Monotonic Smoothing Splines Fitted by Cross Validation

A practical method for calculating monotonic cubic smoothing splines is given. Linear sufficient conditions for monotonicity are employed, and the spline coefficients are obtained using quadratic programming. The method enables efficient cross-validation estimates of the smoothing parameter to be made and confidence intervals to be calculated for the resulting spline. The results are easy to extend to histogram data.

  • [1]  R. K. Beatson and , H. Wolkowicz, Post-processing piecewise cubics for monotonicity, SIAM J. Numer. Anal., 26 (1989), 480–502 10.1137/0726028 90i:65019 0677.65011 LinkISIGoogle Scholar

  • [2]  C. Deboor, A practical guide to splines, Applied Mathematical Sciences, Vol. 27, Springer-Verlag, New York, 1978xxiv+392 80a:65027 0406.41003 CrossrefGoogle Scholar

  • [3]  Paolo Costantini, Co-monotone interpolating splines of arbitrary degree—a local approach, SIAM J. Sci. Statist. Comput., 8 (1987), 1026–1034 89a:65014 0639.65007 LinkISIGoogle Scholar

  • [4]  Peter Craven and , Grace Wahba, Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation, Numer. Math., 31 (1978/79), 377–403 81g:65018 0377.65007 CrossrefISIGoogle Scholar

  • [5]  R. Delbourgo and , J. A. Gregory, $C\sp{2}$ rational quadratic spline interpolation to monotonic data, IMA J. Numer. Anal., 3 (1983), 141–152 85i:65016 0523.65005 CrossrefISIGoogle Scholar

  • [6]  Siegfried Dietze and , Jochen W. Schmidt, Determination of shape preserving spline interpolants with minimal curvature via dual programs, J. Approx. Theory, 52 (1988), 43–57 10.1016/0021-9045(88)90036-6 89g:65013 0662.41008 CrossrefISIGoogle Scholar

  • [7]  Tommy Elfving and , Lars-Erik Andersson, An algorithm for computing constrained smoothing spline functions, Numer. Math., 52 (1988), 583–595 90c:65021 0622.41006 CrossrefISIGoogle Scholar

  • [8]  Randall L. Eubank, Spline smoothing and nonparametric regression, Statistics: Textbooks and Monographs, Vol. 90, Marcel Dekker Inc., New York, 1988xx+438 89g:62055 0702.62036 Google Scholar

  • [9]  F. N. Fritsch and , R. E. Carlson, Monotone piecewise cubic interpolation, SIAM J. Numer. Anal., 17 (1980), 238–246 10.1137/0717021 81g:65012 0423.65011 LinkISIGoogle Scholar

  • [10]  F. N. Fritsch and , J. Butland, A method for constructing local monotone piecewise cubic interpolants, SIAM J. Sci. Statist. Comput., 5 (1984), 300–304 85h:65022 0577.65003 LinkISIGoogle Scholar

  • [11]  Philip E. Gill, Walter Murray and , Margaret H. Wright, Practical optimization, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1981xvi+401 83d:65195 0503.90062 Google Scholar

  • [12]  John A. Gregory, Shape preserving rational spline interpolationRational approximation and interpolation (Tampa, Fla., 1983), Lecture Notes in Math., Vol. 1105, Springer, Berlin, 1984, 431–441 86j:41007 0555.41006 CrossrefGoogle Scholar

  • [13]  J. A. Gregory and , R. Delbourgo, Piecewise rational quadratic interpolation to monotonic data, IMA J. Numer. Anal., 2 (1982), 123–130 83h:65021 0481.65004 CrossrefISIGoogle Scholar

  • [14]  James M. Hyman, Accurate monotonicity preserving cubic interpolation, SIAM J. Sci. Statist. Comput., 4 (1983), 645–654 85a:65021 0533.65004 LinkISIGoogle Scholar

  • [15]  C. Kelly and , J. Rice, Monotone smoothing with application to dose-response curves and assessment of synergism, Biometrics, 46 (1990), 1071–1085 CrossrefISIGoogle Scholar

  • [16]  P. Lancaster and , K. Salkauskas, Curve and surface fitting, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, UK, 1986xii+280 90g:65018 0649.65012 Google Scholar

  • [17]  David F. McAllister and , John A. Roulier, An algorithm for computing a shape-preserving osculatory quadratic spline, ACM Trans. Math. Software, 7 (1981), 331–347 10.1145/355958.355964 82h:65009 0464.65003 CrossrefISIGoogle Scholar

  • [18]  Rossana Morandi and , Paolo Costantini, Piecewise monotone quadratic histosplines, SIAM J. Sci. Statist. Comput., 10 (1989), 397–406 90a:65026 0671.65008 LinkISIGoogle Scholar

  • [19]  William H. Press, Brian P. Flannery, Saul A. Teukolsky and , William T. Vetterling, Numerical recipes in C, Cambridge University Press, Cambridge, UK, 1988xxii+735 89b:65001a 0661.65001 Google Scholar

  • [20]  J. O. Ramsay, Monotone regression splines in action, Statist. Sci., 3 (1988), 425–461 CrossrefGoogle Scholar

  • [21]  C. H. Reinsch, Smoothing by spline functions, Numer. Math., 10 (1967), 177–183 0161.36203 CrossrefISIGoogle Scholar

  • [22]  M. Sakai and , M. C. Lopez De Silanes, A simple rational spline and its application to monotonic interpolation to monotonic data, Numer. Math., 50 (1986), 171–182 88b:65017 0632.65006 CrossrefISIGoogle Scholar

  • [23]  B. W. Silverman, Some aspects of the spline smoothing approach to nonparametric regression curve fitting, J. Roy. Statist. Soc. Ser. B, 47 (1985), 1–52 87i:62110 0606.62038 Google Scholar

  • [24]  F. Utreras, Smoothing noisy data under monotonicity constraint: existence, characterization and convergence rates, Numer. Math., 47 (1985), 611–625 87g:65022 0606.65006 CrossrefISIGoogle Scholar

  • [25]  Miguel Villalobos and , Grace Wahba, Inequality-constrained multivariate smoothing splines with application to the estimation of posterior probabilities, J. Amer. Statist. Assoc., 82 (1987), 239–248 88e:62106 0614.62047 CrossrefISIGoogle Scholar

  • [26]  Grace Wahba, Smoothing noisy data with spline functions, Numer. Math., 24 (1975), 383–393 53:9587 0299.65008 CrossrefISIGoogle Scholar

  • [27]  Grace Wahba, Bayesian “confidence intervals” for the cross-validated smoothing spline, J. Roy. Statist. Soc. Ser. B, 45 (1983), 133–150 84k:62054 0538.65006 Google Scholar

  • [28]  Grace Wahba, Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990xii+169 91g:62028 0813.62001 LinkGoogle Scholar

  • [29]  Simon N. Wood and , Roger M. Nisbet, Estimation of mortality rates in stage-structured population, Lecture Notes in Biomathematics, Vol. 90, Springer-Verlag, Berlin, 1991vi+101 93h:92031 0743.92028 CrossrefGoogle Scholar

  • [30]  Zheng Yan, Piecewise cubic curve-fitting algorithm, Math. Comp., 49 (1987), 203–213 88j:65037 0633.65012 ISIGoogle Scholar