# Graph-Constrained Estimation Calculate coefficient estimates of Grace based on methods described in Li and Li (2008). ```r grace(Y, X, L, lambda.L, lambda.1=0, lambda.2=0, normalize.L=FALSE, K=10, verbose=FALSE) ``` ## Arguments - `Y`: outcome vector. - `X`: matrix of predictors. - `L`: penalty weight matrix L. - `lambda.L`: tuning parameter value for the penalty induced by the L matrix (see details). If a sequence of lambda.L values is supplied, K-fold cross-validation is performed. - `lambda.1`: tuning parameter value for the lasso penalty (see details). If a sequence of lambda.1 values is supplied, K-fold cross-validation is performed. - `lambda.2`: tuning parameter value for the ridge penalty (see details). If a sequence of lambda.2 values is supplied, K-fold cross-validation is performed. - `normalize.L`: whether the penalty weight matrix L should be normalized. - `K`: number of folds in cross-validation. - `verbose`: whether computation progress should be printed. ## Details The Grace estimator is defined as $$ (\hat\alpha, \hat\beta) = \arg\min_{\alpha, \beta}{\|Y-\alpha 1 -X\beta\|_2^2+lambda.L(\beta^T L\beta)+lambda.1\|\beta\|_1+lambda.2\|\beta\|_2^2} $$ In the formulation, L is the penalty weight matrix. Tuning parameters lambda.L, lambda.1 and lambda.2 may be chosen by cross-validation. In practice, X and Y are standardized and centered, respectively, before estimating $\hat\beta$. The resulting estimate is then rescaled back into the original scale. Note that the intercept $\hat\alpha$ is not penalized. The Grace estimator could be considered as a generalized elastic net estimator (Zou and Hastie, 2005). It penalizes the regression coefficient towards the space spanned by eigenvectors of L with the smallest eigenvalues. Therefore, if L is informative in the sense that $L\beta$ is small, then the Grace estimator could be less biased than the elastic net. ## Returns An R `list' with elements: - **intercept**: fitted intercept. - **beta**: fitted regression coefficients. ## References Zou, H., and Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B, 67, 301-320. Li, C., and Li, H. (2008). Network-constrained regularization and variable selection for analysis of genomic data. Bioinformatics, 24, 1175-1182. ## Author(s) Sen Zhao ## Examples ```r set.seed(120) n <- 100 p <- 200 L <- matrix(0, nrow = p, ncol = p) for(i in 1:10){ L[((i - 1) * p / 10 + 1), ((i - 1) * p / 10 + 1):(i * (p / 10))] <- -1 } diag(L) <- 0 ind <- lower.tri(L, diag = FALSE) L[ind] <- t(L)[ind] diag(L) <- -rowSums(L) beta <- c(rep(1, 10), rep(0, p - 10)) Sigma <- solve(L + 0.1 * diag(p)) sigma.error <- sqrt(t(beta) %*% Sigma %*% beta) / 2 X <- mvrnorm(n, mu = rep(0, p), Sigma = Sigma) Y <- c(X %*% beta + rnorm(n, sd = sigma.error)) grace(Y, X, L, lambda.L = c(0.08, 0.12), lambda.2 = c(0.08, 0.12)) ```

grace function