ridge {lessSEM}

ridge

Description

Implements ridge regularization for structural equation models. The penalty function is given by:

p( x_j) = \lambda x_j^2

Note that ridge regularization will not set any of the parameters to zero but result in a shrinkage towards zero.

Usage

ridge(
  lavaanModel,
  regularized,
  lambdas,
  method = "glmnet",
  modifyModel = lessSEM::modifyModel(),
  control = lessSEM::controlGlmnet()
)

Arguments

lavaanModel

model of class lavaan

regularized

vector with names of parameters which are to be regularized. If you are unsure what these parameters are called, use getLavaanParameters(model) with your lavaan model object

lambdas

numeric vector: values for the tuning parameter lambda

method

which optimizer should be used? Currently implemented are ista and glmnet. With ista, the control argument can be used to switch to related procedures (currently gist).

modifyModel

used to modify the lavaanModel. See ?modifyModel.

control

used to control the optimizer. This element is generated with the controlIsta and controlGlmnet functions. See ?controlIsta and ?controlGlmnet for more details.

Details

Identical to regsem, models are specified using lavaan. Currently, most standard SEM are supported. lessSEM also provides full information maximum likelihood for missing data. To use this functionality, fit your lavaan model with the argument sem(..., missing = 'ml'). lessSEM will then automatically switch to full information maximum likelihood as well.

Ridge regularization:

• Hoerl, A. E., & Kennard, R. W. (1970). Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 12(1), 55–67. https://doi.org/10.1080/00401706.1970.10488634

Regularized SEM

• Huang, P.-H., Chen, H., & Weng, L.-J. (2017). A Penalized Likelihood Method for Structural Equation Modeling. Psychometrika, 82(2), 329–354. https://doi.org/10.1007/s11336-017-9566-9

• Jacobucci, R., Grimm, K. J., & McArdle, J. J. (2016). Regularized Structural Equation Modeling. Structural Equation Modeling: A Multidisciplinary Journal, 23(4), 555–566. https://doi.org/10.1080/10705511.2016.1154793

For more details on GLMNET, see:

• Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1–20. https://doi.org/10.18637/jss.v033.i01

• Yuan, G.-X., Chang, K.-W., Hsieh, C.-J., & Lin, C.-J. (2010). A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification. Journal of Machine Learning Research, 11, 3183–3234.

• Yuan, G.-X., Ho, C.-H., & Lin, C.-J. (2012). An improved GLMNET for l1-regularized logistic regression. The Journal of Machine Learning Research, 13, 1999–2030. https://doi.org/10.1145/2020408.2020421

For more details on ISTA, see:

• Beck, A., & Teboulle, M. (2009). A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences, 2(1), 183–202. https://doi.org/10.1137/080716542

• Gong, P., Zhang, C., Lu, Z., Huang, J., & Ye, J. (2013). A General Iterative Shrinkage and Thresholding Algorithm for Non-convex Regularized Optimization Problems. Proceedings of the 30th International Conference on Machine Learning, 28(2)(2), 37–45.

• Parikh, N., & Boyd, S. (2013). Proximal Algorithms. Foundations and Trends in Optimization, 1(3), 123–231.

Value

Model of class regularizedSEM

Examples

library(lessSEM)

# Identical to regsem, lessSEM builds on the lavaan
# package for model specification. The first step
# therefore is to implement the model in lavaan.

dataset <- simulateExampleData()

lavaanSyntax <- "
f =~ l1*y1 + l2*y2 + l3*y3 + l4*y4 + l5*y5 +
     l6*y6 + l7*y7 + l8*y8 + l9*y9 + l10*y10 +
     l11*y11 + l12*y12 + l13*y13 + l14*y14 + l15*y15
f ~~ 1*f
"

lavaanModel <- lavaan::sem(lavaanSyntax,
                           data = dataset,
                           meanstructure = TRUE,
                           std.lv = TRUE)

# Regularization:

lsem <- ridge(
  # pass the fitted lavaan model
  lavaanModel = lavaanModel,
  # names of the regularized parameters:
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 20))

# use the plot-function to plot the regularized parameters:
plot(lsem)

# the coefficients can be accessed with:
coef(lsem)

# elements of lsem can be accessed with the @ operator:
lsem@parameters[1,]


#### Advanced ###
# Switching the optimizer #
# Use the "method" argument to switch the optimizer. The control argument
# must also be changed to the corresponding function:
lsemIsta <- ridge(
  lavaanModel = lavaanModel,
  regularized = paste0("l", 6:15),
  lambdas = seq(0,1,length.out = 20),
  method = "ista",
  control = controlIsta())

# Note: The results are basically identical:
lsemIsta@parameters - lsem@parameters