Solar Data Analysis - A New Approach

July 4, 2010

Abstract

We present a new method for analysing asronomical data havingsome kind of symmetry. As an example we use time series of sun spotnumber and solar flare number which shows some kind of periodic be-haviour.Standars tecniques like correlartion regression analysis, auto-correlation techniques, nonlinear regression analysis, wavelet analysisetc. are being used at present for analysing sunspot data. None ofthese methods are handy enough to get a total handy estimate ofthesymmetry and regularity exhibited by the data. The method basedon approximate entropy calculation which have been extensively usedin physiological time series analysis is found to be suitable for predict-ing the symmetry of solar data as well. Computer analysis is doneusing the MatLab software.The method can be successfully extentedto several astronomical data which shows regular periodicity and canbe considered as an initial attempt to employ methods of regularitystatistics to astronomical data analysis. Since the variation of sunspotand solar flare number is larger compared to physiological and financialtime series the dependence of approximate entropy on the regularityparameter is considerably important. The clculation of approximateentropy is done for various regularity parameter values.

1 Introduction - Origin of Sunspots

Sunspots are dark, planet-sized regions that appear on the surface of theSun. Sunspots are dark because they are cooler than their surroundings. Alarge sunspot might have a central temperature of 4,000 K, much lower thanthe 5,800 K, temperature of the adjacent photosphere. Sunspots are only

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dark in contrast to the bright face of the Sun. If we could cut an averagesunspot out of the Sun and place it elsewhere in the night sky, it would beabout as bright as a full moon. Sunspots have a lighter outer section calledthe penumbra, and a darker central region named the umbra.Sunspots are caused by disturbances in the Sun’s magnetic field welling upto the photosphere, the Sun’s visible surface. The powerful magnetic fieldsin the vicinity of sunspots produce active regions on the Sun, which in turnfrequently spawn disturbances such as solar flares and Coronal Mass Ejec-tions (CMEs). Because sunspots are associated with solar activity, spaceweather forecasters track these features in order to help predict outbursts ofsolar storms. Sunspots form over periods lasting from days to weeks, and canpersist for weeks or even months before dissipating. The average number ofspots visible on the face of the Sun is not constant, but varies in a multi-yearcycle. Historical records of sunspot counts, which go back hundreds of years,verify that this sunspot cycle has an average period of roughly eleven years.Our Sun isn’t the only star with spots. In recent years, astronomers havebeen able to detect starspots - sunspots on other stars Sunspots are causedby extremely strong, localized magnetic fields on the Sun. Jet streams ofplasma (flow of charged particles) that form deep within the Sun’s convec-tive zone produce powerful magnetic fields. When these loops of magnetism,or magnetic ropes, generated by flowing plasma break the visible surface(photosphere) of the Sun, they produce sunspots. Sunspots generally appearin pairs with opposite magnetic polarities; one where the bundle of ropesemerges from the solar surface, and the other where the bundle plunges backdown through the photosphere.The intense magnetic fields at sunspots inhibit mixture of hot plasma fromthe surrounding photosphere into the sunspot regions. Sunspots are thuscooler, and therefore darker, than their surroundings. Magnetic field strengthswithin sunspots range from 1,000 to 4,000 Gauss, and are thousands of timesmore intense than Earth’s average surface field strength of about 0.5 Gauss.The fields within sunspots are also much stronger than the Sun’s globalaverage field, which is around 1 Gauss. Larger sunspots have higher fieldstrengths.Although the details of sunspot formation are not thoroughly understood,scientists believe that the differential rotation of the Sun is the underlyingcause. Since the gaseous sphere of the Sun rotates more quickly at its equa-tor than at its poles, the Sun’s overall magnetic field becomes distorted andtwisted over time. The twisted field lines eventually emerge through the pho-

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tosphere, revealing their presence as sunspots.

Several major solar phenomena are associated with these twisted mag-netic fields. When the tangled fields reach a breaking point, like a rubberband that snaps when wound too tight, huge bursts of energy are releasedas the field lines reconnect. Such sudden shifts in magnetic fields generateenergetic solar flares and vast Coronal Mass Ejections. The twisting of theSun’s global magnetic field also periodically causes the field to reverse itsoverall polarity, giving rise to the 11-year sunspot cycle and the 22-year solarcycle. Both the overall solar field, and the polarities of individual pairs ofsunspots, flip on a periodic basis. Hale’s Polarity Laws specify the polarityof each sunspot in a pair, which depends on the solar hemisphere in whichthe pair lies and on the phase of the solar cycle. Analysis of Solar magneticfields can be done using Zeeman effect spectroscopy which reveals severl in-teresting features of Solar magnetic field

Figure 1: Zeman Splitting of Solar Spectal Lines

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Figure 2: Latest data of variation of Solar Magnetic field

Surprisingly, humans have observed sunspots for a very long time, so his-torical sunspot observations provide us with some of our best long-durationrecords of solar activity. Large sunspots can sometimes be seen with thenaked eye, especially when the Sun is viewed through fog near the horizon atsunrise or sunset. The first written record of sunspots was made by Chineseastronomers around 800 B.C. Court astrologers in ancient China and Korea,who believed sunspots foretold important events, kept sporadic records ofsunspots for hundred of years. An English monk named John of Worcestermade the first drawing of sunspots in December 1128.Soon after the invention of the telescope, several astronomers made the firsttelescopic observations of sunspots. Galileo Galilei in Italy, Johann Goldsmidin Holland, Thomas Harriot in England, and Christoph Scheiner in Germanyall viewed sunspots through telescopes between 1610 and 1613. Astronomersof that era weren’t quite sure what to make of these spots on the Sun; somethought they were silhouettes of previously undiscovered planets transitingthe Sun as they orbited closer than Mercury, while others believed they weredark clouds in the Sun’s atmosphere. The progression of sunspots across theface of the Sun allowed early seventeenth century astronomers to make thefirst estimates of the Sun’s rotation period (about 27 days).

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Figure 3: Observed sunspots

Very few sunspots were seen from about 1645 to 1715, a period referred to asthe Maunder Minimum after the English astronomer Edward W. Maunderwho studied this unusual time of solar inactivity. This period corresponds tothe middle of a series of exceptionally cold winters throughout Europe knownas the Little Ice Age. Scientists still debate whether decreased solar activityhelped cause the Little Ice Age, or if the cold snap coincidentally occurredaround the same time as the Maunder Minimum. Several other less-extremeperiods of decreased sunspot activity have been noted: the Sprer Minimum(1420 to 1570), named after the German astronomer Gustav Sprer; the Dal-ton Minimum of 1790 to 1820; the Wolf Minimum of 1280 to 1340; and theOort Minimum of 1010 to 1050. Indirect evidence from elemental isotopesseemingly indicates that there have been 18 such periods of lessened sunspotsover the last 8,000 years and that the Sun may spend as much as a quarter ofits time in such sunspot minima periods. The Sprer, Wolf, and Oort Minimawere all discovered via such isotope analyses, since they occurred before theera of regular, reliable sunspot observations. In contrast to these periods ofsunspot minima, sunspot counts have been higher than usual since around1900, which has led some scientists to call this time the Modern Maximum.Likewise a period called the Medieval Maximum, which lasted from 1100 to1250, apparently had higher levels of sunspots and associated solar activity,and intriguingly coincides (at least partially) with a period of warmer cli-mates on Earth called the Medieval Warm Period.

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Figure 4: Observed solarflares

In 1843 an amateur German astronomer named Samuel Heinrich Schwabe,using 17 years of his personal sunspot observations, discovered the rise andfall of yearly sunspot counts we now call the sunspot cycle. He initially es-timated the cycle’s length at 10 years. Two French physicists, Louis Fizeauand Lon Foucault, took the first photo of the Sun and sunspots in April1845. Around 1852 four astronomers noted, roughly simultaneously, that theperiod of the sunspot cycle was identical to the period of variation of geo-magnetic activity at Earth, giving birth to the field of study of Sun-Earthconnections we now call space weather.Around 1858 the Englishman Richard C. Carrington and Gustav Sprer in-dependently made two important discoveries: the solar latitude at whichsunspots appear gradually decreases from about 40 to 5 throughout thecourse of a sunspot cycle (now often called Sporer’s Law), and sunspotsat different latitudes move around the Sun at different rates. The latterfact led them to conclude that the Sun does not rotate as a solid sphere,but rather has different rates of rotation at different latitudes (about 30%slower near the poles than near the equator) characteristic of a gaseous body.In 1868 the Swiss astronomer Rudolf Wolf was trying to compare historicalsunspot counts by many different astronomers using various instruments andobserving techniques. He devised a formula, which is still in use today, thatcombined data about counts of individual spots, counts of sunspot groups,

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and a correction factor for each observer. The result of his calculation forany given period is called the Wolf sunspot number.The number of sunspots observed on the surface of the Sun varies from yearto year. This rise and fall in sunspot counts varies in a cyclical way. Thelength of the cycle is around eleven years on average. The cyclical variationin sunspot counts, discovered in 1843 by the amateur German astronomerSamuel Heinrich Schwabe, is called the Sunspot Cycle. A peak in the sunspotcount is referred to as a time of solar maximum (or solar max), whereas aperiod when few sunspots appear is called a solar minimum (or solar min).An example of a recent sunspot cycle spans the years from the solar min in1986, when 13 sunspots were seen, through the solar max in 1989 when morethan 157 sunspots appeared, on to the next solar min in 1996 (ten years afterthe 1986 solar min) when the sunspot count had fallen back down to fewerthan 9. Along with the number of sunspots, the location of sunspots variesthroughout the sunspot cycle. At solar min, sunspots tend to form aroundlatitudes of 30 to 45 North and South of the Sun’s equator. As the solar cycleprogresses through solar max, sunspots tend to appear closer to the equator,around a latitude of 15. Towards the end of a cycle, with solar min once againapproaching, sunspots form quite close to the solar equator, around 7 Northand South latitude. There is often an overlap in this latitudinal migrationtrend around solar min, when sunspots of the outgoing cycle are formingat low latitudes and sunspots of the upcoming cycle begin to form at highlatitudes once again. This gradual equatorward drift of sunspots throughoutthe sunspot cycle, which was first noticed in the early 1860’s by the Germanastronomer Gustav Sprer and the Englishman Richard Christopher Carring-ton, is often called Sprer’s Law. In 1904 another English astronomer, EdwardWalter Maunder, constructed the first butterfly diagram, a graphical plot ofthis sunspot migration trend.The duration of the sunspot cycle is, on average, around eleven years. How-ever, the length of the cycle does vary. Between 1700 and the present, thesunspot cycle (from one solar min to the next solar min) has varied in lengthfrom as short as nine years to as long as fourteen years. Note, however,that of the 26 solar cycles during that three-century span, 21 had a lengthbetween ten and twelve years. Arriving at a precise count of sunspots is notas straightforward as it might appear. Some spots are much larger than oth-ers, some sunspots partially merge together at their edges, and many spotsappear in groups. In 1848 a Swiss astronomer named Rudolf Wolf devisedan algorithm for making consistent counts of sunspots that allows compar-

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isons between data from different observers across the centuries. The sunspotcount derived using Wolf’s formula, now known as the Wolf sunspot number,is still in use today. Wolf used data from earlier astronomers to reconstructsunspot counts as far back as the 1755-1766 cycle, which he dubbed cycle 1.Since then, subsequent cycles have been numbered consecutively, so the cyclethat began with the 1996 solar minimum is cycle 23. The Sun is typicallyvery active when sunspot counts are high. Sunspots are indicators of distur-bances in the Sun’s magnetic field, which can generate energetic solar eventslike solar flares and coronal mass ejections. Since reasonably reliable recordsof sunspot counts extend back to the early 1700s, long before other measuresof solar activity could be observed, sunspot counts serve as a valuable, rela-tively long-term indicator of solar activity. The Sun emits significantly moreradiation than usual in the X-ray and ultraviolet portions of the electromag-netic spectrum during solar max, and this extra energy significantly altersthe uppermost layers of Earth’s atmosphere.The 11-year sunspot cycle is actually half of a longer, 22-year cycle of solaractivity. Each time the sunspot count rises and falls, the magnetic field ofthe Sun associated with sunspots reverses polarity; the orientation of mag-netic fields in the Sun’s northern and southern hemispheres switch. Thus,in terms of magnetic fields, the solar cycle is only complete (with the fieldsback the way they were at the start of the cycle) after two 11-year sunspotcycles. This solar cycle is, on average, about 22 years long - twice the dura-tion of the sunspot cycle. Some scientists believe there is evidence for other,longer-period variations in the sunspot and solar cycles. Other scientists areskeptical about such claims. Most scientists think we need more data, span-ning longer periods of time, to definitively resolve this issue. Besides theseregular cycles, the Sun has exhibited periods of very unusual sunspot counts.Most notably, from about 1645 to 1715 there were very few sunspots - insome years none at all were observed! This period, now called the MaunderMinimum (after E.W. Maunder, who did important pioneering work relatedto this phenomenon), corresponded to an extremely cold spell in Europeknown as the Little Ice Age. In short the regularity of sunspot appearenceand dissappearence is a very important element that could be successfullytackled using regularity statistics like ApEn.

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2 Conventional Solar Data Analysis

The most direct and fundamental procedure for the analysis of sunspot datais to plot the Wolfer number, which measures both the number and the sizeof sunspots with year for the data available for about three hundred years.This can be easily done with the following matlab routine.

load sunspot.dat; year = sunspot(:,1);wolfer = sunspot(:,2); plot(year,wolfer,’k’)

xlabel(’Year’); ylabel(’wolfer Number)title(’Sunspot Data’); grid

Figure 5: Plot of Wolfer Number Vs.Year

It can be seen that the plot clearly exhibits the eleven year periodicity ofsunspots. It also shows prominent sunspot maxima and minima. The solar

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minimum of 1996 as seen in the plot signifies the start of cycle 23 which iscurrently being studied by astrophysicists.The periodicity of sunspot data can be more clearly observed if a plot forfirst fifty years or (for any convenient short duration)is generated. This canbe done by the following addition to the matlab routine.

plot(year(1:50),wolfer(1:50),’k.-’);xlabel(’Year’)

ylabel(’Wolfer Number’)title(’Sunspot Data For Fifty Years’)

Figure 6: Plot of Wolfer Number Vs.Year for first fifty years

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In signal processing Fast Finite Fourier Transform(FFT) is a very usefulaand common tool. In conventional sunspot data analysis FFT of the sunspotdata is usually taken. This can be done by adding the following matlab code.The plot is made in the complex plane.The first component Y(1) representsthe sum of the data and hence it is removed.

Y = fft(wolfer);Y(1)=[];

plot(Y,’k*’)title(’Fourier Coefficients in the Complex Plane’);

xlabel(’Real Axis’);ylabel(’Imaginary Axis’);

The magnitude squared of the parameter Y is usually called the Power. Aplot connecting Power and the frequency is called a periodogram and is com-monly used in conventional time series analysis.A Periodogram of the sunspotdata can be plotted using the following code.

n=length(Y);power = abs(Y(1:floor(n/2))).2;

nyquist = 1/2;freq = (1:n/2)/(n/2)*nyquist;

plot(freq,power)xlabel(’cycles/year’)

ylabel(’Power’)title(’Periodogram’)

The periodogram shows additional peaks for other frequencies also showingthat periodicities other than the normal eleven year periodicity can be presentin the solar cycle. This method is being employed currently to obtain thepresence of other periodicities of the sunspot data.

plot(freq(1:40),power(1:40))xlabel(’cycles/year’)

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Figure 7: Plot of Fourier Coefficients in Complex Plane

The periodogram clearly indicates that the power of the signal(her theWolfer number of sunspots) is maximum for cyles per year near to .1. Thisperiodicity can be made more clear visually if we plot year per cyle (ie) pe-riod along the y axis.

period=1./freq;plot(period,power);

axis(

04002e + 7

);

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Figure 8: Periodogram of the sunspot data

ylabel(’Power’);xlabel(’Period (Years/Cycle)’);

All the above analysis shows a definite period length of about 11 years.Finally, we can fix the cycle length a little more precisely by picking out thestrongest frequency. The dot locates this point.

hold on;index=find(power==max(power));

mainPeriodStr=num2str(period(index));plot(period(index),power(index),’r.’, ’MarkerSize’,25);

text(period(index)+2,power(index),

′Period =′, mainPeriodStr

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Figure 9: Periodogram for Forty years

);hold off;

The completeMatlab Program for conventional sunspot data analysis is thefollowing. The procedure given above can be used to analyse shorter datas aswell. The programs given above forms part of the demonstration programsof matlab and hence the data used is the data provided by them.

3 Method of Approximate entropy

Approximate entropy is a time series regularity indicator similar to autocor-relation index, nonlinear dimensions etc.This description originally appearedthrough the works of Ho, Moody and Peng.ApEn is a “regularity statistic”

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Figure 10: Periodogram

that quantifies the unpredictability of fluctuations in a time series such asan instantaneous heart rate time series,financial time series, stock marketfluctuations, astronomical imageintensity variations,etc. denoted as HR(i).Intuitively, one may reason that the presence of repetitive patterns of fluc-tuation in a time series renders it more predictable than a time series inwhich such patterns are absent. ApEn reflects the likelihood that similarpatterns of observations will not be followed by additional similar observa-tions. A time series containing many repetitive patterns has a relativelysmall ApEn; a less predictable (i.e., more complex) process has a higherApEn.Thus the predictability of a time series is conveniently measured byApproximate Entropy. The algorithm for computing ApEn has been devel-oped by several authors. In this project we use an algorithm which is most

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Figure 11: Periodogram Showing exact length

suitable for matlab programs and applicable for large amount of data usuallyfound in astronomy. Here, we provide a brief summary of the calculations,as applied to a time series of Astronomical measurements, HR(i). Given asequence SN , consisting of N instantaneous measurements HR(1), HR(2),· · ·, HR(N), we must choose values for two input parameters, m and r, tocompute the approximate entropy, ApEn(SN , m, r), of the sequence. Thesecond of these parameters, m, specifies the pattern length, and the third,r, defines the criterion of similarity. We denote a subsequence (or pattern)of m measurements, beginning at measurement i within SN , by the vectorpm(i). Two patterns, pm(i) and pm(j), are similar if the difference betweenany pair of corresponding measurements in the patterns is less than r, i.e., if

|HR(i + k)−HR(j + k)| < r for 0 ≤ k < m (1)

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Now consider the set Pm of all patterns of length m [i.e., pm(1), pm(2), · · · , pm(N−m + 1)], within SN . We may now define

Cim(r) =nim(r)

N −m + 1(2)

where nim(r) is the number of patterns in Pm that are similar to pm(i)(given the similarity criterion r). The quantity Cim(r) is the fraction ofpatterns of length m that resemble the pattern of the same length that beginsat interval i. We can calculate Cim(r) for each pattern in Pm, and we defineCm(r) as the mean of these Cim(r) values. The quantity Cm(r) expresses theprevalence of repetitive patterns of length m in SN . Finally, we define theapproximate entropy of SN , for patterns of length m and similarity criterionr, as

ApEn(SN , m, r) = ln

[Cm(r)

Cm+1(r)

](3)

i.e., as the natural logarithm of the relative prevalence of repetitive pat-terns of length m compared with those of length m + 1.

Thus, if we find similar patterns in a time series, ApEn estimates thelogarithmic likelihood that the next intervals after each of the patterns willdiffer (i.e., that the similarity of the patterns is mere coincidence and lackspredictive value). Smaller values of ApEn imply a greater likelihood thatsimilar patterns of measurements will be followed by additional similar mea-surements. If the time series is highly irregular, the occurrence of similarpatterns will not be predictive for the following measurements, and ApEnwill be relatively large. It should be noted that ApEn has significant weak-nesses, notably its strong dependence on sequence length and its poor self-consistency (i.e., the observation that ApEn for one data set is larger thanApEn for another for a given choice of m and r should, but does not, holdtrue for other choices of m and r). This will invite the necessity for judiciouschoice of pattern length and regularity criterion and comparison with resultsobtained from other statistics.

An example may help to clarify the process of calculating ApEn. Supposethat N = 50, and that the sequence SN consists of 50 samples of the function

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Figure 12: Example of a simple sequence

illustrated above:

SN = {61, 62, 63, 64, 65, 61, 62, 63, 64, 65, 61, ..., 65} (4)

(i.e., the sequence is periodic with a period of 5). Let’s choose m = 5(this choice simplifies the calculations for this example, but similar resultswould be obtained for other nearby values of m) and r = 2 (again, the valueof r can be varied somewhat without affecting the result). This gives us:

cp5(1) = {61, 62, 63, 64, 65 · · · · · ·} (5)

p5(3) = {63, 64, 65, 61, 62} (6)

and so on. The first question to be answered is: how many of the p5(i)are similar to p5(1)? Since we have chosen r = 2 as the similarity criterion,this means that each of the 5 components of p5(i) must be within ±2 unitsof the corresponding component of p1(i). Thus, for example, p5(2) is notsimilar to p5(1), since their last components (61 and 65) differ by more than2 units. The conditions for similarity to p5(1) will be satisfied only by p5(6),

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p5(11), p5(16), p5(21), ..., p5(46), as well as by p5(1) itself [i.e., for 10 of thep5(i)], so we have

n1,5(2) = 10 (7)

Since the total number of p5(i) is N −m + 1 = 50− 5 + 1 = 46,

C1,5(2) =10

46(8)

We can now repeat the above steps to determine how many of the p5(i)are similar to p5(2), p5(3), etc. By the same reasoning, p5(2) is similar top5(7), p5(12), p5(17), ..., p5(41), so that

n2,5(2) = 9 (9)

and in general

ni,5(2) =

{10 if i = 1 modulo 59 otherwise

(10)

(This last statement is true only for the particular example we are con-sidering, since we have specified a sequence with a period of 5, and we havechosen m = 5 as well.)

Hence Ci,5(2) is either 1046

or 946

, depending on ni,5(2), and the mean valueof all 46 of the Ci,5(2) is:

C5(2) =10 · 10

46+ 36 · 9

46

46=

424

2116≈ 0.200378 (11)

In order to obtain ApEn(SN , 5, 2), we need to repeat all of the calculationsabove for m = 6. Doing so, we obtain:

n1,6(2) = 9 (12)

n2,6(2) = 9 (13)

n3,6(2) = 9 · · · (14)

(15)

so that

Ci,6(2) =9

45= 0.2, 1 ≤ i ≤ 45 (16)

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and C6(2) = 0.2. Finally, we calculate that

Apen(SN , 2, 5) = ln

[C5(2)

C6(2)

]≈ 0.00189 (17)

This is a very small value of ApEn, which suggests that the original timeseries is highly predictable (as indeed it is).

4 Matlab Module

Since Astronomical data is large in number programmes having large datahandling capacity is to be employed. A matlab module for doing this isemployedin this project. Following is the abstract of such a module. SinceMatlab uses very efficient algorithms for numerical computations, other al-gorithms available in the literature is not employed.

4.1 Plotting the Example Sequence

PROGRAM FOR PLOTTING EXAMPLE TIME SE-RIES WITH 50 DATA POINTSsequence=[61,62,63,64,65,61,62,63,64,65,61,62,63,64,65,· · ·61,62,63,64,65,61,62,63,64,65,61,62,63,64,65,61,62,63,64,65,· · ·61,62,63,64,65,61,62,63,64,65,61,62,63,64,65];time=[0:49];plot(time,sequence,’k’);xlabel(’TIME’);ylabel(’SEQUENCE’);title(’PLOT FOR EXAMPLE SEQUENCE’)

5 calculation of ApEn

The matlab function file developed for calculating Ap-proximate Entropy of sunspot and solar fare data is re-produced here.

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Figure 13: Example of a simple sequence

function entropy = apen(pre, post, r)%computer approximate entropy for sunspot[N, p] = size(pre);%how many pairs of points are closer than r in the pre spacephiM = 0;%how many are closer in the post valuesphiMplus1 = 0;%will be used in distance calculationfoo = zeros(N,p);% Loop over all the pointsfor k=1:N% fill in matrix foo to contain many replications of the point in questionfor j=1:pfoo(:,j) = pre(k,j);end% calculate the distance

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goo = (abs( foo - pre ) ¡= r );% which ones of them are closer than r using the max normif p == 1closerpre = goo;else closerpre = all(goo’); endprecount = sum(closerpre);phiM = phiM + log(precount);% of the ones that were closer in the pre space, how many are closer% in post alsoinds = find(closerpre);postcount = sum( abs( post(closerpre) - post(k) ) ¡ r );phiMplus1 = phiMplus1 + log(postcount);endentropy = (phiM - phiMplus1)/N;

6 Results for Solarflare Data

Approximate entropy calculated based on Solar flare data for January 1965to June 2006 for different regularity parameters and pattern length m=4 istabulated below (for first three months and total). In calculating the entropydata for each month for first three months and total data is taken seperatelyfor obtaining the pre and post vectors.

Month r ApEn r Apen. r Apen.January 100 0.1386 105 0.1386 200 0.2485February 220 0.1386 305 0.3584 405 0.4836March 220 0.1099 150 0 200 0Total 2000 0.1386 1000 0 2500 0

Similar analysis for the sunspot data also is done.

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Month r ApEn r Apen. r Apen.January 40 0.2225 80 0.2718 100 0.2485February 40 0.1345 80 0.0987 100 .105March 40 0.136 80 .1534 100 .1522Total 80 0.1495 100 .0765 140 0

7 Conclusions

1.The Solar data for about three hundred years is plotted yearwise to observethe common eleven year cycle2.The Finite fast fourier transform of the sunspot number is taken and itsplot in the complex plane is generated.3. The periodogram of sunspot data is plotted and the exact period for themost prominent periodicity is found to be near eleven.Periodicities other thenthe eleven year period can be observed in a smaller scale.4. Approximate entropy for the solar flare data is calculated and the valuesare found to be similar to the nonlinear dimension calculations available inthe literature.5. Approximate entropy for sunspot data is calculated which indicates regu-larity on a monthly and yearly basis.

8 references

1. Ranjit A. Thuraisingham, Georg A. Gottwald, On multiscale entropyanalysis for physiological data, Physica (October 2005)1-10

2. H. Kantz, T. Schreiber, Nonlinear Time Series Analysis, second ed.,Cambridge University Press, Cambridge, 2004.

3. D.E. Lake, J.S. Richmann, M.P. Griffin, J.R. Moorman, Sample en-tropy analysis of neonatal heart rate variability, Am. J. Physiol. Heart Circ.Physiol. 283 (2002) R789R797.

5.The Approximate Entropy of Various Financial Asset Prices, CyprianJ. Bruck Draft Version, August 2005 4. Pincus SM. Approximate entropy asa measure of system complexity. Proc Natl Acad Sci USA 1991;88:2297-2301.

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5. Pincus SM, Goldberger AL. Physiological time-series analysis: Whatdoes regularity quantify? Am J Physiol 1994;266(Heart Circ Physiol):H1643-H1656.

6. Ryan SM, Goldberger AL, Pincus SM, Mietus J, Lipsitz LA. Genderand age-related differences in heart rate dynamics: Are women more complexthan men? J Am Coll Cardiol 1994;24:1700-1707.

7. Ho KKL, Moody GB, Peng CK, Mietus JE, Larson MG, Levy D,Goldberger AL. Predicting survival in heart failure case and control subjectsby use of fully automated methods for deriving nonlinear and conventionalindices of heart rate dynamics. Circulation 1997 (August);96(3):842-848.

8. Richman JS, Moorman JR. Physiological time-series analysis usingapproximate entropy and sample entropy. Am J Physiol Heart Circ Physiol278(6):H2039-H2049 (2000).

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