# Ordinal Smoothing Spline Basis and Penalty Generate the smoothing spline basis and penalty matrix for an ordinal spline. This basis and penalty are for an ordered factor. ```r 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`. ## Returns 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. ## References Gu, C. (2013). Smoothing Spline ANOVA Models. 2nd Ed. New York, NY: Springer-Verlag. tools:::Rd_expr_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. tools:::Rd_expr_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.** tools:::Rd_expr_doi("10.4135/9781526421036885885") ## Author(s) Nathaniel E. Helwig ## 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. ## See Also See `nominal` for a basis and penalty for unordered factors. See `polynomial` for a basis and penalty for numeric variables. ## Examples ```r ######***###### 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)) ```

ordinal function