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Statistical analysis of geomagnetic field reversals and their consequences
Physica A 392 (2013) 6554–6560
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Statistical analysis of geomagnetic field reversals and their consequences Cleiton S. Barbosa a , Douglas S.R. Ferreira a,b , Marco A. do Espírito Santo c , Andrés R.R. Papa a,d,∗ a
Observatório Nacional; R. Gal. José Cristino, 77, 20921-400, Rio de Janeiro, RJ, Brazil
b
Instituto Federal do Rio de Janeiro, Campus Paracambí, RJ, Brazil
c
Instituto Federal do Rio de Janeiro, Campus Volta Redonda, RJ, Brazil
d
Instituto de Física, Universidade do Estado do Rio de Janeiro, Brazil
highlights • • • •
We study geomagnetic reversals through Tsallis statistics. We have extended previous non-extensive analysis of reversals to 160 Ma. We have found q = 1.3 to be the optimum value for data adjustment. Through detrended fluctuation analysis there was found correlation in the series.
article
info
Article history: Received 19 February 2013 Received in revised form 26 July 2013 Available online 27 August 2013 Keywords: Geomagnetic reversals Tsallis statistics Statistical distributions Detrended fluctuation analysis
abstract In this paper we focus on the statistical distribution of time intervals between geomagnetic reversals. Recently the Tsallis distribution was pointed out as a possible alternative to previous proposals. We have performed statistical tests to further prove this and find the parameters of the Tsallis distribution that better fit the data. Additionally we have analyzed the correlation of time intervals between consecutive reversals to show the presence of memory effects on the mechanism that generates them, as could be expected for some Tsallis systems. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Data analysis shows that the paleomagnetic Earth’s magnetic field has reversed its polarity, several times, suddenly and disorderly in the last 160 million years. Recent works on data analysis, theoretical modeling and experimental dynamos have expanded the knowledge on the geomagnetic field. However, some fundamental questions related to polarity transitions remain unanswered: the physical mechanism that gives rise to reversals and the statistical distribution of inter-reversals times, are examples of still open issues. Several contributions indicated that the phenomenon governing the geomagnetic data distribution could be a Poisson process with constant or variable rates. Recent studies have ruled out this possibility. In the case of large inter-reversal time intervals, studies indicate power laws as possible distributions.
∗ Corresponding author at: Observatório Nacional, R. Gal. José Cristino, 77, 20921-400, Rio de Janeiro, RJ, Brazil. Tel.: +55 21 35049142; fax: +55 21 25807081. E-mail address: [email protected] (A.R.R. Papa). 0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.08.025
C.S. Barbosa et al. / Physica A 392 (2013) 6554–6560
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The geomagnetic field is extremely complex in both morphology and temporal dependence. Now we know that the field is generated at the Earth’s liquid metal outer core and that it has intriguing features such as polarity reversals. The first scale of geomagnetic polarities was compiled by Cox et al. [1] and suggested a periodicity of around 106 years (1 My) in geomagnetic reversals. However, shortly after that, new data [2] showed that there were no simple intervals: time intervals between reversals (or what is obviously the same, time intervals with the same polarity) were either too long (∼1 My) or too short (∼0.1 My). It was proposed [1] that inside longer intervals, of the order of 1 My, there were short intervals of approximately 0.1 Ma. Polarity scales have been developed to the extent that new records of reversals have become available. A history of this evolution can be found in the book by Opdike and Channell [3]. There are numerous geomagnetic polarity scales covering different intervals of geological times. They have similarities and discrepancies due to the various methods used to construct them. The largest errors are related to estimates for the age of rocks as well as to the existence of some polarity intervals that can appear in some scales but be absent in others [4]. However, the geomagnetic chronology is continually under construction from new records of reversals. One of the more accepted time scales was proposed by Cande and Kent [5] based on the analysis of marine sediments. In this scale there are records of polarity for the last 80 Ma. In 1995 Cande and Kent [6] published a revised version of their scale (CK95) which presents changes of polarity for the last 160 My. Using CK95, the present study aims to statistically characterize inter-reversals times from the viewpoint of non-extensive statistical mechanics, and to show the existence of temporal correlations in series. 2. Non-extensive statistical mechanics Complex systems, whose ground state is highly degenerate have long-range microscopic memory, have very long relaxation times, or present a mixture of these characteristics, exhibit incompatibility with the extensive statistical mechanics of Boltzmann–Gibbs (BG) [7]. Looking for a solution for those types of systems Tsallis [8] proposed a generalization of the BG entropy. The entropy Sq proposed by Tsallis [8] is given by the following equation: i=w
1− Sq = k
q
pi
i=1
(1)
q−1
where k is a suitable constant, pi are the probabilities and q is the parameter that accounts for the generalization. The Tsallis entropy has many properties in common with the Boltzmann–Gibbs entropy, however, the property of additivity is violated for independent random variables [9], for example: Sq (A, B) = Sq (A) + Sq (B) +
1−q
Sq (A)Sq (B). (2) k The statistic is not extensive and the parameter q can be understood as a measure of the statistics’ non-extensivity. Depending on the value of q we have [10]: – q < 1: superextensivity, since Sq (A, B) > Sq (A) + Sq (B) – q > 1: subextensivity, since Sq (A; B) < Sq (A) + Sq (B) – q = 1: extensivity is recovered. By maximizing the Sq entropy there emerges the non-extensive probability distribution known as Tsallis distribution. This distribution is given by a function known as q-exponential and it is a generalization of the usual exponential. The distribution is given by the following equation: 1
p(x) = C expq (−Bx) = C [1 − (1 − q)Bx] 1−q
(3)
where B is a scale parameter and C is a normalization constant. In general, the distribution used to characterize the frequency of data series can be a cumulative distribution. It can be shown that the cumulative Tsallis distribution is also a q-exponential [10] and its normalized expression is given by: p(y) = expq′ (−B′ y).
(4)
From Eq. (4) it can be noted that the Tsallis distribution has q and B as parameters. A relationship between both parameters can be obtained by applying the q-logarithm to the cumulative Tsallis distribution: lnq p(x) = lnq (expq (−Bx))
(5)
lnq p(x) = −Bx.
(6)
Once estimated the parameter q, we can calculate lnq p(x) =
x1−q − 1 1−q
and from Eq. (6) obtain a linear relationship between lnq p(x) and x, with the slope given by B.
(7)
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Fig. 1. The accumulated frequency distribution function of time intervals between consecutive reversals, ∆t, on a log–log graph (squares). We also show the fitting of a linear trend for large ∆t values (straight line).
3. Optimizing data adjustments Time intervals between consecutive reversals are not randomly distributed [11], but belong to some non-flat statistical distribution. The distribution for these time intervals is shown in Fig. 1 and qualitatively seems to follow an exponential dependence. It was proposed by Constable [12] that these time intervals follow a Poisson statistics, given by: p(∆t ) = λ exp(−λ∆t )
(8)
where λ is the rate of occurrence of reversals. If the reversals follow a Poisson distribution, this would indicate that there are no memory effects in the mechanism that generates reversals, i.e., the reversals are independent events. A statistical test was applied by Jonkers [13] to this case and it was shown that the distribution of polarity of the field does not follow a Poisson distribution [11]. Jonkers [13] proposed that the distribution of time intervals of the same polarity approximates a power law. In Fig. 1 it is also shown the log–log graphic of inter-reversal time intervals and the cumulative frequency distribution function in the form of a power law. We see from Fig. 1 that a power law does not fit the whole data range. However, as shown in Fig. 1, if we ignore short time intervals of the same polarity, there will be a linear relationship between ∆t and the accumulated frequency. Long time intervals of the same polarity are called crons and have duration between 5 × 105 and 5 × 106 years. We see in Fig. 1 that these longer time intervals are distributed in the form of a power law. The possibility that the distribution of crons obeys a power law has been interpreted by several contributors as indicative of a critical phenomenon in the process controlling reversals [14]. So far we have mentioned that statistical distributions as power laws and Poisson’s have been proposed to model the time intervals of the same polarity. However they have not obtained a complete success (that is, a satisfactory fit for all the range of inter-reversal intervals). Recently, Vallianatos [15] applied the non-extensive statistical mechanics to study the reversals. He relied on the work by Carbone et al. [11], where the existence of memory effects and long-range interactions in polarity reversals were found. Thus, Vallianatos [15] proposed that time intervals of the same polarity follow a Tsallis distribution. Vallianatos [15] used the scale of Cande and Kent [5], that covers a period of 80 million years, estimating the distribution parameters q = 1.5 and B = 4.21 Ma−1 . In addition to the work of Vallianatos [15], here we apply the Tsallis distribution for the reviewed scale of Cande and Kent [6] covering a period of 160 million years. Initially we calculate the distribution parameters, show through the Kolmogorov–Smirnov test that the distribution with the calculated parameters fits the data and to finish calculate the degree of correlation between polarity reversals. As a starting point for calculating the parameters for the best fitting distribution we used the results obtained by Vallianatos [15]. He estimated q = 1.5. We assumed that the value of q for our data should be around this. We know that q is between 1 and 3, then, we use q between 1.2 and 1.7. Specifically, we have initially considered values of q = 1.2, 1.3, 1.4, 1.5, 1.6 and 1.7. Using Eq. (6), we have calculated, for each value of q the corresponding value of B and thus obtained a family of Tsallis distributions. For each distribution the Kolmogorov–Smirnov [16–19] test was applied to accept or to reject the hypothesis that the specified distribution fits the data. Through this procedure we aimed to find the values of q and B that better fit the distribution of time intervals between consecutive reversals. The Kolmogorov–Smirnov (KS) test [16–19] unlike more popular tests (for example, the chi-square) does not apply to qualitative data nor to discrete variables, because the table for this test is only accurate if the testing distribution is continuous. However, it has the advantage of not being dependent on rating data that, besides being always somewhat arbitrary, involve information losses. On the other hand, the KS test can only be applied when the distribution indicated in the null hypothesis is completely specified (in our case, the Tsallis distribution). Furthermore, the KS test is applicable to small samples.
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Fig. 2. The q-log of cumulative frequency of actual data (squares). The straight line is the linear fit between the q-log and ∆t.
Let G be the distribution function of the study population and F the proposed distribution function, continuous and completely specified. Hypotheses to test: H0 : G(x) = F (x) for every x H1 : G(x) ̸= F (x) for some x. The KS test compares the accumulated relative frequencies recorded in the sample with those that would be expected if the population distribution was that specified in the null hypothesis. The statistic of the KS test considers the greatest of the differences, in absolute value, between the ratio of observations less than or equal to x, S (x), and the probability of observing a value less than or equal to x if the population distribution is the one specified in H0 , F (x). Since F is a continuous function and S a ladder function, the supreme, dn,obser v ed , occurs at a point where there is a jump of S. Thus, if H0 is true, the maximum vertical distance between the images of both distributions should not be too large, then it is expected that dn,obser v ed take a small value. So, for a significance level α , reject H0 if the observed value is greater than or equal to the critical point Xn,α (critical values Xn,α can be found in a table). Now we illustrate the method for the value q = 1.3 which, as will be shown, was the best value. Let us calculate the log-q for q = 1.3, noting that: lnq F (∆t ) =
∆ t 1 −q − 1 = −B∆t 1−q
ln1.3 F (∆t ) =
∆t 1−1.3 − 1 . 1 − 1.3
(9)
(10)
The dependence between ln1.3 F (∆t ) and ∆t is shown in Fig. 2. The slope of the straight line is −2, 94 and therefore we have B = 2.94 Ma−1 and the Tsallis distribution for q = 1.3 is: F (∆t ) = exp1.3 (−2.94∆t ).
(11)
In Fig. 3 we adjust the data to the Tsallis distribution with the calculated parameters. Let us apply the KS test to that specific distribution. As explained above the idea of testing is to find the maximum vertical distance dn between the empirical distribution Gn (∆t ) and the distribution in which we are testing the hypothesis that fits the data Fn (∆t ). Using the data of CK95 as a function of ∆t the graph of these two functions is shown in Fig. 3: The maximum vertical distance between these functions cannot be greater than a threshold testing, Xn , which in this case is equal to X276 = 0.08174. For each ∆t we have calculated the difference between the Tsallis distribution and the empirical cumulative distribution. That is, calculate d276 = |Gn (∆t ) − Fn (∆t )|. The graph of d276 in terms of ∆t is shown in Fig. 4. From Fig. 4 we see that the maximum difference is d276 = 0; 07325. Then, d276 < X276 => Hypothesis accepted. So, with a confidence level of 5% we can accept the hypothesis that the time intervals between consecutive reversals follow a Tsallis distribution with B = 1.3 Ma−1 and q = 2.94 as parameters. The same process was repeated for other values of q. For all these cases the KS test was applied and the hypothesis was rejected. At this point we have included the q values 1.25 and 1.35 to obtain a more detailed description around the q = 1.3 special value.
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Fig. 3. The cumulative frequency of actual data (squares). The continuous line is the Tsallis distribution using q = 1.3 and B = 2.94. It is remarkable the fit between both distributions.
Fig. 4. Difference between the empirical distribution function and the Tsallis distribution with q = 1.3.
Fig. 5. Maximum difference between the Tsallis distribution function and the empirical distribution function for several q values. The dashed line is the threshold value for the test.
With these results we conclude that the q parameter value that better fit the data is around 1.3. This is shown in Fig. 5, where we have a graph of d276 values for some values of q.
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Fig. 6. Graph of fluctuations depending on the size of the window used to calculate the fluctuation range. For this graph the series was used in chronological order and provided h = 0.92.
We see that around q = 1.3, we have a local minimum for the difference between the proposed distribution and the empirical distribution. The fact of the distribution of polarity intervals follows a Tsallis distribution brings us new insights into the understanding of reversals. Systems that follow Tsallis distributions are systems in which statistical mechanics is non-extensive. 4. Correlation of series There are systems with memory effects and long-range interactions. For geomagnetic reversals this means that a reversal occurred 10 million years ago could be related to a reversal to occur in the future. If indeed the time inter-reversal times follow Tsallis statistics they could be temporally correlated. With this idea in mind we used the method of detrended fluctuation analysis (DFA) [20] to assess the degree of correlation of series. The method consists of dividing the series into windows of size n and calculate the fluctuation range Fdfa . Fdfa is related to the window size. It generally increases when increase the window size, so we have the following relationship between Fdfa and window size: Fdfa ∼ nh
(12)
where h is the Hurst exponent. This exponent is the slope of the log–log dependence of Fdfa with n. The Hurst exponent is of fundamental importance in determining the degree of correlation in series. Depending on its value, we have: – Uncorrelated series: h = 0.5 – Anti correlated series: 0 < h < 0 : 5 – Power law long range temporal correlation: 0 : 5 < h < 1. The last case indicates a series with positive autocorrelation, i.e., that a high (low) value in the series is more likely to be followed by another high (low) value. The window size ranged between 20 < n < 140. The graph of the function of fluctuations is shown in Fig. 6. The Hurst exponent was found to be h = 0.9244, showing that there is a strong correlation between inter-reversal times. We have also followed this procedure for several entangled series obtaining in all cases values for the Hurst exponent 0.5 < h < 0.92 showing that the result on the correlation of the original series is relevant. The mean value for those cases was 0.7 approximately. We believe that we have not obtained a mean value closer to 0.5 (completely uncorrelated case) for entangled cases because the series is not long enough. 5. Conclusions A statistical study was performed on the distribution of time intervals with the same polarity. We mentioned that in the literature there are studies that suggest that these time intervals follow a Poisson distribution and that statistical tests rejected the hypothesis of adjustment for that distribution. Another proposal was a power law distribution, however, as seen in this work, it only fits data for large values of time intervals. Recently there has been proposed the Tsallis distribution as a possible statistic. This work showed through a test of hypothesis that this distribution fits the data and calculated its parameters. As a result we find that the best parameters of the Tsallis distribution which fits the data is q = 1.3 and B = 2, 94. With this, we have calculated the degree of correlation of the lengths of time intervals of the same polarity. The Hurst exponent of the series was 0.92 indicating a strong correlation between time intervals of normal and reversed polarities. Acknowledgments The authors are indebted to two anonymous referees whose criticisms and suggestions have enormously contributed to the improvement of the final form of this work. CSB and ARRP acknowledge their respective fellowships from CNPq (Brazilian Science Funding Agency). ARRP thanks FAPERJ (Rio de Janeiro Science Funding Agency) for partial support.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
A. Cox, R.R. Doeel, G.B. Dalrymple, Nature 198 (1963) 1049. I. McDougall, D.H. Tarling, Nature 200 (1963) 54. N.D. Opdyke, J.E.T. Channell, Magnetic Stratigraphy, Academic Press, San Diego, 1996. R.T. Merrill, M.W. McElhinny, P.L. McFadden, Magnetic Field of the Earth: Paleomagnetism, the Core and the Deep Mantle, Academic Press, San Diego, 1996. S. Cande, D.V. Kent, J. Geophys. Res. 97 (1992) 13917. S. Cande, D.V. Kent, J. Geophys. Res. 100 (1995) 6093. E. Weisstein, http://scienceworld.wolfram.com/physics/, 2013. C. Tsallis, J. Stat. Phys. 52 (1988) 479. H. Touchette, Physica A 305 (2002) 84. C. Tsallis, Introduction to Non-Extensive Statistical Mechanics: Approaching a Complex World, Springer, Berlin, 2009. V. Carbone, L. Sorriso-Valvo, A. Vecchio, F. Lepreti, P. Veltri, P. Harabaglia, I. Guerra, Phys. Rev. Lett. 96 (2006) 128501. C. Constable, Phys. Earth Planet. Int. 118 (2000) 181. A.R.T. Jonkers, Phys. Earth Planet. Int. 135 (2003) 253. Tad J. Ulrych, Robert W. Clayton, J. Geophys. Res. 81 (1976) 1033. F. Vallianatos, Physica A 390 (2011) 1773. W.T. Eadie, D. Drijard, F.E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics, North-Holland, Amsterdam, 1971. A. Kolmogorov, G. Ist. Ital. Attuari 4 (1933) 83. N.V. Smirnov, Ann. Math. Stat. 19 (1948) 279. E.S. Pearson, H.O. Hartley (Eds.), Biometrika Tables for Statisticians, Vol. 2, Cambridge University Press, Cambridge, 1972. C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49 (1994) 1685.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Statistical analysis of geomagnetic field reversals and their consequences Cleiton S. Barbosa a , Douglas S.R. Ferreira a,b , Marco A. do Espírito Santo c , Andrés R.R. Papa a,d,∗ a
Observatório Nacional; R. Gal. José Cristino, 77, 20921-400, Rio de Janeiro, RJ, Brazil
b
Instituto Federal do Rio de Janeiro, Campus Paracambí, RJ, Brazil
c
Instituto Federal do Rio de Janeiro, Campus Volta Redonda, RJ, Brazil
d
Instituto de Física, Universidade do Estado do Rio de Janeiro, Brazil
highlights • • • •
We study geomagnetic reversals through Tsallis statistics. We have extended previous non-extensive analysis of reversals to 160 Ma. We have found q = 1.3 to be the optimum value for data adjustment. Through detrended fluctuation analysis there was found correlation in the series.
article
info
Article history: Received 19 February 2013 Received in revised form 26 July 2013 Available online 27 August 2013 Keywords: Geomagnetic reversals Tsallis statistics Statistical distributions Detrended fluctuation analysis
abstract In this paper we focus on the statistical distribution of time intervals between geomagnetic reversals. Recently the Tsallis distribution was pointed out as a possible alternative to previous proposals. We have performed statistical tests to further prove this and find the parameters of the Tsallis distribution that better fit the data. Additionally we have analyzed the correlation of time intervals between consecutive reversals to show the presence of memory effects on the mechanism that generates them, as could be expected for some Tsallis systems. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Data analysis shows that the paleomagnetic Earth’s magnetic field has reversed its polarity, several times, suddenly and disorderly in the last 160 million years. Recent works on data analysis, theoretical modeling and experimental dynamos have expanded the knowledge on the geomagnetic field. However, some fundamental questions related to polarity transitions remain unanswered: the physical mechanism that gives rise to reversals and the statistical distribution of inter-reversals times, are examples of still open issues. Several contributions indicated that the phenomenon governing the geomagnetic data distribution could be a Poisson process with constant or variable rates. Recent studies have ruled out this possibility. In the case of large inter-reversal time intervals, studies indicate power laws as possible distributions.
∗ Corresponding author at: Observatório Nacional, R. Gal. José Cristino, 77, 20921-400, Rio de Janeiro, RJ, Brazil. Tel.: +55 21 35049142; fax: +55 21 25807081. E-mail address: [email protected] (A.R.R. Papa). 0378-4371/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.08.025
C.S. Barbosa et al. / Physica A 392 (2013) 6554–6560
6555
The geomagnetic field is extremely complex in both morphology and temporal dependence. Now we know that the field is generated at the Earth’s liquid metal outer core and that it has intriguing features such as polarity reversals. The first scale of geomagnetic polarities was compiled by Cox et al. [1] and suggested a periodicity of around 106 years (1 My) in geomagnetic reversals. However, shortly after that, new data [2] showed that there were no simple intervals: time intervals between reversals (or what is obviously the same, time intervals with the same polarity) were either too long (∼1 My) or too short (∼0.1 My). It was proposed [1] that inside longer intervals, of the order of 1 My, there were short intervals of approximately 0.1 Ma. Polarity scales have been developed to the extent that new records of reversals have become available. A history of this evolution can be found in the book by Opdike and Channell [3]. There are numerous geomagnetic polarity scales covering different intervals of geological times. They have similarities and discrepancies due to the various methods used to construct them. The largest errors are related to estimates for the age of rocks as well as to the existence of some polarity intervals that can appear in some scales but be absent in others [4]. However, the geomagnetic chronology is continually under construction from new records of reversals. One of the more accepted time scales was proposed by Cande and Kent [5] based on the analysis of marine sediments. In this scale there are records of polarity for the last 80 Ma. In 1995 Cande and Kent [6] published a revised version of their scale (CK95) which presents changes of polarity for the last 160 My. Using CK95, the present study aims to statistically characterize inter-reversals times from the viewpoint of non-extensive statistical mechanics, and to show the existence of temporal correlations in series. 2. Non-extensive statistical mechanics Complex systems, whose ground state is highly degenerate have long-range microscopic memory, have very long relaxation times, or present a mixture of these characteristics, exhibit incompatibility with the extensive statistical mechanics of Boltzmann–Gibbs (BG) [7]. Looking for a solution for those types of systems Tsallis [8] proposed a generalization of the BG entropy. The entropy Sq proposed by Tsallis [8] is given by the following equation: i=w
1− Sq = k
q
pi
i=1
(1)
q−1
where k is a suitable constant, pi are the probabilities and q is the parameter that accounts for the generalization. The Tsallis entropy has many properties in common with the Boltzmann–Gibbs entropy, however, the property of additivity is violated for independent random variables [9], for example: Sq (A, B) = Sq (A) + Sq (B) +
1−q
Sq (A)Sq (B). (2) k The statistic is not extensive and the parameter q can be understood as a measure of the statistics’ non-extensivity. Depending on the value of q we have [10]: – q < 1: superextensivity, since Sq (A, B) > Sq (A) + Sq (B) – q > 1: subextensivity, since Sq (A; B) < Sq (A) + Sq (B) – q = 1: extensivity is recovered. By maximizing the Sq entropy there emerges the non-extensive probability distribution known as Tsallis distribution. This distribution is given by a function known as q-exponential and it is a generalization of the usual exponential. The distribution is given by the following equation: 1
p(x) = C expq (−Bx) = C [1 − (1 − q)Bx] 1−q
(3)
where B is a scale parameter and C is a normalization constant. In general, the distribution used to characterize the frequency of data series can be a cumulative distribution. It can be shown that the cumulative Tsallis distribution is also a q-exponential [10] and its normalized expression is given by: p(y) = expq′ (−B′ y).
(4)
From Eq. (4) it can be noted that the Tsallis distribution has q and B as parameters. A relationship between both parameters can be obtained by applying the q-logarithm to the cumulative Tsallis distribution: lnq p(x) = lnq (expq (−Bx))
(5)
lnq p(x) = −Bx.
(6)
Once estimated the parameter q, we can calculate lnq p(x) =
x1−q − 1 1−q
and from Eq. (6) obtain a linear relationship between lnq p(x) and x, with the slope given by B.
(7)
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Fig. 1. The accumulated frequency distribution function of time intervals between consecutive reversals, ∆t, on a log–log graph (squares). We also show the fitting of a linear trend for large ∆t values (straight line).
3. Optimizing data adjustments Time intervals between consecutive reversals are not randomly distributed [11], but belong to some non-flat statistical distribution. The distribution for these time intervals is shown in Fig. 1 and qualitatively seems to follow an exponential dependence. It was proposed by Constable [12] that these time intervals follow a Poisson statistics, given by: p(∆t ) = λ exp(−λ∆t )
(8)
where λ is the rate of occurrence of reversals. If the reversals follow a Poisson distribution, this would indicate that there are no memory effects in the mechanism that generates reversals, i.e., the reversals are independent events. A statistical test was applied by Jonkers [13] to this case and it was shown that the distribution of polarity of the field does not follow a Poisson distribution [11]. Jonkers [13] proposed that the distribution of time intervals of the same polarity approximates a power law. In Fig. 1 it is also shown the log–log graphic of inter-reversal time intervals and the cumulative frequency distribution function in the form of a power law. We see from Fig. 1 that a power law does not fit the whole data range. However, as shown in Fig. 1, if we ignore short time intervals of the same polarity, there will be a linear relationship between ∆t and the accumulated frequency. Long time intervals of the same polarity are called crons and have duration between 5 × 105 and 5 × 106 years. We see in Fig. 1 that these longer time intervals are distributed in the form of a power law. The possibility that the distribution of crons obeys a power law has been interpreted by several contributors as indicative of a critical phenomenon in the process controlling reversals [14]. So far we have mentioned that statistical distributions as power laws and Poisson’s have been proposed to model the time intervals of the same polarity. However they have not obtained a complete success (that is, a satisfactory fit for all the range of inter-reversal intervals). Recently, Vallianatos [15] applied the non-extensive statistical mechanics to study the reversals. He relied on the work by Carbone et al. [11], where the existence of memory effects and long-range interactions in polarity reversals were found. Thus, Vallianatos [15] proposed that time intervals of the same polarity follow a Tsallis distribution. Vallianatos [15] used the scale of Cande and Kent [5], that covers a period of 80 million years, estimating the distribution parameters q = 1.5 and B = 4.21 Ma−1 . In addition to the work of Vallianatos [15], here we apply the Tsallis distribution for the reviewed scale of Cande and Kent [6] covering a period of 160 million years. Initially we calculate the distribution parameters, show through the Kolmogorov–Smirnov test that the distribution with the calculated parameters fits the data and to finish calculate the degree of correlation between polarity reversals. As a starting point for calculating the parameters for the best fitting distribution we used the results obtained by Vallianatos [15]. He estimated q = 1.5. We assumed that the value of q for our data should be around this. We know that q is between 1 and 3, then, we use q between 1.2 and 1.7. Specifically, we have initially considered values of q = 1.2, 1.3, 1.4, 1.5, 1.6 and 1.7. Using Eq. (6), we have calculated, for each value of q the corresponding value of B and thus obtained a family of Tsallis distributions. For each distribution the Kolmogorov–Smirnov [16–19] test was applied to accept or to reject the hypothesis that the specified distribution fits the data. Through this procedure we aimed to find the values of q and B that better fit the distribution of time intervals between consecutive reversals. The Kolmogorov–Smirnov (KS) test [16–19] unlike more popular tests (for example, the chi-square) does not apply to qualitative data nor to discrete variables, because the table for this test is only accurate if the testing distribution is continuous. However, it has the advantage of not being dependent on rating data that, besides being always somewhat arbitrary, involve information losses. On the other hand, the KS test can only be applied when the distribution indicated in the null hypothesis is completely specified (in our case, the Tsallis distribution). Furthermore, the KS test is applicable to small samples.
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Fig. 2. The q-log of cumulative frequency of actual data (squares). The straight line is the linear fit between the q-log and ∆t.
Let G be the distribution function of the study population and F the proposed distribution function, continuous and completely specified. Hypotheses to test: H0 : G(x) = F (x) for every x H1 : G(x) ̸= F (x) for some x. The KS test compares the accumulated relative frequencies recorded in the sample with those that would be expected if the population distribution was that specified in the null hypothesis. The statistic of the KS test considers the greatest of the differences, in absolute value, between the ratio of observations less than or equal to x, S (x), and the probability of observing a value less than or equal to x if the population distribution is the one specified in H0 , F (x). Since F is a continuous function and S a ladder function, the supreme, dn,obser v ed , occurs at a point where there is a jump of S. Thus, if H0 is true, the maximum vertical distance between the images of both distributions should not be too large, then it is expected that dn,obser v ed take a small value. So, for a significance level α , reject H0 if the observed value is greater than or equal to the critical point Xn,α (critical values Xn,α can be found in a table). Now we illustrate the method for the value q = 1.3 which, as will be shown, was the best value. Let us calculate the log-q for q = 1.3, noting that: lnq F (∆t ) =
∆ t 1 −q − 1 = −B∆t 1−q
ln1.3 F (∆t ) =
∆t 1−1.3 − 1 . 1 − 1.3
(9)
(10)
The dependence between ln1.3 F (∆t ) and ∆t is shown in Fig. 2. The slope of the straight line is −2, 94 and therefore we have B = 2.94 Ma−1 and the Tsallis distribution for q = 1.3 is: F (∆t ) = exp1.3 (−2.94∆t ).
(11)
In Fig. 3 we adjust the data to the Tsallis distribution with the calculated parameters. Let us apply the KS test to that specific distribution. As explained above the idea of testing is to find the maximum vertical distance dn between the empirical distribution Gn (∆t ) and the distribution in which we are testing the hypothesis that fits the data Fn (∆t ). Using the data of CK95 as a function of ∆t the graph of these two functions is shown in Fig. 3: The maximum vertical distance between these functions cannot be greater than a threshold testing, Xn , which in this case is equal to X276 = 0.08174. For each ∆t we have calculated the difference between the Tsallis distribution and the empirical cumulative distribution. That is, calculate d276 = |Gn (∆t ) − Fn (∆t )|. The graph of d276 in terms of ∆t is shown in Fig. 4. From Fig. 4 we see that the maximum difference is d276 = 0; 07325. Then, d276 < X276 => Hypothesis accepted. So, with a confidence level of 5% we can accept the hypothesis that the time intervals between consecutive reversals follow a Tsallis distribution with B = 1.3 Ma−1 and q = 2.94 as parameters. The same process was repeated for other values of q. For all these cases the KS test was applied and the hypothesis was rejected. At this point we have included the q values 1.25 and 1.35 to obtain a more detailed description around the q = 1.3 special value.
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Fig. 3. The cumulative frequency of actual data (squares). The continuous line is the Tsallis distribution using q = 1.3 and B = 2.94. It is remarkable the fit between both distributions.
Fig. 4. Difference between the empirical distribution function and the Tsallis distribution with q = 1.3.
Fig. 5. Maximum difference between the Tsallis distribution function and the empirical distribution function for several q values. The dashed line is the threshold value for the test.
With these results we conclude that the q parameter value that better fit the data is around 1.3. This is shown in Fig. 5, where we have a graph of d276 values for some values of q.
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Fig. 6. Graph of fluctuations depending on the size of the window used to calculate the fluctuation range. For this graph the series was used in chronological order and provided h = 0.92.
We see that around q = 1.3, we have a local minimum for the difference between the proposed distribution and the empirical distribution. The fact of the distribution of polarity intervals follows a Tsallis distribution brings us new insights into the understanding of reversals. Systems that follow Tsallis distributions are systems in which statistical mechanics is non-extensive. 4. Correlation of series There are systems with memory effects and long-range interactions. For geomagnetic reversals this means that a reversal occurred 10 million years ago could be related to a reversal to occur in the future. If indeed the time inter-reversal times follow Tsallis statistics they could be temporally correlated. With this idea in mind we used the method of detrended fluctuation analysis (DFA) [20] to assess the degree of correlation of series. The method consists of dividing the series into windows of size n and calculate the fluctuation range Fdfa . Fdfa is related to the window size. It generally increases when increase the window size, so we have the following relationship between Fdfa and window size: Fdfa ∼ nh
(12)
where h is the Hurst exponent. This exponent is the slope of the log–log dependence of Fdfa with n. The Hurst exponent is of fundamental importance in determining the degree of correlation in series. Depending on its value, we have: – Uncorrelated series: h = 0.5 – Anti correlated series: 0 < h < 0 : 5 – Power law long range temporal correlation: 0 : 5 < h < 1. The last case indicates a series with positive autocorrelation, i.e., that a high (low) value in the series is more likely to be followed by another high (low) value. The window size ranged between 20 < n < 140. The graph of the function of fluctuations is shown in Fig. 6. The Hurst exponent was found to be h = 0.9244, showing that there is a strong correlation between inter-reversal times. We have also followed this procedure for several entangled series obtaining in all cases values for the Hurst exponent 0.5 < h < 0.92 showing that the result on the correlation of the original series is relevant. The mean value for those cases was 0.7 approximately. We believe that we have not obtained a mean value closer to 0.5 (completely uncorrelated case) for entangled cases because the series is not long enough. 5. Conclusions A statistical study was performed on the distribution of time intervals with the same polarity. We mentioned that in the literature there are studies that suggest that these time intervals follow a Poisson distribution and that statistical tests rejected the hypothesis of adjustment for that distribution. Another proposal was a power law distribution, however, as seen in this work, it only fits data for large values of time intervals. Recently there has been proposed the Tsallis distribution as a possible statistic. This work showed through a test of hypothesis that this distribution fits the data and calculated its parameters. As a result we find that the best parameters of the Tsallis distribution which fits the data is q = 1.3 and B = 2, 94. With this, we have calculated the degree of correlation of the lengths of time intervals of the same polarity. The Hurst exponent of the series was 0.92 indicating a strong correlation between time intervals of normal and reversed polarities. Acknowledgments The authors are indebted to two anonymous referees whose criticisms and suggestions have enormously contributed to the improvement of the final form of this work. CSB and ARRP acknowledge their respective fellowships from CNPq (Brazilian Science Funding Agency). ARRP thanks FAPERJ (Rio de Janeiro Science Funding Agency) for partial support.
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