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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 = "glasso",
  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 (Friedman et al. 2008) .

  2. "ridge": graphical ridge (van Wieringen and Peeters 2016) .

  3. "elnet": graphical elastic net (Zou and Hastie 2005) .

  4. "clime": constrained L1-minimization for inverse (covariance) matrix estimation (Cai et al. 2011) .

  5. "tiger": tuning-insensitive graph estimation and regression (Liu and Wang 2017) , which is only applicable when X is the n-by-p data matrix.

  6. "adapt": adaptive lasso (Zou 2006; Fan et al. 2009) .

  7. "atan": arctangent type penalty (Wang and Zhu 2016) .

  8. "exp": exponential type penalty (Wang et al. 2018) .

  9. "mcp": minimax concave penalty (Zou 2006) .

  10. "scad": smoothly clipped absolute deviation (Fan and Li 2001; Fan et al. 2009) .

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 (Ledoit and Wolf 2004) .

  • "nlshrink": use the precision matrix estimate derived from Ledoit-Wolf non-linear shrinakge estimator of the population covariance matrix (Ledoit and Wolf 2015; Ledoit and Wolf 2017) .

pkgopt

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

  1. For method = "glasso":

  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": the function from glasso.

    • "GLassoElnetFast": the function from gelnet.

    • "glassoFast": the function from glassoFast.

crit

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

  1. "AIC": Akaike information criterion (Akaike 1973) .

  2. "BIC": Bayesian information criterion (Schwarz 1978) .

  3. "EBIC": extended Bayesian information criterion (Foygel and Drton 2010) .

  4. "HBIC": high dimensional Bayesian information criterion (Wang et al. 2013; Fan et al. 2017) .

  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, corresponding to hatOmega.

    gamma

    The actual gamma grid used in the program, corresponding to hatOmega.

    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, corresponding to hatOmega.

    gamma

    The actual gamma grid used in the program, corresponding to hatOmega.

    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

Akaike H (1973). “Information Theory and an Extension of the Maximum Likelihood Principle.” In Petrov BN, Csáki F (eds.), Second International Symposium on Information Theory, 267–281. Akad\'emiai Kiad\'o, Budapest, Hungary.

Cai TT, Liu W, Luo X (2011). “A Constrained \(\ell\)1 Minimization Approach to Sparse Precision Matrix Estimation.” Journal of the American Statistical Association, 106(494), 594–607. doi:10.1198/jasa.2011.tm10155 .

Fan J, Feng Y, Wu Y (2009). “Network Exploration via the Adaptive LASSO and SCAD Penalties.” The Annals of Applied Statistics, 3(2), 521–541. doi:10.1214/08-aoas215 .

Fan J, Li R (2001). “Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties.” Journal of the American Statistical Association, 96(456), 1348–1360. doi:10.1198/016214501753382273 .

Fan J, Liu H, Ning Y, Zou H (2017). “High Dimensional Semiparametric Latent Graphical Model for Mixed Data.” Journal of the Royal Statistical Society Series B: Statistical Methodology, 79(2), 405–421. doi:10.1111/rssb.12168 .

Foygel R, Drton M (2010). “Extended Bayesian Information Criteria for Gaussian Graphical Models.” In Lafferty J, Williams C, Shawe-Taylor J, Zemel R, Culotta A (eds.), Advances in Neural Information Processing Systems 23 (NIPS 2010), 604–612.

Friedman J, Hastie T, Tibshirani R (2008). “Sparse Inverse Covariance Estimation with the Graphical Lasso.” Biostatistics, 9(3), 432–441. doi:10.1093/biostatistics/kxm045 .

Ledoit O, Wolf M (2004). “A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices.” Journal of Multivariate Analysis, 88(2), 365–411. doi:10.1016/S0047-259X(03)00096-4 .

Ledoit O, Wolf M (2015). “Spectrum Estimation: A Unified Framework for Covariance Matrix Estimation and PCA in Large Dimensions.” Journal of Multivariate Analysis, 139, 360–384. doi:10.1016/j.jmva.2015.04.006 .

Ledoit O, Wolf M (2017). “Numerical Implementation of the QuEST Function.” Computational Statistics & Data Analysis, 115, 199–223. doi:10.1016/j.csda.2017.06.004 .

Liu H, Wang L (2017). “TIGER: A Tuning-Insensitive Approach for Optimally Estimating Gaussian Graphical Models.” Electronic Journal of Statistics, 11(1), 241–294. doi:10.1214/16-EJS1195 .

Schwarz G (1978). “Estimating the Dimension of a Model.” The Annals of Statistics, 6(2), 461–464. doi:10.1214/aos/1176344136 .

van Wieringen WN, Peeters CFW (2016). “Ridge Estimation of Inverse Covariance Matrices from High-Dimensional Data.” Computational Statistics & Data Analysis, 103, 284–303. doi:10.1016/j.csda.2016.05.012 .

Wang L, Kim Y, Li R (2013). “Calibrating Nonconvex Penalized Regression in Ultra-High Dimension.” The Annals of Statistics, 41(5), 2505–2536. doi:10.1214/13-AOS1159 .

Wang Y, Fan Q, Zhu L (2018). “Variable Selection and Estimation using a Continuous Approximation to the \(L_0\) Penalty.” Annals of the Institute of Statistical Mathematics, 70(1), 191–214. doi:10.1007/s10463-016-0588-3 .

Wang Y, Zhu L (2016). “Variable Selection and Parameter Estimation with the Atan Regularization Method.” Journal of Probability and Statistics, 2016, 6495417. doi:10.1155/2016/6495417 .

Zou H (2006). “The Adaptive Lasso and Its Oracle Properties.” Journal of the American Statistical Association, 101(476), 1418–1429. doi:10.1198/016214506000000735 .

Zou H, Hastie T (2005). “Regularization and Variable Selection via the Elastic Net.” Journal of the Royal Statistical Society Series B: Statistical Methodology, 67(2), 301–320. doi:10.1111/j.1467-9868.2005.00527.x .