irglmreg_fit function

Internal function for robust penalized generalized linear models

Fit a robust penalized GLM where the loss function is a composite function cfunodfun + penalty. This does computing for irglmreg.

irglmreg_fit(x, y, weights, offset, cfun="ccave", dfun="gaussian", s=NULL, 
             delta=0.1, fk=NULL, iter=10, reltol=1e-5, 
             penalty=c("enet","mnet","snet"), nlambda=100, lambda=NULL, 
             type.path=c("active", "nonactive"), decreasing=TRUE, 
             lambda.min.ratio=ifelse(nobs<nvars,.05, .001), alpha=1, gamma=3,
             rescale=TRUE, standardize=TRUE, intercept=TRUE, 
             penalty.factor= NULL, maxit=1000, type.init=c("bst", "co", "heu"), 
             init.family=NULL, mstop.init=10, nu.init=0.1, 
             eps=.Machine$double.eps, epscycle=10, thresh=1e-6, parallel=FALSE,
             n.cores=2, theta, trace=FALSE, tracelevel=1)

Arguments

• x: input matrix, of dimension nobs x nvars; each row is an observation vector.

• y: response variable. Quantitative for dfun=1 and -1/1 otherwise for classifications.

• weights: observation weights. Can be total counts if responses are proportion matrices. Default is 1 for each observation

• offset: this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases. Currently only one offset term can be included in the formula.

• cfun: character, type of convex cap (concave) function.

  Valid options are:

  • "hcave"
  • "acave"
  • "bcave"
  • "ccave"
  • "dcave"
  • "ecave"
  • "gcave"
  • "tcave"

• dfun: character, type of convex downward function.

  Valid options are:

  • "gaussian"
  • "gaussianC"
  • "binomial"

• s: tuning parameter of cfun. s > 0 and can be equal to 0 for cfun="tcave". If s is too close to 0 for cfun="acave", "bcave", "ccave", the calculated weights can become 0 for all observations, thus crash the program.

• delta: a small positive number provided by user only if cfun="gcave" and 0 < s <1

• fk: predicted values at an iteration in the IRCO algorithm

• nlambda: The number of lambda values - default is 100. The sequence may be truncated before nlambda is reached if a close to saturated model is fitted. See also satu.

• lambda: by default, the algorithm provides a sequence of regularization values, or a user supplied lambda sequence

• type.path: solution path for parallel=FALSE. If type.path="active", then cycle through only the active set in the next increasing lambda sequence. If type.path="nonactive", no active set for each element of the lambda sequence and cycle through all the predictor variables.

• lambda.min.ratio: Smallest value for lambda, as a fraction of lambda.max, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero except the intercept). Note, there is no closed formula for lambda.max. The default of lambda.min.ratio depends on the sample size nobs relative to the number of variables nvars. If nobs > nvars, the default is 0.001, close to zero. If nobs < nvars, the default is 0.05.

• alpha: The $L_2$ penalty mixing parameter, with $0 \le alpha\le 1$. alpha=1 is lasso (mcp, scad) penalty; and alpha=0 the ridge penalty. However, if alpha=0, one must provide lambda values.

• gamma: The tuning parameter of the snet or mnet penalty.

• rescale: logical value, if TRUE, adaptive rescaling of the penalty parameter for penalty="mnet" or penalty="snet" with dfun="binomial". See glmreg_fit

• standardize: logical value for x variable standardization, prior to fitting the model sequence. The coefficients are always returned on the original scale. Default is standardize=TRUE.

• intercept: logical value: if TRUE (default), intercept(s) are fitted; otherwise, intercept(s) are set to zero

• penalty.factor: This is a number that multiplies lambda to allow differential shrinkage of coefficients. Can be 0 for some variables, which implies no shrinkage, and that variable is always included in the model. Default is same shrinkage for all variables.

• type.init: a method to determine the initial values. If type.init="ncl", an intercept-only model as initial parameter and run irglmreg regularization path forward from lambda_max to lambda_min. If type.init="heu", heuristic initial parameters and run irglmreg path backward or forward depending on decreasing, between lambda_min and lambda_max. If type.init="bst", run a boosting model with bst in package bst, depending on mstop.init, nu.init and run irglmreg backward or forward depending on decreasing.

• init.family: character value for initial family, one of "clossR", "closs","gloss","qloss", which can be used to derive an initial estimator, if the selection is different from the default value

• mstop.init: an integer giving the number of boosting iterations when type.init="bst"

• nu.init: a small number (between 0 and 1) defining the step size or shrinkage parameter when type.init="bst".

• decreasing: only used if lambda=NULL, a logical value used to determine regularization path direction either from lambda_max to a potentially modified lambda_min or vice versa if type.init="bst", "heu". Since this is a nonconvex optimization, it is possible to generate different estimates for the same lambda depending on decreasing. The choice of decreasing picks different starting values.

• iter: number of iteration in the IRCO algorithm

• maxit: Within each IRCO algorithm iteration, maximum number of coordinate descent iterations for each lambda value; default is 1000.

• reltol: convergency criteria in the IRCO algorithm

• eps: If a coefficient is less than eps in magnitude, then it is reported to be 0

• epscycle: If nlambda > 1 and the relative loss values from two consecutive lambda values change > epscycle, then re-estimate parameters in an effort to avoid trap of local optimization.

• thresh: Convergence threshold for coordinate descent. Defaults value is 1e-6.

• penalty: Type of regularization

• theta: an overdispersion scaling parameter for family="negbin"

• parallel, n.cores: If TRUE, to compute solution of lambda with parallel computing in number of n.cores. If FALSE, sequential computing. If NULL, still sequential computing with a different convergence criteria based on penalized loss values

• trace, tracelevel: If TRUE, fitting progress is reported. If tracelevel=2, deeper level of fitting progress is reported.

Details

A case weighted penalized least squares or GLM is fit by the iteratively reweighted convex optimization (IRCO), where the loss function is a composite function cfunodfun + penalty. Here convex is the loss function induced by dfun, not the penalty function. The sequence of robust models implied by lambda is fit by IRCO along with coordinate descent. Note that the objective function is

if standardize=FALSE and

if standardize=TRUE.

Returns

An object with S3 class "irglmreg" for the various types of models. - call: the call that produced the model fit

• b0: Intercept sequence of length length(lambda)

• beta: A nvars x length(lambda) matrix of coefficients.

• lambda: The actual sequence of lambda values used

• weights_update: A nobs x length(lambda) matrix of weights computed by the IRCO algorithm. The entry of i-th row and j-th column is the weight for the i-th observation and j-th lambda value.

• decreasing: if lambda is an increasing sequence or not, used to determine regularization path direction either from lambda_max to a potentially modified lambda_min or vice versa if type.init="bst", "heu".

References

Zhu Wang (2024) Unified Robust Estimation, Australian & New Zealand Journal of Statistics. 66(1):77-102.

Author(s)

Zhu Wang zwang145@uthsc.edu

See Also

irglmreg