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A fast and efficient implementation of qualitatively constrained quantile smoothing splines
Pin Ng
Pin Ng is in The WA Franke College of Business, Northern Arizona University, US E-mail: Pin.Ng{at}nau.edu
Martin Maechler
Martin Maechler is in Seminar für Statistik, ETH Zurich, Switzerland
We implement a fast and efficient algorithm to compute qualitatively constrained smoothing and regression splines for quantile regression, exploiting the sparse structure of the design matrices involved in the method. In a previous implementation, the linear program involved was solved using a simplex-like algorithm for quantile smoothing splines. The current implementation uses the Frisch-Newton algorithm, recently described by Koenker and Ng (2005b). It is a variant of the interior-point algorithm proposed by Portnoy and Koenker (1997), which has been shown to out perform the simplex method in many applications. The current R implementation relies on the R package S parse M of Koenker and Ng (2003) which contains a collection of basic linear algebra routines for sparse matrices to exploit the sparse structure of the matrices involved in the linear program to further speed up computation and save memory usage. A small simulation illustrates the superior performance of the new implementation.
Key Words: interior-point • linear program • nonparametric regression • quantile regression • simplex • smoothing spline
References
- Bartels RH and Conn AR (1980) Linearly constrained discrete L1 problems . ACM Transactions on Mathematical Software, 6(4), 594–608 .[CrossRef][Web of Science]
- Becker RA, Chambers JM and Wilks AR (1988) The New S language. Pacific Grove: Wadsworth and Brooks/Cole .
- Chambers JM (1998) Programming with data; a guide to the S language. New York: Springer-Verlag .
- Chen C (2007) A finite smoothing algorithm for quantile regression . Journal of Computational and Graphical Statistics, 16(1), 136–64 .[CrossRef][Web of Science]
- Chen C and Wei Y (2005) Computational issues for quantile regression . Sankhya: The Indian Journal of Statistics, 67(2), 399–417 .
- Frisch R (1955) The logarithmic potential method of convex programming, Technical report, University Institute of Economics, Oslo, Norway .
- Green PJ and Silverman BW (1994) Nonparametric regression and generalized linear models, number 58 in Monographs on statistics and applied probability. Chapman and Hall .
- Härdle W (1990) Applied nonparametric regression, vol. 19 of Econometric Society monographs, Cambridge, UK: Cambridge University Press .
- Hastie TJ and Tibshirani RJ (1990) Generalized additive models, number 43 in Monographs on statistics and applied probability. London: Chapman and Hall .
- He X and Ng P (1999) COBS: qualitatively constrained smoothing via linear programming . Computational Statistics, 14, 315–37 .[CrossRef][Web of Science]
- He X and Shi P (1994) Convergence rate of B-spline estimators of nonparametric conditional quantile functions . Journal of Non parametric Statistics, 3, 299–308 .
- Ihaka R and Gentleman R (1996) R: a language for data analysis and graphics . Journal of Computational and Graphical Statistics, 5(3), 299–314 .[CrossRef]
- Karmarkar N (1984) A new polynomial time algorithm for linear programming . Combinatorica, 4, 373–95 .[CrossRef][Web of Science]
- Koenker R and Bassett G (1978) Regression quantiles . Econometrica, 46, 33–50 .[CrossRef][Web of Science]
- Koenker R, Ng P and Portnoy S (1994) Quantile smoothing splines . Biometrika, 81, 673–80 .[Abstract/Free Full Text]
- Koenker RW and Mizera I (2004) Penalized triograms: total variation regularization for bivariate smoothing . Journal of the Royal Statistical Society (B), 66, 145–63 .[CrossRef]
- Koenker RW and Ng PT (1996) A remark on Bartels and Conn's linearly constrained, discrete L1 problems . ACM Transactions on Mathematical Software, 22(4), 493–95 .[CrossRef][Web of Science]
- Koenker RW and Ng PT (2003) Sparse M: as parse matrix package for R . Journal of Statistical Software, 8(6), 1–9 .
- Koenker RW and Ng PT (2005a) In equality constrained quantile regression . Sankhya: The Indian Journal of Statistics, 67, 418–40 .
- Koenker RW and Ng PT (2005b) A Frisch-Newton algorithm for sparse quantile regression . Acta Mathematicae Applicatae Sinica (English series), 21, 225–36 .[CrossRef]
- Leung D (2005) Cross-validation in nonparametric regression with outliers . The Annals of Statistics, 33, 2291–310 .[CrossRef]
- Maechler M and Ng PT (2006) COBS—Constrained B-splines (Sparse matrix based). R package version 1.1–3 (http://CRAN.R-Project.org/).
- Mammen E, Marron JS, Turlach BA and Wand MP (2001) A general projection frame work for constrained smoothing . Statistical Science, 16, 232–48 .[CrossRef][Web of Science]
- Mehrotra S (1992) On the implementation of a primal-dual interior point method . SIAM Journal on Optimization, 2, 575–601 .[CrossRef]
- Ng PT (1996) An algorithm for quantile smoothing splines , Computational Statistics & Data Analysis, 22(2), 99–118 .[CrossRef][Web of Science]
- Oh H, Nychka D, Brown T and Charbonneau P (2004) Period analysis of variable stars by robust smoothing . Journal of the Royal Statistical Society, Series C, 53, 15–30 .[CrossRef]
- Portnoy S and Koenker R (1997) The Gaussian hare and the Laplacian tortoise: computability of squared-error vs absolute error estimators (with discussion) . Statistical Science, 12, 279–300 .[CrossRef][Web of Science]
- Wang FT and Scott DW (1994) The L1 method for robust nonparametric regression . Journal of the American Statistical Association, 89, 65–76 .[CrossRef][Web of Science]
Statistical Modelling, Vol. 7, No. 4,
315-328 (2007)
DOI: 10.1177/1471082X0700700403
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