Multivariate autoregressive models as tools for UT1-UTC predictions

Transcription

1 Multivariate autoregressive models as tools for UT1-UTC predictions Tomasz Niedzielski 1,2, Wiesław Kosek 1 1 Space Research Centre, Polish Academy of Sciences, Poland 2 Oceanlab, University of Aberdeen, United Kingdom 21st October 2009, Warsaw, Poland

2 Objectives Objectives Data Methods To show the strength of the multivariate autoregressive process (MAR) in modelling and prediction of Earth s rotation rate To test the usefulness of the axial component of the atmospheric angular momentum (AAM χ 3) to predict UT1-UTC by MAR To compare the UT1-UTC predictions derived by MAR technique with forecasts determined by the univariate autoregressive method (AR) To discuss the potential causes of prediction errors

3 Data Objectives Data Methods 1 Length of day without tidal signal ( δ ) Length of day (LOD or ) from to EOPC04 05 and EOPC04 solutions The tidal model (δ ) according to IERS Conventions Axial component of the atmospheric angular momentum (AAM χ 3) Time span from to Ä-äÄ [ms] Time [years] AAM 3[ms] Time [years]

4 Polynomial-harmonic model Objectives Data Methods Polynomial-harmonic model f t (LS; Least Squares) is fit to the time series x t using the least-squares technique f t = s A k sin(ω k t + φ k ) + Bt + γ, k=1 where A 1,..., A s are amplitudes, φ 1,..., φ s are corresponding phases, ω 1,..., ω s correspond to frequencies, B and γ describe a linear trend Application of f t: Calculation of residuals (y t = x t f t) Deterministic prediction (extrapolation of f t)

5 Autoregressive model Objectives Data Methods An autoregressive model of order p (AR(p)): where Z t WN(0, σ 2 ). Y t = φ 1Y t φ py t p + Z t, Order selection - Akaike Information Criterion(AIC): AIC(β) = 2 log L X (β, S X (β)/n) + 2(p + 1), where L X is a likelihood function, β is a vector comprising AR coefficients, S X (β)/n is an estimate of variance, and n is a length of the time series Estimation of AR coefficients φ 1,..., φ p: Yule-Walkera technique (preliminary estimation) and maximum likelihood method (final estimation)

6 Objectives Data Methods Multivariate autoregressive model Multivariate autoregressive model of order p (MAR(p)): Y t = w + A 1Y t A py t p + E, where Y t is a vector of stationary residuals, A r is a matrix of MAR coefficients, E is a white noise vector with mean 0 and a covariance matrix C, w is an intersect term Order selection - Schwarz Bayesian Criterion (SBC): ( SBC(p) = lp 2 1 2p + 1 ) log(n p), n p where l p = log det(n 3p 1))Ĉ, Ĉ is a residual covariance matrix, and n is a length of the time series. Estimation of A 1,..., A p: stepwise least-squares method by Neumaier and Schneider (2001)

7 Prediction procedure Objectives Data Methods (Ä-äÄ) (Ä-äÄ) å(ä-ää) (Ä-äÄ) (Ä-äÄ) å(ä-ää) å(ä-ää)

8 Objectives Data Methods Normal distribution, skewness (S) and kurtosis (K) Normal distribution f (x) = 1 σ exp[ (x 2π µ)2 /2σ 2 ] Skewness = Kurtosis = 1 n 1 n n (x i x) 3 i=1 s 3 n i=1 (x i x) 4 s 4 where x is the mean, s is the standard deviation, and n is the length of time series Remarks Skewness is a measure of the asymmetry of the probability density function Kurtosis is a measure of the flatness of the probability density function The normal (Gaussian) distribution exhibits the zero skewness, and a kurtosis value of 3

9 Predictions of UT1-UTC Potential causes of prediction errors Extrapolation of the polynomial-harmonic LS model Absolute values of differences between UT1-UTC (EOPC04_05) data and their LS prediction Days in the future Starting prediction epoch [years]

10 Predictions of UT1-UTC Potential causes of prediction errors Prediction using LS and AR methods Absolute values of differences between UT1-UTC (EOPC04_05) data and their LS+AR prediction Days in the future Starting prediction epoch [years]

11 Predictions of UT1-UTC Potential causes of prediction errors Prediction using LS and MAR methods Absolute values of differences between UT1-UTC (EOPC04_05) data and their LS+MAR predictions Days in the future Starting prediction epoch [years]

12 Predictions of UT1-UTC Potential causes of prediction errors Prediction using LS and MAR methods Absolute values of differences between UT1-UTC (EOPC04) data and their LS+MAR predictions Days in the future Starting prediction epoch [years]

13 Comparison of prediction errors Predictions of UT1-UTC Potential causes of prediction errors Root mean square error of prediction [ms] LS prediction (EOPC04_05 or EOPC04) LS+AR prediction (EOPC04_05) LS+AR prediction (EOPC04) LS+MAR prediction (EOPC04_05) LS+MAR prediction (EOPC04) Days in the future

14 Predictions of UT1-UTC Potential causes of prediction errors On the probability distribution of δ

15 Predictions of UT1-UTC Potential causes of prediction errors On the probability distribution of UT1-UTC

16 Moment-based analysis Predictions of UT1-UTC Potential causes of prediction errors δ UT1R-TAI S K S K Original EOP data Determination error (1) (1) (1) (1) 1-year residuals (2) 2.89 (2) 2-years residuals (2) 3.03 (2) 3-years residuals (2) 3.02 (2) 4-years residuals (2) 3.08 (2) 5-years residuals (2) 3.17 (2) 6-years residuals (2) 3.31 (2) 7-years residuals (2) 3.26 (2) 8-years residuals (2) 3.13 (2) 9-years residuals (2) 3.04 (2) 10-years residuals (2) 2.99 (2) 15-years residuals years residuals 0.60 (2) 3.91 (2) years residuals 0.32 (2) 3.66 (2) years residuals 0.15 (2) 3.11 (2) years residuals 0.10 (2) 3.27 (2) years residuals 0.30 (2) 3.63 (2) - - (1) Truncated data, , because before data were constant (2) Truncated data, , to avoid misfit of the least squares model

17 The MAR method can be successfully applied for forecasting UT1-UTC The AAM χ 3 is an essential explanatory variable supporting the MAR-based UT1-UTC predictions The MAR method gains the more accurate predictions than the AR technique (particularly if the input comes from EOPC04) The UT1-UTC prediction errors are driven by the El Niño/Southern Oscillation (ENSO) The slight departures from the Gaussian distribution of data may cause the increase of the UT1-UTC prediction errors

18 Thank you for the attention! The research was financed by Polish Ministry of Science and Higher Education through the grant no. N N under leadership of Dr Tomasz Niedzielski at the Space Research Centre of Polish Academy of Sciences