Time Series Prediction

Transcription

1 Sternberg Astronomical Institute Lomonosov Moscow State University Time Series Prediction Leonid Zotov Fulbright Scholar San Juan, Puerto Rico, 8 May 29

2 FROM THE BEGINNING OF THE HISTORY PEOPLE TRIED TO PREDICT FUTURE We remember only past events, not future ones. The Mystery of Time, John Langone Good memory wherewith Nature has endowed us causes everything long past to seem present. Leonardo da Vinci The Past isn't dead. It's not even past. W. Faulkner But I knew, that only that which repeats itself can be grasped by study The future is immanent in the present. Citadelle, Antoine de Saint-Exupery L. Zotov, San Juan, Puerto Rico, 8 May 29

3 PREDICTIVE ABILITY IS ONE OF THE MOST IMPORTANT CHARACTERISTIC OF THE SCIENTIFIC THEORY Weather forecast Climate prediction Future of the Universe L. Zotov, San Juan, Puerto Rico, 8 May 29

4 DETERMINISTIC AND STOCHASTIC COMPONENTS Functional modeling Probabilistic modeling.8 Y-POLE Polynomial trends x,m arcsec year Harmonic trends t,c N.S. Sidorenkov 29 L. Zotov, San Juan, Puerto Rico, 8 May 29

5 MATHEMATICAL AND PHYSICAL MODELING Mathematical approximations Dynamic modeling Least Squares Method u x y Auto Regression with Moving Average Least Squares Collocation Neural networks the flow of cause-effect relationships from the past to the future Kalman filtering L. Zotov, San Juan, Puerto Rico, 8 May 29

6 PREDICTION OF THE ROTATION OF THE EARTH Initial data Series EOP С4 with -day step since 962 г. Series EOP С with.5-year step since 89 г. X-Y POLE Observations from IVS VLBI Laser ranging of Moon and Satellites IGS GPS DORIS.8 UT-UTC.4 sec -.4 X, arcsec Y, arcsec -.8 L. Zotov, San Juan, Puerto Rico, 8 May year

7 ANALYSIS OF THE EARTH ORIENTATION PARAMETERS (EOP) Fourier-spectrogram 2 Wavelet-scalogram X-POLE.. time cycles per year Chandler and annual oscillations TAI-UTC.... E-5 E cycles per year Annual and semiannual modes 8-year harmonic arc sec L. Zotov, San Juan, Puerto Rico, 8 May frequency Singular spectral analysis SSA-decomposition of X-coordinate of the pole Chandler annual trend years

8 TREND MODELING 25 UT-TAI.6 Y-POLE 2.4 sec 5 5 arcsec year year Smooth trend polynomial of the 2-d order LS-based solution L. Zotov, San Juan, Puerto Rico, 8 May 29

9 PERIODIC COMPONENTS MODELING AND PREDICTION.4.4 UT-UTC X-pole.2.2 sec сек arcsec сек дуги MJD MJD Period, years 8.6 9,3 Amplitude, sec,5,4 f c n ( t) = A cos( ω t + ϕ ) i= i i i Period, years,9 Amplitude, arcsec,5,9,5,5,9 Nonlinear LS for phases and frequencies adjustment L. Zotov, San Juan, Puerto Rico, 8 May 29

10 sec.8.4 Burg method Parameters are estimated from the minimization of the sum of squares of the discrepancies between the forward and backward predictions for the time series counts -.4 ACF estimation R AUTOREGRESSION MODELLING xx ( m) N x i = a jxi j + = N biased j = N=5 N m xn+ m n= 2 3 x * n n i Yule-Walker equations solution Parameters are estimated from the first counts of the autocovariance function (ACF) R xx ( m) L. Zotov, San Juan, Puerto Rico, 8 May 29 = N m unbiased N m xn+ m n= year UT-TAI AR model of the 5 order solution with Burg algorithm 2 3 x * n

11 LEAST SQUARES COLLOCATION.3.2 Signal ACF and its forecast N points l = x + x = Hl n Interpolation Filtering Prediction. -. f = Q fl Q ll l N-N points delay N Q xx N-N Q fx Q ext xx = Q Q xx fx N N-N White noise assumption Q nn 2 = σ I Q = Q Q f = Q Q l n ext xx ext ll N L. Zotov, San Juan, Puerto Rico, 8 May 29 nn fx ll

12 NEURAL NETWORK Signal p(k) sequentially comes to the Time Delay Line Amoeba At every iteration vectorial signal pd(k) comes to the neurons of the input layer Neuron with the linear activation function predicts the next value of the signal Neural network is trained with use of the signal of comparisons (taken from the past interval) Levenberg-Marquardt algorithm is used for weights W and bias b tuning L. Zotov, San Juan, Puerto Rico, 8 May 29

13 RESULTS OF EOP PREDICTION CAMPAIGN L. Zotov, San Juan, Puerto Rico, 8 May 29

14 PHYSICAL MODELLING - INPUT EXCITATION RECONSTRUCTION 2 Jefferson-Wilson filter iπfcδt ie iσ Δ = c t χ ( t) m Δt e m Δt σ Δ t+ t c t 2 2 X-POLE.4 Corrective smoothing, regularization χ Excitation reconstructed for the components of SSA-decomposition of X-coordinate of the pole Chandler component annual component ( F( W ) F( m) ) = F reg χ = h u = h (ξ t) u( t dt σ iσ c c hreg ( t, α) = e cos αt α.5. reg reg ) it X-POLE Фильтр Вилсона Регуляризация.2.5 сек дуги arc sec сек дуги годы years годы L. Zotov, San Juan, Puerto Rico, 8 May 29

15 -.2 EXCITATION AND POLE TRAJECTORY FORECAST Excitation forecast Y-POLE.6 Trajectory forecast by Kalman filter Y-POLE сек дуги -.35 сек дуги годы годы L. Zotov, San Juan, Puerto Rico, 8 May 29

16 5-NEURON BRAIN FORECASTS North Atlantic foraminiferal oxygen isotope data -2 Site thousand years 6 Dow Jones Industrial Average Jul-98 Apr- Jan-4 Oct-6 Jul-9 time L. Zotov, San Juan, Puerto Rico, 8 May 29 Jul -

17 Muchas gracias para atencion San Juan, Puerto Rico, 8 May 29