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c 2001 Nonlinear Phenomena in Complex Systems

Testing and Forecasting the Time Series of the Solar Activity by Singular Sp ectrum Analysis
A. Loskutov, I.A. Istomin, K.M. Kuzanyan, and O.L. Kotlyarov
Physics Faculty, Moscow State University, Moscow 119899, Russia E-mail: loskutov@moldyn.phys.msu.su

(Received 5 October 2000)
To study and forecast the solar activity data a quite perspective method of singular spectrum analysis (SSA) is proposed. As known, data of the solar activity are usually presented via the Wolf numbers associated with the effective amount of the sunspots. The advantages and disadvantages of SSA are described by its application to the series of the Wolf numbers. It is shown that the SSA method provides a sufficiently high reliability in the description of the 11-year solar cycle. Moreover, this method is appropriate for revealing longer cycles and forecasting the further solar activity during one and a half of 11-year cycle. Key words: solar activity, singular spectrum analysis, 11-year solar cycle PACS numb ers: 96.60.L; 96.60.R

It has long been observed that solar activity depends on a number of spots visible on its disk. During about 11 years which is called a solar cycle, this number varies over a wide range. Accompanying this process, changes in the structure of magnetic fields of the Sun affect the Earth climate and have a probable connection with the natural catastrophes. In addition, intensity of the solar radiation (the frequency of the Sun flares, etc.) apparently exerts an influence on all areas of human life including socialhistorical activity. Thus, in view of the quite large significance of the magnetic activity of the Sun its prediction is a sub ject of much current interest. At the present time, to describe the dynamics of the solar activity many approaches are used. Among them the Wolf number associated with the effective amount of the sunspots is much more convenience method. The dynamics of the Wolf numbers has more or less quasiperiodic nature but, owing to the facts that models of this process do not take into account many essential factors of the solar magnetic activity, its prediction is difficult. It is known that during last 250 years the cycle period of the sunspots changes its value not more than 20%.

At the same time, the amplitude (i.e. the averaged number of sunspots) has varied over an order or even more. Detailed models of the solar processes do not describe such variations. In the last few years sufficiently many methods devoted to the prediction and reconstruction (to the past) the dynamics of the Wolf numbers have been proposed (see, e.g., [1, 2, 3, 4, 5] and references cited therein). It should be noted however, that these methods have certain limitations and demerits. That is the reason why prediction of the sunspot dynamics based on the data of observations only (i.e. without a modelling of the process) is a quite perspective approach. In this way time-series analysis (see [6, 7, 8, 9, 10, 11]) can give an essential assistance. But in this case there are many problems related to the fact that the Wolf series is not strictly deterministic system, it does not have clearly defined dimension [12, 13], and its length is not so large. In the present paper, for investigations of the Wolf time series a sufficiently efficient method of singular spectrum analysis is proposed. It is shown that this method provides a quite high reliability 47

Nonlinear Phenomena in Complex Systems, 4:1 (2001) 47 - 57

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A. Loskutov et al.: Testing and Forecasting the Time Series . . . sequence as the first column of some matrix X . For the second column, the values of the sequence from x2 to x +1 are chosen. Thus, the last elements of the sequence xn , . . . , xN correspond to the last column with the number n = N - + 1. Therefore the transformed series has the following matrix form: X= x1 x2 x3 . . . x

in the description of amplitudes of the 11-year solar cycle. Moreover, it is appropriate for revealing more long cycles and forecasting the further solar activity during one and a half of 11-year cycle, i.e. the nearest 16 years.

1

Singular sp ectrum analysis

The method of singular spectrum analysis (SSA) [14, 15, 16, 17, 18], which is used in the present article allows us the following. · Recognise certain components in the time series which have been obtained from the observable at regular intervals; · Find periodicities that are not known in advance; · On the basis of the chosen components to smooth out the initial data; · Extract components with the known period; · Predict the further evolution of the observed dependence. The method of SSA is a reasonably new one, but now it is quite clear that it is sufficiently competitive with numerous smoothing methods [19, 20, 21]. Moreover, in certain cases forecasting the system evolution on its basis gives much more reliable results in comparison with the other known algorithms (see [7, 18, 22, 23, 24, 25] and references therein). The SSA method is based on the passage from the investigation of an initial linear series (xi )N to the i=1 analysis of a many-dimensional series consisted of components of some length which contain except for the value xi , certain quantities xi-j , j = 1, . . . , , at the previous instants of time. Let us describe the central steps of the application of SSA to a series (xi )N . i=1 (1) On the first step, a one-dimensional series is transformed into a many-dimensional one. For such a transformation it is necessary to take a certain number of delays [N/2 + 1], where [·] is an integer part, and represent the initial values of the

x2 x3 x4 . . . x
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-1

. . . .

. . . .

. xn . xn+1 . xn+2 . . . . xN

.

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It is obvious that for the constructed matrix the expression ||xij || = xi+j -1 holds. In general, the matrix X is a rectangular one. But, in a limit case, i.e. at = N/2 and an even N , it degenerates into a square matrix. (2) After this transformation, for the matrix X 1 the corresponding covariance matrix C = n X X T should be obtained. (3) Now it is necessary to find eigenvalues and eigenvectors of the matrix C . For this, the matrix C should be factored as follows: C = V V T , where 1 0 . . . 0 0 2 . . . 0 = . . .. . . . . . . . . 0 0 . . . is the diagonal matrix of eigenvalues, V = V 1, V 2, . . . , V
1 2 v1 v1 . . . v1 1 v2 . . . v v2 2 2 . . .. . . . .. . . . 1 2 v v . . . v

=

is the orthogonal matrix of eigenvectors of the matrix C . It is clear that = V C V T , =1 i = i and det C = =1 i . i (4) For the next step, the matrix V of eigenvectors should be presented as a matrix of the conversion to the principal components, Y = V T X = (Y1 , Y2 , . . . , Y ), of the initial series.

Nonlinear Phenomena in Complex Systems Vol. 4, No. 1, 2001

A. Loskutov et al.: Testing and Forecasting the Time Series . . .

49

FIG. 1. The data of the average monthly values of the Wolf numbers.

Here Yi , i = 1, 2, . . . , , are rows of the length n. Therewith, the eigenvalues 1 , 2 , . . . , can be considered as a certain contribution of the principal components to a general information content of the series (xi )N . Then, by means of these principal i=1 components it is possible to reconstruct the initial matrix X : X = V 1, V 2, . . . , V

Vi =

i v1 i v2 . . . i v

Y1T Y2T

. V i YiT .
i=1

. = . . YT

In turn, by the matrix X one can reconstruct the time series (xi )N . It should be noted that i=1 for the reconstruction not all principal components Y1 , Y2 , . . . , Y are usually applied. Only a part of them can be involved. This depends on the goal which we pursue and the informative content of the used components (see [14, 15, 16, 17]). This means that each vector-row Yi can be considered as a result of some pro jection of a -dimensional totality on a direction corresponding to the eigenvector V i . Thus, the series is presented via a set of components Yi . Therewith, the weight of the component Yi in the initial sequence (xi )N can be defined by i=1 the corresponding eigenvalue i which is, in turn, the eigenvalue of the eigenvector V i . Each i-th eigenvector includes components,

Let us construct a dependence of the component i values vk , k = 1, 2, . . . , , as a function of their number: v i = v i (k ). Then using the orthogonality property of eigenvectors, the further analysis of the sequence (xi )N can be performed by means of i=1 diagrams plotted by the analogy with the Lissa jous i figure. Namely, along the axes the components vk , j vk are plotted in pairs. If the constructed diagrams are similar to a circle, then the functions v i = v i (k ), v j = v j (k ) are approximated by certain periodic functions with almost coincided amplitudes and a phase lag about a quarter of the period. Thus, for some pairs of the eigenvectors V i , V j one can find a value which has a meaning of the period. Therefore, such a graphical representation provides a certain estimate of the component frequencies inherent in the initial time series (xi )N . i=1 For the given parameter the number of all possible pairs of principal components is 2 . It is obvious that even for a sufficiently small the analysis of all such pairs is a quite laborious problem.

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A. Loskutov et al.: Testing and Forecasting the Time Series . . .

FIG. 2. An example of the prediction of the solar activity for a length of 216 points (18 years) by the average monthly Wolf numbers. The vertical line corresponds to the boundary value of the removed points. At the expansion, 500 components have been applied; in the reconstruction procedure 150 components have been involved. Numerical analysis has been performed by three stage: after prediction of the next 72 points recalculations have been made.

Moreover, at a large values of only a small part of diagrams has a helical form. Thus, before a graphical analysis it would be reasonable to restrict our search. This can be done if we arrange V i and Yi in order of decreasing their eigenvalues and consider only such pairs of eigenvectors which have close enough eigenvalues . In the diagram = (i) at a quite large these pairs form a decreasing (with the growth of i) step function. (5) Suppose now that for the further reconstruction we have only r leading components. Thus, for the reconstruction of the initial matrix X one should use r leading eigenvectors V i . In this case

~ X = V 1, V 2, . . . , V
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V i Yi ,
i=1

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~ where X is a reconstructed matrix with n columns and rows. Then the initial time series obtained from this matrix is defined as follows: s 1 xi,s-i+1 , ~ s i=1 1 xi,s-i+1 , ~ xs = ~ i=1 N -s+1 1 x ~ N -s+1 i=1

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Described way of the reconstruction is said to be a SSA-smoothing of the initial time series (xi )N i=1 by the leading r components. (6) In the next stage of the SSA application, a prediction procedure of the initial time sequence can be considered (see [25, 26, 27]). This means that the series (xi )N +p which is the extension of the known i=1 data (xi )N , is constructed. In turn, extrapolation i=1 to p points forward is reduced to the application of p times of the prediction procedure to the one point. The basic idea of the computation of the point xN +1 is the following. Consider the sequence x1 , x2 , . . . , xN and construct a sample in the form of matrix X . As a basis of the surface containing this sample one can take the chosen before vectors V 1 , V 2 , . . . , V r of the matrix C . Let us write the parametric equation of this surface as S (P ) =
r

the parameters pi corresponds to the value S (P ) which is a column with elements. In this case the set of parameter values P k = pk , pk , . . . , pk r 12 corresponds to k -th, k = 1, 2, . . . , n, column of the matrix X . Therefore, X 1 = S P 1 , X 2 = S P 2 , . . . , X n = S (P n ). Now, to predict the value xN +1 it is necessary to find the (n + 1)-th column X n+1 which, in turn, fits the parameters P n+1 = pn+1 , pn+1 , . . . , pn+1 . r 1 2 Using the data (xi )N these parameter values can i=1 be obtained from the expression S (P ) =
r

i=1

pi V i , where a set of

Thus, the predicting column is written as follows:

i=1

pi V i .

A. Loskutov et al.: Testing and Forecasting the Time Series . . . X
N +1

51

=S P

n+1

.

FIG. 3. The leading 50 eigenvalues of the covariance matrix obtained at the expansion of the average annual Wolf numbers into 123 components.

Let us introduce the 1 v1 v1 2 V = . . . v p= ~
1 -1

following designations: 2 r v1 . . . v1 2 r v2 . . . v2 . . ; .. . . . . . v
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+1

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+2 +3

;

12 r V = v , v , . . . , v .

predicted point one can completely repeat the SSA algorithm. Then the matrixes V and V will be changed. (7) At the final stage of the SSA application one should dwell on the choice of the main parameter -- the number of delays which are used for the manydimensional sample X . As in the case of selection of the principal components this value essentially depends on the investigated problem. Consider the smoothing procedure of a time series by the SSA method. In this case, as noted above, selection of a principal component is the filtration of the time series with the transition filtering function in the form of the eigenvector of this principal component. If the delay value is the greater, the greater the number of parallel filters, and a bandwidth of each of them is more narrow. Thus, for a large enough we have a sufficiently efficient smoothing of the time series. If it is necessary to define unknown (hidden) periods in the observed sequence then one should take the value of as large as possible. Next, after omitting close to zero eigenvalues, the delay value should be shortened. Suppose now that we should select only the one known periodic component. In this case it is necessary to choose the delay which is equal to the required period. Finally, let us consider the problem of some extension of an observed sequence to a given value, i.e. the problem of forecasting the evolution of the process under investigation. Then one should take a maximum allowable value of the delay and thereafter select the number r.

The set of the values pn+1 , pn+1 , . . . , pn+1 is r 1 2 ~ easily found as a solution of the system V P = Q ~ with respect to P . Thus, the final expression of the forecasting value has the following form: xN
+1

V VT Q = . 1 - V V T

In the simplest ues it is necessary corresponding way V VT 1 - V V T

case, to predict the next valto change the matrix Q in a and multiply it by the value . In addition, however for each

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2

Forecasting the solar activity by SSA

To estimate of the practicality of the SSA method it is necessary to use the time series of a natural origin. In the present article, as such a time series the sequence of the Wolf numbers characterised in a certain way the solar activity was chosen. Taken alone, the Wolf numbers defined by the visible sunspots cannot give a quantitative informa-

52

A. Loskutov et al.: Testing and Forecasting the Time Series . . .

FIG. 4. Some of components corresponding to eigenvalues shown in Fig.3. Their percentage in the initial series is denoted by the parentheses.

tion related to the solar activity. However there is a large enough correlation between the Wolf numbers and the F 10.3 emission. That is the reason why investigations of the relative variations in this sequence can give a specific information concerning the solar activity. For the first time, in 1848 a Swiss astronomer R.Wolf proposed that a measure of the solar activity can be characterised by the number of sunspots. To this end he recommended to consider the union of the total number of spots visible on the Sun and tenfold numbers of regions in which these spots are placed. This last summand should coordinate the results of measurements performed under several conditions. Thus, since the year 1849 the results of daily measurements have become available. Using more earlier observations and various sources, R.Wolf has reconstructed the data of the solar activity (with an admissible accuracy and negligible gaps) up to the year 1818. Now the averaged number of the sunspots is called the Wolf number. Later the average monthly values of the Wolf numbers up to the year 1749 (namely this series is

used in the present paper) and their average annual values up to the year 1700 have been reconstructed. In the last case however, the error can reach more than 10 percent. The chosen data covers the wide time interval without gaps and a quite high time resolution. The investigated sequence is shown in Fig.1. The time with the interval of one month is plotted on the abscissa. The corresponding Wolf numbers are laid as ordinates. On the first step of the application of SSA it is necessary to take the maximum permissible value of parameter . For our investigations we used = 500. This value allows us to make up periodicities up to period of 42 years. The use of the larger values leads to essential computer problems. Moreover, increasing the parameter (up to 600) does not yield any significant change in the results of the first expansion in principal components. Owing to a large value of the dependence of the roots of eigenvalues of the covariance matrix (ordered in decreasing) decays exponentially. In addition, the sum of the leading five eigenvalues exceeds 99% of their sum total. Coupled with a quite large

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A. Loskutov et al.: Testing and Forecasting the Time Series . . .

53

FIG. 5. Dependencies of the 2-nd and the 3-rd components (on the left-hand side) and the 6-th and the 7-th components (on the right-hand side) shown in Fig.4. The left and the right parts of the Figure correspond to the 1-st and the 3-rd step-like data of eigenvalues, respectively.

numbers of initial points, this leads to the fact that the first principal component yields a small enough smoothing of the initial series, and by means of the leading four-five components one can reconstruct this series. Moreover, at small values of the parameter , say at = 5, the first component weakly changes its form. This is due to the fact that the SSA method is stable with respect to this parameter. Therefore, application of a sufficiently large is justified only for the prediction. To check possibilities of the prediction let us cut off the initial sequence of the averaged monthly values for a length of 216 points (18 years) and reconstruct it by the following way. Evaluate optimal parameter values of the reconstruction algorithm by means of an additional truncation and resolve the obtained series into components. Therewith it is necessary to choose such a number of the leading components at which we get the best coincidence of the reconstructed values with the additionally truncated data. Then using the obtained parameters, we will find the initially shortened part of 216 points. By means of the direct exhaustion one can define that the best results we get at r = 150 (the number of the chosen components). Once again let us take the reduced sequence (for a length of 18 years) and

apply the chosen r for the prediction. One can further improve the quality of the prediction if we will decompose the predicted interval (216 points) into subintervals and after prediction of each part calculate the principal components. In the ideal case, after the prediction of each point it is necessary to realise this process. However, such an approach leads to an increase of the computer time. Fig.2. illustrates the reconstructed series at which recalculations have been realised every 72 points (i.e. 3 times). However, as it turns out, this result is almost identical to the prediction of the whole interval of 216 points, without decomposition into subintervals. Theoretically, one can analyse the series components with the aim of detection the existence of the known and any other periods. However a number of these periods and thus, the resemblance of the components Yi of the initial series makes this analysis a quite cumbersome procedure. In addition, selection of sufficiently short periodicities (about several months) is a very difficult computer task. For these reasons, the choosing a large enough subinterval of time is the more simply problem. Let us consider now a series with the average annual Wolf numbers. This series contains only 248

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54

A. Loskutov et al.: Testing and Forecasting the Time Series . . .

FIG. 6. Dependencies of the 2-nd and the 3-rd components (on the left-hand side) and the 4-th and the 7-th components (on the right-hand side) at = 80.

points. Thus, the maximum allowable value of is 123. Choose this value as an initial one. The corresponding eigenvalues is shown in Fig.3. The first number presents the principal component related to a trend. The following step-like data form the pairs of components with the numbers 23, 45, 67, 89 and 1112. Beginning with 14-th number this dependence gives way to an exponential tail. Eigenvectors for the pairs 23, 45, 89 and 1112 (see Fig.4) fit periodicities with 11-year period. The corresponding helical dependence for the components 2 and 3 is shown in the left side of Fig.5. It should be noted that except for the evident the eleven-year solar cycle it is possible to guess a supposed eighty-year Gleisberg cycle (see Fig.5). Here we keep in mind the pair of the eigenvectors 6 and 7. The corresponding eigenvalues is not exactly equal to each other, i.e. the step-like data is oblique (see Fig.3) and the phase lag is not /2. That is the reason why the form of the function diagram is not helical. In spite of this fact and a small enough eigenvalues, it is quite possible to trace a periodicity. For the best selection of the eighty-year dynamics one can identify parameter . By the numerical analysis we have found that = 80 is the most suitable value. In this case, vectors 4 and 7 are associated with such a periodicity. The diagram for

4-th and 7-th components is shown in Fig.6. A possible eighty-year solar cycle obtained by the reconstruction only via these two components is shown in Fig.7. Consider now a much more interesting problem concerning the possibility of the prediction of the average annual Wolf numbers by the SSA method. Shorten this series from the right hand side for a length of 18 points (that means years) and reconstruct it. To make this let us truncate additionally an interval of 11 points and try to restore it in the best way. Choosing the suited numbers of eigenvectors we will use such a procedure in various parts of the series. For the series of 219 points (219 = 248 - 18 - 11, where 248 is the number of all points, see above) the maximal possible = 109. As follows from numerical analysis, prediction for 11 points strongly depends on the picking of the components. At the same time, in a sufficiently wide range a qualitative guide of the prediction is not bad. However, for the quantitative prediction the given is too large. For the lesser values of it is quite possible to find the necessary number of components. For example, for = 33 and choosing the leading 11-th components we get fairly good results of the prediction. Thus, let us use these parameters for the pre-

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A. Loskutov et al.: Testing and Forecasting the Time Series . . . diction of the removed 18 points. In contrast to the case of the average monthly data and = 500 now, at = 33 we have a possibility to recalculate eigenvectors with consideration of the last predicted point. This recalculations can be made at the stage of the selection of vectors as well as the prediction. The result of the prediction with the correction at each step is shown in Fig.8.
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choose for the prediction the leading eleven of them. The result of the prediction is shown in Fig.9. As follows from this figure, in the nearest future, in comparison with the two previous maxima, the Sun will be in a relatively quiescent state. In addition, the level of the forthcoming maximum in the year 2011 will not be so high. On the basis of our numerical investigations concerning the application of SSA, the prediction during more than one and a half of 11-th year solar cycle is not justified. Nevertheless, we believe that the obtained values of the Wolf numbers shown in Fig.9 are a quite reliable. For comparison, a series of the average monthly Wolf numbers has been analysed. However, its investigation did not yield an essential modifications to the obtained results.

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FIG. 7. Reconstruction of the 4-th and the 7-th components corresponding 80-year solar cycle

In addition, we have studied a sequence of natural logarithms obtained from the initial series. Taking the logarithm is often used at the data processing (for example, in the case of the correlation analysis) and allows to get more interesting and complete results. However, eigenvalues and eigenvectors after such an operation have not principally changed. Thus, in this case taking the logarithm is not necessary because the basic information can be obtained from the analysis of the initial series. In the closing stage let us consider application of SSA for a real prediction of the solar activity. For this purpose, the average annual sequence of the Wolf numbers from the year 1748 to the year 1996 has been chosen. The end of this series corresponds to a minimum of the solar activity. Therefore it is interesting to describe the further activity of the Sun and predict its two next maxima. To realise this idea it is necessary to extend the average annual sequence for a length of 18 year points. We resolved the series into 33 components and

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Conclusion

It seems to be true that in the nearest future investigations of time series by means of SSA will occupy a more important and deserving place among different ways related to the processing and forecasting many experimentally obtained sequences. Resolving the initial series into components the analytical form of which is not fixed, this method allows us on a quite high level to reveal periodic subsequences and forecast the dynamics of this series. Therewith, restrictions to the numbers of points and characteristic periods in the investigated data as a rule, is essentially less than at applications of the other methods (for example, at correlation or Fourier analyses). In the present article a possibility of the SSA application to the sequence of the Wolf numbers characterising the solar activity, is considered. In spite of a relatively small length of this sequence, SSA allows to reveal the components corresponding the known solar cycles and, on the basis of only certain constituents, admits reconstruction of the data. Moreover, it is found that by means of SSA it is possible to extend short time series with an acceptable accuracy. Like any other method, SSA is not devoid of drawbacks. First, there is a certain difficulty in the problem of revealing unknown frequencies in the

56

A. Loskutov et al.: Testing and Forecasting the Time Series . . .

FIG. 8. Prediction of the solar activity the each-step correction. 11 in 33 components are chosen. The vertical line marks the boundary value of the removed 18 points.

FIG. 9.Prediction of the solar activity up to 2015 year.

investigated sequence. These frequencies can be more easily obtained by the Fourier analysis. Secondly, SSA does not contain clear rules concerning the choice of components, especially in the case of forecasting. Finally, SSA does not give the correct

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prediction of the cycle period. Therefore in the predicted sequence a systematic phase lag is accumulated. Nevertheless, as follows from the performed analysis it is a useful supplement to the existing methods of the experimental data processing. In the forthcoming investigations devoted to the analysis of the solar activity, we will consider a possibility of a certain correction of the cycle phase by means of additional methods and via special feature of the sunspot dynamics. Besides, to resolve this problem one can use an empirical dependence of the amplitude and the phase of the solar cycle [28]. Moreover, it seems to be reasonable that it is quite possible to improve control parameters which are used in SSA. On the whole we can say that the described method of singular spectrum analysis is a sufficiently advantageous and promising way for the prediction of dynamics of the solar activity.

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References
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