ssBasis {bigsplines}

Smoothing Spline Basis for Polynomial Splines

Description

Generate the smoothing spline basis matrix for a polynomial spline.

Usage

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.

Value

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.

Note

Inputs x and knots should be within the interval [xmin, xmax].

Author(s)

Nathaniel E. Helwig <[e-mail]>

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 Unpublished doctoral dissertation.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. 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.

Examples

##########   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")