Journal of Geosciences and Geomatics
ISSN (Print): 2373-6690 ISSN (Online): 2373-6704 Website: Editor-in-chief: Maria TSAKIRI
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Journal of Geosciences and Geomatics. 2017, 5(3), 109-118
DOI: 10.12691/jgg-5-3-2
Open AccessArticle

Performance Evaluation of Multivariate Adaptive Regression Splines (MARS) and Multiple Linear Regression (MLR) for Forward Conversion of Geodetic Coordinates (ϕ, λ, h) to Cartesian Coordinates (X, Y, Z)

M.S. Peprah1, , I.O. Mensah1 and J. A. Akresi1

1Department of Geomatic Engineering, University of Mines and Technology, Ghana

Pub. Date: May 03, 2017

Cite this paper:
M.S. Peprah, I.O. Mensah and J. A. Akresi. Performance Evaluation of Multivariate Adaptive Regression Splines (MARS) and Multiple Linear Regression (MLR) for Forward Conversion of Geodetic Coordinates (ϕ, λ, h) to Cartesian Coordinates (X, Y, Z). Journal of Geosciences and Geomatics. 2017; 5(3):109-118. doi: 10.12691/jgg-5-3-2


In Ghana’s local Geodetic Reference Network, the standard forward transformation equation has played a major role in coordinate transformation between World Geodetic System 1984 (WGS84) and local geodetic datum. Thus, it is an initial step in forward conversion of geodetic coordinates (ϕ, λ, h) to Cartesian coordinates (X, Y, Z) in transformation from global to local datum and vice versa. Several studies in the recent decades have been conducted on converting Cartesian coordinates to geodetic coordinates (reverse procedure) through the utilisation of iterative, approximate, closed form, vector-based and computational intelligence algorithms. However, based on the existing literature covered pertaining to this present study, it was found that the existing knowledge do not fully adhere to the issue of evaluating alternative techniques in the case of the forward conversion. Hence, the aim of this present study was to explore the coordinate conversion performance of the Multivariate Adaptive Regression Splines (MARS) and Multiple Linear Regression (MLR). The performance of each model was assessed based on statistical indicators of Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Bias Error (MBE), Mean Absolute Error (MAE), Standard Deviation (SD), Noise to Signal Ratio (NSR), Correlation Coefficient (R), and Correlation of Determination (R2). The statistical findings revealed that the MARS and MLR offered satisfactory prediction of Cartesian coordinates. However, the MLR compared to MARS showed better stability and more accurate prediction results. From the results of this present study, the main conclusion drawn is that, MLR provides a promising alternative in the forward conversion of geodetic coordinates into Cartesian coordinates. Therefore, the capability of MLR as a powerful tool for solving majority of function approximation problems in mathematical geodesy has been demonstrated in this present study.

Multivariate Adaptive Regression Splines Multiple Linear Regression forward transformation equation coordinate conversion

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[1]  Ziggah, Y. Y., Youjian, H., Yu, X., & Basommi, L. P., “Capability of Artificial Neural Network for forward Conversion of Geodetic Coordinates (Ф, λ, h) to Cartesian Coordinates (X, Y, Z)”, Math Geosci, 48, 687-721, 2016.
[2]  Civicioglu, P., “Transforming geocentric Cartesian coordinates to geodetic coordinates by using differential search algorithm”, Comput Geosci, 46, 229-247, 2012.
[3]  Ligas, M., and Banasik, P., “Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations”, Geod Cartogr, 60(2), 145-159, 2011.
[4]  Shu, C., and Li, F., “An iterative algorithm to compute geodetic coordinates”, Comput Geosci, 36, 1145-1149, 2010.
[5]  Zhu, J., “Conversion of earth-centered earth-fixed coordinates to geodetic coordinates”, IEEE Trans Aerosp Electron Syst, 30(3), 957-961, 1994.
[6]  Vanicek, P., and Steeves, R. R., “Transformation of coordinates between two horizontal geodetic datums”, J Geod, 70, 740-745, 1996.
[7]  Cai, G., Chen, B. M. and Lee, T. H., Unmanned rotorcraft systems, Advances in industrial control, Springer-Verlag London Limited, 2011, 23-34.
[8]  Andrei, O. C., “3D affine coordinate transformations”, Masters of Science Thesis in Geodesy, No. 3091 TRITA-GIT EX 06-004, School of Architecture and the Built Environment, Royal Institute of Technology (KTH), 100 44 Stockholm, 2006, 1-4.
[9]  Tierra, A., Dalazoana, R., and De Freitas, S., “Using an artificial neural network to improve the transformation of coordinates between classical geodetic reference frames”, Comput Geosci, 34, 181-189, 2008.
[10]  Ge, Y., Yuan, Y., and Jia, N., “More efficient methods among commonly used robust estimation methods for GPS coordinate transformation”, Surv Rev, 45(330), 229-234, 2013.
[11]  Pan, G., Zhou, Y., Sun, H., and Guo, W., “Linear observation based total least squares”, Surv Rev, 47(340), 18-27, 2015.
[12]  Solomon, M., “Determination of transformation parameters for Montserrado County, Republic of Liberia”, Masters Thesis, Faculty of Civil and Geomatic Engineering, College of Engineering, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana, 2013.
[13]  Zeng, H.E., “Geodetic datum transformation and inverse transformation”, Appl Mech Mater, 501(504), 2154-2157, 2014.
[14]  Zeng, H. E., “Analytical algorithm of weighted 3D datum transformation using the constraint of orthonormal matrix”, Earth Planets Space, 67, 1-10, 2015.
[15]  Ziggah, Y. Y., Youjian, H., Odutola, A. C., and Fan, D.L., “Determination of GPS coordinate transformation parameters of geodetic data between reference datums: a case study of Ghana geodetic reference network”, Int J Eng Sci Res Technol, 2(4), 2277-9655, 2013.
[16]  Annan, R. F., Ziggah, Y. Y., Ayer, J., Odutola, C. A., “A Hybridized Centroid Technique for 3D Molodensky-Badekas Coordinate Transformation in the Ghana Reference Network using Total Least Squares Approach”, South African Journal of Geomatics, 5(3), 269-284, 2016.
[17]  Hoar, G. J., “Satellite surveying”, Magnavox Advanced Products and Systems Company, 2829 Maricopa Street. Torrance, California, 1982, 1-3.
[18]  Leick, A., “GPS satellite surveying”, Wiley, Hoboken, NJ, 2004, 1-5.
[19]  Schofield, W., “Engineering surveying: theory and examination problems for students”, 5th edn. Butterworth-Heinemann, Linacre House, Jordan Hill, Oxford OX2 8DP, UK, 2001.
[20]  Sickle, J. V., “Basic GIS coordinates”, 2nd edn. CRC Press, Taylor and Francis Group, New York, 2010.
[21]  Gullu, M., “Coordinate transformation by radial basis function neural network”, Sci Res Essays, 5(20), 3141-3146, 2010.
[22]  Gullu, M., Yilmaz, M., Yilmaz, I., and Turgut, B., “Datum transformation by artificial neural networks for geographic information systems applications”, In: International symposium on environmental protection and planning: geographic information systems (GIS) and remote sensing (RS) applications (ISEPP), Izmir-Turkey, 2011, 13-19.
[23]  Lin, L. S., and Wang, Y. J., “A study on cadastral coordinate transformation using artificial neural network”, In: Proceedings of the 27th Asian conference on remote sensing, Ulaanbaatar, Mongolia, 2006, 1-5.
[24]  Mihalache, R. M., “Coordinate transformation for integrating map information in the new geocentric European system using artificial neural networks”, GeoCAD, 2012, 1-9.
[25]  Tierra, A. R., De Freitas, S. R. C., and Guevara, P. M., “Using an artificial neural network to transformation of coordinates from PSAD56 to SIRGAS95”, Geodetic reference frames, international association of geodesy symposia, Springer, 134, 173-178, 2009.
[26]  Tierra, A., and Romero, R., “Planes coordinates transformation between PSAD56 to SIRGAS using a multilayer artificial neural network”, Geod Cartogr, 63(2)199-209, 2014.
[27]  Turgut, B., “A back-propagation artificial neural network approach for three-dimensional coordinate transformation”, Sci Res Essays, 5(21), 3330-3335, 2010.
[28]  Yilmaz, I., Gullu, M., “Georeferencing of historical maps using back propagation artificial neural network”, Exp Tech, 36(5), 15-19, 2012.
[29]  Zaletnyik, P., “Coordinate transformation with neural networks and with polynomials in Hungary”, International Symposium on Modern Technologies, Education and Professional Practice in Geodesy and Related Fields, Sofia, Bulgaria, 2004, 471-479.
[30]  Lee, T. S., and Chen, I. F., “A two-stage hybrid credit scoring model using artificial neural networks and multivariate adaptive regression splines”, Expert Syst Appl, 28, 743-752, 2005.
[31]  Samui, P., “Multivariate Adaptive Regression Spline (MARS) for prediction of Elastic Modulus of jointed Rock Mass”, Geotech Geol Eng, 31, 249-253, 2013.
[32]  Feltens, J., “Vector methods to compute azimuth, elevation, ellipsoidal normal, and the Cartesian (X, Y, Z) to geodetic (ϕ, λ, h) transformation”, J Geod, 82(8), 493-504, 2007.
[33]  Feltens, J., “Vector method to compute the Cartesian (X, Y, Z) to geodetic (ϕ, λ, h) transformation on a triaxial ellipsoid”, J Geod, 83(2), 129-137, 2009.
[34]  Fok, S. H., and Iz, H. B., “A comparative analysis of the performance of iterative and non-iterative solutions to the Cartesian to geodetic coordinate transformation”, J Geospatial Eng, 5(2), 61-74, 2003.
[35]  Gerdan, G. P., and Deakin, R. E., “Transforming Cartesian Coordinates (X, Y, Z) to Geographic Coordinates (ϕ, λ, h)”, Aust Surv, 44(1), 53-63, 1999.
[36]  Shu, C., and Li, F., “An iterative algorithm to compute geodetic coordinates”, Comput Geosci, 36, 1145-1149, 2010.
[37]  Friedman, J. H., “Multivariate adaptive regression splines”, Annals Statistics, 19, 1-67, 1991.
[38]  Leathwick, J. R, Rowe, D., Richardson, J., Elith, J., and Hastie, T., “Using multivariate adaptive regression splines to predict the distributions of New Zealand’s freshwater diadromous fish”, Freshw Biol, 50, 2034-2051, 2005.
[39]  Alreja, J., Parab, S., Mathur, S., and Samui, P., “Estimating hysteretic energy demand in steel moment resisting frames using Multivariate Adaptive Regression Spline and Least Square Support Vector Machine”, Ains Shams Engineering Journal, 2015, 1-7.
[40]  Lall, U., Sangoyomi, T., Abarbanel, H. D. I., “Nonlinear dynamics of the Great Salt Lake: nonparametric short term forecasting”, Water Resour Res, 32, 975-985, 1996.
[41]  Attoh-Okine, N. O., Mensah, S., Nawaiseh, M., “A new technique for using multivariate adaptive regression splines (MARS) in pavement roughness prediction”, Proc ICE Trans, 156(1), 51-55, 2003.
[42]  Attoh-Okine, N. O., Cooger, K., Mensah, S., “Multivariate Adaptive Regression (MARS) and hinged hyperplanes (HHP) for doweled pavement performance modelling”, Constr Build Mater, 23(9), 3020-3023, 2009.
[43]  Wang, L. J., Guo, M., Sawada, K., Lin, J., and Zhang, L., “Landslide Susceptibility mapping in Mizunami city, Japan: A comparison between logistic regression, bivariate statistical analysis and multivariate adaptive regression spline models”, Journal of Catena, 135, 271-282, 2015.
[44]  Kisi, O., and Parmar, K. S., “Application of Least Square Support Vector Machine and Multivariate Adaptive Regression Spline Models in Long Term Prediction of River Water Pollution”, Journal of Hydrology, 2015, 1-28, Accessed: February 10, 2017.
[45]  Samui, P. and Kim, D., “Modelling of reservoir-induced earthquakes: a multivariate adaptive regression spline”, Journal of Geophysics and Engineering, 9, 494-497, 2012.
[46]  Laurin, G. V., Puletti, N., Chen, Q., Piermaria, C., Papale, D., and Valentini, R., “Above ground biomass and tree species richness estimation with airborne Lidar in tropical Ghana forest”, International Journal of applied Earth Observation and Geoinformation, 52, 371-379, 2016.
[47]  Durmaz, M. and Karslioglu, M. O., “Non-parametric regional VTEC modelling with Multivariate Adaptive Regression B-Splines”, Advances in Space Research, 48, 1523-1530, 2011.
[48]  Durmaz, M., Karslioglu, M. O., and Nohutcu, M., “Regional VTEC modelling with multivariate adaptive regression splines”, Advances in Space Research. 46, 180-189, 2010.
[49]  Chen, M., Tompson, A. F. B., Mellors, R. S., Ramirez, A. L., Dyer, K. M., Yang, X., and Wagoner, J. L., “An efficient Bayesian inversion of a geothermal prospect using a multivariate adaptive regression spline method”, Journal of Applied Energy, 136, 619-627, 2014.
[50]  Dawod, G. M., Mirza, N. M., Al-Ghamdi, A. K., “Simple precise coordinate transformations for geomatics applications in Makkah metropolitan area, Saudi-Arabia”, Bridging the gap between cultures FIG working week, Marrakech Morocco, 2010, 18-22.
[51]  Featherstone, W. E., “A comparison of existing co-ordinate transformation models and parameters in Australia”, Cartogr, 26(1), 13-26, 1997.
[52]  Ziggah, Y. Y., Hu, Y., and Odutola, A. C., “Regression models for 2-dimensional Cartesian coordinates prediction: a case study at University of Mines and Technology (UMaT), Tarkwa-Ghana”, Int J Comput Sci Eng Surv, 3(6), 61-79, 2012.
[53]  Odutola, A. C., Beiping, W., and Ziggah, Y. Y., “Testing simple regression model for coordinate transformation by comparing its predictive result for two regions”, Acad Res Int, 4(6), 540-550, 2013.
[54]  Forson, K. I., “Design of distribution network for University of Mines and Technology”, Unpublished BSc Project Report, University of Mines and Technology, Tarkwa, Ghana, 2006, 1-10.
[55]  Seidu, M., “GIS as a Tool in Water Monitoring for Public Health and Safety Management”, Unpublished BSc Report, University of Mines and Technology (UMaT), Tarkwa, Ghana, 2004, pp. 1-10.
[56]  Dreiseitl, S., and Ohno-Machado, L., “Logistic regression and artificial neural network classification models: a methodology review”, J Biomed Inf, 35(5-6), 352-359, 2002.
[57]  Ismail, S., Shabri, A., and Samsudin, R., “A hybrid model of self-organizing maps and least square support vector machine for river flow forecasting”, Hydrol Earth Syst Sci, 16, 4417-4433, 2012.
[58]  Yakubu, I., and Kumi-Boateng, B., “Control position fix using single frequency global positioning system receiver technique-a case study”, Res J Environ Earth Sci, 3(1), 32-37, 2011.
[59]  Konaté, A. A., Pan, H., Khan, N., and Ziggah, Y. Y., “Prediction of porosity in crystalline rocks using artificial neural networks: an example from the Chinese continental scientific drilling main hole”, Stud Geophys Geod, 59(1), 113-136, 2015.
[60]  Zabihi, M., Pourghasemi, H. R., Pourtjhi, Z. S., and Behzadfar, M., “GIS-based multivariate adaptive regression spline and random forest models for groundwater potential mapping in Iran”, Environ Earth Sci, 75(665), 646-665, 2016.
[61]  Samui, P. and Kothari, D. P., “A Multivariate Adaptive Regression Spline Approach for Prediction of Maximum Shear Modulus (Gmax) and Minimum Damping Ratio (£min), Engineering Journal, 16(5), 1-10, 2012.
[62]  Craven, P. and Wahba, G., “Smoothing noisy data with spline function: estimating the correct degree of smoothing by the method of generalized cross-validation”, Numer Math, 31, 317-403, 1979.