Sshaped (version 1.1)

Sshaped-package: tools:::Rd_package_title("Sshaped")

Description

Estimation of an S-shaped function and its corresponding inflection point via a least squares approach. A sequential mixed primal-dual bases algorithm is implemented for the fast computation of the estimator. The same algorithm can also be used to solve other shape-restricted regression problems, such as convex regression. See Fraser and Massam (1989) and Feng et al. (2022).

Author

• Oliver Y. Feng

• Yining Chen

• Qiyang Han

• Raymond J. Carroll

• Richard J. Samworth

Details

Consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimisation problem, since the inflection point is unknown. Nevertheless, the estimator may be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation.

In the current version of the package, we use this algorithm to implement the least squares regression estimator under the following shape-restrictions: S-shaped functions, i.e. increasing convex to the left of the inflection point and increasing concave to the right of the inflection point; and increasing and convex functions (as a by-product of the former). The corresponding plot and predict methods are also included. In the future, we plan to also include the estimation of additive S-shaped functions, where the covariates are multivariate for the regression.

References

Fraser, D. A. S. and Massam, H. (1989). A mixed primal-dual bases algorithm for regression under inequality constraints. Application to concave regression. Scandinavian Journal of Statistics, Volume 16, Pages 65-74.

Feng, O. Y., Chen, Y., Han, Q., Carroll, R. J. and Samworth, R. J. (2022). Nonparametric, tuning-free estimation of S-shaped functions. Journal of the Royal Statistical Society: Series B (Statistical Methodology), Volume 84, Issue 4, Pages 1324-1352. <doi:10.1111/rssb.12481>

Examples

# Generate data
set.seed(1)
x <- seq(-1,1,0.005)
y <- sin(x*pi/2) + rnorm(length(x))

# Fit S-shape
output <- sshapedreg(x,y)

# Plot
plot(output)

# prediction at new design points
xnew=rnorm(5)
predict(output,xnew)