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Forecast for the solar activity based on the autoregressive desciption of the sunspot number time series R. Werner Solar Terrestrial Influences Institute.

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Presentation on theme: "Forecast for the solar activity based on the autoregressive desciption of the sunspot number time series R. Werner Solar Terrestrial Influences Institute."— Presentation transcript:

1 Forecast for the solar activity based on the autoregressive desciption of the sunspot number time series R. Werner Solar Terrestrial Influences Institute - BAS

2 In the last year we have learned about some basics of the time-series analysis by descriptive and inference statistics Descriptive statistics: We have been acquainted with definitons for the time series, arithmetic mean, variance, correlation and auto-correlation, co-variance and cross-correlation We have decomposed the time series into a trend, a seasonal and a rest component We have examined problems to estimate the trend component and we have learned basic methods such as average moving, linear and polynomial regression, analysis and the harmonic analysis to determine the seasonal component. We have used the phase average method, the periodogram We have learned the Box/Cox transformation as a method to stabilize the variance Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

3 Inference statistics: We have learned about a very important condition as the stationarity, and the weakly stationarity. We have presented auto-regression models and average moving models and have shown some important characteristics of AR and AM models of first and second order. We have shown properties of the auto-correlation func- tion (ACF) and of the partial auto-correlation function (PACF) We have presented the Yule Walker equation. We have demonstrated how we can determine the auto- regressive model using the ACF and the PACF for the time series of the sunspot number We have learned about the principles of the dynamic regression, of some simple models, the Koyke transformation and the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

4 Today We will analyse the sunspot number time series in more detail with the main goal to make forecasts for the next solar cycle activity using the Box/Jenkins methodology for the model identification. Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

5 The forecasts for the solar activity are very important:  for the satellite drag  the telecommunication outages  for hazards in connection with the occurrence of strong solar wind streams producing the blackout of power plants.  for manned space flights, for the prognosis of the radiation risk  High powerful radiation can lead to computer upsets and computer memory failures Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

6 Pesnell, Solar Phys. (2008) 252:209-220 Solar activity predictions of the R 24 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

7 Regardless of the advance in the application of physical methods for the purpose of forecasting, the results are very inconsistently spread and substantiate the application of statistical methods. Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

8 Box/Jenkins method 1. Box/Jenkins model identification 1.1 Stationarity  Box/Cox transformation 1.2 Seasonality 1.3 Auto-correlation and partial auto-correlation plots 1.4 Determination of the type of the process and its order 2. Estimation of the model parameters 3. Model diagnostics 4. Forecasting Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

9 Yule, 1927 Stationarity? Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

10 Yule, 1927 The means are not significantly different, however, the standard deviations depend on the means. Therefore, the series is not stationary!  Box-Cox transformation Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

11 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

12 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

13 AR(2)-model Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

14 AR(2)-model a t error term: white noise φ i have to be determined by the Yule-Walker equations YuleIn this work φ1φ1 1.34251.3571 φ2φ2 -0.6550-0.6601 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

15 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

16 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

17 1749-19241749-2008 φ1 φ1 1.211202 1.203376 φ2 φ2 -0.458256-0.493566 φ3 φ3 -0.124498-0.108303 φ4 φ4 0.149170 0.206869 φ5 φ5 -0.131423-0.223224 φ6 φ6 -0.020802 0.057670 φ7 φ7 0.098886 0.098426 φ8 φ8 -0.103147-0.176114 φ9 φ9 0.189920 0.286768 1848-2008 1.043363 -0.336989 -0.176310 0.198759 -0.215978 0.060787 0.061170 -0.245499 0.401278 1909-2008 1.095090 -0.416573 -0.159289 0.219609 -0.243758 0.088986 0.105020 -0.298126 0.435216 YuleIn this work φ1φ1 1.34251.3571 φ2φ2 -0.6550-0.6601 AR(9) model Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

18 Forecast (ex-post-prognosis, prognosis of known values of the past) One-step prognosis For an AR(2) Two-step prognosis mean squared forecast error: p: model order Which is the optimal model? For example, minimization of the Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

19 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

20 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

21 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

22 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

23 Prognosis horizon 1749- 2008 1848- 2008 1908- 2008 115.815.417.4 226.723.827.7 333.329.433.2 436.431.034.2 The standard deviations for the 1848-2008 series are the smallest ones, unfortunatelly the deviations in the solar activity maxima during this period are greater than the ones for the 1749-2008 series. Standard deviations Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

24 For an AR(2) t=1,…,n h: horizon prognosis Forecast (ex-ante-prognosis) Prognosis of the future value, based on the last and the next to last series value, and so on Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

25 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

26 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

27 I would like to acknowledge the support of this work by the Ministry of Education, Science and Youth under the DVU01/0120 Contract Acknowledgement Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

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