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Estimation

Source: R/estimation.R
estimation.Rd

This function performs alternating direction method of multipliers optimization for a variety of loss functions to estimate the differential network given two samples of multivariate normal data.

Usage

estimation(
  X,
  Y,
  lambdas = NULL,
  lambda_min_ratio = 0.3,
  nlambda = 10,
  a = NULL,
  loss = "lasso",
  tuning = "none",
  perturb = FALSE,
  stop_tol = 1e-05,
  max_iter = 500,
  correlation = FALSE,
  Delta_init = NULL,
  rho = NULL,
  gamma = NULL,
  cores = 1,
  verbose = FALSE
)

Arguments

X

The first multivariate normal sample.

Y

The second multivariate normal sample.

lambdas

Optional parameter - A list of the regularization values to be used within the loss functions.

lambda_min_ratio

Optional parameter - Defines the smallest regularization values as this proportion of the largest regularization value. Defaults to 0.3.

nlambda

Optional parameter - The number of regularization values considered. Defaults to 10.

a

Optional parameter - The thresholding parameter used in SCAD and MCP loss functions. Defaults to 3.7 with SCAD, and 3 with MCP respectively.

loss

Optional parameter - The loss function of choice to implement. The function allows for four choices, namely "lasso", "scad", "mcp" and "d-trace". Defaults to "lasso".

tuning

Optional parameter - The tuning method selected to determine the optimal value for the regularization parameter. Options are "none", "AIC", "BIC" and "EBIC". Defaults to "none".

perturb

Optional parameter - When set to TRUE perturbation as done by the CLIME software to improve performance is implemented. Options are TRUE or FALSE, with the function defaulting to FALSE.

stop_tol

Optional parameter - The stop tolerance to determine whether convergence has occurred. Defaults to 1e-5.

max_iter

Optional parameter - The maximum number of iterations that can be perform for any one regularization value. Defaults to 100.

correlation

Optional parameter - Determines whether the sample correlation matrices should be used in the place of the sample covariance matrices. Choices are TRUE and FALSE with the function defaulting to FALSE.

Delta_init

Optional parameter - Allows for the algorithm to provided an initial estimate of the differential network to ease computation.

rho

Optional parameter - Allows the user to adjust the ADMM step-size. Defaults to 1.

gamma

Optional parameter - Allows the user to adjust the EBIC value when EBIC is the selected tuning method. Defaults to 0.5.

cores

Optional parameter - Allows the user to specify the number of cores used by the optimization. Defaults to 1, i.e sequential solving however appropriate values range from 2 to the minimum of the number of lambdas and the total number of available cores.

verbose

Optional parameter - Allows the user to obtain a summary of the estimation results. Options are TRUE or FALSE, where FALSE indicates the summary is not provided. Defaults to FALSE.

Value

A list of various outputs, namely:

  • n_X - The number of observations in X.

  • n_Y - The number of observations in Y.

  • Sigma_X - The covariance matrix of X.

  • Sigma_Y - The covariance matrix of Y.

  • loss - The loss function implemented.

  • tuning - The tuning method utilized.

  • lip - The value of the lipschitz constant.

  • iter - The iterations until convergence for each of the regularization values.

  • elapse - The total system time (in seconds) elapsed from initialization to completion of the optimization.

  • lambdas - The regularization parameter values used.

  • sparsity - The level of sparsity of the differential network for each regularization value.

  • path - The set of all differential networks for all regularization values considered.

  • ic - The output obtained from any possible tuning.

  • ic_index - The index at which the tuning is optimized.

  • ic_value - The tuning method optimal value.

  • chosen_lambda_ic - The regularization value that occurs at ic_index.

  • loss_index - The index at which the loss function is optimized.

  • loss_value - The loss function optimal value.

  • chosen_lambda_loss - The regularization value that occurs at loss_index.

References

Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J., 2011. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends® in Machine learning, 3(1), pp.1-122.

Chen, J. and Chen, Z., 2008. Extended Bayesian information criteria for model selection with large model spaces. Biometrika, 95(3), pp.759-771.

Friedman, J., Hastie, T. and Tibshirani, R., 2008. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3), pp.432-441.

Tang, Z., Yu, Z. and Wang, C., 2020. A fast iterative algorithm for high-dimensional differential network. Computational Statistics, 35(1), pp.95-109.

Yuan, H., Xi, R., Chen, C. and Deng, M., 2017. Differential network analysis via lasso penalized D-trace loss. Biometrika, 104(4), pp.755-770.

Zhang, T. and Zou, H., 2014. Sparse precision matrix estimation via lasso penalized D-trace loss. Biometrika, 101(1), pp.103-120.

Examples

data <- data_generator(n_X = 100, p = 50, seed = 123)
X <- data$X
Y <- data$Y

# Sequential (default)
result_seq <- estimation(X, Y, nlambda = 5, cores = 1)
#> --------------------------------------------------------------------------------
#> 
#> --------------------------------------------------------------------------------
result_seq$elapse
#> elapsed 
#>   0.727 

# \donttest{
# Parallel - set cores to a value greater than 1
# (detectCores() - 1 is recommended on your own machine)
result_par <- estimation(X, Y, nlambda = 5, cores = 2)
#> --------------------------------------------------------------------------------
#> 
#> --------------------------------------------------------------------------------
result_par$elapse
#> elapsed 
#>   1.153 
# }