Nat Hazards (2012) 62:723–731 DOI 10.1007/s11069-012-0104-x ORIGINAL PAPER
Examining Granger causality between atmospheric parameters and radon Alessandro Attanasio • Maurizio Maravalle • Giulia Fioravanti
Received: 7 November 2011 / Accepted: 25 January 2012 / Published online: 11 February 2012 Springer Science+Business Media B.V. 2012
Abstract In this paper, we study the relationship between atmospheric parameters (i.e., temperature and humidity) and radon data. We use the linear Granger causality in order to observe possible connections, on short and mid time scale periods, between radon time series and meteorological parameters that strongly influence radon emissions. The analysis suggests radon emission is not affected by these atmospheric parameters on short periods, while there is an evidence of Granger causality on mid periods. Keywords
Radon Time Series Granger causality
1 Introduction Radon is an inert and radioactive soil gas, generated in the Earth crust, which moves upward toward the surface and diffuses into air. Radon’s properties have led to its use for geophysical purposes, as tracer for locating buried faults and geological structures, in exploring for uranium and for predicting earthquakes or volcanic eruptions. It is known that Rn emissions are affected by environmental parameters, such as temperature, relative humidity, elevation, air drafts. However, it has to be elucidated how variation in these factors affects the exhalation process. One of the current problems is the assessment of the role of atmospheric parameters on radon emission, such as seasonal and daily changes in A. Attanasio (&) Department of Pure and Applied Mathematics, Universita` di L’Aquila, via Vetoio 1, Coppito, 67010 L’Aquila, Italy e-mail:
[email protected] M. Maravalle Department of Economics, University of L’Aquila, via G. Falcone 25, 67010 L’Aquila, Italy e-mail:
[email protected] G. Fioravanti Department of Chemistry, Chemical Engineering and Materials, University of L’Aquila, via Campo di Pile, Pile, 67010 L’Aquila, Italy e-mail:
[email protected]
123
724
Nat Hazards (2012) 62:723–731
atmospheric factors (Baykut et al. 2010; Groves-Kirkby et al. 2006; Omori et al. 2009; Singh et al. 2004; Sundal et al. 2008; Winkler et al. 2001). In this paper, we analyze the relationship between temperature–humidity and radon using the concept of Granger causality. This method has recently been employed to detect the connection between complex processes in Earth sciences such as the global warming (Kaufmann and Stern 1997; Kodra et al. 2011; Triacca 2005). The paper is organized as following: A review of Granger causality is given in the next section; We present the data in Sect. 3; A preliminary study of the time series is explained in Sect. 4; The results of Granger causality analysis are discussed in Sect. 5; A brief conclusion is drawn in Sect. 6.
2 Granger causality testing Granger (1969) has defined a concept of causality which is most popular in econometrics. To illustrate this idea, two variables x and y are considered. Let Itx;y ¼ fx0 ; . . .; xt ; y0 ; . . .; yt g be the information set. We will say that x Granger cause y if the mean square error (MSE) of the forecast of yt?1 based on Ix,y t is smaller than the mean square error of the forecast of yt?1 that use Iyt = Ix,y t - {xs, s B t}. Then, past observations of x contain information useful for forecasting y because the future values of y can be better predicted employing the past values of x and y rather than only past values of y. Granger causality can be also used in a trivariate (or multivariate) system. Let zt ¼ ðyt ; x1t ; x2t Þ be a 3 9 1 time series. The variables x1 and x2 Granger cause y if we can better predict the future values of y considering the information set Itx1 ;x2 ;y than only Iyt . If we are interested in testing the null hypothesis that ðx1 ; x2 Þ do not Granger cause y, we can consider the unrestricted regression model yt ¼ a 1 þ
p X
/k ytk þ
k¼1
p X
wk x1;tk þ
k¼1
p X
ck x2;tk þ et
ð1Þ
k¼1
and the restricted regression model yt ¼ a 2 þ
p X
bk ytk þ ut
ð2Þ
k¼1
where a1 and a2 are constants, /, w, c and b are thecoefficients of the models, et and ut are univariate white noise. If w1 ; . . .; wp ; c1 ; . . .; cp is equal to the zero vector, then ðx1 ; x2 Þ do not Granger cause y. Therefore, we perform a test of the following hypotheses H0 : w1 ¼ ¼ wp ¼ c1 ¼ ¼ cp ¼ 0
ð3Þ
versus H1 : wi ; cj 6¼ 0
for some i; j
ð4Þ
We estimate the parameters of (1) and (2) by OLS, and we use the test F F¼
ðRSSr RSSu Þ=q RSSu =ðT 3p 1Þ
ð5Þ
where RSSr and RSSu are, respectively, the sum of the squared residuals of restricted and unrestricted models, q corresponds to the number of coefficients restricted to zero, and T is
123
Nat Hazards (2012) 62:723–731
725
the number of observations. Under the assumption that the time series are stationary, the null hypothesis can be tested using a standard Wald statistic. An insignificant statistics means there is not Granger causality, while a significant statistics implies the null hypothesis H0 is rejected. When the time series are not stationary, the procedure described in Toda and Yamamoto (1995) can be employed to test the null hypothesis (3). In order to apply the Wald statistic, Toda and Yamamoto suggest to overfitting the model (1) by d extra lags, where d is the maximum order of integration of the series1. Then, the 3(p ? d) ? 1 parameters of the Eq. 1 and the p ? d ? 1 parameters of the Eq. 2 are estimated by OLS, and the null hypothesis H0 in (3) (ignoring the coefficients of the augmented lags) can be tested using a standard Wald statistic.
3 Data In this paper, we deal with the following three hourly time series: • Radon emissions fyt gt . Data given us by Mr. Gioacchino Giuliani. • Temperature fx1t gt . It is given us by CETEMPS (Center of Excellence for the integration of remote sensing TEchniques and numerical Modelling for the Prediction of Severe weather). • Humidity fx2t gt . It is given us by CETEMPS. The radon detector consists of a plastic scintillator (NE 110 or NE 102) coupled with two or four photomultipliers (Photonis XP3462b). It provides an indirect radon measurement, based on detection of radon daughters (214Pb and 214Bi, with their characteristic gamma emission at 351 and 609 KeV, respectively). Background is often important enough in routine usage so that the majority of radiation detectors are provided with some degree of external shielding to effect a reduction in the measured level. External sources of background are natural radioactivity that comes from the detector itself (contamination in the materials), the earth’s surface, building material, etc. or from the air (airborne radioactivity). Another source of background is due to cosmic radiation (primary and secondary components). The background may vary in time, and thus we performed background measurements regularly for good measurements practice. The detector is inserted in a 7-cm-thick stable lead box, which provides a passive shielding, in order to minimize the natural radioactivity coming from the environment. Lead is the most widely used material for its high density and large atomic number, which make photoelectric absorption predominant for gamma rays up to 0.5 MeV. The lead thickness of 7 cm is also reasonably effective at reducing cosmic radiation, while the use of lead from very old sources or highpurity (i.e., stable) contains low natural radioactivity with reference to ordinary lead. The lead container is covered with a thin layer of Mylar (40 lm), used as a reflective coating with over 98% of reflectivity, in order to improve the device’s performance. Moreover, the detector is placed in closed spaces, located 3 meters underground (to reduce cosmic-rayinduced background); it is necessary to avoid ventilation systems that would cause unwanted drastic variation in radon concentration due to external influences. The characteristics of the commercial scintillator used in the equipment are a time of response of 2.4 ns, and light emission at 423 nm, which lies at the lower end of the violet. The 1 In particular, a time series wt is integrated of order d if Dd wt is stationary, where Dh wt is non-stationary for h \ d. For example, if wt is non-stationary but its first difference Dwt is stationary, the series is said to be integrated of order one or I(1).
123
726
Nat Hazards (2012) 62:723–731
response in light is of 65%, compared with anthracene, organic crystal, which has the characteristic of having the highest response among all the organic scintillators. The received analogical signal is converted into a digital one by the discriminator and sent to an adapter board (NIM-TTL-NIM) that forwards it to the acquisition system, consisting of a scaler board interfaced with a computer, which acquires the output in real time. The detector automatically takes measurements at closely spaced time intervals (10 min) and gives output signals as counts per hour (e.g., photons per hour). The passive lead shield thickness of about 7 cm is suitable to reduce the gamma background by a factor of about 6, in the interest energy window (0.13–24 MeV), as shown in Fig. 2, where the integral background (counts/s versus energy plot) was reduced from 2.80 to 0.48 counts/s. The reliability of the device is established by comparing it with that of a commercial instrument, and measurements were simultaneously taken from radon detector and a commercial digital Rad7 (manufactured by Durridge Co., Bedford, MA, USA) operating in the immediate vicinity, for a period of one month. The Rad7 detector is a solid state alpha counter, and the operation principle of the device is based on the spectral analysis of alphaemitters, and it measures the radon concentration in the environment. The Rad7 measurement conditions were T = 16 C, p = 945 mbar and rH = 50% (relative humidity), with uncertainty of 150 ± 50 Bq/m3 on average values, as reported by manufacturer’s manual. By correlating the Rad7 and radon detector time series (for 1 month data), it was found a correlation coefficient of 0.59, as shown in Fig. 1. The results of simultaneous measurements show a good positive correlation between the two methodologies (by considering the fact that Rad7 is an alpha device, while radon detector measures gamma radiation), and a variation of 1 Bq/m3 in Rad7 corresponds to a variation of 4.86 in the radon detector. The output signals are counts per second for both instruments, with an acquisition time of 7,200 s (2 h, counts/2 h), and readout units are calculated on the counts sum any 2 h. The correlation line equation extrapolated from the two time series is y = 166415 (±123) ? 4.86 (±0.37) x, where on y axes are reported radon counts/2 h taken from the detector and on x axes Bq/m3 any 2 h from Rad7. The intercept represents an intrinsic background, at a value of 166,415 counts/2 h. Again, the detector is located near L’Aquila city (Coppito Radon Station; Lat. 42.366, Long. ?13.343), far away from the Cetemps weather station approximately 600 m (Cetemps Meteorological Station; Lat. ?42.367, Long. ?13.350; font: http://cetemps. aquila.infn.it), both at an altitude of 685 m above the sea level. The two sites proximity prevents local effects due to measurements site difference. Fig. 1 Effect of lead thickness on radon measurements
123
Nat Hazards (2012) 62:723–731
727
Fig. 2 Cross correlation between Rad7 and radon detector
We are interested in testing the null hypothesis that temperature and humidity do not Granger cause radon, in this indoor methodology of measurement. The period of study ranged from March 2008 to May 2010. There are some missing values in the radon time series, caused by external supply interruption. After interruption, the system need to be recalibrated, and data are not soon reliable. Also, in the temperature and humidity series, there are some missing values, so we study the Granger causality over continuous subperiod whose number of observations is greater than 144 (6 days). This lower bound is selected in order to obtain consistent estimators in our models. The following thirteen subperiods {Spi}13 i=1 are considered: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Sp1 (454 obs): from 7 p.m. of the 2008/03/18 to 4 p.m. of the 2008/04/06. Sp2 (189 obs): from 9 p.m. of the 2008/06/05 to 5 p.m. of the 2008/06/13. Sp3 (1218 obs): from 3 p.m. of the 2008/10/02 to 8 a.m. of the 2008/11/22. Sp4 (193 obs): from 8 p.m. of the 2008/12/06 to 8 p.m. of the 2008/12/14. Sp5 (1564 obs): from 2 a.m. of the 2009/01/11 to 5 p.m. of the 2009/03/17. Sp6 (176 obs): from 6 p.m. of the 2009/03/29 to 1 a.m. of the 2009/04/06. Sp7 (748 obs): from 1 p.m. of the 2009/07/27 to 4 p.m. of the 2009/08/27. Sp8 (400 obs): from 11 p.m. of the 2009/09/01 to 2 p.m. of the 2009/09/18. Sp9 (154 obs): from 10 p.m. of the 2009/11/04 to 7 p.m. of the 2009/11/11. Sp10 (463 obs): from 1 p.m. of the 2009/12/12 to 7 p.m. of the 2009/12/24. Sp11 (217 obs): from 4 p.m. of the 2010/01/07 to 4 p.m. of the 2010/01/16. Sp12 (240 obs): from 7 p.m. of the 2010/02/02 to 6 p.m. of the 2010/02/12. Sp13 (1007 obs): from 11 p.m. of the 2010/04/19 to 9 p.m. of the 2010/05/31.
In Fig. 3, for example, is shown the plots of radon, Temperature and Humidity over the subperiod Sp3.
4 Preliminary analysis Time series can be described in many ways. For example, it can be stationary or integrated of order one or characterized by evident cyclical component. In this section, we analyze the structures of radon, temperature and humidity time series. Let us consider the radon series yt. We focus on the presence or absence of unit roots in yt, and we utilize the Augmented Dickey–Fuller test (Dickey and Fuller 1981) and the
123
728
Nat Hazards (2012) 62:723–731
Radon
Temperature
73000
25
72500
20
72000
15
71500
10
71000
5
70500
0
70000
0
-5
204 407 610 813 1016 1219
0
204 407 610 813 1016 1219
Humidity 100 90 80 70 60 50 40 30 20
0
204 407 610 813 1016 1219
Fig. 3 Sketch of the radon, temperature and humidity time series over the subperiod Sp3
KPSS test (Kwiatowski et al. 1992). The model of the Augmented Dickey–Fuller (ADF) test is specified as
Dyt ¼ a þ uyt1 þ
p X
nj Dytj þ vt
ð6Þ
j¼1
where vt is a white noise. If u ¼ 0 and a = 0 then yt has a unit root. Alternatively, if u\0 then the series is stationary. We select p* suggesting by the Schwartz’s Bayesian criterion (SBC). The model for the KPSS test is yt ¼ lt þ gt
lt ¼ lt1 þ et
ð7Þ
where lt is a random walk, and gt is a stationary process. If re = 0, then lt is a constant and yt is stationary. Alternatively, yt has a stochastic trend. Then, the KPSS test differs from the ADF test because under the null hypothesis, the time series is stationary. We use the Newey–West procedure to correct for serial correction, and the lag troncation is chosen employing an automated bandwidth estimator that uses the Bartlett kernel. Using the same procedure described in Hamilton (1994), the ADF test shows that yt is often integrated of order zero so the structure of this series seems totally stationary. The results are not similar employing the KPSS test. In fact, the null hypothesis of stationary is rejected over 11 subperiods and the series can be characterized by stochastic trend. Only over 6 intervals, the two tests have the same outcomes. The results are illustrated in Table 1. The time series x1t and x2t have a periodicity of 24-h due to the day/night cycle which is shown on Fig. 3. A spectral analysis based on Bartlett estimator confirms this periodicity because in every subperiods, the maximum value of the estimated spectrum has a period of 24-h. This cyclical component is not regular to suggest us to eliminate it. Indeed, there are cyclical time series whose periodicity does not produce non-stationarity and they can be studied using their level or first difference (Lutkepohl and Kratzig 2004). Therefore, the
123
Nat Hazards (2012) 62:723–731
729
Table 1 Univariate tests for order of integration of radon using a level of significativity equal to 0.05
Subperiod
Result ADF
Result KPSS
Sp1
I(0)
I(0)
Sp2
I(0)
I(1)
Sp3
I(0)
I(1)
Sp4
I(1)
I(1)
Sp5
I(0)
I(1)
Sp6
I(0)
I(1)
Sp7
I(0)
I(0)
Sp8
I(0)
I(1)
Sp9
I(1)
I(1)
Sp10
I(0)
I(1)
Sp11
I(0)
I(0)
Sp12
I(1)
I(1)
Sp13
I(0)
I(1)
temperature and humidity time series are sometimes described by AR(2) model that can explain the periodicity.
5 Results We are interested in testing the null hypothesis of non-causality from temperature and humidity to radon. We apply the procedure of Toda and Yamamoto. The order p of the model (1) is selected by considering the values of the SBC using a maximum order equal to 12. Considering the analysis of the previous section, the maximum order of integration d is taken equal to 1. Then, we estimate the Eqs. 1 and 2 using the order p ? 1. In this way, we avoid the possibility that the distribution of the test (5) is not standard. Table 2 Results of Granger non-causality test from temperature and humidity to radon Subperiod
p
d
p value
Bootstrap p value
Conclusion
Sp1
2
1
0.619
0.613
NGC
Sp2
2
1
0.363
0.374
NGC
Sp3
3
1
0.020**
0.022**
GC
Sp4
3
1
0.347
0.348
NGC
Sp5
3
1
0.004*
0.002*
GC
Sp6
2
1
0.149
0.156
NGC
Sp7
2
1
0.039**
0.040**
GC
Sp8
2
1
0.043**
0.043**
GC
Sp9
2
1
0.206
0.217
NGC
Sp10
2
1
0.921
0.922
NGC
Sp11
2
1
0.296
0.298
NGC
Sp12
2
1
0.305
0.304
NGC
Sp13
3
1
0.002*
0.003*
GC
NGC implies that the null hypothesis of Granger non-causality cannot be rejected at 10% significance level; GC implies that there is Granger causality; * significant at the 1% level; ** significant at the 5% level
123
730
Nat Hazards (2012) 62:723–731
The Toda and Tamamoto method can suffer from size distortion and low power especially for small samples. Then, we also apply bootstrap method based on residuals (Giles 1997; Mantalos 2000; Lukasz 2010) in order to control the robustness of our results. The bootstrap p value is calculated by means of 10,000 replications in radon time series. From our analysis, there is not a detectable Granger causality from temperature and humidity to radon. In particular, we observe that Granger causality is absent over nine subperiods which have a number of observations in the range [144,454]. Then, the influence of temperature and humidity on radon is low on short time scale periods (from days to weeks). A strong evidence from temperature and humidity to radon appears only over Sp5 and Sp13 (p value \0.01). In these subperiods, there are a lot of observations and the relationship between temperature–humidity is evident on mid-term period (from weeks to months). Short- and mid-term periods are only related to the length of the observation period, since the measurements are hourly. We do not intend short and term forecasting as in Gelper et al. (2007). Then, we detect more easily Granger causality if we have more data. A weak Granger causality is shown over the subperiods Sp3, Sp7 and Sp8. These results are also confirmed by bootstrap method (Table 2). Moreover, it is interesting to observe that the radon time series has a periodicity of 24-h over the intervals which have an evidence (strong or low) of causality.
6 Conclusion This analysis suggests that temperature and humidity do not have an important role in the radon time series by considering short-term monitoring. Indeed, past observations of temperature and humidity do not improve the radon forecasting using linear models. Otherwise, a linear Granger causality is detectable over mid-term monitoring. Our results may be influenced by omitted variables but, unfortunately, only temperature and humidity time series are available. Acknowledgments The authors are grateful to Mr. G. Giuliani for radon time series, to CETEMPS for temperature-humidity time series, and to Prof. F. Battaglia for the useful comments on a previous version of this paper. G. Fioravanti is grateful to the Italian National Project PRIN2009 (Prot. 2009N9N8RX ‘‘Chemistry in Motion’’), for the financial support. All computations were done using gretl.
References Baykut S, Akgul T, Inan T, Seyis C (2010) Observation and removal of daily quasi-periodic components in soil radon data. Radiat Meas 45:872–879 Dickey D, Fuller WA (1981) Likelihood ratio statistic for autoregressive time series with a unit root. Econometrica 49:427–431 Gelper S, Lemmens A, Croux C (2007) Consumer sentiment and consumer spending: decomposing the granger causal relationship in the time domain. Appl Econ 39:1–11 Giles DEA (1997) Causality between the measured and underground economies in New Zealand. Appl Econ Lett 4:63–67 Granger CWJ (1969) Investigating causal relations by econometric methods and cross-spectral methods. Econometrica 34:424–438 Hamilton JD (1994) Time series analysis. Princeton University Press, Princeton Kaufmann RK, Stern DI (1997) Evidence for human influence on climate from hemispheric temperature relations. Nature 388:39–44 Lukasz L (2010) Application of bootstrap methods in investigation of size of the Granger causality test for integrated VAR systems. Manag Glob Transit 8:167–186
123
Nat Hazards (2012) 62:723–731
731
Groves-Kirkby CJ, Denman AR, Crockett RGM, Phillips PS, Woolridge AC, Gillmore GK (2006) Timeintegrating radon gas measurements in domestic premise: comparison of short-, medium- and longterm exposures. J Environ Radioactiv 86:92–109 Kodra E, Chatterjee S, Ganguly AR (2011) Exploring Granger causality between global average observed time series of carbon dioxide and temperature. Theor Appl Climatol 104:325–335 Kwiatowski D, Phillips PCB, Schmidt P, Shin Y (1992) Testing the null hypothesis of stationarity against the alternative of a unit root: how sure are we that economic time series have a unit root. J Econom 54:159–178 Lutkepohl H, Kratzig M (2004) Applied time series econometrics. Cambridge University Press, Cambridge Mantalos P (2000) A graphical investigation of the size and power of the Granger-causality tests in integrated-cointegrated VAR systems. Stud Nonlinear Dyn E 4:17–33 Omori Y, Tohbo I, Nagahama H, Ishikawa Y, Takahashi M, Sato H, Sekine T (2009) Variation of atmospheric radon concentration with bimodal seasonality. Radiat Meas 44:1045–1050 Singh K, Singh M, Singh S, Sahota HS, Papp Z (2004) Variation of radon (222Rn) progeny concentrations in outdoor air as a function of time, temperature and relative humidity. Radiat Meas 39:213–217 Sundal AV, Valen V, Soldan O, Strand T (2008) The influence of meteorological parameters on soil radon levels in permeable glacial sediments. Sci Total Environ 389:418–428 Toda HY, Yamamoto T (1995) Statistical inference in vector autoregression with possibly integrated processes. J Econom 66:225–250 Triacca U (2005) Is Granger causality analysis appropriate to investigate the relationship between atmospheric concentration of carbon dioxide and global surface air temperature?. Theor Appl Climatol 81:133–135 Winkler R, Ruckerbauer F, Bunzl K (2001) Radon concentration in soil gas: a comparison of the variability resulting from different methods, spatial heterogeneity and seasonal fluctuations. Sci Total Environ 272:273–282
123
