This is the main model fitting function of the package. Function bamlss() is a wrapper function that parses the data and the model formula, or extended bamlss.formula, as well as the bamlss.family into a bamlss.frame. The bamlss.frame then holds all model matrices and information that is needed for setting up estimation engines. The model matrices are based on mgcv infrastructures, i.e., smooth terms are constructed using smooth.construct and smoothCon. Therefore, all mgcv model term constructors like s, te, t2 and ti can be used. Identifiability conditions are imposed using function gam.side.

After the bamlss.frame is set up function bamlss() applies optimizer and/or sampling functions. These functions can also be provided by the user. See the details below on how to create new engines to be used with function bamlss().

Finally, the estimated parameters and/or samples are used to create model output results like summary statistics or effect plots. The computation of results may also be controlled by the user.

bamlss(formula, family = "gaussian", data = NULL,
start = NULL, knots = NULL, weights = NULL,
subset = NULL, offset = NULL, na.action = na.omit,
contrasts = NULL, reference = NULL, transform = NULL,
optimizer = NULL, sampler = NULL, samplestats = NULL,
results = NULL, cores = NULL, sleep = NULL,
combine = TRUE, model = TRUE, x = TRUE,
light = FALSE, ...)

## Arguments

formula A formula or extended formula, i.e., the formula can be a list of formulas where each list entry specifies the details of one parameter of the modeled response distribution, see bamlss.formula. For incorporating smooth terms, all model term constructors implemented in mgcv such as s, te and ti can be used, amongst others. A bamlss.family object, specifying the details of the modeled distribution such as the parameter names, the density function, link functions, etc. Can be a character without the "_bamlss" extension of the bamlss.family name. A data.frame or list containing the model response variable(s) and covariates specified in the formula. By default the variables are taken from environment(formula): typically the environment from which bamlss is called. A named numeric vector containing starting values to be send to the optimizer and/or sampler function. For a possible naming convention for the parameters see function parameters, but this is not restrictive and engine specific. An optional list containing user specified knots, see the documentation of function gam. Prior weights on the data. An optional vector specifying a subset of observations to be used in the fitting process. Can be used to supply model offsets for use in fitting. A function which indicates what should happen when the data contain NA's. The default is set by the na.action setting of options, and is na.omit if set to NULL. An optional list. See the contrasts.arg of model.matrix.default. A character specifying a reference category, e.g., when fitting a multinomial model. A transformer function that is applied on the bamlss.frame. See, e.g., function randomize and bamlss.engine.setup. An optimizer function that returns, e.g., posterior mode estimates of the parameters as a named numeric vector. The default optimizer function is bfit. If set to FALSE, no optimizer function will be used. A sampler function that returns a matrix of samples, the columns represent the parameters, the rows the iterations. The returned matrix must be coerced to an object of class "mcmc", see as.mcmc. The default sampler function is GMCMC. If set to FALSE, no sampler function will be used. A function computing statistics from samples, per default function samplestats is used. If set to FALSE, no samplestats function will be used. Note that this option is crucial for very large datasets, as computing statistics from samples this way may be very time consuming! A function computing results from the parameters and/or samples, e.g., for creating effect plots, see function link{results.bamlss.default}. If set FALSE no results function will be used. An integer specifying the number of cores that should be used for the sampler function. This is based on function mclapply of the parallel package. Time the system should sleep before the next core is started. If samples are computed on multiple cores, should the samples be combined into one mcmc matrix? If set to FALSE the model frame used for modeling is not part of the return value. If set to FALSE the model matrices are not part of the return value. Should the returned object be lighter, i.e., if light = TRUE the returned object will not contain the model.frame and design and penalty matrices are deleted. Arguments passed to the transformer, optimizer, sampler, results and samplestats function.

## Details

The main idea of this function is to provide infrastructures that make it relatively easy to create estimation engines for new problems, or write interfaces to existing software packages.

The steps that are performed within the function are:

• First, the function parses the data, the formula or the extended bamlss.formula as well as the bamlss.family into a model frame like object, the bamlss.frame. This object holds all necessary model matrices and information that is needed for subsequent model fitting engines. Per default, all package mgcv smooth term constructor functions like s, te, t2 and ti can be used (see also function smooth.construct), however, even special user defined constructors can be included, see the manual of bamlss.frame.

• In a second step, the bamlss.frame can be transformed, e.g., if a mixed model representation of smooth terms is needed, see function randomize.

• Then an optimizer function is started, e.g., a function that finds posterior mode estimates of the parameters. A convention for model fitting engines is that such functions should have the following arguments:

optimizer(x, y, family, start, weights, offset, ...)

Internally, function bamlss() will send the x object that holds all model matrices, the response y object, the family object, starting values for the parameters, possible weights and offsets of the created bamlss.frame to the optimizer function (see the manual of bamlss.frame for more details on the x, y and other objects). The job of the optimizer is to return a named numeric vector of optimum parameters. The names of the parameters should be such that they can be uniquely mapped to the corresponding model matrices in x. See function parameters for more details on parameter names. The default optimizer function is bfit. The optimizer can return more information than only the optimum parameters. It is possible to return a list, the convention here is that an element named "parameters" then holds the named vector of estimated parameters. Possible other return values could be fitted values, the Hessian matrix, information criteria or information about convergence of the algorithm, etc. Note that the parameters are simply added to the bamlss.frame in an (list) entry named parameters.

• After the optimization step, a sampler function is started. The arguments of such sampler functions are the same as for the optimizer functions

sampler(x, y, family, start, weights, offset, ...)

Sampler functions must return a matrix of samples, each row represents one iteration and the matrix can be coerced to mcmc objects. The function may return a list of samples, e.g., if multiple chains are returned each list entry then holds one sample matrix of one chain. The column names of the sample matrix should be the same as the names of estimated parameters. For a possible naming convention see function parameters, which ensures unique mapping of samples with the model matrices in the x object of the bamlss.frame. The samples are added to the bamlss.frame in an (list) entry named samples.

• Next, the samplestats function is applied. This function can compute any quantity from the samples and the x object, the arguments of such functions are

samplestats(samples, x, y, family, ...)

where argument samples are the samples returned from the sampler function, and x, y and family are the same objects as passed to the optimizer and or sampler functions. For example, the default function in bamlss() for this task is also called samplestats and returns the mean of the log-likelihood and the log-posterior computed of all samples, as well as the DIC.

• The last step is to compute more complex information about the model using the results function. The arguments of such results functions are

results(bamlss.frame, ...)

here, the full bamlss.frame including possible parameters and samples is passed to the function within bamlss(). The default function for this task is results.bamlss.default which returns an object of class "bamlss.results" for which generic plotting functions are and a summary function is provided. Hence, the user can control the output of the model, the plotting and summary statistics, too.

Note that function transform(), optimizer(), sampler(), samplestats() and results() can be provided from the bamlss.family object, e.g., if a bamlss.family object has an element named "optimizer", which represents a valid optimizer function such as bfit, exactly this optimizer function will be used as a default when using the family.

## Value

An object of class "bamlss". The object is in principle only a slight extension of a bamlss.frame, i.e., if an optimizer is applied it will hold the estimated parameters in an additional element named "parameters". If a sampler function is applied it will additionally hold the samples in an element named "samples". The same mechanism is used for results function.

If the optimizer function computes additional output next to the parameters, this will be saved in an element named "model.stats". If a samplestats function is applied, the output will also be saved in the "model.stats" element.

Additionally, all functions that are called are saved as attribute "functions" in the returned object.

## References

Umlauf N, Klein N, Zeileis A (2019). BAMLSS: Bayesian Additive Models for Location, Scale and Shape (and Beyond). Journal of Computational and Graphical Statistics, 27(3), 612--627. doi: 10.1080/10618600.2017.1407325

Umlauf N, Klein N, Simon T, Zeileis A (2019). “bamlss: A Lego Toolbox for Flexible Bayesian Regression (and Beyond).” arXiv:1909.11784, arXiv.org E-Print Archive. https://arxiv.org/abs/1909.11784

bamlss.frame, family.bamlss, bamlss.formula, randomize, bamlss.engine.setup, bfit, GMCMC, continue, coef.bamlss, parameters, predict.bamlss, plot.bamlss

## Examples

# NOT RUN {
## Simulated data example.
d <- GAMart()
f <- num ~ s(x1) + s(x2) + s(x3) + te(lon, lat)
b <- bamlss(f, data = d)
summary(b)
plot(b)
plot(b, which = 3:4)
plot(b, which = "samples")

## Use of optimizer and sampler functions:
## * first run optimizer,
b1 <- bamlss(f, data = d, optimizer = bfit, sampler = FALSE)
print(b1)
summary(b1)

## * afterwards, start sampler with staring values,
b2 <- bamlss(f, data = d, start = coef(b1), optimizer = FALSE, sampler = GMCMC)
print(b2)
summary(b2)

## Continue sampling.
b3 <- continue(b2, n.iter = 12000, burnin = 0, thin = 10)
plot(b3, which = "samples")
plot(b3, which = "max-acf")
plot(b3, which = "max-acf", burnin = 500, thin = 4)
# }