Citation: Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, M. Ganzha, L. Maciaszek, M. Paprzycki (eds). ACSIS, Vol. 11, pages 537–541 (2017)
Abstract. There are many practical applications based on the Least Square Error (LSE) method approximation. It is based on a square error minimization ``on a vertical'' axis. The LSE method is simple and easy also for analytical purposes. However, if data span is large over several magnitudes or non-linear LSM is used, severe numerical instability can be expected. The presented contribution describes a simple method for large span of data LSE computation. It is especially convenient if large span of data are to be processed, when the ``standard'' pseudoinverse matrix is ill conditioned. It is actually based on a LSE solution using orthogonal basis vectors instead of orthonormal basis vectors. The presented approach has been used for a linear regression as well as for approximation using radial basis functions.
- Abatzoglou,T., Mendel,J. 1987. Constrained total least squares, IEEE Conf. Acoust., Speech, Signal Process. (ICASSP’87), Vol. 12, 1485–1488.
- Alciatore,D., Miranda,R. 1995. The best least-square line fit, Graphics Gems V (Ed.Paeth,A.W.), 91-97, Academic Press.
- Amiri-Simkooei,A.R.,Jazaeri,S. 2012.Weighted total least squares formulated by standard least squares theory,J.Geodetic Sci.,2(2):113-124.
- Charpa,S., Canale,R. 1988. Numerical methods for Engineers, McGraw-Hill.
- Chatfield,C. 1970. Statistics for technology, Penguin Book.
- de Groen,P. 1996 An introduction to total least squares, Nieuw Archief voor Wiskunde,Vierde serie, deel 14, 237–253
- DeGroat,R.D., Dowling,E.M. 1993 The data least squares problem and channel equalization. IEEE Trans. Signal Processing, Vol. 41(1), 407–411.
- Golub,G.H., Van Loan,C.F. 1980. An analysis of the total least squares problem. SIAM J. on Numer. Anal., 17, 883–893.
- Jo,S., Kim,S.W. 2005. Consistent normalized least mean square filtering with noisy data matrix. IEEE Trans. Signal Proc.,Vol. 53(6), 2112–2123.
- Kryszig,V. 1983. Advanced engineering mathematics, John Wiley & Sons.
- Lee,S.L. 1994. A note on the total least square fit to coplanar points, Tech.Rep. ORNL-TM-12852, Oak Ridge National Laboratory.
- Levy,D., 2010. Introduction to Numerical Analysis, Univ.of Maryland
- Nievergelt,Y. 1994. Total least squares: State of the Art regression in numerical mathematics, SIAM Review, Vol.36, 258-264
- Nixon,M.S., Aguado,A.S. 2012. Feature extraction & image processing for computer vision, Academic Press.
- Markowsky,I., VanHueffel,S., 2007. Overview of total least square methods, Signal Processing, 87 (10), 2283-2302.
- Pearson,K., 1901. On line and planes of closest fit to system of points in space, Phil.Mag., Vol.2, 559-572
- Skala,V., 2016.Total Least Square Error Computation in E2: A New Simple, Fast and Robust Algorithm, CGI 2016 Proc., ACM, pp.1-4, Greece
- Skala,V., 2016. A new formulation for total Least Square Error method in d-dimensional space with mapping to a parametric line, ICNAAM 2015, AIP Conf. Proc.1738, pp.480106-1 - 480106-4, Greece
- Skala,V., 2017. Least Square Error method approximation and extended cross product using projective representation, ICNAAM 2016 conf., to appear in ICNAAM 2016 proceedings, AIP Press
- Skala,V., 2008. Barycentric Coordinates Computation in Homogeneous Coordinates, Computers & Graphics, Elsevier, , Vol.32, No.1, pp.120-127
- Van Huffel,S., Vandewalle,J 1991. The total least squares problems: computational aspects and analysis. SIAM Publications, Philadelphia PA.
- van Huffel,S., Lemmerling,P. 2002. Total least squares and errors-in-variables modeling: Analysis, algorithms and applications. Dordrecht, The Netherlands: Kluwer Academic Publishers
- Perpendicular regression of a line (download 2015-12-20) http://www.mathpages.com/home/kmath110.htm
- Skala,V., 2017. RBF Interpolation with CSRBF of Large Data Sets, ICCS 2017, Procedia Computer Science, Vol.108, pp. 2433-2437, Elsevier
- Skala,V., 2017. High Dimensional and Large Span Data Least Square Error: Numerical Stability and Conditionality, accepted to ICPAM 2017