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Statistical methods for estimating precision matrix

Source: R/fcstat.R
fcstat.Rd

Provide a collection of statistical methods to estimate a precision matrix.

Usage

fcstat(
  X,
  method,
  base = "cov",
  n = NULL,
  lambda = NULL,
  nlambda = 20,
  lambda.min.ratio = 0.01,
  gamma = NA,
  initial = "glasso",
  pkgopt = "glassoFast_cold_diagture",
  crit = "CV",
  fold = 5,
  ebic.tuning = 0.5,
  cores = 1
)

Arguments

X

  1. An n-by-p data matrix with sample size n and dimension p.

  2. A p-by-p sample covariance/correlation matrix with dimension p.

method

A character string specifying the statistical method for estimating precision matrix. Available options include:

  1. "glasso": graphical lasso friedman2008sparsefcstat.

  2. "ridge": graphical ridge vanwieringen2016ridgefcstat.

  3. "elnet": graphical elastic net zou2005regularizationfcstat.

  4. "clime": constrained L1-minimization for inverse (covariance) matrix estimation cai2011aconstrainedfcstat.

  5. "tiger": tuning-insensitive graph estimation and regression liu2017tigerfcstat, which is only applicable when X is the n-by-p data matrix.

  6. "adapt": adaptive lasso zou2006adaptive,fan2009networkfcstat.

  7. "atan": arctangent type penalty wang2016variablefcstat.

  8. "exp": exponential type penalty wang2018variablefcstat.

  9. "mcp": minimax concave penalty zou2006adaptivefcstat.

  10. "scad": smoothly clipped absolute deviation fan2001variable,fan2009networkfcstat.

base

A character string (default = "cov") specifying the calculation base, either the covariance matrix ("cov") or the correlation matrix ("cor"). This is only applicable when X is the n-by-p data matrix.

n

An integer (default = NULL) specifying the sample size. This is only required when the input matrix X is a p-by-p sample covariance/correlation matrix with dimension p.

lambda

Grid of non-negative scalars for the regularization parameter. The default is NULL, which generates its own lambda sequence based on nlambda and lambda.min.ratio.

nlambda

An integer (default = 20) specifying the number of lambda values to be generated when lambda = NULL.

lambda.min.ratio

A scalar (default = 0.01) specifying the fraction of the maximum lambda value \(\lambda_{max}\) to generate the minimum lambda \(\lambda_{min}\). If lambda = NULL, the program automatically generates a lambda grid as a sequence of length nlambda in log scale, starting from \(\lambda_{min}\) to \(\lambda_{max}\).

gamma

Grid of scalars specifying the hyperparameter for the chosen method. Default values:

  1. "elnet": a sequence from 0.1 to 0.9 with increments of 0.1

  2. "adapt": 0.5

  3. "atan": 0.005

  4. "exp": 0.01

  5. "scad": 3.7

  6. "mcp": 3

initial

A p-by-p matrix or a p-by-p-by-npara (the number of all combinations of lambda and gamma) array specifying the initial estimate for method set to "atan", "exp", "scad", and "mcp"; or specifying \(\tilde{\Omega}\) of the adaptive weight for method = "adapt", calculated as \(|\tilde{\omega}_{ij}|^{-\gamma}\), where \(\tilde{\Omega} := (\tilde{\omega}_{ij})\). Some options are also offered when a character string is provided (default "linshrink"), including:

  • "glasso": use the precision matrix estimate derived from the graphical lasso.

  • "invS": use the inverse calculation base matrix if the matrix is invertible.

  • "linshrink": use the precision matrix estimate derived from Ledoit-Wolf linear shrinakge estimator of the population covariance matrix ledoit2004wellfcstat.

  • "nlshrink": use the precision matrix estimate derived from Ledoit-Wolf non-linear shrinakge estimator of the population covariance matrix ledoit2015spectrum,ledoit2017numericalfcstat.

pkgopt

A character string specifying the package option to use. The available options depend on the selected method:

  1. For method = "glasso":

    • "ADMMsigma_diagtrue": the function from ADMMsigma with warm-starts, and the diagonal elements of the estimated precision matrix are penalized.

    • "ADMMsigma_diagfalse": the function from ADMMsigma with warm-starts, and the diagonal elements of the estimated precision matrix are not penalized.

    • "CovTools": the function from PreEst.glasso.

    • "CVglasso_diagtrue": the function from CVglasso with warm-starts, and the diagonal elements of the estimated precision matrix are penalized.

    • "CVglasso_diagfalse": the function from CVglasso with warm-starts, and the diagonal elements of the estimated precision matrix are not penalized.

    • "glasso_cold_diagtrue": the function from glasso, and the diagonal elements of the estimated precision matrix are penalized.

    • "glasso_cold_diagfalse": the function from glasso, and the diagonal elements of the estimated precision matrix are not penalized.

    • "glasso_warm_diagtrue": the function from glasso with warm-starts, and the diagonal elements of the estimated precision matrix are penalized.

    • "glasso_warm_diagfalse": the function from glasso with warm-starts, and the diagonal elements of the estimated precision matrix are not penalized.

    • "GLassoElnetFast_cold_diagtrue": the function from gelnet, and the diagonal elements of the estimated precision matrix are penalized.

    • "GLassoElnetFast_cold_diagfalse": the function from gelnet, and the diagonal elements of the estimated precision matrix are not penalized.

    • "GLassoElnetFast_warm_diagtrue": the function from gelnet with warm-starts, and the diagonal elements of the estimated precision matrix are penalized.

    • "GLassoElnetFast_warm_diagfalse": the function from gelnet with warm-starts, and the diagonal elements of the estimated precision matrix are not penalized.

    • "glassoFast_cold": the function from glassoFast, and the diagonal elements of the estimated precision matrix are penalized.

    • "glassoFast_warm": the function from glassoFast with warm-starts, and the diagonal elements of the estimated precision matrix are penalized.

    • "huge": the function from huge.glasso, and the diagonal elements of the estimated precision matrix are penalized.

  2. For method = "ridge":

    • "ADMMsigma": the function from RIDGEsigma.

    • "GLassoElnetFast": the function from gelnet.

    • "porridge": the function from ridgePgen.

    • "rags2ridges": the function from ridgeP.

  3. For method = "elnet":

    • "ADMMsigma": the function from ADMMsigma.

    • "GLassoElnetFast": the function from gelnet.

  4. For method = "clime":

    • "clime": the function from clime.

    • "flare": the function from sugm.

  5. For method = "tiger":

    • "flare": the function from sugm.

    • "huge": the function from huge.tiger.

  6. For method set to "adapt", "atan", "exp", "scad", and "mcp":

    • "glasso_cold": the function from glasso.

    • "glasso_warm": the function from glasso with warm-starts.

    • "GLassoElnetFast_cold": the function from gelnet.

    • "GLassoElnetFast_warm": the function from gelnet with warm-starts.

    • "glassoFast_cold": the function from glassoFast.

    • "glassoFast_warm": the function from glassoFast with warm-starts.

crit

A string (default = "CV") specifying the parameter selection method to use. Available options include:

  1. "AIC": Akaike information criterion akaike1973informationfcstat.

  2. "BIC": Bayesian information criterion schwarz1978estimatingfcstat.

  3. "EBIC": extended Bayesian information criterion foygel2010extendedfcstat.

  4. "HBIC": high dimensional Bayesian information criterion wang2013calibrating,fan2017highfcstat.

  5. "CV": k-fold cross validation with negative log-likelihood loss.

fold

An integer (default = 5) specifying the number of folds used for crit = "CV".

ebic.tuning

A scalar (default = 0.5) specifying the tuning parameter to calculate for crit = "EBIC".

cores

An integer (default = 1) specifying the number of cores to use for parallel execution.

Value

  • For crit = "CV", an object with S3 class "fcstat" containing the following components:

    hatOmega_opt

    The estimated precision matrix.

    lambda_opt

    The optimal regularization parameter.

    gamma_opt

    The optimal hyperparameter.

    loss_opt

    The optimal k-fold loss.

    hatOmega

    A list of estimated precision matrices for lambda grid and gamma grid.

    lambda

    The actual lambda grid used in the program.

    gamma

    The actual gamma grid used in the program.

    loss.mean

    The mean of k-fold loss for each parameter grid value.

    loss.sd

    The standard deviation of k-fold loss for each parameter grid value.

  • For other criteria, an object with S3 class "fcstat" containing the following components:

    hatOmega_opt

    The estimated precision matrix.

    lambda_opt

    The optimal regularization parameter.

    gamma_opt

    The optimal hyperparameter.

    score_opt

    The optimal information criterion score.

    hatOmega

    A list of estimated precision matrices for lambda grid and gamma grid.

    lambda

    The actual lambda grid used in the program.

    gamma

    The actual gamma grid used in the program.

    score

    The information criterion score for each parameter grid value.

Note

For the method tiger, the estimation process solely relies on the raw n-by-p data X and does not utilize the argument base. This argument is not applicable for tiger and will have no effect if provided.

References