Statistical application of barycentric rational interpolants: an alternative to splines

Spline curves, originally developed by numerical analysts for interpolation, are widely used in statistical work, mainly as regression splines and smoothing splines. Barycentric rational interpolants have recently been developed by numerical analysts, but have yet seen very few statistical applications. We give the necesssary information to enable the reader to use barycentric rational interpolants, including a suggestion for a Bayesian prior distribution, and explore the possible statistical use of barycentric interpolants as an alternative to splines. We give the all the necessary formulae, compare the numerical accuracy to splines for some Monte-Carlo datasets, and apply both regression splines and barycentric interpolants to two real datasets. We also discuss the application of these interpolants to data smoothing, where smoothing splines would normally be used, and exemplify the use of smoothing interpolants with another real dataset. Our conclusion is that barycentric interpolants are as accurate as splines, and no more difficult to understand and program. They offer a viable alternative methodology.

[1]  Frank E. Harrell,et al.  Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis , 2001 .

[2]  H. Akaike.  A new look at the statistical model identification , 1974 .

[3]  S. Thompson,et al.  Quantifying heterogeneity in a meta‐analysis , 2002, Statistics in medicine.

[4]  Riaz A. Usmani,et al.  Inversion of a tridiagonal jacobi matrix , 1994 .

[5]  Jack Bowden,et al.  A sensitivity analysis framework for the treatment effect measure used in the meta-analysis of comparative binary data from randomised controlled trials , 2012, Statistics in medicine.

[6]  E. Steyerberg,et al.  [Regression modeling strategies]. , 2011, Revista espanola de cardiologia.

[7]  Kai Hormann,et al.  Barycentric rational interpolation with no poles and high rates of approximation , 2007, Numerische Mathematik.

[8]  Jean-Paul Berrut,et al.  Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation , 2007 .

[9]  Jean-Paul Berrut,et al.  Linear Rational Finite Differences from Derivatives of Barycentric Rational Interpolants , 2012, SIAM J. Numer. Anal..

[10]  Kai Hormann,et al.  On the Lebesgue constant of barycentric rational interpolation at equidistant nodes , 2012, Numerische Mathematik.

[11]  W. J. Taylor.  Method of Lagrangian curvilinear interpolation , 1945 .

[12]  Jean-Paul Berrut,et al.  Linear barycentric rational quadrature , 2012 .

[13]  A. F. Möbius,et al.  Der barycentrische Calcul : ein neues Hülfsmittel zur analytischen Behandlung der Geometrie , 1827 .

[14]  DeAngelis,et al.  Multipole strength in 12C from the (e,e' alpha ) reaction for momentum transfers up to 0.61 fm-1. , 1995, Physical review. C, Nuclear physics.

[15]  Paul H. C. Eilers,et al.  Flexible smoothing with B-splines and penalties , 1996 .

[16]  P. Royston,et al.  Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling. , 1994 .

[17]  B. Ripley,et al.  Semiparametric Regression: Preface , 2003 .

[18]  Ian G. McHale,et al.  How good is Tiger Woods , 2012 .

[19]  Jean-Paul Berrut,et al.  Rational functions for guaranteed and experimentally well-conditioned global interpolation , 1988 .

[20]  Stefan Güttel,et al.  Convergence of Linear Barycentric Rational Interpolation for Analytic Functions , 2012, SIAM J. Numer. Anal..

[21]  Georges Klein.  An Extension of the Floater-Hormann Family of Barycentric Rational Interpolants , 2013, Math. Comput..

[22]  William H. Press,et al.  Numerical recipes , 1990 .

[23]  I. W. Wright.  Splines in Statistics , 1983 .

[24]  Lloyd N. Trefethen,et al.  Barycentric Lagrange Interpolation , 2004, SIAM Rev..

[25]  B. Bushman,et al.  Egos inflating over time: a cross-temporal meta-analysis of the Narcissistic Personality Inventory. , 2008, Journal of personality.

[26]  David Ruppert,et al.  Semiparametric Regression: Author Index , 2003 .

[27]  Dan Jackson,et al.  Inference for meta-analysis with a suspected temporal trend. , 2010, Biometrical journal. Biometrische Zeitschrift.

[28]  Göran Kauermann,et al.  Data-driven selection of the spline dimension in penalized spline regression , 2010 .

[29]  Gerald E. Farin,et al.  Automatic fairing algorithm for B-spline curves , 1990, Comput. Aided Des..

[30]  G. Wahba.  Spline models for observational data , 1990 .

[31]  Brian Mullen,et al.  Sufficiency and stability of evidence for public health interventions using cumulative meta-analysis. , 2006, American journal of public health.