{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:50:45Z","timestamp":1760151045935,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,9]],"date-time":"2022-02-09T00:00:00Z","timestamp":1644364800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Curve fitting discrete data (x, y) with a smooth function is a complex problem when faced with sharply oscillating data or when the data are very large in size. We propose a straightforward method, one that is often overlooked, to fit discrete data (s, ys) with rational functions. This method serves as a solid data fitting choice that proves to be fast and highly accurate. Its novelty lies on scaling positive explanatory data to the interval [0, 1], before solving the associated linear problem Ax=0. A rescaling is performed once the fitting function is derived. Each solution in the null space of A provides a rational fitting function. Amongst them, the best is chosen based on a pointwise error check. This avoids solving an overdetermined nonhomogeneous linear system Ax=b with a badly conditioned and scaled matrix A. With large data, the latter can lack accuracy and be computationally expensive. Furthermore, any linear combination of at least one solution in the basis of the null space produces a new fitting function, which gives the flexibility to choose the best rational function that fits the constraints of specific problems. We tested our method with many economic variables that experienced sharp oscillations owing to the effects of COVID-19-related shocks to the economy. Such data are intrinsically difficult to fit with a smooth function. Deriving such continuous model functions over a desired period is important in the analysis and prediction algorithms of such economic variables. The method can be expanded to model behaviors of interest in other applied sciences, such as electrical engineering, where the method was successfully fitted into network scattering parameter measurements with high accuracy.<\/jats:p>","DOI":"10.3390\/a15020057","type":"journal-article","created":{"date-parts":[[2022,2,9]],"date-time":"2022-02-09T21:19:06Z","timestamp":1644441546000},"page":"57","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Data Fitting with Rational Functions: Scaled Null Space Method with Applications of Fitting Large Scale Shocks on Economic Variables and S-Parameters"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0857-5696","authenticated-orcid":false,"given":"Indrit","family":"Hoxha","sequence":"first","affiliation":[{"name":"School of Business Administration, Pennsylvania State University Harrisburg, Middletown, PA 17057, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0184-1269","authenticated-orcid":false,"given":"Taoufik","family":"Meklachi","sequence":"additional","affiliation":[{"name":"School of Science, Engineering and Technology, Pennsylvania State University Harrisburg, Middletown, PA 17057, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ba\u00f1uelos-Cabral, G., and Gustavsen, B. (2017). Rational Fitting Techniques for the Modeling of Electric Power Components and Systems Using MATLAB Environment, IntechOpen.","DOI":"10.5772\/intechopen.71358"},{"key":"ref_2","unstructured":"Bode, H. (1945). Network Analysis and Feedback Amplifier Design, Van Nostrand."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1109\/TAC.1959.6429401","article-title":"Complex curve fitting","volume":"AC-4","author":"Levy","year":"1959","journal-title":"IRE Trans. Autom. Control"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bj\u00f6rck, A. (1996). Numerical Methods for Least Squares Problems, SIAM. [1st ed.].","DOI":"10.1137\/1.9781611971484"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Dennis, J.E., and Schnabel, R. (1996). Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM. [1st ed.].","DOI":"10.1137\/1.9781611971200"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1109\/TAC.1963.1105517","article-title":"Transfer function synthesis as a ratio of two complex polynomials","volume":"8","author":"Sanathanan","year":"1963","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1109\/61.772353","article-title":"Rational Approximation of frequency domain responses by vector fitting","volume":"14","author":"Gustavsen","year":"1999","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1587","DOI":"10.1109\/TPWRD.2005.860281","article-title":"Improving the Pole Relocating Properties of Vector Fitting","volume":"21","author":"Gustavsen","year":"2006","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1090\/qam\/10666","article-title":"A method for the solution of certain problems in least squares","volume":"2","author":"Levenberg","year":"1944","journal-title":"Q. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/0111030","article-title":"An algorithm for least squares estimation on nonlinear parameters","volume":"11","author":"Marquardt","year":"1963","journal-title":"SIAM J. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1023\/A:1022365107361","article-title":"Accuracy and Stability of the Null Space Method for Solving the Equality Constrained Least Squares Problem","volume":"39","author":"Cox","year":"1999","journal-title":"BIT Numer. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"838","DOI":"10.4169\/amer.math.monthly.119.10.838","article-title":"The Extraordinary SVD","volume":"119","author":"Martin","year":"2012","journal-title":"Am. Math. Mon."},{"key":"ref_13","first-page":"1","article-title":"Prediction Functions","volume":"6","author":"Mathiasen","year":"1978","journal-title":"Scand. J. Stat."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/S0893-6080(98)00116-6","article-title":"On the momentum term in gradient descent learning algorithms","volume":"12","author":"Qian","year":"1999","journal-title":"Neural Netw."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1681","DOI":"10.1111\/j.1540-6261.1996.tb05222.x","article-title":"Momentum strategies","volume":"51","author":"Chan","year":"1996","journal-title":"J. Financ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1111\/1468-0297.00466","article-title":"Methodical madness: Technical analysis and the irrationality of exchange-rate forecasts","volume":"109","author":"Chang","year":"1999","journal-title":"Econ. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.rfe.2008.10.001","article-title":"Profitability of technical stock trading: Has it moved from daily to intraday data?","volume":"18","author":"Schulmeister","year":"2009","journal-title":"Rev. Financ. Econ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1017\/S0022109000001782","article-title":"Stock Market Uncertainty and the Stock-Bond Return Relation","volume":"40","author":"Connolly","year":"2005","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"742","DOI":"10.1093\/rapstu\/raaa008","article-title":"The Unprecedented Stock Market Reaction to COVID-19","volume":"10","author":"Baker","year":"2020","journal-title":"Rev. Asset Pricing Stud."},{"key":"ref_20","unstructured":"(2021, September 03). Federal Reserve Economic Data of Federal Reserve Bank of St. Louis. Available online: https:\/\/fred.stlouisfed.org."},{"key":"ref_21","unstructured":"(2021, December 26). Mathworks. Available online: https:\/\/www.mathworks.com\/help\/rf\/ref\/rationalfit.html?searchHighlight=rationalfit&s_tid=srchtitle_rationalfit_2."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mavaddat, R. (1996). Network Scattering Parameters. Advanced Series in Circuits and Systems, World Scientific Publishing.","DOI":"10.1142\/2791"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zeng, R., and Sinsky, J. (2006, January 11\u201316). Modified Rational Function Modeling Technique for High-Speed Circuits. Proceedings of the 2006 IEEE MTT-S International Microwave Symposium Digest, San Francisco, CA, USA.","DOI":"10.1109\/MWSYM.2006.249816"},{"key":"ref_24","unstructured":"Zielesny, A. (2016). From Curve Fitting to Machine Learning an Illustrative Guide to Scientific Data Analysis and Computational Intelligence, Springer International Publishing. [2nd ed.]."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/2\/57\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:17:10Z","timestamp":1760134630000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/2\/57"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,9]]},"references-count":24,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["a15020057"],"URL":"https:\/\/doi.org\/10.3390\/a15020057","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2022,2,9]]}}}