ordinal {npreg}

Ordinal Smoothing Spline Basis and Penalty

Description

Generate the smoothing spline basis and penalty matrix for an ordinal spline. This basis and penalty are for an ordered factor.

Usage

basis.ord(x, knots, K = NULL, intercept = FALSE, ridge = FALSE)

penalty.ord(x, K = NULL, xlev = NULL)

Arguments

x

Predictor variable (basis) or spline knots (penalty). Ordered factor or integer vector of length n.

knots

Spline knots. Ordered factor or integer vector of length r.

K

Number of levels of x. If NULL, this argument is defined as K = length(unique(x)).

xlev

Factor levels of x (for penalty). If NULL, the levels are defined as levels(as.ordered(x)).

intercept

If TRUE, the first column of the basis will be a column of ones.

ridge

If TRUE, the basis matrix is post-multiplied by the inverse square root of the penalty matrix. See Note and Examples.

Details

Generates a basis function or penalty matrix used to fit ordinal smoothing splines.

With an intercept included, the basis function matrix has the form

X = [X_0, X_1]

where matrix X_0 is an n by 1 matrix of ones, and X_1 is a matrix of dimension n by r. The X_0 matrix contains the "parametric part" of the basis (i.e., the intercept). The matrix X_1 contains the "nonparametric part" of the basis, which consists of the reproducing kernel function

\rho(x, y) = 1 - (x \vee y) + (1 / 2K) * ( x(x-1) + y(y-1) ) + c

evaluated at all combinations of x and knots. The notation (x \vee y) denotes the maximum of x and y, and the constant is c = (K-1)(2K-1) / (6K).

The penalty matrix consists of the reproducing kernel function

\rho(x, y) = 1 - (x \vee y) + (1 / 2K) * ( x(x-1) + y(y-1) ) + c

evaluated at all combinations of x.

Value

Basis: Matrix of dimension c(length(x), df) where df = length(knots) + intercept.

Penalty: Matrix of dimension c(r, r) where r = length(x) is the number of knots.

Note

If the inputs x and knots are factors, they should have the same levels.

If the inputs x and knots are integers, the knots should be a subset of the input x.

If ridge = TRUE, the penalty matrix is the identity matrix.

Author(s)

Nathaniel E. Helwig <[e-mail]>

References

Gu, C. (2013). Smoothing Spline ANOVA Models. 2nd Ed. New York, NY: Springer-Verlag. doi:10.1007/978-1-4614-5369-7

Helwig, N. E. (2017). Regression with ordered predictors via ordinal smoothing splines. Frontiers in Applied Mathematics and Statistics, 3(15), 1-13. doi:10.3389/fams.2017.00015

Helwig, N. E. (2020). Multiple and Generalized Nonparametric Regression. In P. Atkinson, S. Delamont, A. Cernat, J. W. Sakshaug, & R. A. Williams (Eds.), SAGE Research Methods Foundations. doi:10.4135/9781526421036885885

See Also

See nominal for a basis and penalty for unordered factors.

See polynomial for a basis and penalty for numeric variables.

Examples

######***######   standard parameterization   ######***######

# generate data
set.seed(0)
n <- 101
x <- factor(sort(rep(LETTERS[1:4], length.out = n)))
knots <- LETTERS[1:3]
eta <- 1:4
y <- eta[x] + rnorm(n, sd = 0.5)

# ordinal smoothing spline basis
X <- basis.ord(x, knots, intercept = TRUE)

# ordinal smoothing spline penalty
Q <- penalty.ord(knots, K = 4)

# pad Q with zeros (for intercept)
Q <- rbind(0, cbind(0, Q))

# define smoothing parameter
lambda <- 1e-5

# estimate coefficients
coefs <- solve(crossprod(X) + n * lambda * Q) %*% crossprod(X, y)

# estimate eta
yhat <- X %*% coefs

# check rmse
sqrt(mean((eta[x] - yhat)^2))



######***######   ridge parameterization   ######***######

# generate data
set.seed(0)
n <- 101
x <- factor(sort(rep(LETTERS[1:4], length.out = n)))
knots <- LETTERS[1:3]
eta <- 1:4
y <- eta[x] + rnorm(n, sd = 0.5)

# ordinal smoothing spline basis
X <- basis.ord(x, knots, intercept = TRUE, ridge = TRUE)

# ordinal smoothing spline penalty (ridge)
Q <- diag(rep(c(0, 1), times = c(1, ncol(X) - 1)))

# define smoothing parameter
lambda <- 1e-5

# estimate coefficients
coefs <- solve(crossprod(X) + n * lambda * Q) %*% crossprod(X, y)

# estimate eta
yhat <- X %*% coefs

# check rmse
sqrt(mean((eta[x] - yhat)^2))