S.R. Gámiz-Fortis(1), D. Pozo-Vázquez(2), M.J. Esteban-Parra(1), and Y. Castro-Díez(1) (1)Dept. Applied Physics, University of Granada, E-18071, Granada, Spain. (2)Dept. Physics, University of Jaén, E-23071, Jaén, Spain. University of Granada DATA METHODOLOGY RESULTS OBTAINED BY FOURIER ANALYSIS Ø Winter NAO index (December-January-February): Ø Analysis of the NAO index by the Singular Spectral T = 2-3 T = 4.8 T= 5.8 T = 7.8 T = 23.5 Trend Analysis (SSA) method and the Monte Carlo method (MC). (years) (years) (years) (years) (years) ü Hurrell et al. (1995): Difference of monthly pressure values between Lisbon and Iceland. Period 1865-1997. Ø Parameters: 5-6 NAO-LI 10-12 __ 9-11-13 2-3 7-8 1-4 ü Jones et al. (1997): Difference of monthly pressure values 1. Lagged-covariance matrix: Broomhead and King between Gibraltar and Iceland. Period 1865-1997. method. 14-15 2. Window length: M = 40 3-4 ü Barnston y Livezey (1987): Obtained by a PCA of SLP data. 3. Monte Carlo test basis: data Period: 1951-1997. 4. Number of red noise processes generated by MC: p = NAO-GI 7-10 11-12 8-9 1-2 __ 5-6 10000. 13-14 Ø Normalisation Period : 1951-1980. Ø Forecasting of the NAO index using models ARMA. Periods (in years) corresponding to different oscillatory modes represented by pairs of PCs 4 280 180 260 PC 1 PCs 1 & 2 3 160 240 PCs 2 & 3 Power espectrum for the PCs 3 & 4 220 PCs 5 & 6 2 PC 4 140 200 pairs of oscillatory PCs PC7 180 PCs 5 & 6 120 PC 8 1 obtained for the NAO- 160 PC 9 PCs 7 & 8 100 140 PC10 0 PCs 9&11&13 LISBON index 120 80 PCs 11 & 12 NAO PCs 10 & 12 Periodogram 100 PC 13 & 14 -1 60 periodogram 80 PCs 14 &15 PC15 60 Power espectrum for the 40 -2 40 pairs of oscillatory PCs 20 20 NAO-LISBON -3 0 NAO-GIBRALTAR obtained for the NAO- 0 0 0,1 0,2 0,3 0,4 0,5 0,6 NAO-BARNSTON 0 0,1 0,2 0,3 0,4 0,5 0,6 -4 GIBRALTAR index 1865 1885 1905 1925 1945 1965 1985 Frequency (cycles/year) Frequency (cycles/year) YEARS l FIRST MC TEST RESULTS. NH: Serie = AR(1) 5,5 Eigenvalues NAO-LISBON l FIRST MC TEST RESULTS. NH: Serie = Percentiles 97.5 & 2.5 Ø Significant EOFs at the M = 40 NH = AR(1) 4,5 4,0 AR(1) 97.5 level: 1, 2, 5-6. Eigenvalues NAO-GIBRALTAR 3,5 Percentiles 97.5 & 2.5 Ø Significant EOFs at the 97.5 3,5 3,0 M = 40 NH = AR(1) 2,5 Ø True confidence level to 2,5 level: 1-2, 3-4. 2,0 Power of EOF-k 1,5 reject this AR(1) noise null 1,5 Ø True confidence level to 1,0 0,5 Power of EOF-k hypothesis: 92% 0,5 reject this AR(1) noise null -0,5 ª NAO-LISBON: 0,0 0 5 10 15 20 25 30 35 40 -0,5 4 eigenvalue rank k 5 0,5 5,5 10,5 15,5 20,5 25,5 30,5 35,5 40,5 hypothesis: 91.5% Eigenvalues NAO-LISBON Eigenvalues NAO-GIBRALTAR Percentiles 97.5 & 2.5 eigenvalues rank k ü Percentiles 97.5 & 2.5 M = 40 NH = AR(1) 11 Reconstructed Components. ¨ EOF 1: F @ 0.019 years-1 (T @ 4 3 M = 40 NH = AR(1) 3 ü -1 It appears an oscillation with period 23 2 53y) ¨ EOFs 1-2: F @ 0.132 years (T 2 Power of EOF-k years (but it is not significant). @ 8y) Power of EOF-k - 1 ¨ EOFs 2-3: F @ 0.13years 1 1 ª -1 0 NAO-GIBRALTAR: ¨ @ ([email protected]) 0,0 0,1 0,2 0,3 0,4 0,5 EOFs 3-4: F 0.419 years 0 0,0 0,1 0,2 0,3 0,4 0,5 -1 Frq assoc with EOF-k (years ) ([email protected]) Frq assoc with EOF-k (years-1 ) ¨ EOFs 5-6: F @ 0.42years- ü 14 Reconstructed Components. 1 @ (T 2.4y) ü It appears a significant oscillation with l FOURTH MC TEST RESULTS.(Excluding EOFs 1-2,3-4,5-6,7-9- l FOURTH MC TEST RESULTS. (Excluding EOFs 1-4, 2-3, 5-6, 9-11- period 4.6 years, that it is not related with the 10, 8,11-12,13-14 ) oscillation with period 5.8 years in Lisbon. 13, 10-12) CNH3:Serie = Oscillation (T=8y) + Oscillation (T=2.4y) + Trend CNH3: Serie =Trend (>50y) + Oscillation (T=8y) + Oscillation (T=2.4y) + ª The filtered index has got the same phase (>50y) + Oscillation (T=2.3y) + Oscillation (T=5.6y) + Oscillation Oscillation (T=5.8y) + oscillation (T=2.2y) + AR(1) than the non filtered index for the most cases, (T=4.5y) + Oscillation (T=2.9y)4,0 + AR(1) 3,5 Eigenvalues NAO-GIBRALTAR Percentiles 97.5 & 2.5 5,5 for both series. 3,0 M = 40 CNH3 Eigenvalues NAO-LISBON 2,5 4,5 Percentiles 97.5 & 2.5 2,0 M = 40 CNH3 ª Filtered indices represent very well the 1,5 3,5 1,0 extreme values series, specially in the minimum Power of EOF-k 2,5 0,5 case. 0,0 1,5 Power of EOF-k -0,5 0 5 10 15 20 25 30 35 40 eigenvalue rank k 0,5 ª The using two different and quasi Ø -0,5 independent data sets Þ a major confidence in Significant EOFs at the 97.5 level: 19 0 5 10 15 20 25 30 35 40 eigenvalue rank k the results obtained. Ø True confidence level to reject this composite null Ø Significant EOFs at the 97.5 level: NONE hypothes: 71.4% Ø True confidence level to reject this composite null hypothes: 70% METHOD CNH3 is accepted § NAO-GIBRALTAR MODEL: CNH3 is accepted •Modelling of the winter-annual NAO index § NAO-LISBON MODEL: based in models ARMA. ¯An oscillation of period 7.8 years,(EOFs 1-2). ¯A non linear trend, that contains the variability with •Adjustment period of the model: 1865-1985 ¯Three oscillations of periods 2.4, 2.3 and 2.9 years periods of more than 50 years and indicated by EOFs 1 (EOFs 3-4, 7-9-10 and 13-14 respectively), (QBO). and 4. •Checking period: 1986-1997 ¯A no linear trend, that contains the variability with ¯ An oscillation of period 7.8 years (EOFs 2-3) •Forecasting period: 1998-2010 periods of more than 50 years, indicated by the EOFs 5- ¯Two oscillations of periods 2.4 year (EOFs 5-6) and 2.23 6. years (EOFs 10-12), (QBO). ¯Two oscillations of periods 4.5 and 5.6 years (indicated Y ¯An oscillation of period 5.8 years (EOFs 9-11-13), that NAO LISBON : by the EOFs 11-12 and 8 respectively), (ENSO). can be associated with the influence of the ENSO on the © 5 5 Model: ARMA(6, 2) ¯A red noise process AR(1).. NAO-GIBRALTAR RCs 1 & 4 4 NAO. RCS 5-6 4 RC1-13 ©Akaike information criterion (AIC): 223 RCS 1-14 NAO-LISBON 3 ¯3A red NAO-BARNSTONnoise process AR(1). NAO-BARNSTON © 2 2 Absolute mean error (period 1865-1985):0.49 1 1 ©Explained variance : 50% 0 0 Y -1 -1 NAO GIBRALTAR : -2 -2 ©Model: ARMA(9, 5) -3 -3 1865 1885 1905 1925 1945 1965 1985 1865 1885 1905 1925 1945 1965 1985 Years Years ©AIC: 190 RCs for NAO-GI RCs for NAO-LI ©Absolute mean error (period 1865-1985): 0.38 index index 4 3 ©Explained variance: 45% OBSERVED VALUE LISBON ONE STEP FORECASTING 3 2 1986-2010 FORECASTING 2 1 1 0 0 -1 Index Index CONCLUSION -1 -2 -3 ª OBSERVED VALUES GIBRALTAR -2 Quantitative forecasts can’t be done using ARMA lineal models. ONE STEP FORECASTING -4 FORECASTIN 1986-2010 FORECASTING FORECASTIN -3 ª Qualitative forecasts can be done about the oscillation phase a year -5 1960 1970 1980 1990 2000G 2010 1960 1970 1980 1990 G 2000 2010 Time (years) early. Time (years).