# 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. ```r ## 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`. ## Returns 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**. tools:::Rd_expr_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**. tools:::Rd_expr_doi("10.1007/s11634-015-0205-y") ## See Also `s`, `smooth.construct` ## Examples ```r ## Not run: ## 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))) ## Response. set.seed(123) y <- bmu[id] + rnorm(length(id), sd = exp(bsigma[id])) ## 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) ## 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) ## Plot information criterion and coefficient paths. lasso_plot(b1, which = 1) lasso_plot(b1, which = 2) lasso_plot(b1, which = 2, model = "mu", name = "mu.s.la(id).id") lasso_plot(b1, which = 2, model = "sigma", name = "sigma.s.la(id).id") ## Extract coefficients for optimum Lasso parameter. coef(b1, mstop = lasso_stop(b1)) ## Predict with optimum Lasso parameter. p1 <- predict(b1, mstop = lasso_stop(b1)) ## 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) summary(b2) plot(b2) ci <- confint(b2, model = "mu", pterms = FALSE, sterms = TRUE) lasso_plot(b1, which = 2, model = "mu", name = "mu.s.la(id).id", spar = FALSE) for(i in 1:8) { abline(h = ci[i, 1], lty = 2, col = "red") abline(h = ci[i, 2], lty = 2, col = "red") } ## End(Not run) ```

lasso function