# Nonnegative Garrote Method with Hierarchical Structures `nnGarrote()` implements the nonnegative garrote method, as described in Yuan et al. (2009), for selecting important variables while preserving hierarchical structures. The method begins by obtaining the least squares estimates of the regression parameters under a linear model. These initial estimates are then used in the nonnegative garrote to perform variable selection. This function supports prediction based on the linear model fitted with the selected variables and their nonnegative garrote estimates. Note that this method is suitable only when the number of observations is much larger than the number of variables, ensuring that the least squares estimation remains reliable. ```r nnGarrote(U, y, new_U = NULL, heredity = "weak") ``` ## Arguments - `U`: An $n \times P$ model matrix, where $n$ is the number of data and $P$ is the number of potential variables. The inclusion of potential variables supports only up to second-order interactions. Three-order and higher order interactions are not supported. The colon symbol ":" must be included in the names of a second-order interaction for separating its parent variables. Please see the example for the naming format. - `y`: A vector for the responses. - `new_U`: Optional. A matrix or data frame of the new model matrix for prediction. - `heredity`: Choice of heredity principles: weak or strong. The default is weak. ## Returns If `new_U` is `NULL`, the function returns a vector for the nonnegative garrote estimates of the identified variables. If `new_U` is not `NULL`, the function returns a list with: * `beta_nng`: a vector for the nonnegative garrote estimates of the identified variables. * `pred`: predictions for the output corresponding to `new_U`. ## Examples ```r x1 <- runif(1000) x2 <- runif(1000) x3 <- runif(1000) error <- rnorm(1000) X <- data.frame(x1, x2, x3) U_all <- data.frame(model.matrix(~. + x1:x2 + x1:x3 + x2:x3 + I(x1^2) + I(x2^2) + I(x3^2), X)) colnames(U_all) <- c("X.Intercept.", "x1", "x2", "x3", "x1:x1", "x2:x2", "x3:x3", "x1:x2", "x1:x3", "x2:x3") # ":" is required for detecting the parent variables of a second-order interaction. new_idx <- sample(1:1000, 800) new_U <- U_all[new_idx,] U_idx <- setdiff(1:1000, new_idx) U <- U_all[U_idx,] y_all <- 20*U_all$x1 + 15*U_all$`x1:x1` + 10*U_all$`x1:x2` + error y <- y_all[U_idx] nnGarrote(U, y, new_U) ``` ## References Yuan, M., Joseph, V. R., and Zou H. (2009) "Structured Variable Selection and Estimation," The Annals of Applied Statistics, 3(4):1738–1757.

nnGarrote function