Test data interval		Test seasons
	Start date	End date	Start date	End date	Winter	Spring	Summer	Autumn
AQU	2019-06-29	2021-06-30	2021-06-30	2021-12-31
BADI	2014-08-28	2019-06-11	2019-06-11	2020-06-01
BAT1	2018-01-13	2020-12-08	2020-12-08	2021-06-29
BAT2	2018-01-12	2021-01-11	2021-01-11	2021-07-06
CDCA	2014-01-23	2018-11-22	2018-11-22	2020-06-03
CMDO	2018-02-23	2021-03-14	2021-03-14	2021-12-31
CMPL	2015-08-01	2018-09-30	2018-09-30	2019-06-25
CTTR	2014-01-01	2020-06-18	2020-06-19	2021-12-02
DUR	2019-04-16	2021-04-30	2021-04-30	2021-12-31
FRME_E146	2014-01-01	2016-07-03	2016-07-03	2017-01-14
FRME_H146	2017-08-05	2021-02-12	2021-02-12	2021-12-31
GUAR2	2018-02-03	2021-03-29	2021-03-29	2021-12-31
ISAR	2018-12-11	2021-06-04	2021-06-04	2021-12-31
MMN	2014-01-01	2020-01-25	2020-01-25	2021-09-03
MMNG	2014-01-01	2019-10-12	2019-10-13	2021-11-26
MTR1	2018-07-04	2021-04-27	2021-04-27	2021-12-31
NRCA	2016-08-26	2021-01-05	2021-01-05	2021-12-31
SSFR	2014-01-01	2019-02-01	2019-02-01	2020-05-31

Statistical tests

The MAPE values on the test datasets were compared using two different statistical tests: the Smirnov test (known also as two-sample Kolmogorov-Smirnov test; KS), and the Wilcoxon signed rank test.

The Smirnov test is a nonparametric statistical test³³ that quantifies the difference between two Empirical Cumulative Distribution Functions (ECDFs). In this study, we applied it to compare the ECDF of the MAPE values from the meteorological model with the ECDF of the MAPE values from each dummy model. By measuring the maximum absolute difference between the two ECDFs, the test helps assess whether the error distributions of the dummy models deviate significantly from that of the meteorological model. Mathematically, given two cumulative distribution functions and , the Smirnov statistics is defined as:

8

The null hypothesis states that the two distributions are identical. The test returns a p-value that determines whether to reject at a given significance level (0.05 in this case). Since the Smirnov test is sensitive to differences in both the location and shape of distributions, it is useful for detecting deviations in the overall error structure.

The Wilcoxon signed-rank test is a widely used nonparametric procedure³⁴, but instead of comparing full distributions, it assesses a zero-median difference between two sampled populations with paired observations. In the two-tailed version of the test, the null hypothesis states that the median of the differences between the two paired samples is zero, meaning there is no significant difference in the central tendency between the two distributions. In our study, the Wilcoxon two-tailed test was used to compare the MAPE values of the meteorological model with those of each dummy-based model. If the p-value from the test is below the significance threshold (0.05), it indicates that there is a significant difference between the two models.

For this study, we also applied the one-tailed (right-sided) version of the Wilcoxon test to determine whether the dummy models outperforms the meteorological one, with the null hypothesis stating that the median of the dummy model errors is more than or equal to the median of the meteorological model errors. A low p-value (below 0.05) would indicate that the meteorological model produces significantly higher errors than the dummy models, whereas a higher p-value would suggest no strong evidence against equivalence.

Each of these tests contributes a different perspective: the Smirnov statistics checks for overall distribution differences, while Wilcoxon one focuses on median differences. Together, they offer a comprehensive statistical evaluation of whether the dummy models can be considered interchangeable with the meteorological model.

Author contributions

The entire article, along with the detailed conclusions, resulted from a collective brainstorming effort, and the writing and revision processes were equally collaborative, as was the code implemented to perform the analysis. More specifically, A.P. conceived the core idea of the article and, regarding the coding, focused on generalizing the code for regression, while V.V. adapted it to the specific analysis and added the section related to plots and statistical testing. G.R. contributed to the review of the state of the art and the comparison of different methodological approaches, and worked on the automated procedures for assessing data coverage and continuity. G.R. and V.V. ensured the selection of stations without data gaps and prepared the tables and figures included in the paper.

Data availability

The IRON radon time series used for this study are available at https://doi.org/10.6084/m9.figshare.27292554.v1. Please refer to the corresponding author for additional requests.

Declarations

Competing interests

The authors declare that they have no competing financial or non-financial interests that could have influenced the work presented in this study.