# la

##### 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.

- Keywords
- regression

##### Usage

```
## Smooth constructor function.
la(formula, type = c("single", "multiple"), ...)
```## Single Lasso smoothing parameter optimizer.
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 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 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

`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`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`lasso()`

optimizer function. For the optimizer`lasso()`

arguments passed to function`bfit`

.

##### Value

For function `la()`

, similar to function `s`

a simple smooth
specification object.

For function `lasso()`

a list containing the following objects:

A named list of the fitted values based on the last lasso iteration of the modeled parameters of the selected distribution.

A matrix, each row corresponds to the parameter values of one boosting iteration.

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*.
https://doi.org/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*.
http://dx.doi.org/10.1007/s11634-015-0205-y

##### See Also

##### Examples

```
# 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 = 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")
}
# }
```

*Documentation reproduced from package bamlss, version 1.1-2, License: GPL-2 | GPL-3*