ANALYSIS OF THE TERRESTRIAL POLE COORDINATES USING REGRESSION DYNAMIC MODELING

Abstract

For processing observations of the terrestrial pole dynamics, the regression dynamic modeling (RDM) approach was used. With the RDM software package the models describing the dynamics of the terrestrial polar coordinates were built. This approach provides accurate combined models of observations which describe to some extent causal and deterministic communication and provide forecast of characteristics. A comparison of observations and predicted values of the terrestrial polar coordinates obtained with the RDM approach and by other researchers is performed. To solve the problem, the expansion of the RDM automated systems was made. The basic version is supplemented by new software modules for the developed technique and geophysical observations features. The RDM software package intended for processing geophysical characteristics contains the modules as follows: 1) A spectral window to transform uneven observations to even ones; 2) Cross-spectral analysis to identify common significant harmonics of two observations; 3) Kalman Filter to eliminate noise from the residues of a model; 4) Fractal analysis to verify the series on trendstability; 5) Set of wavelets; 6) Processing scenarios to build the best on the "external" standard deviation model of (SD) model of Time series (TS) automatically.

Headnote

ABSTRACT

For processing observations of the terrestrial pole dynamics, the regression dynamic modeling (RDM) approach was used. With the RDM software package the models describing the dynamics of the terrestrial polar coordinates were built. This approach provides accurate combined models of observations which describe to some extent causal and deterministic communication and provide forecast of characteristics. A comparison of observations and predicted values of the terrestrial polar coordinates obtained with the RDM approach and by other researchers is performed. To solve the problem, the expansion of the RDM automated systems was made. The basic version is supplemented by new software modules for the developed technique and geophysical observations features. The RDM software package intended for processing geophysical characteristics contains the modules as follows: 1) A spectral window to transform uneven observations to even ones; 2) Cross-spectral analysis to identify common significant harmonics of two observations; 3) Kalman Filter to eliminate noise from the residues of a model; 4) Fractal analysis to verify the series on trendstability; 5) Set of wavelets; 6) Processing scenarios to build the best on the "external" standard deviation model of (SD) model of Time series (TS) automatically.

Keywords: terrestrial pole coordinates, regression dynamic modeling, software package

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

The method of TS processing includes the steps as follows: 1) transformation of the uneven TS to even ones with the method of spectral windows or by data averaging [1]; 2) fractal analysis of TS, trend selection [2]; 3) spectral and wavelet analysis; 4) harmonic components selection, application of the Kalman filter [3]; 5) building Generalized AutoRegressive Conditional Heteroskedasticity (GARCH model); 6) Autoregressive moving-average model (ARMA model) construction [4]; 7) smoothing of residues using the martingale approximation [5]. At each step reducing the "internal" and "external" SD, the significance of the error changes is controlled, and analysis of the TS model quality is performed [6].

SUBJECT OF RESEARCH: MODELING OF TS OF THE EARTH POLE DYNAMICS

The dynamic series of coordinates of the North Pole of the Earth from January 1980 to October 2010 (http://www.iers.org) are investigated [7].

Model the dynamics of the X coordinate of the North Pole

At the first stage of data analysis, within the DRM-approach, the hypothesis on series stationarity is rejected with a probability of 0.95.

The centering of the original series is performed (s = 0,1323, sΔ = 0,14).

According to the results of spectral analysis [8], there is a confirmed presence of harmonic components. To determine the carrier harmonics, the method of stepwise regression is used. 4 significant harmonics with periods of 363, 388, 433, 490 days are distinguished (s = 0,0831, sΔ = 0,0962).

ARMA model (6,0) with s= 0.0002 and sΔ = 0,0094 is built. The diagram for the combined model is given in Fig. 1, its form is demonstrated below.

The final model is represented as the sum of periodic component and the ARMA (6.0) model:

...

where X2(t) are residue after removal of the harmonic component, X3(t) are residue after combined model.

Model the dynamics of the Y coordinate of the North Pole

The tested hypothesis of series stationarity is rejected with a probability of 0.95.

The centering of the original series is performed (s = 0,1306, sΔ = 0,11).

To determine the carrier harmonics, the method of stepwise regression is used [9]. This method selects four significant harmonics with periods of 363, 388, 433, 490 days. The DS of the models are respectively: s = 0,044, sΔ = 0,043.

ARMA (6,0) model is built with s= 0.000135, and with sΔ = 0,0039. The diagram of the complex model is given in Fig. 2. The final model is represented as the sum of the periodic component and the ARMA (6.0) model:

...

DISCUSSIONS & RESULTS: PREDICTION OF THE DYNAMICS OF THE EARTH POLE COORDINATES

Using these models the prediction diagrams for X and Y coordinates of the North Pole of the Earth at 90 and 365 days are built.

Comparison of the obtained predictions with real data for the period from November 1, 2010 to February 2, 2011 (Fig. 3) is made.

Similarly, a comparison of observations and predicted values of the polar coordinates of the Earth obtained by other researchers (http://maia.usno.navy.mil/conv2010) [10] is made. Some predictions for the X coordinate are shown in Fig. 4, for the coordinate Y - in Fig. 5.

CONCLUSION

The deterministic mathematical model allows for a prediction of the studied characteristic value for future moments of time. Attempts to build such models have been made repeatedly, however, their predictive values turn out to be low. The development of statistical methods for time series (TS) modeling allows us to hope for the successful application of statistical models in the geophysical systems in the description of latitude variability in time.

Unlike deterministic models, statistical (regression) ones do not remain constant in structure and parameter values for the entire period of use. After getting the forecast for a step or several steps of discreteness in the future, the model is "updated" according to the current latitude values.

The regression dynamic modeling (RDM - approach) is a special case of the adaptive regression modeling approach (ARM approach) [14]. With its application, a complex TS model is formed, consisting of a set of optimal mathematical structures, each describing the dependence of the "remnants" of its step on time. Let us call such a model of the regression dynamic modeling, keeping in mind that time is the main argument, and the final form of RDM is formed as a result of computational adaptation to the properties of the residuals of one or another step and to violations of the conditions of application of the least squares method (LSM).

The aim of the research was to expand the notion of the variability of geographical latitude based on the application of the DRM approach to one-dimensional time series of its values. At the same time, we hoped to reveal a stable polyharmonic structure that contains, apart from the two main harmonics, other ones that can be explained. An important practical result was the discovery of predictive value of the model.

The methodology and software allow us to construct better models of speed (several orders) and provides more (up to ten times) prediction accuracy compared to standard observations treatment package.

Judging from the forecasts, we can conclude that the models provide high accuracy of approximation and prediction in the range of 60-80 days. Comparison with the work of other researchers of the North Pole dynamics has shown that the proposed models in the application of RDM-approach allows for a more accurate prediction of the coordinate Y while maintaining the accuracy of the coordinate X.

The results obtained in the paper confirm the promise of using the so-called adaptive dynamic regressions, first proposed in [15] and being developed at the present time, for describing changes in latitudes. Their advantages, compared with the traditional approaches to the analysis of time series, in particular, to the analysis of the variability of geographical latitude, are: 1) expansion of the concept of the structure of the mathematical model describing the dynamics, 2) isolation of time-stable harmonics of oscillations, 3) several times increased accuracy of forecasting the changes on a certain time interval forward, which can have practical consequences.

ACKNOWLEDGEMENTS

The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University. We would also like to especially acknowledge the contributions of S.Valeev for the materials were provided for the article. This work was partially supported by scholarship of the President of the Russian Federation to young scientists and post-graduate students number CΠ-3225.2018.3. This work was partially supported by the Russian Foundation for Basic Research, grant nos. 16-32-60071mol_a_dk, 18-32-00895 mol_a and Russian Science Foundation 18-72-10037.

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