1. (2) Space Research Centre, Polish Academy of Sciences, Warsaw, Poland
Pole coordinates data prediction by combination of least squares extrapolation and double autoregressive forecast
Wiesław Kosek
(1) Environmental Engineering and Land Surveying, University of Agriculture in Krakow, Poland
(2) Space Research Centre, Polish Academy of Sciences, Warsaw, Poland
Pole coordinates data prediction by combination of least squares extrapolation and double autoregressive prediction
EGU General Assembly 2016, EGU

2. Pole coordinates data prediction by combination of least squares extrapolation and double autoregressive forecast
Wiesław Kosek
Contents:
Pole coordinates data and their analysis
LS+AR prediction algorithm
Time series of differences between pole coordinates data and their LS+AR predictions
- statistics
- the FTBPF amplitude spectra
- the NMWT amplitudes
AR predictions of differences between pole coordinates data and their LS+AR predictions
LS+2AR prediction algorithm
DISCUSSION
One of the main problem in Earth rotation studies is computation of the most accurate prediction of pole coordinates data.
As we may expect the differences between predictions of pole coordinate data and their future values increase with the prediction length and mostly depend on starting prediction epochs no meter what prediction method is applied. In this case the applied prediction method is the combination of the least squares extrapolation and autoregressive prediction. The time series of the differences between pole coordinates data and their predictions for particular prediction length can be analyzed and predict to improve the prediction accuracy of pole coordinates data.
Time-frequency amplitudes computed by the normalized wavelet transform of these differences for particular prediction lengths show that there exists remaining signal in them corresponding to the residual prograde Chandler and annual oscillations as well as the signal corresponding to chaotic high frequency oscillations. The signal corresponding to the Chandler and annual frequency band occurs even in prediction differences at few days in the future.
The autoregresive prediction of these differences shows that it is possible to predict them. The autoregressive prediction method applied for the second time is more tuned to lower frequency variations, while the autoregressive prediction applied the first time in the combination of least squares extrapolation and autoregressive prediction is mostly tuned to shorter period variations of pole coordinates data.
EGU General Assembly 2016, EGU

3. *
DATA
x, y from IERS: EOPC04_IAU now ( now), Δt = 1 day,
Long term earth orientation data EOP C01 IAU2000 ( now), Δt = 0.05 years
The following data sets were used in the analysis: the IERS x, y pole coordinates data from now, with 1 day sampling interval. Long term earth orientation data since 1890.

4. *
This presentation is focused on the pole coordinates data prediction, for which errors increase with prediction length. This map show time variable amplitude spectrum of complex-valued pole coordinates data computed by the Fourier transform band pass filter. The Chandler and annual oscillations are the most energetic however there are shorter period oscillations with variable amplitudes. Irregular character of all these oscillations can the cause of increase of polar motion prediction errors with the prediction length.
Time variable amplitude spectrum of complex-valued pole coordinates data computed by the Fourier transform band pass filter

5. *
To find possible causes of increase of EOP prediction errors variable amplitudes and phases of the most energetic oscillation were computed using the combination of complex-demodulation and the Fourier transform band pass filter. These graphs show the amplitudes and phases of the most energetic prograde oscillations in pole coordinates data. The variations can be responsible for increase of polar motion prediction errors.
Amplitudes and phases of the Chandler (green) and Annual (x-blue, y-red) oscillations computed by combination of complex demodulation and the Fourier transform band pass filter

6. *
These graphs show first differences of amplitudes in red and orange and the products of first differences of phases and corresponding to them amplitudes in blue and navy blue for the Chandler, annual and semi-annual oscillations. For oscillation with small amplitude the phase can take any value and to show how variable phases cause irregular variations similar to those created by irregular amplitude variations the first differences of phases were multiplied by the corresponding amplitude.
It is clearly seen than the contribution of the irregular change of the phase of Chandler and annual oscillations has greater impact on prediction errors than the amplitude change of these oscillations. The contribution of the semiannual oscillation to the irregular changes of pole coordinates data is one order less than the contribution of irregular changes of the Chandler and annual oscillations.
First differences of amplitudes (x-red, y-orange) and the products of amplitudes and phase differences (x-navy blue, y-blue) of the Chandler, annual and semi-annual oscillations computed by the CD+FTLPF combination.

7. Prediction of x, y by combination of the LS+AR method x, y*
Prediction of x, y by combination of the LS+AR method
x, y LS model fit
to last 10 years of PM data
x, y
LS residuals
x, y AR LS
LS+AR prediction
of x, y
LS extrapolation
of x, y
AR prediction of x, y
residuals fit to last
850 days of these residuals
DATA PREDICTION
To compute the prediction of x,y pole coordinates data by combination of the least-squares (LS) and autocovariance prediction (AR) first the LS model is fit to the last 10 years of pole coordinates data. The differences between the pole coordinates data and this least-squares model are x,y LS residuals, which are predicted using the autoregressive prediction method adopted to complex-valued time series in which the autoregressive order is computed by Akaike’s goodness of fit criterion. The sum of the autoregressive prediction of the LS residuals and the LS extrapolation model is called the LS+AR prediction
last data point
first prediction point

8. * Standard deviation (SDE) Mean absolute error (MAE) Skewness (SKE)
To analyze the differences between x,y pole coordinates data and their predictions some basic statistics were computed as a function of prediction length. The prediction accuracy can be expressed by standard deviation and mean absolute errors. To check if these prediction differences satisfy normal distribution the skewness and kurtosis were computed. Additionally, the statistical errors of all these statistics were computed to show error bars of these estimates. Skewness is a measure of the asymmetry of the probability distribution and for normal distribution it should be equal to zero. Kurtosis is a measure of the "peakedness" of the probability distribution and If it is equal to 3 then the differences between pole coordinate data and their predictions follow normal distribution. If it is less than 3 then the probability distribution is more flat (platykurtic). If its greater than 3 then the probability distribution is more peaked (leptokurtic).
Kurtosis (KUR)

9. * Standard deviation error Mean absolute error Skewness Kurtosis
As can be noticed the mean absolute error is smaller than the standard deviation error. The skewness oscillates near zero and the kurtosis is close to 3 for prediction lengths greater than 50 days. For smaller prediction lengths the distribution of prediction errors for x and y pole coordinates is more letokurting than normal.
Kurtosis

10. *
The maps show the differences between pole coordinate data and their predictions up to 1 year in the future computed by combination of least-squares extrapolation and autoregressive prediction. These differences increase with prediction length and depend mostly on starting prediction epochs. The prediction errors increased almost every year but the highest prediction differences were noticed in 1982, 2006 and beginning of 2007 as well as in 2012.
Two plots below show time series of these differences for 2 (purple) and 4 (grren) weeks in the future. To detect what oscillations are present in these differences the spectral analysis and time frequency spectral analyses algorithms were applied.
The maps of differences between x,y pole coordinates data and their LS+AR predictions and time series of these differences for 2 (purple) and 4 (green) weeks in the future.

11. *
Two plots show time series of differences between x,y pole coordinate data and their LS+AR predictions for 1 day (blue) and one week (pink) in the future. It is clearly seen that before 1984 the prediction difference have smaller values which is due to smoother pole coordinates data at that time.
The differences between the x,y pole coordinates data and their LS+AR predictions for 1 day (blue) and 1 week (pink) in the future. 11

12. The Fourier Transform Band Pass Filter (FTBPF)*
where:
complex-valued time series
- broadband oscillation with central frequency ω
- parabolic transmittance function
- half of the bandwidth
- sampling interval of data
To analyze the differences between pole coordinates data and their LS+AR predictions the FTBPF method was applied to compute time frequency spectra of these residuals. This method enales computation of the time variable amplitude spectrum.
T – mean period of broadband oscillation
Time variable FTBPF amplitude spectrum: 12

13. *
The maps of time variable amplitude spectra computed by the same method for prediction differences at 1 day, 1 week, 2 and 4 weeks in the future look very similar. They show maxima for prograde oscillations in the Chandler and annual frequency bands. The highest maxima occured in 2006 to beginning of 2007 and in 2011 to beginning of These maxima correspond to time where prediction errors of pole coordinates data were bigger what suggests that the reason of these prediction errors are irregular variations of the Chandler and annual oscillations.
Time variable FTBPF amplitude spectra (λ=0.001) of the differences between the x-iy pole coordinates data and their LS+AR predictions at 1 day as well as 1, 2 and 4 weeks in the future.

14. *
These graph shows the amplitude spectra computed by the Fourier transform band pass filter of time series being the difference between pole coordinates data and their predictions at 2, 4 and 8 weeks in the future. It can be noticed that the spectra are very similar and show maxima for the prograde Chandler and annual oscillations. Prograde character of these oscillations suggests that they were mismodelled in the combination of the least-squares extrapolation and autoregressive prediction. To compute the least squares extrapolation model the Chandler and annual oscillation are treated as they have constant amplitudes and phases. Besides the autoregressive model cannot model these two oscillations properly since it is more tuned to shorter period oscillations.
The mean FTBPF amplitude spectra (λ=0.0003) of the differences between the x-iy pole coordinates data and their LS+AR predictions at 2, 4 and 8 weeks in the future.

15. Normalized Morlet Wavelet Transform (NMWT)*
where translation (or time) parameter and
- dilation (or period) parameter,
- DFT or FFT of complex-valued time series
is the continuous FT of the complex-valued modified Morlet wavelet function,
- the decay parameter which controls the frequency resolution
The instant amplitudes as a function of a frequency can be estimated by the normalized Morlet wavelet transform realized in the frequency domain.
The instantaneous NMWT amplitude: 15

16. *
Maps show the amplitudes of oscillations with periods ranging from 4 to 700 days computed b y the normalized Morlet wavelet transform of the differences between pole coordinates data and their predictions computed by combination of least squares and autoregressive prediction for 1 day, 1,2 and 4 weeks in the future. As can be noticed there are remaining signals in these differences which corresponds to the residual Chandler and annual frequencies which were mismodelled in the prediction model computed by combination of LS extrapolation and autoregressive prediction. In the frequency band corresponding to hight frequency oscillations there is a remaining signal corresponding to chaotic variations with period upo to 200 days.
The NMWT amplitudes as a function of periods T (σ=1) of the differences between the x-iy pole coordinates data and their LS+AR predictions at 1 day as well as 1, 2 and 4 weeks in the future.

17. *
The mean NMWT amplitudes computed in the differences between pole coordinates data and their LS+AR predictions for 1,2, 4 and 8 weeks in the future show the signal corresponding to the residual Chandler and annual oscillations as well as high frequency signal corresponding to chaotic oscillations with periods less than about 200 days.
The mean NMWT amplitudes as a function of periods T (σ=1) of the differences between the x-iy pole coordinates data and their LS+AR predictions at 1, 2, 4 and 8 weeks in the future.

18. *
Graphs show the time series of differences between pole coordinates data and their predictions computed by the combination of the least-squares extrapolation and autoregressive prediction at 14th day in the future. These differences were predicted using autoregressive prediction at different starting prediction epochs to show if the follow the time series of these differences.
100-day autoregressive predictions (red) at different starting prediction epochs of the differences between x-iy pole coordinates data and their LS+AR predictions at 2 weeks in the future (grey). The length of these differences to fit the autoregressive coefficients is equal to 6 years.

19. *
100-day autoregressive predictions (red) at different starting prediction epochs of the differences between x-iy pole coordinates data and their LS+AR predictions at 4 weeks in the future (grey). The length of these differences to fit the autoregressive coefficients is equal to 6 years.

20. *
100-day autoregressive predictions (red) at different starting prediction epochs of the differences between x-iy pole coordinates data and their LS+AR predictions at 8 weeks in the future (grey). The length of these differences to fit the autoregressive coefficients is equal to 6 years.

21. *
Comparison of 100-day autoregressive (red) and autocovariance (blue) predictions at different starting prediction epochs of the differences between x-iy pole coordinates data and their LS+AR predictions at 8 weeks in the future. The lengths of these differences to fit the autoregressive coefficients and to fit the autocovariance prediction model are equal to 6 and 18 years, respectively.

22. *
The standard deviations of the AR
predictions of the differences
between x (blue) ,y (red) pole coordinates data and their LS+AR predictions at 2 (thin line) 4 (dotted line) and 8 (dashed line) weeks in the future.

23. *
Mean AR prediction error of the differences between x,y data
and their LS+AR prediction for 8 weeks in the future is smaller
than mean LS+AR prediction error of x,y for prediction lengths
greater than L=70 days.
The SDE of LS+AR predictions of x (blue) and y (red) pole coordinates data (dashed line). The SDE of AR predictions of time series of the differences between x,y pole coordinates data and their LS+AR predictions at 8 weeks in the future.

24. Prediction of x, y by combination of the LS+2AR method*
x,y pole coordinates
LS model fit to last 10 years of x,y data
x,y LS residuals AR LS
LS+AR prediction
of x, y
LS extrapolation
of x, y
AR prediction of x, y
LS residuals fit to last
850 days of these residuals +
AR prediction of differences
between x,y and their
LS+AR predictions fit to
last 6 years of them
LS+2AR prediction
of x, y
Differences between x,y
and their LS+AR predictions = AR
LS+AR prediction at L day in the future
LS+AR PREDICTIONS
first prediction point
AR prediction of the differences between x,y pole coordinate and their LS+AR predictions at “L”-th day in the future (L>70 days) is weighted according to prediction length and added to each LS+AR prediction point
DATA
LS+2AR PREDICTIONS
last data point

25. *
Comparison of the mean LS+AR and LS+2AR prediction errors (SDE) of x,y pole coordinates data (L=100 days).

26. *
DISCUSSION
The increase of differences values between pole coordinates data and their LS+AR predictions with the prediction length is caused by mismodelling of the irregular Chandler and annual oscillations in the forecast model.
Time frequency analysis by the FTBPF and NMWT of the differences between pole coordinates data and their LS+AR predictions for 1 day as well as 1, 2, 4 and 8 weeks in the future show wideband signal corresponding to the residual Chandler and annual oscillations.
The skewness values of the differences between pole coordinates data and their LS+AR predictions for different prediction lengths are close to zero which means that they follow normal distribution. The kurtosis values of these differences are close to 3 for prediction lengths greater than about 50 days which means that they follow normal distribution and for shorter prediction lengths the probability distribution becomes more leptokurtic than normal.
The standard deviation of the AR prediction of the differences between pole coordinates data and their LS+AR predictions at 1, 2, 4 and 8 weeks in the future decrease with prediction length. These standard deviations decrease faster when the LS+AR prediction lengths become shorter. Thus, the differences of pole coordinate and their LS+AR predictions at 8 weeks in the future are the most optimum to correct the LS+AR predictions of pole coordinates data. The mean prediction errors of the AR prediction of these differences become less than the mean LS+AR prediction errors of pole coordinates data for prediction lengths greater than about 70 days.
The increase of the differences between pole coordinates data and their prediction with the prediction length is caused by mismodelling of the irregular Chandler and annual oscillations in the LS+AR forecast models.
Time frequency analysis of the differences between pole coordinates data and their LS+AR predictions for 1 day and 1, 2 and 4 weeks in the future show residual wideband signal corresponding to the Chandler and annual oscillations.
The skewness and kurtosic values of the differences between pole coordinates data and their predictions for different prediction lengths are close to 0 and 3, respectively which means that they follow normal distribution.

27. 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., 2002, Autocovariance prediction of complex-valued polar motion time series, Advances of Space Research, 30,

28. Acknowledgments *
Paper was supported by the Polish Ministry of Science and Education, project 2012/05/B/ST10/02132 under the leadership of Prof. A. Brzeziński.
Time frequency analysis FTBPF and NMWT algorithms were prepared by W. Kosek and W. Popiński and autoregressive coefficients estimation procedure was adopted to complex-valued time series by A. Brzeziński.
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.
Time frequency analysis FTBPF and NMWT softwares were elaborated by W. Kosek and W. Popiński and autoregressive coefficients estimation procedure was adopted to complex-valued time series by A. Brzeziński.

29. Autoregressive prediction*
Autoregressive coefficients:
Autoregressive order:
Akaike godness-of-fit criterion:
In the autoregressive prediction the method the linear autoregressive model is used in which the current time series value depends on its previous values multiplied by constant autoregressive coefficients. Thus, the prediction points satisfy this autoregressive model in which noise part is replaced by is expected value equal to zero and autoregressive coefficients are replace by their estimates. The autoregressive order is estimated by the goodness of fit criterion where sigma is the estimation of noise variance. The autoregressive coefficients are computed from the autocovariance estimate of time series for each autoregressive order M. The final autoregressive coefficients are computed when the function P(M) is a minimum.
where 29

30. *
The NMWT amplitudes were computed for complex-valued differences between x,y pole coordinates data and their LS+AR predictions for 1 day as well as 1, 2, 4 and 8 weeks in the future. It is clearly seen that these prediction differences contain the signal corresponding to the residual Chandler and annual oscillations. The maxima of these amplitude spectra correspond to time moments in which the differences between pole coordinates data and their LS+AR predictions attained the biggest values. There also appear shorter period variations with periods less than 100 days with chaotic polarization but noticeable amplitudes, especially in prediction differences for 2 and 4 weeks in the future.
The NMWT amplitudes as a function of periods T (σ=1) of the differences between the x-iy pole coordinates data and their LS+AR predictions at 1 day as well as 1, 2, 4 and 8 weeks in the future.