Lasso Smooth Constructor

Smooth constructors and optimizer for Lasso penalization with bamlss. The penalization is based on a Taylor series approximation of the Lasso penalty.

## Smooth constructor function.
la(formula, type = c("single", "multiple"), ...)

## Single Lasso smoothing parameter optimizer.
opt_lasso(x, y, start = NULL, adaptive = TRUE, lower = 0.001, upper = 1000,
  nlambda = 100, lambda = NULL,  multiple = FALSE, verbose = TRUE,
  digits = 4, flush = TRUE, nu = NULL, stop.nu = NULL,
  ridge = .Machine$double.eps^0.5, zeromodel = NULL, ...)

lasso(x, y, start = NULL, adaptive = TRUE, lower = 0.001, upper = 1000,
  nlambda = 100, lambda = NULL,  multiple = FALSE, verbose = TRUE,
  digits = 4, flush = TRUE, nu = NULL, stop.nu = NULL,
  ridge = .Machine$double.eps^0.5, zeromodel = NULL, ...)

## Lasso transformation function to set
## adaptive weights from an unpenalized model.
lasso_transform(x, zeromodel, nobs = NULL, ...)

## Plotting function for opt_lasso() optimizer.
lasso_plot(x, which = c("criterion", "parameters"),
  spar = TRUE, model = NULL, name = NULL, mstop = NULL,
  retrans = FALSE, color = NULL, show.lambda = TRUE,
  labels = NULL, digits = 2, ...)

## Extract optimum stopping iteration for opt_lasso() optimizer.
## Based on the minimum of the information criterion.
lasso_stop(x)

## Extract retransformed Lasso coefficients.
lasso_coef(x, ...)

Arguments

formula: A formula like ~ x1 + x2 + ... + xk of variables which should be penalized with Lasso.
type: Should one single penalty parameter be used or multiple parameters, one for each covariate in formula.
x: For function opt_lasso() and lasso_transform() the x list, as returned from function bamlss.frame, holding all model matrices and other information that is used for fitting the model. For the plotting function and lasso_stop()/lasso_coef() the corresponding bamlss object fitted with the opt_lasso() optimizer.
y: The model response, as returned from function bamlss.frame.
start: A vector of starting values. Note, Lasso smoothing parameters will be dropped.
adaptive: Should adaptive weights be used for fused Lasso terms?
lower: Numeric. The minimum lambda value.
upper: Numeric. The maximum lambda value.
nlambda: Integer. The number of smoothing parameters for which coefficients should be estimated, i.e., the vector of smoothing parameters is build up as a sequence from lower to upper with length nlambda.
lambda: Numeric. A sequence/vector of lambda parameters that should be used.
multiple: Logical. Should the lambda grid be exapnded to search for multiple lambdas, one for each distributional parameter.
verbose: Print information during runtime of the algorithm.
digits: Set the digits for printing when verbose = TRUE. If the optimum lambda value is plotted, the number of decimal decimal places to be used within lasso_plot().
flush: use flush.console for displaying the current output in the console.
nu: Numeric or logical. Defines the step length for parameter updating of a model term, useful when the algorithm encounters convergence problems. If nu = TRUE the step length parameter is optimized for each model term in each iteration of the backfitting algorithm.
stop.nu: Integer. Should step length reduction be stopped after stop.nu iterations of the Lasso algorithm?
ridge: A ridge penalty parameter that should be used when finding adaptive weights, i.e., parameters from an unpenalized model. The ridge penalty is used to stabilize the estimation in complex models.
zeromodel: A model containing the unpenalized parameters, e.g., for each la() terms one can place a simple ridge penalty with la(x, ridge = TRUE, sp = 0.1). This way it is possible to find the unpenalized parameters that can be used as adaptive weights for fusion penalties.
nobs: Integer, number of observations of the data used for modeling. If not supplied nobs is taken from the number of rows from the model term design matrices.
which: Which of the two provided plots should be created, character or integer 1 and 2.
spar: Should graphical parameters be set by the plotting function?
model: Character selecting for which model the plot shpuld be created.
name: Character, the name of the coefficient group that should be plotted. Note that the string provided in name will be removed from the labels on the 4th axis.
mstop: Integer vector, defines the path length to be plotted.
retrans: Logical, should coefficients be re-transformed before plotting?
color: Colors or color function that creates colors for the group paths.
show.lambda: Logical. Should the optimum value of the penalty parameter lambda be shown?
labels: A character string of labels that should be used on the 4 axis.
...: Arguments passed to the subsequent smooth constructor function. lambda controls the starting value of the penalty parameter, const the constant that is added within the penalty approximation. Moreover, fuse = 1 enforces nominal fusion of categorical variables and fuse = 2 ordered fusion within la() Note that la() terms with and without fusion should not be mixed when using the opt_lasso() optimizer function. For the optimizer opt_lasso() arguments passed to function bfit.

Value

For function la(), similar to function s a simple smooth specification object.

For function opt_lasso() a list containing the following objects:

fitted.values: A named list of the fitted values based on the last lasso iteration of the modeled parameters of the selected distribution.
parameters: A matrix, each row corresponds to the parameter values of one boosting iteration.
lasso.stats: A matrix containing information about the log-likelihood, log-posterior and the information criterion for each lambda.

References

Andreas Groll, Julien Hambuckers, Thomas Kneib, and Nikolaus Umlauf (2019). Lasso-type penalization in the framework of generalized additive models for location, scale and shape. Computational Statistics & Data Analysis. doi:10.1016/j.csda.2019.06.005

Oelker Margreth-Ruth and Tutz Gerhard (2015). A uniform framework for combination of penalties in generalized structured models. Adv Data Anal Classif. doi:10.1007/s11634-015-0205-y

Examples

if (FALSE) ## Simulated fusion Lasso example.
bmu <- c(0,0,0,2,2,2,4,4,4)
bsigma <- c(0,0,0,-2,-2,-2,-1,-1,-1)
id <- factor(sort(rep(1:length(bmu), length.out = 300)))
#> Error in eval(expr, envir, enclos): object 'bmu' not found

## Response.
set.seed(123)
y <- bmu[id] + rnorm(length(id), sd = exp(bsigma[id]))
#> Error in eval(expr, envir, enclos): object 'bmu' not found

## Estimate model:
## fuse=1 -> nominal fusion,
## fuse=2 -> ordinal fusion,
## first, unpenalized model to be used for adaptive fusion weights.
f <- list(y ~ la(id,fuse=2,fx=TRUE), sigma ~ la(id,fuse=1,fx=TRUE))
b0 <- bamlss(f, sampler = FALSE)
#> Error in eval(predvars, data, env): object 'y' not found

## Model with single lambda parameter.
f <- list(y ~ la(id,fuse=2), sigma ~ la(id,fuse=1))
b1 <- bamlss(f, sampler = FALSE, optimizer = opt_lasso,
  criterion = "BIC", zeromodel = b0)
#> Error in eval(bf, envir = parent.frame()): object 'b0' not found

## Plot information criterion and coefficient paths.
lasso_plot(b1, which = 1)
#> Error in eval(expr, envir, enclos): object 'b1' not found
lasso_plot(b1, which = 2)
#> Error in eval(expr, envir, enclos): object 'b1' not found
lasso_plot(b1, which = 2, model = "mu", name = "mu.s.la(id).id")
#> Error in eval(expr, envir, enclos): object 'b1' not found
lasso_plot(b1, which = 2, model = "sigma", name = "sigma.s.la(id).id")
#> Error in eval(expr, envir, enclos): object 'b1' not found

## Extract coefficients for optimum Lasso parameter.
coef(b1, mstop = lasso_stop(b1))
#> Error in eval(expr, envir, enclos): object 'b1' not found

## Predict with optimum Lasso parameter.
p1 <- predict(b1, mstop = lasso_stop(b1))
#> Error in eval(expr, envir, enclos): object 'b1' not found

## Full MCMC, needs lasso_transform() to assign the
## adaptive weights from unpenalized model b0.
b2 <- bamlss(f, optimizer = FALSE, transform = lasso_transform,
  zeromodel = b0, nobs = length(y), start = coef(b1, mstop = lasso_stop(b1)),
  n.iter = 4000, burnin = 1000)
#> Error in eval(bf, envir = parent.frame()): object 'b1' not found
summary(b2)
#> Error in eval(expr, envir, enclos): object 'b2' not found
plot(b2)
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'x' in selecting a method for function 'plot': object 'b2' not found

ci <- confint(b2, model = "mu", pterms = FALSE, sterms = TRUE)
#> Error in eval(expr, envir, enclos): object 'b2' not found
lasso_plot(b1, which = 2, model = "mu", name = "mu.s.la(id).id", spar = FALSE)
#> Error in eval(expr, envir, enclos): object 'b1' not found
for(i in 1:8) {
  abline(h = ci[i, 1], lty = 2, col = "red")
  abline(h = ci[i, 2], lty = 2, col = "red")
}
#> Error in eval(expr, envir, enclos): object 'ci' not found

Arguments

Value

References

See also

Examples