# Smoothing Spline Basis for Polynomial Splines Generate the smoothing spline basis matrix for a polynomial spline. ```r ssBasis(x, knots, m=2, d=0, xmin=min(x), xmax=max(x), periodic=FALSE, intercept=FALSE) ``` ## Arguments - `x`: Predictor variable. - `knots`: Spline knots. - `m`: Penalty order. 'm=1' for linear smoothing spline, 'm=2' for cubic, and 'm=3' for quintic. - `d`: Derivative order. 'd=0' for smoothing spline basis, 'd=1' for 1st derivative of basis, and 'd=2' for 2nd derivative of basis. - `xmin`: Minimum value of 'x'. - `xmax`: Maximum value of 'x'. - `periodic`: If `TRUE`, the smoothing spline basis is periodic w.r.t. the interval [`xmin`, `xmax`]. - `intercept`: If `TRUE`, the first column of the basis will be a column of ones. ## Returns - **X**: Spline Basis. - **knots**: Spline knots. - **m**: Penalty order. - **d**: Derivative order. - **xlim**: Inputs `xmin` and `xmax`. - **periodic**: Same as input. - **intercept**: Same as input. ## References Gu, C. (2013). **Smoothing spline ANOVA models, 2nd edition**. New York: Springer. Helwig, N. E. (2013). **Fast and stable smoothing spline analysis of variance models for large samples with applications to electroencephalography data analysis**. Unpublished doctoral dissertation. University of Illinois at Urbana-Champaign. Helwig, N. E. (2017). [Regression with ordered predictors via ordinal smoothing splines](http://dx.doi.org/10.3389/fams.2017.00015). Frontiers in Applied Mathematics and Statistics, 3(15), 1-13. Helwig, N. E. and Ma, P. (2015). Fast and stable multiple smoothing parameter selection in smoothing spline analysis of variance models with large samples. **Journal of Computational and Graphical Statistics, 24**, 715-732. ## Author(s) Nathaniel E. Helwig ## Note Inputs `x` and `knots` should be within the interval [`xmin`, `xmax`]. ## Examples ```r ########## EXAMPLE ########## # define function and its derivatives n <- 500 x <- seq(0, 1, length.out=n) knots <- seq(0, 1, length=20) y <- sin(4 * pi * x) d1y <- 4 * pi * cos(4 * pi * x) d2y <- - (4 * pi)^2 * sin(4 * pi * x) # linear smoothing spline linmat0 <- ssBasis(x, knots, m=1) lincoef <- pinvsm(crossprod(linmat0$X)) %*% crossprod(linmat0$X, y) linyhat <- linmat0$X %*% lincoef linmat1 <- ssBasis(x, knots, m=1, d=1) linyd1 <- linmat1$X %*% lincoef # plot linear smoothing spline results par(mfrow=c(1,2)) plot(x, y, type="l", main="Function") lines(x, linyhat, lty=2, col="red") plot(x, d1y, type="l", main="First Derivative") lines(x, linyd1, lty=2, col="red") # cubic smoothing spline cubmat0 <- ssBasis(x, knots) cubcoef <- pinvsm(crossprod(cubmat0$X)) %*% crossprod(cubmat0$X, y) cubyhat <- cubmat0$X %*% cubcoef cubmat1 <- ssBasis(x, knots, d=1) cubyd1 <- cubmat1$X %*% cubcoef cubmat2 <- ssBasis(x, knots, d=2) cubyd2 <- cubmat2$X %*% cubcoef # plot cubic smoothing spline results par(mfrow=c(1,3)) plot(x, y, type="l", main="Function") lines(x, cubyhat, lty=2, col="red") plot(x, d1y, type="l", main="First Derivative") lines(x, cubyd1, lty=2, col="red") plot(x, d2y, type="l", main="Second Derivative") lines(x, cubyd2, lty=2, col="red") # quintic smoothing spline quimat0 <- ssBasis(x, knots, m=3) quicoef <- pinvsm(crossprod(quimat0$X)) %*% crossprod(quimat0$X, y) quiyhat <- quimat0$X %*% quicoef quimat1 <- ssBasis(x, knots, m=3, d=1) quiyd1 <- quimat1$X %*% quicoef quimat2 <- ssBasis(x, knots, m=3, d=2) quiyd2 <- quimat2$X %*% quicoef # plot quintic smoothing spline results par(mfrow=c(1,3)) plot(x, y, type="l", main="Function") lines(x, quiyhat, lty=2, col="red") plot(x, d1y, type="l", main="First Derivative") lines(x, quiyd1, lty=2, col="red") plot(x, d2y, type="l", main="Second Derivative") lines(x, quiyd2, lty=2, col="red") ```

ssBasis function