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Analysis of a GPS Network Based on Functional Data Analysis


This paper demonstrates the usefulness of approaching the dynamic study of the precise positioning of a network of permanent global positioning system (GPS) stations through functional data analysis. The displacement data for each GPS station, obtained from observations of the global navigation satellite system, are a discrete sample of the positioning curve. The aim of this paper is to reconstruct the original functions in order to use them as functional data. In the method presented in this paper, the geodetic series are obtained first by processing the GPS data with respect to a reference station. Second, for each station, a cleaning process is applied to eliminate the values considered as outliers, and the missing values are imputed by using a Kalman filter. Finally, the original functions are reconstructed by using smoothing techniques and by evaluating several bases of functions. Moreover, these functions are treated with statistical techniques for functional data. This procedure is applied to the permanent stations of the south of the Iberian peninsula and the north of Africa (SPINA) network. The topocentric series: east, north and up are analysed. In the analysis of the positioning curves, there is observed a synchronized behaviour of the functions in those periods of time with important seismic activity. This behaviour also appears in the analysis of the second principal component of the East and up dimensions. Furthermore, the first two principal components of the East coordinate enable us to make a classification of the stations in the SPINA network. The classification made is consistent with the previous knowledge of the tectonic plates in the studied area.

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  • Besse P, Ramsay J (1986) Principal components analysis of sampled functions. Psychometrika 51:285–311

    Article  Google Scholar 

  • Craven P, Wahba G (1979) Smoothing noisy data with splines functions: estimating the correct degree of smoothing by the method of generalized cross-validation. Numer Math 31:377–403

    Article  Google Scholar 

  • Dach R, Hugentobler U, Walser P (2011) Bernese GPS software version 5.0.

  • Dauxois J, Romain Y (1982) Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J Multivar Anal 12:54–61

    Article  Google Scholar 

  • Febrero-Bande M, de la Fuente MO (2012) Statistical computing in functional data analysis: the r package fda.usc. J Stat Softw 51:1–28

    Article  Google Scholar 

  • Fernández JA, Muniz CD, Nieto PG (2012) Detection of outliers in water quality monitoring samples using functional data analysis in san esteban estuary (northern spain). Sci Total Environ 439:54–61

    Article  Google Scholar 

  • Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Springer, Berlin

    Google Scholar 

  • Gutscher M, Malod J, Rehault J, Contrucci I, Klingelhoefer F, Mendes-Victor L, Spakman W (2002) Evidence for active subduction beneath gibraltar. Geology 30:1071–1074

    Article  Google Scholar 

  • Harris P, Brunsdon C, Charlton M, Juggins S, Clarke A (2014) Multivariate spatial outlier detection using robust geographically weighted methods. Math Geosci 46:1–31

    Article  Google Scholar 

  • Hofer V (2011) Functional methods for classification of different petrographic varieties by means of reflectance spectra. Math Geosci 43:165–181

    Article  Google Scholar 

  • Mancilla F, Stich D, Berrocoso M, Martín R, Morales J, Ros AF, Páez R, Pérez-Pen̄a A (2013) Delamination in the betic range: deep structure, seismicity, and gps motion. Geology 41(3):307–310.

    Article  Google Scholar 

  • Ostini L (2012) Analysis and quality assessment of GNSS derived parameter time series. PhD thesis, University of Bern

  • Quintela-Del-Río A, Ferraty F, Vieu P (2011) Analysis of time of occurrence of earthquakes: a functional data approach. Math Geosci 43:695–719

    Article  Google Scholar 

  • Ramsay J, Silverman B (2005) Functional data analysis. Springer, Berlin

    Book  Google Scholar 

  • Rosado B, Barbero I, Jiménez A, Páez R, Prates G, Fernández-Ros A, Gárate J, Berrocoso M (2017) SPINA region (South of Iberian Peninsula, North of Africa) GNSS geodynamic model. Springer, Berlin.

  • Sancho J, Iglesias C, Pin̄eiro J (2016) Study of water quality in a spanish river based on statistical process control and functional data analysis. Math Geosci 48:163–186

    Article  Google Scholar 

  • Yeh T, Chiung Y, Wu C, Wang C, Zhang K, Chen C (2012) Identifying the relationship between gps data quality and positioning precision: case study on IGS tracking stations. J Surv Eng 138(3):136–142

    Article  Google Scholar 

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Correspondence to Sonia Pérez-Plaza.

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Pérez-Plaza, S., Fernández-Palacín, F., Berrocoso, M. et al. Analysis of a GPS Network Based on Functional Data Analysis. Math Geosci 50, 659–677 (2018).

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