- Email: [email protected]
Application of the Grey Topological Theory in the Prediction of Yearly Mean Sunspot Numbers1, 2
CHINESE ASTRONOMY AND ASTROPHYSICS Chinese Astronomy and Astrophysics 39 (2015) 45–53
Application of the Grey Topological Theory in the Prediction of Yearly Mean Sunspot Numbers† TANG Jie School of Physics and Telecommunication Engineering, Shaanxi University of Technology, Hanzhong 723001
Abstract The topological prediction theory of grey system is introduced in this paper, and the grey topological prediction model has been founded by using a series of smoothed yearly mean values of sunspot numbers from 1944 to 2008. Then this grey topological prediction model is applied to the prediction of the smoothed yearly mean sunspot numbers for the solar cycles 24, 25, and 26 from 2009 to 2039, respectively. The results indicate that the maximum values of sunspot numbers will most probably appear in 2014, 2023, and 2033, and the peak values are about 90, 110, and 130, respectively. And the predicted minimum values of about 20, 20, and 10 will occur around 2017, 2025, and 2039, respectively. Key words
sunspots—sun: activity–methods: statistical 1.
INTRODUCTION
Solar activity has extremely closed relations with the human life, national economy, and national defence, the prediction of solar activity is the urgent necessity in many fields, such as the spaceflight, communication, electricity, navigation, meteorology, hydrology, and so on[1−4] . The prediction of solar activity is generally classified into the short-term prediction from several hour to several day in advance, the mid-term prediction of several month in advance, and the long-term prediction of several year in advance, and the content to be predicted includes the sunspot number, solar flare, solar radio flux, solar proton event, †
Supported by the National Natural Science Foundation (2013JM1021) of Shaanxi Province, and the
Research Foundation of Shaanxi University of Technology (slgky13-50) Received 2013–04–07; revised version 2013–09–02
A translation of Acta Astron. Sin. Vol. 55, No. 2, pp. 137–143, 2014 [email protected]
0275-1062/15/$-see © 2015 Elsevier B.V. Science All rightsB.reserved. c 2015 Elsevier V. All rights reserved. 0275-1062/01/$-seefront frontmatter matter doi:10.1016/j.chinastron.2015.01.002 PII:
46
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
coronal mass ejection, and other solar activity phenomena[5] . Among these solar activity phenomena, the sunspot number is most easy to be observed, and closely related with other activity phenomena, the long-term prediction of solar activity indicates mainly the prediction on the overall solar activity level represented by the monthly or yearly mean values of relative sunspot numbers, especially the predictions on the maximum and minimum values of relative sunspot numbers, and the time of their occurrence[6] . The long-term prediction of relative sunspot numbers is always a difficult topic in the research of long-term prediction of solar activity, many authors in the world have developed various prediction methods in their practice of long-term prediction of sunspot numbers, and obtained many good results. At present, the common-used methods of sunspot prediction are the time-series method, the parametric method of activity cycle, autoregressive model, long-period method, neural network prediction method, precursor method, etc.[5] . Besides the precursor method, these methods are all based on the data of previous sunspot numbers, which contain the basic information of future sunspot numbers, as well as the statistical regularity of the long-term variation of the sunspot number, and they are widely used up to now. But these methods generally need a large number of history data to establish a statistical model, and in the modeling the selection of parameters will unavoidably have certain subjectivity and empiricism. Not based on the statistical regularity, the precursor method, such as the solar activity precursor method and earth parameter precursor method, considers that the sunspot activity is strongly correlated with the solar magnetic activity parameter and geophysical parameter, taking these parameters as the prediction factors the maximum and minimum sunspot numbers can be derived. This method has performed rather well in the 21th and 22th solar cycles, but the predicted time is generally limited in one solar cycle. Because of the uncertainty of the physical process of the long-term variation of solar activity, and the very limited information available, to make the long-term prediction on the sunspot activity is very difficult. The grey prediction theory developed in recent years provides a new idea for the longterm prediction of sunspot numbers. From the random, disorderly, and unsystematic data, the grey prediction looks for the intrinsic regularity hidden in the data, it has solved many practical problems, and achieved significant successes in the society, meteorology, hydrology, economy, power source, and other fields[8] . The traditional method of probability and statistics requires a large number of sample data, otherwise the statistical regularity can hardly be found, however the grey prediction theory has no strict requirement on the number and regularity of the sample data, it especially suits for the indefinite system with the unclear physical type and insufficient information that can hardly be treated by the probability and statistics theory[9] , for example the solar activity. The common-used grey prediction model GM(1,1) has a good effect for the simple exponential curve, the yearly mean value of relative sunspot numbers varies irregularly, to predict the sunspot number by using directly the GM(1,1) model can hardly obtain an ideal result, for the data with large and frequent
47
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
fluctuations the grey topological prediction is a better choice[8,10] . With the data of yearly mean sunspot numbers from 1944 to 2008, this paper makes the prediction by adopting the grey topological method which predicts the future waveform from the existing waveform. 2.
BASIC PRINCIPLE OF THE GREY TOPOLOGICAL PREDICTION
Since the grey system theory was proposed by Prof. Deng Ju-long in 1982, it has rapidly received attentions, and successfully been applied to the grey systems with partial known information and partial unknown information[9] . The grey system theory takes all the systems with known information as the white system, all the systems without known information are defined as the black-box system, and the transitional system between the two is the grey system. The grey prediction predicts the future by digging out the intrinsic regularity that hidden in the grey series. The GM(1,1) model is the kernel of the grey prediction, suitable for the prediction of the grey series with a rather strong exponential law, and the grey topological prediction makes the prediction by using the GM(1,1) model group[8−10] . 2.1
GM(1,1) Model
Denote the original observed data to be x (0) , namely x(0) = x(0) (1), x(0) (2), ..., x(0) (n) ,
(1)
to make the corresponding 1st-order accumulation, we obtain the series k x(1) = x(1) (1), x(1) (2), ..., x(1) (n) , x(1) (k) = x(0) (i) .
(2)
i=1
The series x (1) satisfies the following 1st-order linear differential equation: dx(1) + ax(1) = u , dt
(3)
in which a is the development grey number, u is the endogenous controlling grey number. The two parameters can be derived by the least square method, inserting these parameters into the 1st-order linear differential equation, its solution is: u −ak u e (4) + . xˆ(1) (k + 1) = x(0) (1) − a a To make the degressive reduction on the above equation, we obtain the grey prediction model: u xˆ(0) (k + 1) = x(0) (1) − (1 − ea ) e−ak . (5) a This grey prediction model cannot be used to practical prediction until the precision test is performed.
48
2.2
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
Grey Topological Prediction
The grey topological prediction is called also the waveform prediction, it predicts the waveform of the future development and the variation of the system from the waveform of existing observed data. Because of the complexity of the predicted system, the topological prediction consists of many models, and one threshold value corresponds to one model. To plot the analyzed data into a diagram of curve, and select a group of threshold values according to the requirement of the research, then to find out the respective point of intersection for the every threshold value on the diagram, the distance between the projection of the point of intersection on the abscissa and the origin is taken as the original data for the GM(1,1) modeling of this threshold value, and the GM(1,1) models are built for these threshold values. Each threshold value corresponds to one GM(1,1) model, then this group of models are used to predict the time that each threshold value may probably occur in the future, to have the predicted time values and the corresponding threshold values connected, the obtained curve is the trend curve of future waveform variation of the topological prediction.
3.
APPLICATION OF THE GREY TOPOLOGICAL PREDICTION METHOD TO THE PREDICTION OF YEARLY MEAN SUNSPOT NUMBERS
In many astronomical phenomena exists periodicity[11−14] , for example the periodicity of solar activity. The relative sunspot numbers are generally used to study the variation of solar activity. Since the record of sunspot number started in 1700, the observed data have accumulated for more than 300 years. Beginning from 1755, the 11 yr average period of the relative sunspot numbers has been taken as one solar cycle of solar activity[15] , in 2008 the 24th solar cycle began[15] , and the reference [16] suggested also that the 24th solar cycle began in Nov. 2008. Since the grey topological prediction has the advantage that the effect of new information on the knowledge is greater than the old information, and that unlike the method of probability and statistics, it does not need a large sample, with the yearly mean sunspot number data of 6 solar cycles from 1944 to 2088, this paper adopts the grey topological prediction method to predict the yearly mean values of sunspot numbers in the 24th, 25th, and 26th solar cycles. According to the grey topological prediction theory, the data of yearly mean sunspot numbers from 1944 to 2008 are used to establish the group of grey topological prediction models. From Fig.1 we can find that the yearly mean value of relative sunspot numbers always increases gradually from a minimum value approximately to be zero to the maximum value, then gradually decreases to another minimum value, the minimum value differs with the solar cycle not very large, but the maximum value differs with the solar cycle greatly, the greatest maximum value 184.8 is 63.7 times of the smallest minimum value 2.9, the amplitude of fluctuations is quite large, and with an approximately 11 yr period the fluctuations are
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
rather frequent. According to the range of the maximum and minimum relative sunspot numbers, in order to guarantee each threshold value have more than 4 points of intersection with the historic curve, we adopt 14 threshold values in space of 10, and draw the isolines, from the points of intersection between these isolines and the variation curve of the relative sunspot numbers from 1944 to 2008 we can obtain 14 threshold value series. The abscissaes of these points of intersection are taken as the original abscissaes minus 1944, then the converted threshold value series are used as the original data for modeling. The group of grey topological prediction models have been built for the every threshold value series, as shown in Table 1, since this paper is purposed for making long-term prediction, the grey system theory requires that the prediction model can be used for the medium-and-long-term prediction only if its development coefficient C ≤0.3, and fortunately the development coefficients of the group of models obtained by this paper are all far less than 0.3 (the smaller the value of C, the better the model), indicating that the built model group can serve very well the purpose of long-term prediction.
200 observed value
180
Group sunspot numbers
160 140 120 100 80 60 40 20 0 1940
Fig. 1
1950
1960
1970
1980 Year
1990
2000
2010
The smoothed yearly mean sunspot numbers and threshold values from 1944 to 2008
In the modeling process we have to test if the precision of the prediction model satisfies the requirement, the grey topological prediction generally evaluates the quality of the prediction model by two specifications: a posteriori error ratio, which expresses the dispersion of prediction errors, and the small error probability. A posteriori error ratio is required to be less than 0.35, the smaller the better; the small error probability should be greater than 0.8, the greater the better, indicating the probability of small prediction error is large, and the prediction accuracy is high. The two parameters of this paper are all good enough to satisfy the requirements, and the precision of the built model group is first class, with a rather high accuracy. To compare the predicted results given by the models with the observed values, the absolute errors between the predicted series and the observed series are quite small, and
49
50
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
Table 1 The GM(1,1) prediction model of the smoothed yearly mean sunspot numbers Threshold value
a
C
Prediction model (0)
449.67e0.1010k
10
-0.1010
x1
− 413.42
0.2367
20
-0.0560
x2 (0) (k + 1) = 623.09e0.0560k − 591.02
0.0620
(0)
(k + 1) =
631.83e0.0552k
30
-0.0552
x3
− 599.81
0.0669
40
-0.0550
x4 (0) (k + 1) = 630.63e0.0550k − 598.90
0.0716
(0)
(k + 1) =
631.07e0.0548k
50
-0.0548
x5
− 599.45
0.0751
60
-0.0546
x6 (0) (k + 1) = 631.81e0.0546k − 600.28
0.0787
(0)
(k + 1) =
626.04e0.0549k
70
-0.0549
x7
− 594.66
0.0827
80
-0.0547
x8 (0) (k + 1) = 627.11e0.0547k − 595.79
0.0873
(0)
(k + 1) =
641.09e0.0537k
90
-0.0537
x9
− 609.72
0.0867
100
-0.0542
x10 (0) (k + 1) = 635.38e0.0542k − 604.04
0.0987
(0)
(k + 1) =
633.26e0.0583k
110
-0.0583
x11
− 600.95
0.1802
120
-0.0719
x12 (0) (k + 1) = 462.72e0.0719k − 433.39
0.1436
(0)
(k + 1) =
354.98e0.0829k
130
-0.0829
x13
− 329.29
0.1866
140
-0.0727
x14 (0) (k + 1) = 459.02e0.0727k − 430.28
0.1974
150
-0.1061
x15
(0)
(k + 1) =
(k + 1) =
274.82e0.1061k
− 249.25
0.1365
the relative errors are even small, not greater than 15%. Table 2 shows part of the predicted values and observed values. Hence, by using the built model group to make the long-term prediction, the prediction accuracy is very high, and the effect of prediction is fairly ideal. Using the built prediction model group, we have predicted the times that different sunspot numbers probably occur in the period from 2009 to 2039, the result is given in Table 3, and shown as the diagram in Fig.2. According to the predicted result listed in Table 3, the 24th solar cycle may occur from 2008 to 2017, and it is predicted that the maximum yearly mean sunspot number will occur in 2014, the maximum value is 90, the minimum value will occur in 2017, the minimum value is 20, and the time span of this solar cycle is 9 yr. The 25th solar cycle probably occurs from 2017 to 2026, it is predicted that the maximum yearly mean sunspot number will occur in 2023, the maximum value is probably greater than that of the 24th cycle, to be 110, the minimum value will occur in 2025, the minimum value is also 20, and the time span of this solar cycle is also 9 yr. The 26th solar cycle probably occurs from 2026 to 2040, it is predicted that the maximum value will occur in 2033, the maximum value 130 is greater than those of the 24th and 25th cycles, the minimum value 10 will occur in 2039, and the time span of this solar cycle will be as long as 14 yr.
51
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
Table 2 Part of the observed values and predicted values Threshold Observed Predicted Error Observed Predicted Error Observed Predicted Error value
value
value
/(%)
value
value
/(%)
value
value
/(%)
10
45.07
47.76
-5.98
58.45
58.45
0
63.74
64.66
-1.45
20
37.36
37.90
-1.46
53.00
53.02
-0.04
62.73
62.71
0.02
30
40.14
40.07
0.17
52.97
52.82
0.28
58.15
58.99
-1.44
40
42.35
42.01
0.80
50.34
49.54
1.59
57.79
58.42
-1.08
50
39.59
39.65
-0.15
52.49
52.15
0.67
60.65
61.46
-1.33
60
36.75
37.45
-1.91
49.99
49.20
1.59
57.59
57.96
-0.64
70
39.27
39.39
-0.31
49.89
49.05
1.67
54.94
54.75
0.36
80
36.56
37.27
-1.94
44.54
43.92
1.38
54.89
54.68
0.38
90
36.65
37.35
-1.91
49.61
48.86
1.50
57.18
57.41
-0.41
100
39.28
39.44
-0.41
44.52
43.95
1.28
54.80
54.59
0.38
110
42.93
42.73
0.46
54.55
53.96
1.08
57.45
57.20
0.43
120
36.30
37.05
-2.06
40.21
39.81
0.99
49.99
49.39
1.21
130
32.62
33.32
-2.16
36.55
36.20
0.95
46.97
46.42
1.17
140
32.89
33.55
-2.03
36.85
36.38
1.27
46.96
46.39
1.21
150
34.04
34.20
-0.48
41.98
42.29
-0.73
46.57
47.02
-0.97
Table 3 The topological prediction results of the smoothed yearly mean sunspot numbers from 2009 to 2039 Threshold value
Primitive number(Year)
10
96.84(2039)
20
66.32(2009)
74.17(2017)
30
73.57(2016)
82.16(2025)
40
72.78(2015)
81.24(2024)
95.80(2038)
50
68.58(2011)
72.44(2015)
80.83(2023)
84.38(2027)
60
68.27(2011)
72.11(2015)
80.43(2023)
94.74(2037)
70
72.02(2015)
80.37(2023)
84.91(2027)
80
71.90(2014)
75.94(2018)
80.21(2023)
84.73(2027) 93.12(2036)
82.95(2025)
90
71.18(2014)
75.11(2018)
79.26(2022)
100
75.57(2018)
79.78(2022)
93.87(2036)
110
76.56(2019)
80.15(2023)
86.03(2029)
120
87.76(2030)
130
90.08(2033)
4.
95.26(2038)
94.53(2037)
91.19(2034)
DISCUSSION
The precise prediction of the variation of sunspot numbers has a strong applicability and great practical value, the authors around the world have done a lot of research work for the
52
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
2023.15
120 2014.18
100 Group Sunspot Numbers
predicted value
2033.08
80
60
40
20 2039.84 2008.00 0 2000
Fig. 2
2005
2017.17 2010
2015
2020
2025.95 2025 Year
2030
2035
2040
2045
2050
The topological prediction of the smoothed yearly mean sunspot numbers from 2009 to 2039
prediction method of sunspot numbers[2−4,15−16] , and obtained a series of results, however, because of the complexity and uncertainty of the variation of sunspot numbers, most predicted values have a large difference from the observed values, the quality of predictions remains to improve. For example, Pesnell[17] analyzed the 54 predicted results on the sunspot numbers of the 24th solar cycle, and found that the predicted results obtained from different prediction methods differ quite large, the effects of predictions are not very good, and these methods basically make predictions only for nearly one solar cycle, the predicted time is rather short, and a large number of historic data of sunspot numbers are required. Kane[18] collected also 43 predicted values on the sunspot numbers of the 24th solar cycle, and found also that these predicted values are extremely scattered. The grey topological prediction method applied in this paper does not need a large number of data, the procedures of calculation is not very many, the restriction on the data distribution is very small, from limited known information the valuable information can be extracted to give us a better knowledge of the implicit regularity of the analyzed object. With the grey topological prediction method this paper has predicted the yearly mean sunspot numbers of the 24th, 25th, and 26th solar cycles, the errors of the predicted data derived from the observed data and the model group formula are relatively small. For the prediction of solar activity cycle, the literature generally gives the monthly mean value, and the maximum yearly mean value is generally larger than the maximum monthly mean value, in order to make comparison with other literature, this paper will neglect this difference. The maximum sunspot number of the 24th solar cycle is probably 90, this conclusion is
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
consistent with the conclusion of the reference [16] that the solar activity level of the 24th cycle will be 30% weaker than the 23th solar activity cycle, and also very consistent with our result obtained by using the prediction model based on the radial basis functional neutral network[7]. This conclusion is supported by the prediction group of 24th solar activity cycle of the US Space Environment Central Organization. The United States National Aeronautics and Space Administration and the National Oceanic and Atmospheric Administration suggested this value being about 90, identical to our conclusion, and similar results are also obtained in part of 54 papers collected by Pasnell[17]. This paper has predicted that the maximum value of the 24th solar cycle will appear in about 2014, this is also supported by the result about the maximum of the 24th solar activity cycle summarized from many pieces of literature by Kane[18] . It is predicted that the maximum sunspot number of the 25th solar activity cycle will be slightly larger than that of the 24th solar cycle, and that the maximum value of the 26th solar cycle will increase to 130, a little larger than the predicted maximum value 119.6 of the 23th solar activity cycle. The accuracy of these predicted results should be verified by later observations. ACKNOWLEDGEMENTS We thank the referee for valuable opinion and Prof. Wu Xue-bing of Astronomy Department of Beijing University for the support and instruction to this paper. References 1
Gan W. Q., Huang Y., Yan Y. H., Science in China: Physics, Mechanics, Astronomy, 2012, 42, 1274
2
Li K. J., BAA, 2009, 9, 959
3
Li K. J., Qiu J., Xiang F. Y., et al., New A, 2005, 10, 209
4
Wang J. L., Miao J., Liu S. Q., et al., ScChG, 2008, 51, 1938
5
Lin Y. Z., Introduction of Solar Physics, Beijing: Science Press, 2000, 575-590
6
Zhao H. J., Wang J. L., Zong W. G., et al., Chinese Journal of Geophysics, 2010, 51, 31
7
Tang J., Zhang X., Acta Physica Sinica, 2012, 61, 169601
8
Liu S. F., Guo T. B., Dang Y. G., Grey System Theory and Its Applications, Beijing: Science Press, 2000, 144-150
9
Deng J. L., Grey System Theory and Its Applications, Beijing: Scientific and Technical Documentation Press, 1992, 168-196
10
Fu L., Grey Theory and Its Applications, Beijing: Scientific and Technical Documentation Press, 1992, 150-157
11
Tang J., AcASn, 2012, 53, 1
12
Tang J., ChA&A, 2012, 36, 229
13
Tang J., Zhang X. J., Pang Q., et al., AcASn, 2009, 50, 364
14
Tang J., Zhang X. J., Pang Q., et al., ChA&A, 2010, 34, 121
15
Li K. J., Feng W., Liang H. F., et al., AnGeo, 2011, 29, 341
16
Li K. J., Feng W., Liang H. F., Science in China: Physics, Mechanics, Astronomy, 2010, 40, 1293
17
Pasnell W. D., SoPh, 2008, 252, 209
18
Kane R. P., SoPh, 2007, 243, 205
53
Application of the Grey Topological Theory in the Prediction of Yearly Mean Sunspot Numbers† TANG Jie School of Physics and Telecommunication Engineering, Shaanxi University of Technology, Hanzhong 723001
Abstract The topological prediction theory of grey system is introduced in this paper, and the grey topological prediction model has been founded by using a series of smoothed yearly mean values of sunspot numbers from 1944 to 2008. Then this grey topological prediction model is applied to the prediction of the smoothed yearly mean sunspot numbers for the solar cycles 24, 25, and 26 from 2009 to 2039, respectively. The results indicate that the maximum values of sunspot numbers will most probably appear in 2014, 2023, and 2033, and the peak values are about 90, 110, and 130, respectively. And the predicted minimum values of about 20, 20, and 10 will occur around 2017, 2025, and 2039, respectively. Key words
sunspots—sun: activity–methods: statistical 1.
INTRODUCTION
Solar activity has extremely closed relations with the human life, national economy, and national defence, the prediction of solar activity is the urgent necessity in many fields, such as the spaceflight, communication, electricity, navigation, meteorology, hydrology, and so on[1−4] . The prediction of solar activity is generally classified into the short-term prediction from several hour to several day in advance, the mid-term prediction of several month in advance, and the long-term prediction of several year in advance, and the content to be predicted includes the sunspot number, solar flare, solar radio flux, solar proton event, †
Supported by the National Natural Science Foundation (2013JM1021) of Shaanxi Province, and the
Research Foundation of Shaanxi University of Technology (slgky13-50) Received 2013–04–07; revised version 2013–09–02
A translation of Acta Astron. Sin. Vol. 55, No. 2, pp. 137–143, 2014 [email protected]
0275-1062/15/$-see © 2015 Elsevier B.V. Science All rightsB.reserved. c 2015 Elsevier V. All rights reserved. 0275-1062/01/$-seefront frontmatter matter doi:10.1016/j.chinastron.2015.01.002 PII:
46
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
coronal mass ejection, and other solar activity phenomena[5] . Among these solar activity phenomena, the sunspot number is most easy to be observed, and closely related with other activity phenomena, the long-term prediction of solar activity indicates mainly the prediction on the overall solar activity level represented by the monthly or yearly mean values of relative sunspot numbers, especially the predictions on the maximum and minimum values of relative sunspot numbers, and the time of their occurrence[6] . The long-term prediction of relative sunspot numbers is always a difficult topic in the research of long-term prediction of solar activity, many authors in the world have developed various prediction methods in their practice of long-term prediction of sunspot numbers, and obtained many good results. At present, the common-used methods of sunspot prediction are the time-series method, the parametric method of activity cycle, autoregressive model, long-period method, neural network prediction method, precursor method, etc.[5] . Besides the precursor method, these methods are all based on the data of previous sunspot numbers, which contain the basic information of future sunspot numbers, as well as the statistical regularity of the long-term variation of the sunspot number, and they are widely used up to now. But these methods generally need a large number of history data to establish a statistical model, and in the modeling the selection of parameters will unavoidably have certain subjectivity and empiricism. Not based on the statistical regularity, the precursor method, such as the solar activity precursor method and earth parameter precursor method, considers that the sunspot activity is strongly correlated with the solar magnetic activity parameter and geophysical parameter, taking these parameters as the prediction factors the maximum and minimum sunspot numbers can be derived. This method has performed rather well in the 21th and 22th solar cycles, but the predicted time is generally limited in one solar cycle. Because of the uncertainty of the physical process of the long-term variation of solar activity, and the very limited information available, to make the long-term prediction on the sunspot activity is very difficult. The grey prediction theory developed in recent years provides a new idea for the longterm prediction of sunspot numbers. From the random, disorderly, and unsystematic data, the grey prediction looks for the intrinsic regularity hidden in the data, it has solved many practical problems, and achieved significant successes in the society, meteorology, hydrology, economy, power source, and other fields[8] . The traditional method of probability and statistics requires a large number of sample data, otherwise the statistical regularity can hardly be found, however the grey prediction theory has no strict requirement on the number and regularity of the sample data, it especially suits for the indefinite system with the unclear physical type and insufficient information that can hardly be treated by the probability and statistics theory[9] , for example the solar activity. The common-used grey prediction model GM(1,1) has a good effect for the simple exponential curve, the yearly mean value of relative sunspot numbers varies irregularly, to predict the sunspot number by using directly the GM(1,1) model can hardly obtain an ideal result, for the data with large and frequent
47
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
fluctuations the grey topological prediction is a better choice[8,10] . With the data of yearly mean sunspot numbers from 1944 to 2008, this paper makes the prediction by adopting the grey topological method which predicts the future waveform from the existing waveform. 2.
BASIC PRINCIPLE OF THE GREY TOPOLOGICAL PREDICTION
Since the grey system theory was proposed by Prof. Deng Ju-long in 1982, it has rapidly received attentions, and successfully been applied to the grey systems with partial known information and partial unknown information[9] . The grey system theory takes all the systems with known information as the white system, all the systems without known information are defined as the black-box system, and the transitional system between the two is the grey system. The grey prediction predicts the future by digging out the intrinsic regularity that hidden in the grey series. The GM(1,1) model is the kernel of the grey prediction, suitable for the prediction of the grey series with a rather strong exponential law, and the grey topological prediction makes the prediction by using the GM(1,1) model group[8−10] . 2.1
GM(1,1) Model
Denote the original observed data to be x (0) , namely x(0) = x(0) (1), x(0) (2), ..., x(0) (n) ,
(1)
to make the corresponding 1st-order accumulation, we obtain the series k x(1) = x(1) (1), x(1) (2), ..., x(1) (n) , x(1) (k) = x(0) (i) .
(2)
i=1
The series x (1) satisfies the following 1st-order linear differential equation: dx(1) + ax(1) = u , dt
(3)
in which a is the development grey number, u is the endogenous controlling grey number. The two parameters can be derived by the least square method, inserting these parameters into the 1st-order linear differential equation, its solution is: u −ak u e (4) + . xˆ(1) (k + 1) = x(0) (1) − a a To make the degressive reduction on the above equation, we obtain the grey prediction model: u xˆ(0) (k + 1) = x(0) (1) − (1 − ea ) e−ak . (5) a This grey prediction model cannot be used to practical prediction until the precision test is performed.
48
2.2
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
Grey Topological Prediction
The grey topological prediction is called also the waveform prediction, it predicts the waveform of the future development and the variation of the system from the waveform of existing observed data. Because of the complexity of the predicted system, the topological prediction consists of many models, and one threshold value corresponds to one model. To plot the analyzed data into a diagram of curve, and select a group of threshold values according to the requirement of the research, then to find out the respective point of intersection for the every threshold value on the diagram, the distance between the projection of the point of intersection on the abscissa and the origin is taken as the original data for the GM(1,1) modeling of this threshold value, and the GM(1,1) models are built for these threshold values. Each threshold value corresponds to one GM(1,1) model, then this group of models are used to predict the time that each threshold value may probably occur in the future, to have the predicted time values and the corresponding threshold values connected, the obtained curve is the trend curve of future waveform variation of the topological prediction.
3.
APPLICATION OF THE GREY TOPOLOGICAL PREDICTION METHOD TO THE PREDICTION OF YEARLY MEAN SUNSPOT NUMBERS
In many astronomical phenomena exists periodicity[11−14] , for example the periodicity of solar activity. The relative sunspot numbers are generally used to study the variation of solar activity. Since the record of sunspot number started in 1700, the observed data have accumulated for more than 300 years. Beginning from 1755, the 11 yr average period of the relative sunspot numbers has been taken as one solar cycle of solar activity[15] , in 2008 the 24th solar cycle began[15] , and the reference [16] suggested also that the 24th solar cycle began in Nov. 2008. Since the grey topological prediction has the advantage that the effect of new information on the knowledge is greater than the old information, and that unlike the method of probability and statistics, it does not need a large sample, with the yearly mean sunspot number data of 6 solar cycles from 1944 to 2088, this paper adopts the grey topological prediction method to predict the yearly mean values of sunspot numbers in the 24th, 25th, and 26th solar cycles. According to the grey topological prediction theory, the data of yearly mean sunspot numbers from 1944 to 2008 are used to establish the group of grey topological prediction models. From Fig.1 we can find that the yearly mean value of relative sunspot numbers always increases gradually from a minimum value approximately to be zero to the maximum value, then gradually decreases to another minimum value, the minimum value differs with the solar cycle not very large, but the maximum value differs with the solar cycle greatly, the greatest maximum value 184.8 is 63.7 times of the smallest minimum value 2.9, the amplitude of fluctuations is quite large, and with an approximately 11 yr period the fluctuations are
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
rather frequent. According to the range of the maximum and minimum relative sunspot numbers, in order to guarantee each threshold value have more than 4 points of intersection with the historic curve, we adopt 14 threshold values in space of 10, and draw the isolines, from the points of intersection between these isolines and the variation curve of the relative sunspot numbers from 1944 to 2008 we can obtain 14 threshold value series. The abscissaes of these points of intersection are taken as the original abscissaes minus 1944, then the converted threshold value series are used as the original data for modeling. The group of grey topological prediction models have been built for the every threshold value series, as shown in Table 1, since this paper is purposed for making long-term prediction, the grey system theory requires that the prediction model can be used for the medium-and-long-term prediction only if its development coefficient C ≤0.3, and fortunately the development coefficients of the group of models obtained by this paper are all far less than 0.3 (the smaller the value of C, the better the model), indicating that the built model group can serve very well the purpose of long-term prediction.
200 observed value
180
Group sunspot numbers
160 140 120 100 80 60 40 20 0 1940
Fig. 1
1950
1960
1970
1980 Year
1990
2000
2010
The smoothed yearly mean sunspot numbers and threshold values from 1944 to 2008
In the modeling process we have to test if the precision of the prediction model satisfies the requirement, the grey topological prediction generally evaluates the quality of the prediction model by two specifications: a posteriori error ratio, which expresses the dispersion of prediction errors, and the small error probability. A posteriori error ratio is required to be less than 0.35, the smaller the better; the small error probability should be greater than 0.8, the greater the better, indicating the probability of small prediction error is large, and the prediction accuracy is high. The two parameters of this paper are all good enough to satisfy the requirements, and the precision of the built model group is first class, with a rather high accuracy. To compare the predicted results given by the models with the observed values, the absolute errors between the predicted series and the observed series are quite small, and
49
50
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
Table 1 The GM(1,1) prediction model of the smoothed yearly mean sunspot numbers Threshold value
a
C
Prediction model (0)
449.67e0.1010k
10
-0.1010
x1
− 413.42
0.2367
20
-0.0560
x2 (0) (k + 1) = 623.09e0.0560k − 591.02
0.0620
(0)
(k + 1) =
631.83e0.0552k
30
-0.0552
x3
− 599.81
0.0669
40
-0.0550
x4 (0) (k + 1) = 630.63e0.0550k − 598.90
0.0716
(0)
(k + 1) =
631.07e0.0548k
50
-0.0548
x5
− 599.45
0.0751
60
-0.0546
x6 (0) (k + 1) = 631.81e0.0546k − 600.28
0.0787
(0)
(k + 1) =
626.04e0.0549k
70
-0.0549
x7
− 594.66
0.0827
80
-0.0547
x8 (0) (k + 1) = 627.11e0.0547k − 595.79
0.0873
(0)
(k + 1) =
641.09e0.0537k
90
-0.0537
x9
− 609.72
0.0867
100
-0.0542
x10 (0) (k + 1) = 635.38e0.0542k − 604.04
0.0987
(0)
(k + 1) =
633.26e0.0583k
110
-0.0583
x11
− 600.95
0.1802
120
-0.0719
x12 (0) (k + 1) = 462.72e0.0719k − 433.39
0.1436
(0)
(k + 1) =
354.98e0.0829k
130
-0.0829
x13
− 329.29
0.1866
140
-0.0727
x14 (0) (k + 1) = 459.02e0.0727k − 430.28
0.1974
150
-0.1061
x15
(0)
(k + 1) =
(k + 1) =
274.82e0.1061k
− 249.25
0.1365
the relative errors are even small, not greater than 15%. Table 2 shows part of the predicted values and observed values. Hence, by using the built model group to make the long-term prediction, the prediction accuracy is very high, and the effect of prediction is fairly ideal. Using the built prediction model group, we have predicted the times that different sunspot numbers probably occur in the period from 2009 to 2039, the result is given in Table 3, and shown as the diagram in Fig.2. According to the predicted result listed in Table 3, the 24th solar cycle may occur from 2008 to 2017, and it is predicted that the maximum yearly mean sunspot number will occur in 2014, the maximum value is 90, the minimum value will occur in 2017, the minimum value is 20, and the time span of this solar cycle is 9 yr. The 25th solar cycle probably occurs from 2017 to 2026, it is predicted that the maximum yearly mean sunspot number will occur in 2023, the maximum value is probably greater than that of the 24th cycle, to be 110, the minimum value will occur in 2025, the minimum value is also 20, and the time span of this solar cycle is also 9 yr. The 26th solar cycle probably occurs from 2026 to 2040, it is predicted that the maximum value will occur in 2033, the maximum value 130 is greater than those of the 24th and 25th cycles, the minimum value 10 will occur in 2039, and the time span of this solar cycle will be as long as 14 yr.
51
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
Table 2 Part of the observed values and predicted values Threshold Observed Predicted Error Observed Predicted Error Observed Predicted Error value
value
value
/(%)
value
value
/(%)
value
value
/(%)
10
45.07
47.76
-5.98
58.45
58.45
0
63.74
64.66
-1.45
20
37.36
37.90
-1.46
53.00
53.02
-0.04
62.73
62.71
0.02
30
40.14
40.07
0.17
52.97
52.82
0.28
58.15
58.99
-1.44
40
42.35
42.01
0.80
50.34
49.54
1.59
57.79
58.42
-1.08
50
39.59
39.65
-0.15
52.49
52.15
0.67
60.65
61.46
-1.33
60
36.75
37.45
-1.91
49.99
49.20
1.59
57.59
57.96
-0.64
70
39.27
39.39
-0.31
49.89
49.05
1.67
54.94
54.75
0.36
80
36.56
37.27
-1.94
44.54
43.92
1.38
54.89
54.68
0.38
90
36.65
37.35
-1.91
49.61
48.86
1.50
57.18
57.41
-0.41
100
39.28
39.44
-0.41
44.52
43.95
1.28
54.80
54.59
0.38
110
42.93
42.73
0.46
54.55
53.96
1.08
57.45
57.20
0.43
120
36.30
37.05
-2.06
40.21
39.81
0.99
49.99
49.39
1.21
130
32.62
33.32
-2.16
36.55
36.20
0.95
46.97
46.42
1.17
140
32.89
33.55
-2.03
36.85
36.38
1.27
46.96
46.39
1.21
150
34.04
34.20
-0.48
41.98
42.29
-0.73
46.57
47.02
-0.97
Table 3 The topological prediction results of the smoothed yearly mean sunspot numbers from 2009 to 2039 Threshold value
Primitive number(Year)
10
96.84(2039)
20
66.32(2009)
74.17(2017)
30
73.57(2016)
82.16(2025)
40
72.78(2015)
81.24(2024)
95.80(2038)
50
68.58(2011)
72.44(2015)
80.83(2023)
84.38(2027)
60
68.27(2011)
72.11(2015)
80.43(2023)
94.74(2037)
70
72.02(2015)
80.37(2023)
84.91(2027)
80
71.90(2014)
75.94(2018)
80.21(2023)
84.73(2027) 93.12(2036)
82.95(2025)
90
71.18(2014)
75.11(2018)
79.26(2022)
100
75.57(2018)
79.78(2022)
93.87(2036)
110
76.56(2019)
80.15(2023)
86.03(2029)
120
87.76(2030)
130
90.08(2033)
4.
95.26(2038)
94.53(2037)
91.19(2034)
DISCUSSION
The precise prediction of the variation of sunspot numbers has a strong applicability and great practical value, the authors around the world have done a lot of research work for the
52
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
2023.15
120 2014.18
100 Group Sunspot Numbers
predicted value
2033.08
80
60
40
20 2039.84 2008.00 0 2000
Fig. 2
2005
2017.17 2010
2015
2020
2025.95 2025 Year
2030
2035
2040
2045
2050
The topological prediction of the smoothed yearly mean sunspot numbers from 2009 to 2039
prediction method of sunspot numbers[2−4,15−16] , and obtained a series of results, however, because of the complexity and uncertainty of the variation of sunspot numbers, most predicted values have a large difference from the observed values, the quality of predictions remains to improve. For example, Pesnell[17] analyzed the 54 predicted results on the sunspot numbers of the 24th solar cycle, and found that the predicted results obtained from different prediction methods differ quite large, the effects of predictions are not very good, and these methods basically make predictions only for nearly one solar cycle, the predicted time is rather short, and a large number of historic data of sunspot numbers are required. Kane[18] collected also 43 predicted values on the sunspot numbers of the 24th solar cycle, and found also that these predicted values are extremely scattered. The grey topological prediction method applied in this paper does not need a large number of data, the procedures of calculation is not very many, the restriction on the data distribution is very small, from limited known information the valuable information can be extracted to give us a better knowledge of the implicit regularity of the analyzed object. With the grey topological prediction method this paper has predicted the yearly mean sunspot numbers of the 24th, 25th, and 26th solar cycles, the errors of the predicted data derived from the observed data and the model group formula are relatively small. For the prediction of solar activity cycle, the literature generally gives the monthly mean value, and the maximum yearly mean value is generally larger than the maximum monthly mean value, in order to make comparison with other literature, this paper will neglect this difference. The maximum sunspot number of the 24th solar cycle is probably 90, this conclusion is
Tang Jie / Chinese Astronomy and Astrophysics 39 (2015) 45–53
consistent with the conclusion of the reference [16] that the solar activity level of the 24th cycle will be 30% weaker than the 23th solar activity cycle, and also very consistent with our result obtained by using the prediction model based on the radial basis functional neutral network[7]. This conclusion is supported by the prediction group of 24th solar activity cycle of the US Space Environment Central Organization. The United States National Aeronautics and Space Administration and the National Oceanic and Atmospheric Administration suggested this value being about 90, identical to our conclusion, and similar results are also obtained in part of 54 papers collected by Pasnell[17]. This paper has predicted that the maximum value of the 24th solar cycle will appear in about 2014, this is also supported by the result about the maximum of the 24th solar activity cycle summarized from many pieces of literature by Kane[18] . It is predicted that the maximum sunspot number of the 25th solar activity cycle will be slightly larger than that of the 24th solar cycle, and that the maximum value of the 26th solar cycle will increase to 130, a little larger than the predicted maximum value 119.6 of the 23th solar activity cycle. The accuracy of these predicted results should be verified by later observations. ACKNOWLEDGEMENTS We thank the referee for valuable opinion and Prof. Wu Xue-bing of Astronomy Department of Beijing University for the support and instruction to this paper. References 1
Gan W. Q., Huang Y., Yan Y. H., Science in China: Physics, Mechanics, Astronomy, 2012, 42, 1274
2
Li K. J., BAA, 2009, 9, 959
3
Li K. J., Qiu J., Xiang F. Y., et al., New A, 2005, 10, 209
4
Wang J. L., Miao J., Liu S. Q., et al., ScChG, 2008, 51, 1938
5
Lin Y. Z., Introduction of Solar Physics, Beijing: Science Press, 2000, 575-590
6
Zhao H. J., Wang J. L., Zong W. G., et al., Chinese Journal of Geophysics, 2010, 51, 31
7
Tang J., Zhang X., Acta Physica Sinica, 2012, 61, 169601
8
Liu S. F., Guo T. B., Dang Y. G., Grey System Theory and Its Applications, Beijing: Science Press, 2000, 144-150
9
Deng J. L., Grey System Theory and Its Applications, Beijing: Scientific and Technical Documentation Press, 1992, 168-196
10
Fu L., Grey Theory and Its Applications, Beijing: Scientific and Technical Documentation Press, 1992, 150-157
11
Tang J., AcASn, 2012, 53, 1
12
Tang J., ChA&A, 2012, 36, 229
13
Tang J., Zhang X. J., Pang Q., et al., AcASn, 2009, 50, 364
14
Tang J., Zhang X. J., Pang Q., et al., ChA&A, 2010, 34, 121
15
Li K. J., Feng W., Liang H. F., et al., AnGeo, 2011, 29, 341
16
Li K. J., Feng W., Liang H. F., Science in China: Physics, Mechanics, Astronomy, 2010, 40, 1293
17
Pasnell W. D., SoPh, 2008, 252, 209
18
Kane R. P., SoPh, 2007, 243, 205
53