Distribution of the daily Sunspot Number variation for the last 14 solar cycles
aa r X i v : . [ a s t r o - ph . S R ] O c t Distribution of the daily Sunspot Number variation for thelast 14 solar cycles
Mihail-Ioan Pop Department of Physics, Transilvania University of Brasov, email: [email protected]
Abstract.
The difference between consecutive daily Sunspot Numbers was analysed.Its distribution was approximated on a large time scale with an exponential law. Inorder to verify this approximation a Maximum Entropy distribution was generated bya modified version of the Simulated Annealing algorithm. The exponential approximationholds for the generated distribution too. The exponential law is characteristic for timescales covering whole cycles and it is mostly a characteristic of the Sunspot Numberfluctuations and not of its average variation.
Keywords:
Solar Cycle, Models; Sunspots, Statistics
1. Introduction
The oldest and most used index for solar activity is the Sunspot Number( SN ) or Wolf number describing the number of sunspots S and groups ofsunspots G that can be observed on the Sun: SN = 10 G + S . The SN indexis computed daily and there is a complete record of values SN i since the1850’s, spanning Solar Cycles 10-23. For the following we used the differencebetween daily SN numbers: DSN i = SN i − SN i − , (1)where i is the count of days. The SN values have been obtained from theSolar Influences Data Analysis Center of the Royal Observatory of Belgium(SIDC, 2010). The graph of DSN as a function of time is presented inFigure 1. The distribution of
DSN is linked to the mechanisms that generatesunspots with a periodicity of about 11 years. While there are deterministicmechanisms involved, there are also some stochastic mechanisms, responsibleespecially for the high day to day variability of SN . Thus, a statistical analy-sis of DSN may reveal some characteristics of the latter type of mechanism.We will further refer to
DSN values and their characteristics determinedfrom measured SN values as ”empirical” values or characteristics.The distribution of DSN has been reconstructed both as a histogram andas a Maximum Entropy Method (MEM) distribution (Jaynes, 1957a,1957b,1963; Uffink, 1995). The MEM distribution was built by imposing somelocal conditions and was determined using a modified form of the SimulatedAnnealing (SA) heuristic search algorithm (Kirkpatrick, Gelatt, and Vecchi,
Accepted for publication in Solar Physics.
Figure 1.
Evolution of
DSN (black points) along solar cycles (average SN - grey line, notto scale). The graph starts in the middle of the 12 th cycle and spans for the last 14 cycles. DSN as computed from SN values are integer numbers.However, we argue that both SN and DSN values can be thought as realnumbers, where the fractional part expresses the degree with which a cer-tain ”average sunspot” is realised. Thus, the maximum entropy distributionwas extended into the set of real numbers. The distributions obtained wereapproximated on certain ranges of
DSN with exponential functions of theform: ρ ( DSN ) = 1 Z exp (cid:18) − DSNm (cid:19) , (2)where m is a distribution parameter and Z is a constant such that ρ hasintegral equal to 1.The SN time series have both a deterministic component, which may betaken as the monthly or yearly average of SN , and a stochastic component,the latter appearing as daily fluctuations of SN around the average. Bothcomponents have characteristic distributions. The basic models used in ex-plaining the long-term (on scales of months and above) behaviour of the solarcycle are the dynamo models (Proctor and Gilbert, 1994). It was pointed outthat this model may exhibit chaotic behaviour. Letellier et al. (2006) discusssuch a chaotic mechanism. Their conclusions are that it has low dimension-ality and similar behaviour to a R¨ossler system of differential equations. Onthe other hand, Aguirre, Letellier, and Maquet (2008) model the solar cycle,transformed into a symmetrical space, through an autoregressive model withdeterministic terms and stochastic residuals with Gaussian distribution. Themodel obtained has a chaotic attractor similar to that of the R¨ossler system.Aguirre, Letellier, and Maquet point out that the sunspot number timeseries is nonstationary. Nordemann (1992) approximated the time series ofyealy averages of SN with an exponential function separately for the risingand declining phases. The exponential dependence indicates a power lawdistribution of the SN yearly averages for each phase. For larger time scalesthe distribution may deviate from the power law.The stochastic part of the SN time series has been analysed both inconnection to a chaotic dynamo and as a separate term. In many studiesexponential distributions and dependences were found. Pontieri et al. (2003)build a dynamo model with fractional Brownian motion in order to explainthe stochastic character of the SN time series. They determine the Hurstexponent of SN to be H ≈ .
76 on time scales from 20 to 350 days. Thisindicates a Brownian motion with positive autocorrelation at work at thesetime scales. Noble and Wheatland (2011) develop a model for the stochas-tic part of the SN time series which is represented as a diffusion processdescribed by the Fokker-Planck equation. The solution of this equation hasan approximate exponential form. They find that at solar minima the SN index has a distribution with exponential tail. They also point out thatthe daily SN distribution is approximately exponential for the period 1850-2010. Greenkorn (2009) looked at daily SN data in order to determine thetype of flow in the Sun’s convective layer. The SN values indicate stochasticbehaviour for Cycles 10-19 and chaotic behaviour for Cycles 20-23, the last ofwhich presents an increase in the stochastic character compared to previouscycles.Salakhutdinova (1998; 1999) analysed the SN time series by separating itinto a regular (deterministic) low frequency component and a stochastic highfrequency component. The stochastic component has properties of white(Gaussian) noise at time scales lower than two months and for time scalesbetween two months and two years it has properties similar to pink (flicker)noise. The two kinds of properties may characterize two sub-components thatcombine to form the stochastic part of SN . Generally, for large time scalesthe stochastic component can be associated with a chromatic noise withsharp maxima. The regular component behaves as a nonlinear oscillator,with similarities to a R¨ossler system. Lepreti et al. (2000) compute the Hurstindex for the SN index and the H α flare index for the period 1976-1996.The two indices have similar Hurst values (0 .
76 and 0 .
74) on intervals of20-350 days and 24-450 days respectively. These show the presence of long-range autocorrelations for the two solar activity indices for the period 1976-1996. The authors determined the distribution of normalised fluctuationsof the SN and H α flare indices. The fluctuations were computed as 1-daydifferences of the respective indices. Next, these differences were normalisedby subtracting their average value and then dividing by their root meansquare value. For both indices, the probability density functions are stretchedexponentials of the form ρ ( x ) = A exp ( − a | x | r ). For SN they found r ≈ . r ≈ . et al. (2005) analyse the coronal emission time series at 530.3nm from 1949 to 1996 with the Proper Orthogonal Decomposition method.The method uses a series expansion of the coronal emission relative to somebasis functions by means of coefficients which depend on time. The authorsdetermined the distribution of the 1-day difference of each coefficient’s val-ues. The probability density function is similar to a stretched exponentialfor the central part of the distribution, where r is close to 1.Kanazir and Wheatland (2010) model the flare energy distribution andwaiting times for two solar active regions spanning each about one month.They find that the flare frequency-size index has a power law distribution,while the waiting times have an exponential distribution. The parameter ofthe exponential distribution may suffer a jump in time, as found for one ofthe two active regions. This jump is explained in a jump-transition model bycorresponding step changes in the rates of energy input and flare transition.Nevertheless, between step changes the waiting times distribution is a stableexponential distribution.
2. Empirical Data Approximations
In order to obtain the distribution of
DSN , daily data from Cycles 10 -23 were used. The values of
DSN range from -91 to 112. The empiricalvalues of
DSN were computed from the registered SN values and then theempirical probability p ( DSN ) was determined by counting the occurrenceof each value of
DSN . The probability for each
DSN is presented in Figure2. By applying a least squares fit to ln p ( DSN ), it was found that p ( DSN )can be approximated well with exponential functions (2) with the followingparameters: m = − .
32 for
DSN ∈ [ − −
10] and m = 9 .
37 for
DSN ∈ [10; 60]. The goodness of fit is characterised by the correlation coefficient R ≈ .
99 in both cases. Outside these intervals p ( DSN ) diverges from theexponential fit. A special case is
DSN = 0, for which the probability ismuch bigger than probabilities associated to other values of
DSN . Thus, itappears that
DSN = 0 represents a special state. We propose a model todescribe the general behaviour of
DSN , according to which the probabilitydensity function (pdf) of values of
DSN in the set of real numbers is of theform: ρ ( DSN ) = p ( DSN = 0) δ ( DSN ) + p ( DSN < ρ − ( DSN )+ p ( DSN > ρ + ( DSN ) , (3) Figure 2. (left) Empirical and Maximum Entropy distributions of the values of
DSN .Note the high pdf value associated with
DSN = 0; (right) Natural logarithm of the
DSN distributions with exponential approximations of the Maximum Entropy pdf computedfor | DSN | ∈ [10; 60]. where δ ( · ) is the Dirac distribution, ρ − ( · ), ρ + ( · ) are probability density func-tions associated with negative and positive values of DSN respectively and p ( · ) are probabilities associated with the respective intervals of DSN . Fromthe empirical distribution determined above, p ( DSN = 0) = 0 . p ( − DSN ) ≈ p ( DSN ), we take p ( DSN <
0) = p ( DSN >
0) = 0 . ρ − , ρ + have been determined by the Maximum EntropyMethod (see Appendix A). Note that the empirical probabilities p ( DSN )can be taken as (smoothed) pdf values associated with the correspondingvalues of
DSN since pdf is probability divided by corresponding intervaland p ( DSN ) corresponds to an interval of
DSN of length 1.
3. Results
Maximum Entropy distribution approximations
We collected 55 882 empirical values of
DSN . In order to determine thepdfs ρ − ( · ), ρ + ( · ) all values of 0 were eliminated, according to model (3). Weused N − = 23 587 DSN values for ρ − ( · ) and N + = 22 696 values for ρ + ( · ).The pdf estimate obtained is presented in Figure 2; also, its logarithm isrepresented along exponential fits computed by the least squares method for DSN ∈ [ − −
10] and
DSN ∈ [10; 60] respectively. The fits have param-eters m = − .
51 for
DSN ∈ [ − −
10] and m = 9 .
60 for
DSN ∈ [10; 60].The goodness of fit is characterised by R ≈ . DSN .From both the empirical values and the MEM distribution it can beseen that the distribution of
DSN changes abruptly from the exponentialapproximations for small | DSN | . There are two local maxima at about DSN = − DSN between 7 and 8. There may be an observationalcause for this characteristic. When the SN index is computed from sunspotobservations, each group of sunspots counts as 10 individual sunspots, thusit may be easier to observe a higher value, e.g. SN ≥ SN = 0 to SN = 7 and back may bemore probable than other jumps around the cycles’ minima.For values of | DSN | < | DSN | = 1 . | DSN | >
60 the MEM pdf diverges from the exponentialapproximations, which is supported by the empirical distribution of
DSN .At the edge of the interval for
DSN considered for building the MEM pdf,a sharp increase appears. The latter is a spurious feature. In this area theconditions have very small values, such that the entropy exceeds the valueof the quadratic error E (see Appendix). Thus, the optimisation is doneby maximising the entropy, which shifts the distribution toward a uniformdistribution. The entropy may also affect the divergence from the exponen-tial law for | DSN | >
60; thus, we consider that the MEM pdf gives onlya qualitative description for very high values of | DSN | . This is not verysignificant though, as the values of the pdf are below 10 − in this region, farsmaller than those close to DSN = 0.3.2.
Time-scale Localisation of the Exponential Approximation
In order to determine the time scale at which the exponential law is valid, themoving average and standard deviation of
DSN were computed for differenttime intervals and their values were compared to theoretical values computedwith (2) and m = ± .
35, where the sign was taken + for
DSN < − for DSN >
0. All computations were carried out for | DSN | ∈ [10; 60].The results are shown in Figure 3. The best localisation, i.e. the smallesttime interval for which the distribution of
DSN is close to the theoreticaldistribution (2), is obtained for a time span of about 11 years or approxi-mately one solar cycle period. Cycles 10 and 11 stand out as anomalies, sincetheir standard deviation is very different from the theoretical one even forlarge time intervals. For the Cycles 12-23, the standard deviation is close tothe theoretical value with displacements of about 10% from the theoreticalstandard deviation. The results obtained for a 1 year time interval showwhere deviations from the exponential law occur. The average
DSN hassharp peaks during cycles’ minima. The standard deviation has maximaand minima corresponding to the maxima and minima of solar cycles. Itsmaxima are close to the theoretical standard deviation, while the minimaare far from it. It appears thus that deviations from the exponential lawoccur around minima of the solar cycles. This is partly a consequence ofthe fact that the minima are populated with only a few values of | DSN | above 10; thus, the sharp behaviour is partly generated by fluctuations inthe average and standard deviation of | DSN | .3.3. The Origin of the Exponential Law: Fluctuations vs.Deterministic SN
The difference
DSN between SN consecutive values at a fixed date can bereduced to two components: a smooth deterministic variation DSN det andan error (or fluctuation) term
DSN err , such that
DSN = DSN det + DSN err . (4)A similar relation holds between the variances of the terms implied: V ar ( DSN ) =
V ar ( DSN det ) +
V ar ( DSN err )+ Cov ( DSN det , DSN err ) , (5)where V ar ( · ) represents the variance and Cov ( · , · ) the covariance of therespective quantities. We tested to see whether one of the terms DSN det , DSN err is far more important than the other. If this were the case, it isexpected that the negligible term would have negligible statistical charac-teristics and its distribution would have a negligible contribution in the
DSN overall distribution. Since the averages of the
DSN components above areclose to 0, the test was carried on their variances. Moving variances onintervals of one year were used. The results are shown in Figure 4. It canbe seen that
V ar ( DSN ) is at least 100 times greater than
V ar ( DSN det )or
Cov ( DSN det , DSN err ), thus by far the most important component of
DSN appears to be the fluctuation
DSN err . The exponential distributioncan be assumed to be given by
DSN fluctuations and not its deterministiccomponent.
4. Conclusions
The most important characteristic of the distribution of
DSN is its almostperfect symmetry with respect to the vertical axis. The small differences inthe MEM pdf between negative and positive values of
DSN are not wellsupported by empirical values and we consider them spurious phenomena,due to the stochastic nature of the Simulated Annealing algorithm and tonoise in the initial data.
Figure 3.
Moving averages and standard deviations of
DSN (black lines) compared to the1 year moving average SN shape (grey line - not to scale). Only values | DSN | ∈ [10; 60]were considered in the computations. The statistical characteristics of
DSN were com-puted on time intervals of (a) 1 year, (b) 5.5 years and (c) 11 years. The dashed linesrepresent theoretical average and standard deviation computed with (2) and m = ± . DSN . The average SN describes the smoothed solar cycles’evolution. Overall, the distribution of
DSN has several areas of specific variation:(i) For low values of | DSN | the pdf decreases as | DSN | increases, probablyexponentially, but this is uncertain because of the small number of points.Around | DSN | = 7 the pdf has a slight but abrupt rise. (ii) For | DSN | Figure 4.
The time evolution of the variance components of
DSN : variance of total
DSN ,variance of deterministic
DSN (det) component as estimated from the one year average SN and the covariance of the deterministic (det) and error (err) components of DSN .The difference between the total variance and the latter two components represents thevariance of the error component (not represented), which was found to be superposed onthe total variance. All values were computed on moving one year intervals. between 10 and 60 it obeys an exponential law of an approximate form: ρ ( DSN ) = 1 Z exp (cid:18) − | DSN | m (cid:19) , (6)with m ≈ .
35 experimentally and m ≈ .
56 in the MEM pdf; Z is aconstant. (iii) For very large | DSN | the pdf decreases somewhat faster thanexponential with increasing | DSN | .The distribution (6) appears to be valid only when whole cycles areconsidered. For small time intervals, the DSN distribution may deviatefrom the exponential law. This happens especially around minima of thesolar cycles.The presence of the exponential law dependence is interesting in itself.This result was previously found by Lepreti et al. (2000) for SN valuesspanning a 20 year period. Here the result is investigated in more detailand for a longer time period. The formula (6) represents a Laplace distri-bution centered at DSN = 0 (Kotz, Kozubowski, and Podgorski, 2001).The deviations from this law happen for very low values of | DSN | , whichoccur especially (but not only) in the minima of solar cycles, and also forvery high values of | DSN | , which occur at the maxima of solar cycles. Thelatter case happens only for a few values of the solar cycles analysed, i.e. for | DSN | >
60, as can be seen from Figure 1. Thus, it can be concludedthat overall the exponential law holds for all values of | DSN | of statisticalsignificance, except values around DSN = 0. The failure of the exponential0law for small | DSN | may be due to the way the Sunspot Number ( SN ) iscomputed by counting both individual sunspots and groups of sunspots. Thisis especially significant for solar cycles minima, where abrupt jumps from SN = 0 to SN ≈ DSN = 0stays above the Laplace pdf (6).The obtained results can be applied in the field of stochastic predictivemodels for solar activity and space weather. The daily stochastic behaviouris superposed over a long-term cyclic characteristic. These daily fluctuationsmay be represented as a stochastic process with jumps distributed accordingto (6). Nevertheless, such a stochastic process would give only an average,long-term representation. It must be completed with additional terms thatgive the exact distribution at smaller time scales. These terms are yet to bedetermined.The Laplace distribution of
DSN is linked to the exponential distributionof SN , as evidenced by Noble and Wheatland (2011). Indeed, by assumingthat daily SN values are independent and identically distributed randomvariables with exponential distributions, the difference of any two SN vari-ables is a random variable with a Laplace distribution. However, the SN values are not independent on a day-to-day basis. Similarly, the Laplacedistribution does not appear at a daily level, but rather at whole cyclelevels. We hypothesize that the Laplace distribution of DSN appears attime intervals for which the beginning and end values of SN are completelyindependent. It is expected that the distribution is stationary at this timescale. This seems to be the case for 11 year intervals, as seen from Figure3, where both the average and standard deviation tend to become stablein time. Nevertheless, the standard deviation still has strong jumps even atthis time scale, possibly showing abrupt changes in the Laplace distributionparameter m from one cycle to the next. The long time span at whichstability occurs may be a characteristic of the magnetic flux tubes lyingin the convective layer, which is preserved at least until the magnetic fieldchanges polarity. As they emerge through the photosphere, the flux tubesreveal this characteristic in the form of the Laplace distribution of DSN .Nevertheless, other more complex processes are at work for small time scales,where the distribution is highly irregular.
Appendix A. MEM Distributions
The Maximum Entropy approach for probability distributions consists indetermining the distribution that maximises the entropy with some imposedconstraints. We present the way it was applied in the present work. Eachpdf ρ − ( DSN ), ρ + ( DSN ) was determined separately as sets of probabilities p − ( DSN i ), p + ( DSN j ) associated with corresponding values DSN i < DSN j > i, j = 1 , , ..., N P , which were next transformed into pdf values.This is a general approach to building a continuous pdf which eliminatesthe need for a prior model to fit ρ − ( · ), ρ + ( · ). The number N P of values of DSN considered in each case is much greater than the empirical number ofprobability values. The values
DSN i , DSN j where chosen at equal distancesin the corresponding intervals.We illustrate below the path followed for DSN <
0; for
DSN > H = − N P X i =1 p − ( DSN i ) ln p − ( DSN i ) . (7)Next, some local constraints were imposed on the distribution. These werebuilt with a Gaussian function: f k ( x ) = exp (cid:18) − ( x − x − k ) σ (cid:19) , k = 1 , , ..., N C . (8)The points x − k are equally distanced in the interval of DSN <
0. The methodof Kernel Density Estimation uses similar functions for building a pdf esti-mate with good results (Rosenblatt, 1956; Parzen, 1962; Cranmer, 2000; Wuand Mielniczuk, 2002). The functions f k were averaged, yielding empiricalvalues < f k > emp computed over N − empirical values of x = DSN < < f k > sim computed with the values x = DSN i and their associated probabilities p − ( DSN i ): < f k > emp = 1 N − X x =empirical DSN f k ( x ) , (9) < f k > sim = 1 N P N P X i =1 p − ( DSN i ) f k ( DSN i ) . (10)A quadratic error was used: E = N C X k =1 ( < f k > emp − < f k > sim ) . (11)2When the error E is minimised, the obtained distribution presents a lot ofnoise. In order to smooth it, the entropy H was maximised along with theminimisation of E. Thus, the function to be minimised was taken of theform: E H = E − kH, (12)where k is a constant that regulates the degree of smoothing. For big k the distribution obtained is close to the uniform distribution, while for verysmall k the distribution has a lot of noise. Based on tests carried on someknown distributions, a value of k = 10 − was found to be sufficient.In order to find the minimum of E H a modified version of the SimulatedAnnealing (SA) algorithm was used. In the original algorithm, a temperatureparameter is decreased during the run in order to control the convergencetowards the absolute minimum. We found this to be unnecessary, thus weset the temperature to 0. Also, in the original algorithm, at each step, thesolution vector ( p − ( DSN i )) i is constructed by modifying all its elements atrandom at each step. We found this to be detrimental and the modificationwas carried on only one element chosen at random at each step. The rate ofchange of the vector elements was decreased as the algorithm proceeded.A number of N C = 101 conditions were used in each case. An equal num-ber of values of x − k , x + k were generated in the intervals [ − −
1] and [1; 120]respectively. In each case the solution was a vector of N P = 1 000 probabili-ties ( p − ( DSN i )) i , ( p + ( DSN j )) j , corresponding to equally distanced valuesof DSN set in the above intervals. Thus, in each interval there are about 10times more simulated probability values for the pdf than empirical values.The conditions’ parameter σ was chosen such that each condition covers asignificant part of the interval of values of DSN . It vas taken equal to 1 / f k of about 10 units of DSN and a base coverage of about 20 unitsof
DSN . The functions f k have maxima distanced at about 1 unit of DSN ,thus yielding a dense set of conditions. This value of σ was tested on someknown distributions and was found satisfactory. Acknowledgements
We are grateful to Professor Gelu M. Nita from theCenter for Solar-Terrestrial Research of the New Jersey Institute of Tech-nology for helpful discussions and suggestions regarding this work. We arealso grateful to an unknown reviewer for advice that helped us significantlyimprove this paper.
References
Aguirre, L.A., Letellier, C., Maquet, J.: 2008, Forecasting the Time Series of SunspotNumbers,
Solar Phys. . Cerny, V.: 1985, Thermodynamical approach to the traveling salesman problem: Anefficient simulation algorithm,
Journal of Optimization Theory and Applications .Cranmer, K. S.: 2000, Kernel Estimation in High-Energy Physics, Computer PhysicsCommunications .Greenkorn, R.A.: 2009, Analysis of Sunspot Activity Cycles,
Solar Phys. .Ingber, L.: 1993, Simulated annealing: Practice versus theory ,
Mathematical andComputer Modelling .Jaynes, E. T.: 1957a, Information Theory and Statistical Mechanics, Phys. Rev. .Jaynes, E. T.: 1957b, Information Theory and Statistical Mechanics II,
Phys. Rev. .Jaynes, E. T.: 1963, Information Theory and Statistical Mechanics, in
Statistical Phys. ,K. Ford (ed.), Benjamin, New York.Kanazir, M., Wheatland, M.S.: 2010, Time-Dependent Stochastic Modeling of Solar ActiveRegion Energy,
Solar Phys. .Kirkpatrick, S., Gelatt, C. D., Vecchi, M. P.: 1983, Optimization by simulated annealing,
Science .Kotz, S., Kozubowski, T.J., Podgorski, K.: 2001, The Laplace distribution and general-izations: a revisit with applications to Communications, Economics, Engineering andFinance, Birkh¨auser.Lepreti, F., Fanello, P.C., Zaccaro, F., Carbone, V.: 2000, Persistence of solar activity onsmall scales: Hurst analysis of time series coming from H flares,
Solar Phys. .Letellier, C., Aguirre, L.A., Maquet, J., Gilmore, R.: 2006, Evidence for low dimensionalchaos in sunspot cycles,
Astron. Astrophys. .Noble, P.L., Wheatland, M.S.: 2011, Modeling the Sunspot Number Distribution with aFokker-Planck Equation,
Astrophys. J. .Nordemann, D. J. R.: 1992, Sunspot number time series - Exponential fitting and solarbehavior,
Solar Phys. .Parzen, E.: 1962, On Estimation of a Probability Density Function and Mode, Ann.
Math.Statist., .Pontieri, A., Lepreti, F., Sorriso-Valvo, L., Vecchio, A., Carbone, V.: 2003, A Simple Modelfor the Solar Cycle, Solar Phys. .Proctor, M.R.E., Gilbert, A.D. (eds.): 1994, Lectures on Solar and Planetary Dynamos,Cambridge.Rosenblatt, M.: 1956, Remarks on Some Nonparametric Estimates of a Density Function,
Ann. Math. Statist. .Salakhutdinova, I. I.: 1998, A Fractal Structure of the Time Series of Global Indices ofSolar Activity, Solar Phys. .Salakhutdinova, I. I.: 1999, Identifying the quasi-regular and stochastic components ofsolar cyclicity and their properties,
Solar Phys. .SIDC-team, World Data Center for the Sunspot Index, Royal Observatory of Belgium,
Monthly Report on the International Sunspot Number , online catalogue of the sunspotindex: , 1848-2010.Uffink, J.: 1995, Can the Maximum Entropy Principle be explained as a consistencyrequirement?,
Studies in History and Philosophy of Modern Physics .Vecchio, A., Primavera, L., Carbone, V., Sorriso-Valvo, L.: 2005, Periodic Behavior andStochastic Fluctuations of Solar Activity: Proper Orthogonal Decomposition Analysis,
Solar Phys. .Wu, W. B., Mielniczuk, J.: 2002, Kernel density estimation for linear processes,
Ann.Statist.30