Skip to contents

Sparse Precision (Inverse Covariance) Estimation

Source: R/spice.R
spice.Rd

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

Usage

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

Arguments

X

  1. An \(n \times p\) data matrix with sample size \(n\) and dimension \(p\).

  2. A \(p \times p\) 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 \(n \times p\) 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:

  1. "cov": The covariance matrix.

  2. "cor": The correlation matrix.

This is only applicable when X is the \(n \times p\) data matrix.

n

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

lambda

A non-negative numeric vector specifying the grid for the regularization parameter. The default is NULL, which generates its own lambda sequence based on nlambda and lambda.min.ratio. For method = "clime" combined with pkg = "clime", the lambda sequence is based on nlambda, lambda.min and lambda.max.

nlambda

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

lambda.min.ratio

A numeric value > 0 (default = 0.01) specifying the fraction of the maximum lambda value \(\lambda_{\max}\) to generate the minimum lambda \(\lambda_{\min}\). If lambda = NULL, a lambda grid of length nlambda is automatically generated on a log scale, ranging from \(\lambda_{\max}\) down to \(\lambda_{\min}\).

gamma

A numeric value specifying the additional parameter 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. "mcp": 3

  6. "scad": 3.7

initial

A \(p \times p\) matrix or a \(p \times p \times \mathrm{npara}\) (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 \(\lvert\tilde{\omega}_{ij}\rvert^{-\gamma}\), where \(\tilde{\Omega} := (\tilde{\omega}_{ij})\). Some options are also offered when a character string is provided (default = "glasso"), 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 shrinkage estimator of the population covariance matrix (Ledoit and Wolf 2004) .

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

pkg

A character string specifying the package option to use. The available options depend on the chosen 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 character 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 numeric value in [0, 1] (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

An object with S3 class "spice" containing the following components:

hatOmega_opt

The estimated precision matrix.

lambda_opt

The optimal regularization parameter.

gamma_opt

The optimal hyperparameter.

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.

CV.loss

Matrix of CV losses, with rows for CV folds and columns for parameter combinations, when crit = "CV".

IC.score

The information criterion score for each parameter combination when crit is set to "AIC", "BIC", "EBIC", or "HBIC".

Note

For method = "tiger", the estimation process solely relies on the raw \(n \times p\) data X and does not utilize the argument base. This argument is not applicable for method = "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 .