Fit BAMLSS with Backfitting

Description

This optimizer function is a generic tool for fitting BAMLSS using a backfitting algorithm. The backfitting procedure is based on iteratively weighted least squares (IWLS) for finding posterior mode estimates, however, the updating methods for model terms can be more general, see the details section. In addition, the default IWLS updating scheme implements optimum smoothing variance selection based on information criteria using a stepwise approach.

Usage

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## Optimizer function:
bfit(x, y, family, start = NULL, weights = NULL, offset = NULL,
  update = "iwls", criterion = c("AICc", "BIC", "AIC"),
  eps = .Machine$double.eps^0.25, maxit = 400,
  outer = FALSE, inner = FALSE, mgcv = FALSE,
  verbose = TRUE, digits = 4, flush = TRUE, ...)

## Model term updating functions:
bfit_iwls(x, family, y, eta, id, weights, criterion, ...)
bfit_iwls_Matrix(x, family, y, eta, id, weights, criterion, ...)
bfit_iwls_spam(x, family, y, eta, id, weights, criterion, ...)
bfit_lm(x, family, y, eta, id, weights, criterion, ...)
bfit_optim(x, family, y, eta, id, weights, criterion, ...)

Arguments

x

For function bfit() 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 updating functions an object as returned from function smooth.construct or smoothCon.

y

The model response, as returned from function bamlss.frame.

family

A bamlss family object, see family.bamlss.

start

A named numeric vector containing possible starting values, the names are based on function parameters.

weights

Prior weights on the data, as returned from function bamlss.frame.

offset

Can be used to supply model offsets for use in fitting, returned from function bamlss.frame.

update

Sets the updating function for model terms, e.g. for a term s(x) in the model formula. Per default this is set to "iwls", a character pointing to the set of updating functions, see above. Other options are "optim" and "lm" etc., however, this is more experimental and should not be set by the user. Another option is to pass a full updating function which should be used for each model term, the structure of updating functions is described in the details below. Model terms may also have different updating functions, see the example section implementing a new model term constructor for Gompertz growth curves using this feature.

criterion

Set the information criterion that should be used, e.g., for smoothing variance selection. Options are the corrected AIC "AICc", the "BIC" and "AIC".

eps

The relative convergence tolerance of the backfitting algorithm.

maxit

The maximum number of iterations for the backfitting algorithm

outer

Should the current working observations and weights be computed in one outer iteration, otherwise the working observations are computed anew for each model term updating step.

inner

Should the model terms for one parameter of the modeled distribution be fully updated until convergence in an inner iteration, i.e., the algorithm waits until coefficients for the current distribution parameter do not change anymore before updating the next parameter.

mgcv

Should the mgcv gam function be used for computing updates in an inner iteration with working observations provided in an outer iteration.

verbose

Print information during runtime of the algorithm.

digits

Set the digits for printing when verbose = TRUE.

flush

use flush.console for displaying the current output in the console.

eta

The current value of the predictors, provided as a named list, one list entry for each parameter. The names correspond to the parameter names in the family object, see family.bamlss. E.g., when using the gaussian_bamlss family object, the current values for the mean can be extracted by eta\$mu and for the standard deviation by eta\$sigma.

id

Character, the name of the current parameter for which the model term should be updated.

...

For function bfit(), arguments passed to function bamlss.engine.setup. For updating functions, within the dots argument the actual iteration number of the backfitting algorithm, the actual total number of equivalent degrees of freedom edf and vectors z and hess only if argument outer = TRUE are provided.

Details

This algorithm is based on iteratively weighted least squares (IWLS) for BAMLSS, i.e., a Newton-Raphson or Fisher scoring algorithm is applied, similar to Rigby~and~Stasinopoulos~(2005). The algorithm utilizes the chain rule for computing derivatives of the log-posterior w.r.t. regression coefficients, therefore, to compute the working observations and weights only the derivatives of the log-likelihood w.r.t. the linear predictors are required.

It is assumed that the provided family object holds functions for computing the first and second order derivatives of the log-likelihood w.r.t. the linear predictors. These Functions are provided within the named lists "score" and "hess" within the family object. See the documentation of family.bamlss and the code of the provided families, e.g. gaussian_bamlss, for examples of the required structure.

The algorithm either updates each model term over all distributional parameters sequentially, or does a full update until convergence for model terms for one distributional parameter before updating the next parameter, see argument inner. Additionally, working observations and weights can be computed only once in an outer iteration.

Starting values of regression coefficients and smoothing variances can be supplied, moreover, if a family object holds functions for initializing the distributional parameters, see also family.bamlss, starting values are based on the initialize functions.

The default updating function for model terms is based on IWLS, which is assigned by function bamlss.engine.setup, however, special updating functions can be used. This is achieved by providing an updating function to argument update, which should be used for all model terms. Another option is to set the updating function within the xt argument of the mgcv smooth term constructor functions, see e.g. function s. If the xt list then holds an element named "update", which is a valid updating function, this updating function is used for the corresponding model term. This way it is possible to call different (special) updating functions for specific terms, e.g., that do not fit in the IWLS scheme. See the examples below. Note that this does not work if mgcv = TRUE, since the gam function assumes a strict linear representation of smooth terms.

A model term updating function has the following arguments:

update(x, family, y, eta, id, weights, criterion, ...)

Here x is an object as returned from function smooth.construct or smoothCon. The x object is preprocessed by function bamlss.engine.setup, i.e., an element called "state" is assigned. The state element represents the current state of the model term holding the current values of the parameters with corresponding fitted values, as well as equivalent degrees of freedom, see also the values that are returned by such functions below. The backfitting algorithm uses the state of a model term for generating updates of the parameters. Note that for special model terms the state list should already be provided within the call to the corresponding smooth constructor function, see the growth curve example below.

In addition, for special model terms the fitted values may not be computed by a linear combination of the design matrix and the coefficients. Therefore, the x object should hold an element named "fit.fun" which is a function for computing the fitted values. See also smooth.construct.bamlss.frame and predict.bamlss that use this setup. The arguments of fitting functions are

fit.fun(X, b, ...)

where X is the design matrix and b is the vector of coefficients. Hence, for usual IWLS updating the fitted values are computed by X %*% b. For special terms like nonlinear growth curves this may not be the case, see the example below. The fitting functions are assigned by bamlss.engine.setup, unless the function is already provided after calling the constructor function smooth.construct or smoothCon. Note that the dots argument is usually not needed by the user.

The default updating function is bfit_iwls(). Functions bfit_iwls_Matrix() and bfit_iwls_spam() use the sparse matrix infrastructures of package Matrix and spam. The Matrix package and bfit_iwls_Matrix() is used for model terms where the maximum number of non-zero entries in the design matrix is less than half of the total number of columns, if an additional argument force.spam is set to TRUE in the bfit() call. Similarly the spam package and bfit_iwls_spam() is used if force.spam = TRUE is supplied in the bfit() call.

The IWLS updating functions find optimum smoothing variances according to an information criterion using a stepwise approach, i.e., in each iteration and for each model term update the updating functions try to find a better smoothing variance to control the trade-off between over-smoothing and nonlinear functional estimation. The search interval is centered around the current state of the smoothing variances, hence, in each iteration only a slight improvement is achieved. This algorithm is based on Belitz~and~Lang~(2008) and can also be viewed as a boosting approach for optimization.

Value

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

fitted.values

A named list of the fitted values of the modeled parameters of the selected distribution.

parameters

The estimated set regression coefficients and smoothing variances.

edf

The equivalent degrees of freedom used to fit the model.

logLik

The value of the log-likelihood.

logPost

The value of the log-posterior.

IC

The value of the information criterion.

converged

Logical, indicating convergence of the backfitting algorithm.

For updating functions a list providing the current state
fitted.values

The resulting fitted values after updating.

parameters

The resulting named numeric vector of updated model term parameters. Coefficients should be named with "b1", ..., "bk", where k is the total number of coefficients. Smoothing variances should be named with "tau21", ..., "tau2m", where m is the total number of smoothing variances assigned to the model term.

edf

The equivalent degrees of freedom used to produce the fitted values.

hessian

Optional, the coefficient Hessian information

log.prior

Optional, the value of the log-prior of the model term.

References

Belitz C, Lang S (2008). Simultaneous Selection of Variables and Smoothing Parameters in Structured Additive Regression Models. Computational Statistics \& Data Analysis, 53, pp 61-81.

Umlauf N, Klein N, Zeileis A (2016). Bayesian Additive Models for Location Scale and Shape (and Beyond). (to appear)

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape, (with discussion), Appl. Statist., 54, part 3, pp 507-554.

See Also

bamlss, bamlss.frame, bamlss.engine.setup, set.starting.values, s2

Examples

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## Not run: ## Simulated data example illustrating
## how to call the optimizer function.
## This is done internally within
## the setup of function bamlss().
d <- GAMart(n = 200)
f <- num ~ s(x1) + s(x2) + s(x3)
bf <- bamlss.frame(f, data = d, family = "gaussian")
opt <- with(bf, bfit(x, y, family))
print(str(opt))

## Same with bamlss().
b <- bamlss(f, data = d, family = "gaussian", sampler = FALSE)
plot(b)
summary(b)

## Use of different updating function.
b <- bamlss(f, data = d, family = "gaussian",
  sampler = FALSE, update = bfit_lm)
plot(b)

## Use mgcv gam() function for updating.
b <- bamlss(f, data = d, family = "gaussian",
  sampler = FALSE, mgcv = TRUE)
plot(b)

## Special smooth constructor including updating/sampler
## function for nonlinear Gompertz curves.
## Note: element special.npar is needed here since this
##       function has 3 parameters but the design matrix only
##       one column!
smooth.construct.gc.smooth.spec <- function(object, data, knots) 
{
  object$X <- matrix(as.numeric(data[[object$term]]), ncol = 1)
  center <- if(!is.null(object$xt$center)) {
    object$xt$center
  } else FALSE
  object$by.done <- TRUE
  if(object$by != "NA")
    stop("by variables not supported!")
  object$fit.fun <- function(X, b, ...) {
    f <- b[1] * exp(-b[2] * exp(-b[3] * drop(X)))
    if(center)
      f <- f - mean(f)
    f
  }
  object$update <- bfit_optim
  object$propose <- GMCMC_slice
  object$prior <- function(b) { sum(dnorm(b, sd = 1000, log = TRUE)) }
  object$fixed <- TRUE
  object$state$parameters <- c("b1" = 0, "b2" = 0.5, "b3" = 0.1)
  object$state$fitted.values <- rep(0, length(object$X))
  object$state$edf <- 3
  object$special.npar <- 3 ## Important!
  class(object) <- c("gc.smooth", "no.mgcv", "special")
  object
}

## Work around for the "prediction matrix" of a growth curve.
Predict.matrix.gc.smooth <- function(object, data, knots) 
{
  X <- matrix(as.numeric(data[[object$term]]), ncol = 1)
  X
}

## Heteroscedastic growth curve data example.
set.seed(123)

d <- data.frame("time" = 1:30)
d$y <- 2 + 1 / (1 + exp(0.5 * (15 - d$time))) +
  rnorm(30, sd = exp(-4 + 0.1 * d$time))

## Special model terms must be called with s2()!
f <- list(
  "mu" = y ~ s2(time, bs = "gc"),
  "sigma" = ~ s(time)
)

## Fit model with special model term.
b <- bamlss(f, data = d,
  optimizer = bfit, sampler = GMCMC)

## Plot the fitted curves.
plot(b)

## Predict with special model term.
nd <- data.frame("time" = seq(1, 30, length = 100))
p <- predict(b, newdata = nd, model = "mu", FUN = c95)
plot(d, ylim = range(c(d$y, p)))
matplot(nd$time, p, type = "l",
  lty = c(2, 1, 2), col = "black", add = TRUE)

## End(Not run)

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