Forecast for the solar activity based on the autoregressive desciption of the sunspot number time series R. Werner Solar Terrestrial Influences Institute - BAS
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
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
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
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
Pesnell, Solar Phys. (2008) 252: Solar activity predictions of the R 24 Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
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
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
Yule, 1927 Stationarity? Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
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
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
AR(2)-model Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
AR(2)-model a t error term: white noise φ i have to be determined by the Yule-Walker equations YuleIn this work φ1φ φ2φ Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
φ1 φ φ2 φ φ3 φ φ4 φ φ5 φ φ6 φ φ7 φ φ8 φ φ9 φ YuleIn this work φ1φ φ2φ AR(9) model Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
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
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
Prognosis horizon The standard deviations for the series are the smallest ones, unfortunatelly the deviations in the solar activity maxima during this period are greater than the ones for the series. Standard deviations Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
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
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010
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
2010