Comparison of the autoregressive and autocovariance prediction results on different stationary time series Wiesław Kosek University of Agriculture in Krakow, Poland Abstract. The advantages and disadvantages of the autoregressive and autocovariance prediction methods are presented using different model time series similar to the observed geophysical ones, e.g. Earth orientation parameters or sea level anomalies data. In the autocovariance prediction method the first predicted value is determined by the principle that the autocovariances estimated from the extended by the first prediction value series coincide as closely as possible with the autocovariances estimated from the given series. In the autoregressive prediction method the autoregressive model is used to estimate the first prediction value which depends on the autoregressive order and coefficients computed from the autocovariance estimate. In both methods the autocovariance estimations of time series must be computed, thus application of them makes sense when these series are stationary. However, the autoregressive prediction is more suitable for less noisy data and can be applied to short time span series. The autocovariance prediction is recommended for longer time series and but unlike autoregressive method can be applied to more noisy data. The autoregressive method can be applied for time series having close frequency oscillations while the autocovariance prediction is not suitable for such data. In case of the autocovariance prediction the problem of estimation of the appropriate forecast amplitude is also discussed. 1/2 European Geosciences Union General Assembly 2015, Vienna | Austria | 12 – 17 April 2015 Autocovariance prediction Autoregressive prediction Results Conclusions

Comparison of the autoregressive and autocovariance prediction results on different stationary time series Wiesław Kosek University of Agriculture in Krakow MODEL DATA 2/2 Autocovariance prediction Autoregressive prediction Results Conclusions PeriodsAmplitudesPhasesNoise MODEL 1 L=3 M=100 20, 30, 50All equal to 1.0All equal to 0.0standard deviations: 0, 3 MODEL 2 L=3 M=100 50, 57, 60All equal to 1.0All equal to 0.0standard deviations: 0, 1, 2, 3 MODEL 3 L=9 M=100 10, 15, 20, 25, 40, 60, 90, 120, 180 All equal to 1.0All equal to 0.0standard deviations: 0, 1, 2, 3, 5 MODEL 4 L=1 M= Random walk computed by integration of white nose with standard deviations equal to 1 o,2 o, 3 o standard deviation: 0.1 MODEL 5 L=2 M= , , 0.5Random walk computed by integration of white nose with standard deviations equal to 2 o standard deviation: 0.03 MODEL 6 L=2 M= , 0.016Random walk computed by integration of white nose with standard deviations equal to 2 o standard deviation:

Autocovariance prediction - complex-valued stationary time series with - prediction - the number of data - biased autocovariance estimate next slide ▼

The biased autocovariances of a complex-valued stationary time series can be expressed by the real-valued auto/cross-covariances of the real and imaginary parts: After the time series is extended by the first prediction point computed by: a new estimation of the autocovariance can be computed using the previous one by the following recursion formula: where and it can be used to compute the next prediction point etc. results

Autoregressive prediction next slide ▼ Autoregressive order: Akaike godness-of-fit criterion: Autoregressive coefficients: where

Uncorrected autocovariance predictions (red) of the deterministic and noise data next slide ▼

Correction to amplitudes of the autocovariance prediction 1.n=0.7×N where N is the total number of data 2.computation of the autocovariance c k of x t time series for k=0,1,…,n-1 3.computation of uncorrected autocovariance predictions x n+m for m=1,2,…,N-n+1 4.computation of the autocovariance c k of prediction time series x n+m for k=0,1,…,m-1 5.computation of the amplitude coefficient β= sqrt[( | c 1 | + | c 2 | +…+ | c 8 | )/( | c 1 | + | c 2 | +…+ | c 8 | )] 6. computation of corrected autocovariance predictions β×x N+L for L=1,2,….M β×x N+L x n+m signal next slide ▼ time series

Autocovariance (red) and autoregressive (green) predictions of the model data [T=20, 30, 50; A=1, 1, 1] next slide ▼

Autocovariance (red) and autoregressive (green) predictions of the model data with close frequencies (T=50,57,60, A=1,1,1, noise std. dev.: sd=0,1,2,3) next slide ▼

Autocovariance predictions (red) of the deterministic model data with close frequencies [T=50, 57, 60; A=1, 1, 1] [sd=0.0] next slide ▼

Autocovariance (red) and autoregressive (green) predictions of the model data with many frequencies T=10, 15, 20, 25, 40, 60, 90, 120, 180; A=1 (all); f=0 (all) (noise sd=0,1,2) next slide ▼

Autocovariance predictions (red) of the model data with big number of frequencies T=10, 15, 20, 25, 40, 60, 90, 120, 180; A=1 (all); f=0 (all) next slide ▼

Autocovariance (red) and autoregressive (green) predictions of the seasonal model data with random walk phase: Random walk computed by integration of white noise (sd=1 o, 2 o, 3 o ) next slide ▼

Autocovariance (red) and autoregressive (green) predictions of the model data with random walk phase. Random walk computed by integration of white noise (sd= 2 o ) next slide ▼

Autocovariance (red) and autoregressive (green) predictions of noise data [sd=1.0] next slide ▼

Conclusions The input time series for computation of autocovariance and autoregressive predictions should be stationary, because both methods need autocovarince estimates that should be functions of time lag only. The autocovariance prediction formulae do not able to estimate the appropriate value of prediction amplitude, so it must be rescaled using constant value of the amplitude coefficient β estimated empirically. The accuracy of the autocovariance predictions depend on the length of time series and noise level in data. The predictions may become unstable and when the length of time series decreases, the noise level is big or the frequencies of oscillations are too close. The autoregressive prediction is not recommended for noisy time series, but it can be applied when oscillation frequencies are close. The autocovariance prediction method can be applied to noisy time series if their length is long enough, but it is not recommended if frequencies of oscillations are close. The autocovariance predictions of noise data are similar to noise with smaller standard deviations and autoregressive predictions are close to zero. next slide

References Barrodale I. and Erickson R. E., 1980, Algorithms for least-squares linear prediction and maximum entropy spectral analysis - Part II: Fortran program, Geophysics, 45, Brzeziński A., 1994, Algorithms for estimating maximum entropy coefficients of the complex valued time series, Allgemeine Vermessungs-Nachrichten, Heft 3/1994, pp , Herbert Wichman Verlag GmbH, Heidelberg. Kosek W., 1993, The Autocovariance Prediction of the Earth Rotation Parameters. Proc. 7th International Symposium ”Geodesy and Physics of the Earth” IAG Symposium No. 112, Potsdam, Germany, Oct. 5-10, H. Montag and Ch. Reigber (eds.), Springer Verlag, Kosek W., 1997, Autocovariance Prediction of Short Period Earth Rotation Parameters, Artificial Satellites, Journal of Planetary Geodesy, 32, Kosek W., 2002, Autocovariance prediction of complex-valued polar motion time series, Advances of Space Research, 30, next slide

Acknowledgments Paper was supported by the Polish Ministry of Science and Education, project UMO-2012/05/B/ST10/02132 under the leadership of Prof. A. Brzeziński.