mboostLSS: Fitting GAMLSS by Boosting

Description

Functions for fitting GAMLSS (generalized additive models for location, scale and shape) using boosting techniques. Two algorithms are implemented: (a) The cyclic algorithm iteratively rotates between the distribution parameters, updating one while using the current fits of the others as offsets (for details see Mayr et al., 2012). (b) The noncyclic algorithm selects in each step the update of a base-learner for the distribution parameter that best fits the negative gradient (algorithm with inner loss of Thomas et al., 2018).

Usage

mboostLSS(formula, data = list(), families = GaussianLSS(),
          control = boost_control(), weights = NULL, 
          method = c("cyclic", "noncyclic"), ...)
glmboostLSS(formula, data = list(), families = GaussianLSS(),
            control = boost_control(), weights = NULL, 
            method = c("cyclic", "noncyclic"), ...)
gamboostLSS(formula, data = list(), families = GaussianLSS(),
            control = boost_control(), weights = NULL, 
            method = c("cyclic", "noncyclic"), ...)
blackboostLSS(formula, data = list(), families = GaussianLSS(),
              control = boost_control(), weights = NULL,
              method = c("cyclic", "noncyclic"), ...)
## fit function:
mboostLSS_fit(formula, data = list(), families = GaussianLSS(),
              control = boost_control(), weights = NULL,
              fun = mboost, funchar = "mboost", call = NULL, method, ...)

Value

An object of class mboostLSS or nc_mboostLSS (inheriting from class mboostLSS) for models fitted with method = "cyclic"

and method = "non-cyclic", respectively, with corresponding methods to extract information. A mboostLSS model object is a named list with one list entry for each modelled distribution parameter. Special "subclasses" inheriting from mboostLSS exist for each of the model-types (with the same name as the function, e.g., gamboostLSS).

Arguments

formula: a symbolic description of the model to be fit. See mboost for details. If formula is a single formula, the same formula is used for all distribution parameters. formula can also be a (named) list, where each list element corresponds to one distribution parameter of the GAMLSS distribution. The names must be the same as in the family (see example for details).
data: a data frame containing the variables in the model.
families: an object of class families. It can be either one of the pre-defined distributions that come along with the package or a new distribution specified by the user (see Families for details). Per default, we use the two-parametric GaussianLSS family.
control: a list of parameters controlling the algorithm. For more details see boost_control.
weights: a numeric vector of weights (optional).
method: fitting method, currently two methods are supported: "cyclic" (see Mayr et al., 2012) and "noncyclic" (algorithm with inner loss of Thomas et al., 2018). The latter requires a one dimensional mstop value.
fun: fit function. Either mboost, glmboost, gamboost or blackboost. Specified directly via the corresponding LSS function. E.g. gamboostLSS() calls mboostLSS_fit(..., fun = gamboost).
funchar: character representation of fit function. Either "mboost", "glmboost", "gamboost" or "blackboost". Specified directly via the corresponding LSS function.
call: used to forward the call from mboostLSS, glmboostLSS, gamboostLSS and blackboostLSS. This argument should not be directly specified by users!
...: Further arguments to be passed to mboostLSS_fit. In mboostLSS_fit, ... represent further arguments to be passed to mboost and mboost_fit. So ... can be all arguments of mboostLSS_fit and mboost_fit.

Details

For information on GAMLSS theory see Rigby and Stasinopoulos (2005) or the information provided at https://www.gamlss.com/. For a tutorial on gamboostLSS see Hofner et al. (2016). Thomas et al. (2018) developed a novel non-cyclic approach to fit gamboostLSS models. This approach is suitable for the combination with stabsel and speeds up model tuning via cvrisk (see also below).

glmboostLSS uses glmboost to fit the distribution parameters of a GAMLSS -- a linear boosting model is fitted for each parameter.

gamboostLSS uses gamboost to fit the distribution parameters of a GAMLSS -- an additive boosting model (by default with smooth effects) is fitted for each parameter. With the formula argument, a wide range of different base-learners can be specified (see baselearners). The base-learners imply the type of effect each covariate has on the corresponding distribution parameter.

mboostLSS uses mboost to fit the distribution parameters of a GAMLSS. The type of model (linear, tree-based or smooth) is specified by fun.

blackboostLSS uses blackboost to fit the distribution parameters of a GAMLSS -- a tree-based boosting model is fitted for each parameter.

mboostLSS, glmboostLSS, gamboostLSS and blackboostLSS all call mboostLSS_fit while fun is the corresponding mboost function, i.e., the same function without LSS. For further possible arguments see these functions as well as mboost_fit. Note that mboostLSS_fit is usually not called directly by the user.

For method = "cyclic" it is possible to specify one or multiple mstop and nu values via boost_control. In the case of one single value, this value is used for all distribution parameters of the GAMLSS model. Alternatively, a (named) vector or a (named) list with separate values for each component can be used to specify a separate value for each parameter of the GAMLSS model. The names of the list must correspond to the names of the distribution parameters of the GAMLSS family. If no names are given, the order of the mstop or nu values is assumed to be the same as the order of the components in the families. For one-dimensional stopping, the user therefore can specify, e.g., mstop = 100 via boost_control. For more-dimensional stopping, one can specify, e.g., mstop = list(mu = 100, sigma = 200) (see examples).

If method is set to "noncyclic", mstop has to be a one dimensional integer. Instead of cycling through all distribution parameters, in each iteration only the best base-learner is used. One base-learner of every parameter is selected via RSS, the distribution parameter is then chosen via the loss (in Thomas et. al., 2018, called inner loss). For details on the noncyclic fitting method see Thomas et. al. (2018).

To (potentially) stabilize the model estimation by standardizing the negative gradients one can use the argument stabilization of the families. See Families for details.

References

B. Hofner, A. Mayr, M. Schmid (2016). gamboostLSS: An R Package for Model Building and Variable Selection in the GAMLSS Framework. Journal of Statistical Software, 74(1), 1-31.

Available as vignette("gamboostLSS_Tutorial").

Mayr, A., Fenske, N., Hofner, B., Kneib, T. and Schmid, M. (2012): Generalized additive models for location, scale and shape for high-dimensional data - a flexible approach based on boosting. Journal of the Royal Statistical Society, Series C (Applied Statistics) 61(3): 403-427.

M. Schmid, S. Potapov, A. Pfahlberg, and T. Hothorn. Estimation and regularization techniques for regression models with multidimensional prediction functions. Statistics and Computing, 20(2):139-150, 2010.

Rigby, R. A. and D. M. Stasinopoulos (2005). Generalized additive models for location, scale and shape (with discussion). Journal of the Royal Statistical Society, Series C (Applied Statistics), 54, 507-554.

Buehlmann, P. and Hothorn, T. (2007), Boosting algorithms: Regularization, prediction and model fitting. Statistical Science, 22(4), 477--505.

Thomas, J., Mayr, A., Bischl, B., Schmid, M., Smith, A., and Hofner, B. (2018), Gradient boosting for distributional regression - faster tuning and improved variable selection via noncyclical updates. Statistics and Computing. 28: 673-687. tools:::Rd_expr_doi("10.1007/s11222-017-9754-6")
(Preliminary version: https://arxiv.org/abs/1611.10171).

Examples

Run this code


### Data generating process:
set.seed(1907)
x1 <- rnorm(1000)
x2 <- rnorm(1000)
x3 <- rnorm(1000)
x4 <- rnorm(1000)
x5 <- rnorm(1000)
x6 <- rnorm(1000)
mu    <- exp(1.5 +1 * x1 +0.5 * x2 -0.5 * x3 -1 * x4)
sigma <- exp(-0.4 * x3 -0.2 * x4 +0.2 * x5 +0.4 * x6)
y <- numeric(1000)
for( i in 1:1000)
    y[i] <- rnbinom(1, size = sigma[i], mu = mu[i])
dat <- data.frame(x1, x2, x3, x4, x5, x6, y)

### linear model with y ~ . for both components: 400 boosting iterations
model <- glmboostLSS(y ~ ., families = NBinomialLSS(), data = dat,
                     control = boost_control(mstop = 400),
                     center = TRUE)
coef(model, off2int = TRUE)


### estimate model with different formulas for mu and sigma:
names(NBinomialLSS())      # names of the family

### Do not test the following code per default on CRAN as it takes some time to run:
# Note: Multiple formulas must be specified via a _named list_
#       where the names correspond to the names of the distribution parameters
#       in the family (see above)
model2 <- glmboostLSS(formula = list(mu = y ~ x1 + x2 + x3 + x4,
                                    sigma = y ~ x3 + x4 + x5 + x6),
                     families = NBinomialLSS(), data = dat,
                     control = boost_control(mstop = 400, trace = TRUE),
                     center = TRUE)
coef(model2, off2int = TRUE)
### END (don't test automatically)



### Offset needs to be specified via the arguments of families object:
model <- glmboostLSS(y ~ ., data = dat,
                     families = NBinomialLSS(mu = mean(mu),
                                             sigma = mean(sigma)),
                     control = boost_control(mstop = 10),
                     center = TRUE)
# Note: mu-offset = log(mean(mu)) and sigma-offset = log(mean(sigma))
#       as we use a log-link in both families
coef(model)
log(mean(mu))
log(mean(sigma))

### Do not test the following code per default on CRAN as it takes some time to run:
### use different mstop values for the two distribution parameters
### (two-dimensional early stopping)
### the number of iterations is passed to boost_control via a named list
model3 <- glmboostLSS(formula = list(mu = y ~ x1 + x2 + x3 + x4,
                                    sigma = y ~ x3 + x4 + x5 + x6),
                     families = NBinomialLSS(), data = dat,
                     control = boost_control(mstop = list(mu = 400,
                                                          sigma = 300),
                                             trace  = TRUE),
                     center = TRUE)
coef(model3, off2int = TRUE)

### Alternatively we can change mstop of model2:
# here it is assumed that the first element in the vector corresponds to
# the first distribution parameter of model2 etc.
mstop(model2) <- c(400, 300)
par(mfrow = c(1,2))
plot(model2, xlim = c(0, max(mstop(model2))))
## all.equal(coef(model2), coef(model3)) # same!
### END (don't test automatically)

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