# Graph-constrained regression with addition of a small ridge term to handle the non-invertibility of a graph Laplacian matrix Graph-constrained regression with addition of a diagonal matrix multiplied by a predefined (small) scalar to handle the non-invertibility of a graph Laplacian matrix (see: References). Bootstrap confidence intervals computation is available (not set as a default option). ```r riPEERc(Q, y, Z, X = NULL, lambda.2 = 0.001, compute.boot.CI = FALSE, boot.R = 1000, boot.conf = 0.95, boot.set.seed = TRUE, boot.parallel = "multicore", boot.ncpus = 4, verbose = TRUE) ``` ## Arguments - `Q`: graph-originated penalty matrix $(p \times p)$; typically: a graph Laplacian matrix - `y`: response values matrix $(n \times 1)$ - `Z`: design matrix $(n \times p)$ modeled as random effects variables (to be penalized in regression modeling); assumed to be already standarized - `X`: design matrix $(n \times k)$ modeled as fixed effects variables (not to be penalized in regression modeling); should contain colum of 1s if intercept is to be considered in a model - `lambda.2`: (small) scalar value of regularization parameter for diagonal matrix by adding which the `Q` matrix is corrected (note: correction is done **before** $\lambda_Q$ regularization parameter value estimation; in other words: $\lambda_Q$ estimation is done for the corrected `Q` matrix) - `compute.boot.CI`: logical whether or not compute bootstrap confidence intervals for $b$ regression coefficient estimates - `boot.R`: number of bootstrap replications used in bootstrap confidence intervals computation - `boot.conf`: confidence level assumed in bootstrap confidence intervals computation - `boot.set.seed`: logical whether or not set seed in bootstrap confidence intervals computation - `boot.parallel`: value of `parallel` argument in `boot` function in bootstrap confidence intervals computation - `boot.ncpus`: value of `ncpus` argument in `boot` function in bootstrap confidence intervals computation - `verbose`: logical whether or not set verbose mode (print out function execution messages) ## Returns - **b.est**: vector of $b$ coefficient estimates - **beta.est**: vector of $\beta$ coefficient estimates - **lambda.Q**: $\lambda_Q$ regularization parameter value - **lambda.R**: `lambda.Q` * `lambda.2` value - **lambda.2**: `lambda.2` supplied argument value - **boot.CI**: data frame with two columns, `lower` and `upper`, containing, respectively, values of lower and upper bootstrap confidence intervals for $b$ regression coefficient estimates ## Examples ```r set.seed(1234) n <- 200 p1 <- 10 p2 <- 90 p <- p1 + p2 # Define graph adjacency matrix A <- matrix(rep(0, p*p), nrow = p, ncol = p) A[1:p1, 1:p1] <- 1 A[(p1+1):p, (p1+1):p] <- 1 L <- Adj2Lap(A) # Define Q penalty matrix as graph Laplacian matrix normalized) Q <- L2L.normalized(L) # Define Z,X design matrices and aoutcome y Z <- matrix(rnorm(n*p), nrow = n, ncol = p) b.true <- c(rep(1, p1), rep(0, p2)) X <- matrix(rnorm(n*3), nrow = n, ncol = 3) beta.true <- runif(3) intercept <- 0 eta <- intercept + Z %*% b.true + X %*% beta.true R2 <- 0.5 sd.eps <- sqrt(var(eta) * (1 - R2) / R2) error <- rnorm(n, sd = sd.eps) y <- eta + error ## Not run: riPEERc.out <- riPEERc(Q, y, Z, X) plt.df <- data.frame(x = 1:p, y = riPEERc.out$b.est) ggplot(plt.df, aes(x = x, y = y, group = 1)) + geom_line() + labs("b estimates") ## End(Not run) ## Not run: # riPEERc with 0.95 bootstrap confidence intervals computation riPEERc.out <- riPEERc(Q, y, Z, X, compute.boot.CI = TRUE, boot.R = 500) plt.df <- data.frame(x = 1:p, y = riPEERc.out$b.est, lo = riPEERc.out$boot.CI[,1], up = riPEERc.out$boot.CI[,2]) ggplot(plt.df, aes(x = x, y = y, group = 1)) + geom_line() + geom_ribbon(aes(ymin=lo, ymax=up), alpha = 0.3) ## End(Not run) ``` ## References Karas, M., Brzyski, D., Dzemidzic, M., J., Kareken, D.A., Randolph, T.W., Harezlak, J. (2017). Brain connectivity-informed regularization methods for regression. doi: https://doi.org/10.1101/117945

riPEERc function