# fit a complex-valued graphical lasso Fit a complex-valued graphical lasso for spectral precision matrix (inverse spectral density matrix) via a complex variable-wise coordinate descent algorithm for classo with covariates update. ```r cglasso( S, D = NULL, type = c("I", "II"), nobs, lambda = NULL, nlambda = 50, lambda.min.ratio = ifelse(nobs < p, 0.01, 1e-04), W.init = NULL, stopping_rule = TRUE, stop_criterion = c("EBIC", "AIC", "RMSE"), maxit = 500, thresh = 1e-04, trace.it = 0, ... ) ``` ## Arguments - `S`: p x p-dimensional symmetric spectral density (or spectral coherence) matrix. S is considered as being computed by average smoothed periodogram (the bandwidth is computed by using the given nobs). - `D`: The p x p-dimensional diagonal matrix with spectral densities as the diagonal entries. Default is `NULL`. If D is not provided, diagonals of S are chosen. - `type`: A logical flag to choose the formulation to solve. Default is `I`. If type is `I`, the algorithm solves CGLASSO-I in the reference, $$ D^{-1/2} \left( \arg\min_{\Theta} \operatorname{Tr} \left[ \hat{R} \hat{\Theta} \right] - \log \det \Theta + \sum_{i \ne j} \left| \Theta_{ij} \right| \right) D^{-1/2} $$ for the given D. If type is `II`, the algorithm solves CGLASSO-II in the reference. It is for each iterative classo with covariate update, the squared-root of scale matrix $ D^{-1/2}$ is multiplied. Please refer to the equation (5.2) in the reference for the details. - `nobs`: Number of observations used in computation of the spectral density matrix S. This quantity is need to compute the Fourier frequency, extended BIC, and bandwidth for the average smoothed periodogram. - `lambda`: A user supplied `lambda` sequence. Typical usage is to have the program compute its own `lambda` sequence based on `nlambda` and `lambda.min.ratio`. Supplying a value of `lambda` overrides this. WARNING: use with care. Avoid supplying a single value for `lambda` - `nlambda`: The number of `lambda` values - default is 50. - `lambda.min.ratio`: Smallest value for `lambda`, as a fraction of `lambda.max`, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero). The default depends on the sample size `nobs` relative to the number of variables `nvars`. If `nobs > p`, the default is `0.0001`, close to zero. If `nobs < p`, the default is `0.01`. - `W.init`: Logical flag whether the initially estimated spectral density matrix is given. Default is `NULL`. - `stopping_rule`: Logical flag if the algorithm is terminated by stopping rule. If the algorithm is early terminated ,not all estimates for initially designated lambdas are explored. - `stop_criterion`: Stopping criterion for early termination. Default is `EBIC` (Extended BIC). Alternatively, `AIC` (AIC) and `RMSE` (root mean squared error between two consecutive estimates) can be used. - `maxit`: Maximum number of iterations of both outer and inner loops. Default 500. - `thresh`: Convergence threshold for coordinate descent. Default is 1e-4. - `trace.it`: If `trace.it=1`, then a progress bar is displayed; - ``...``: Other arguments that can be passed to `cglasso` useful for big models that take a long time to fit. ## Returns An object with class "cglassofit" and "cglasso". - **stop_arr**: Sequence of values of information criterion for a fixed lambda. - **stop_criterion**: Stopping criterion used. - **min_index**: The index for lambda that minimizes the value of the information criterion. - **lambda_grid**: Sequence of lambdas used. - **Theta_list**: Estimated inverse spectral matrix for each fixed lambda. It is provided in the list. - **type**: Type of the formulation used, either CGALSSO-I or CGLASSO-II. - **scale**: Whether the spectral density matrix (covariance) or spectral coherence (coherence) is given. - **D**: Used scale diagonal matrix. ## Details The sequence of models implied by `lambda` is fit by coordinate descent. ## Examples ```r p <- 30 n <- 500 C <- diag(0.7, p) C[row(C) == col(C) + 1] <- 0.3 C[row(C) == col(C) - 1] <- 0.3 Sigma <- solve(C) set.seed(1010) m <- floor(sqrt(n)); j <- 1 X_t <- mvtnorm::rmvnorm(n = n, mean = rep(0, p), sigma = Sigma) d_j <- dft.X(X_t,j,m) f_j_hat <- t(d_j) %*% Conj(d_j) / (2*m+1) fit <- cglasso(S=f_j_hat, nobs=n) ``` ## References Deb, N., Kuceyeski, A., Basu, S. (2024) **Regularized Estimation of Sparse Spectral Precision Matrices**, [https://arxiv.org/abs/2401.11128](https://arxiv.org/abs/2401.11128). ## Author(s) Navonil Deb, Younghoon Kim, Sumanta Basu Maintainer: Younghoon Kim yk748@cornell.edu

cglasso function