# Bernstein Basis Functions Basis functions defining a polynomial in Bernstein form ```r Bernstein_basis(var, order = 2, ui = c("none", "increasing", "decreasing", "cyclic", "zerointegral", "positive", "negative", "concave", "convex"), extrapolate = FALSE, log_first = FALSE, fun = NULL) ``` ## Arguments - `var`: a `numeric_var` object - `order`: the order of the polynomial, one defines a linear function - `ui`: a character describing possible constraints - `extrapolate`: logical; if `TRUE`, the polynomial is extrapolated linearily outside `support(var)`. In particular, the second derivative of the polynomial at `support(var)` is constrained to zero. - `log_first`: logical; the polynomial in Bernstein form is defined on the log-scale if `TRUE`. It makes sense to define the support as `c(1, q)`$, ie putting the first basis function of the polynomial on `log(1)`. - `fun`: a list of functions for transforming the input data, typically of length three: the monotone transformation and its first two derivatives. ## Details `Bernstein_basis` returns a function for the evaluation of the basis functions with corresponding `model.matrix` and `predict` methods. ## References Rida T. Farouki (2012), The Bernstein Polynomial Basis: A Centennial Retrospective, **Computer Aided Geometric Design**, 29 (6), 379--419, tools:::Rd_expr_doi("10.1016/j.cagd.2012.03.001") . ## Examples ```r ### set-up basis bb <- Bernstein_basis(numeric_var("x", support = c(0, pi)), order = 3, ui = "increasing") ### generate data + coefficients x <- as.data.frame(mkgrid(bb, n = 100)) cf <- c(1, 2, 2.5, 2.6) ### evaluate basis (in two equivalent ways) bb(x[1:10,,drop = FALSE]) model.matrix(bb, data = x[1:10, ,drop = FALSE]) ### check constraints cnstr <- attr(bb(x[1:10,,drop = FALSE]), "constraint") all(cnstr$ui %*% cf > cnstr$ci) ### evaluate and plot Bernstein polynomial defined by ### basis and coefficients plot(x$x, predict(bb, newdata = x, coef = cf), type = "l") ### evaluate and plot first derivative of ### Bernstein polynomial defined by basis and coefficients plot(x$x, predict(bb, newdata = x, coef = cf, deriv = c(x = 1)), type = "l") ### illustrate constrainted estimation by toy example N <- 100 order <- 10 x <- seq(from = 0, to = pi, length.out = N) y <- rnorm(N, mean = -sin(x) + .5, sd = .5) if (require("quadprog")) { prnt_est <- function(ui) { xv <- numeric_var("x", support = c(0, pi)) xb <- Bernstein_basis(xv, order = 10, ui = ui) X <- model.matrix(xb, data = data.frame(x = x)) uiM <- as(attr(X, "constraint")$ui, "matrix") ci <- attr(X, "constraint")$ci if (all(is.finite(ci))) parm <- solve.QP(crossprod(X), crossprod(X, y), t(uiM), ci)$solution else parm <- coef(lm(y ~ 0 + X)) plot(x, y, main = ui) lines(x, X %*% parm, col = col[ui], lwd = 2) } ui <- eval(formals(Bernstein_basis)$ui) col <- 1:length(ui) names(col) <- ui layout(matrix(1:length(ui), ncol = ceiling(sqrt(length(ui))))) tmp <- sapply(ui, function(x) try(prnt_est(x))) } ```

Bernstein_basis function