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# Part of the rstanarm package for estimating model parameters
# Copyright (C) 2016 Simon N. Wood
# Copyright (C) 2015, 2016, 2017 Trustees of Columbia University
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU General Public License
# as published by the Free Software Foundation; either version 3
# of the License, or (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program; if not, write to the Free Software
# Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
#' Bayesian generalized linear additive models with optional group-specific
#' terms via Stan
#'
#' \if{html}{\figure{stanlogo.png}{options: width="25px" alt="http://mc-stan.org/about/logo/"}}
#' Bayesian inference for GAMMs with flexible priors.
#'
#' @export
#' @templateVar fun stan_gamm4
#' @templateVar pkg gamm4
#' @templateVar pkgfun gamm4
#' @template return-stanreg-object
#' @template see-also
#' @template args-prior_intercept
#' @template args-priors
#' @template args-prior_aux
#' @template args-prior_smooth
#' @template args-prior_PD
#' @template args-algorithm
#' @template args-adapt_delta
#' @template args-QR
#' @template args-sparse
#'
#' @param formula,random,family,data,knots,drop.unused.levels Same as for
#' \code{\link[gamm4]{gamm4}}. \emph{We strongly advise against
#' omitting the \code{data} argument}. Unless \code{data} is specified (and is
#' a data frame) many post-estimation functions (including \code{update},
#' \code{loo}, \code{kfold}) are not guaranteed to work properly.
#' @param subset,weights,na.action Same as \code{\link[stats]{glm}},
#' but rarely specified.
#' @param ... Further arguments passed to \code{\link[rstan:stanmodel-method-sampling]{sampling}} (e.g.
#' \code{iter}, \code{chains}, \code{cores}, etc.) or to
#' \code{\link[rstan:stanmodel-method-vb]{vb}} (if \code{algorithm} is \code{"meanfield"} or
#' \code{"fullrank"}).
#' @param prior_covariance Cannot be \code{NULL}; see \code{\link{decov}} for
#' more information about the default arguments.
#'
#' @details The \code{stan_gamm4} function is similar in syntax to
#' \code{\link[gamm4]{gamm4}} in the \pkg{gamm4} package. But rather than performing
#' (restricted) maximum likelihood estimation with the \pkg{lme4} package,
#' the \code{stan_gamm4} function utilizes MCMC to perform Bayesian
#' estimation. The Bayesian model adds priors on the common regression
#' coefficients (in the same way as \code{\link{stan_glm}}), priors on the
#' standard deviations of the smooth terms, and a prior on the decomposition
#' of the covariance matrices of any group-specific parameters (as in
#' \code{\link{stan_glmer}}). Estimating these models via MCMC avoids
#' the optimization issues that often crop up with GAMMs and provides better
#' estimates for the uncertainty in the parameter estimates.
#'
#' See \code{\link[gamm4]{gamm4}} for more information about the model
#' specicification and \code{\link{priors}} for more information about the
#' priors on the main coefficients. The \code{formula} should include at least
#' one smooth term, which can be specified in any way that is supported by the
#' \code{\link[mgcv]{jagam}} function in the \pkg{mgcv} package. The
#' \code{prior_smooth} argument should be used to specify a prior on the unknown
#' standard deviations that govern how smooth the smooth function is. The
#' \code{prior_covariance} argument can be used to specify the prior on the
#' components of the covariance matrix for any (optional) group-specific terms.
#' The \code{\link[gamm4]{gamm4}} function in the \pkg{gamm4} package uses
#' group-specific terms to implement the departure from linearity in the smooth
#' terms, but that is not the case for \code{stan_gamm4} where the group-specific
#' terms are exactly the same as in \code{\link{stan_glmer}}.
#'
#' The \code{plot_nonlinear} function creates a ggplot object with one facet for
#' each smooth function specified in the call to \code{stan_gamm4} in the case
#' where all smooths are univariate. A subset of the smooth functions can be
#' specified using the \code{smooths} argument, which is necessary to plot a
#' bivariate smooth or to exclude the bivariate smooth and plot the univariate
#' ones. In the bivariate case, a plot is produced using
#' \code{\link[ggplot2]{geom_contour}}. In the univariate case, the resulting
#' plot is conceptually similar to \code{\link[mgcv]{plot.gam}} except the
#' outer lines here demark the edges of posterior uncertainty intervals
#' (credible intervals) rather than confidence intervals and the inner line
#' is the posterior median of the function rather than the function implied
#' by a point estimate. To change the colors used in the plot see
#' \code{\link[bayesplot:bayesplot-colors]{color_scheme_set}}.
#'
#' @references
#' Crainiceanu, C., Ruppert D., and Wand, M. (2005). Bayesian analysis for
#' penalized spline regression using WinBUGS. \emph{Journal of Statistical
#' Software}. \strong{14}(14), 1--22.
#' \url{https://www.jstatsoft.org/article/view/v014i14}
#'
#' @seealso The vignette for \code{stan_glmer}, which also discusses
#' \code{stan_gamm4}. \url{http://mc-stan.org/rstanarm/articles/}
#'
#' @examples
#' # from example(gamm4, package = "gamm4"), prefixing gamm4() call with stan_
#' \donttest{
#' dat <- mgcv::gamSim(1, n = 400, scale = 2) ## simulate 4 term additive truth
#' ## Now add 20 level random effect `fac'...
#' dat$fac <- fac <- as.factor(sample(1:20, 400, replace = TRUE))
#' dat$y <- dat$y + model.matrix(~ fac - 1) %*% rnorm(20) * .5
#'
#' br <- stan_gamm4(y ~ s(x0) + x1 + s(x2), data = dat, random = ~ (1 | fac),
#' chains = 1, iter = 500) # for example speed
#' print(br)
#' plot_nonlinear(br)
#' plot_nonlinear(br, smooths = "s(x0)", alpha = 2/3)
#' }
#'
stan_gamm4 <-
function(formula,
random = NULL,
family = gaussian(),
data,
weights = NULL,
subset = NULL,
na.action,
knots = NULL,
drop.unused.levels = TRUE,
...,
prior = default_prior_coef(family),
prior_intercept = default_prior_intercept(family),
prior_smooth = exponential(autoscale = FALSE),
prior_aux = exponential(autoscale=TRUE),
prior_covariance = decov(),
prior_PD = FALSE,
algorithm = c("sampling", "meanfield", "fullrank"),
adapt_delta = NULL,
QR = FALSE,
sparse = FALSE) {
data <- validate_data(data, if_missing = list())
family <- validate_family(family)
if (!is.null(random)) {
fake.formula <- as.character(mgcv::interpret.gam(formula)$fake.formula)
form <- paste(fake.formula[2], fake.formula[1], fake.formula[3],
"+", random[2], collapse = " ")
glmod <- lme4::glFormula(as.formula(form), data, family = gaussian,
subset, weights, na.action,
control = make_glmerControl(
ignore_x_scale = prior$autoscale %ORifNULL% FALSE
)
)
data <- glmod$fr
weights <- validate_weights(glmod$fr$weights)
}
else {
weights <- validate_weights(weights)
glmod <- NULL
}
if (family$family == "binomial") {
data$temp_y <- rep(1, NROW(data)) # work around jagam bug
temp_formula <- update(formula, temp_y ~ .)
jd <- mgcv::jagam(formula = temp_formula, family = gaussian(), data = data,
file = tempfile(fileext = ".jags"), weights = NULL,
na.action = na.action, offset = NULL, knots = knots,
drop.unused.levels = drop.unused.levels, diagonalize = TRUE)
if (!is.null(random)) {
y <- data[, as.character(formula[2L])]
} else {
y <- eval(formula[[2L]], data)
}
if (binom_y_prop(y, family, weights)) {
y1 <- as.integer(as.vector(y) * weights)
y <- cbind(y1, y0 = weights - y1)
weights <- double(0)
}
} else {
jd <- mgcv::jagam(formula = formula, family = gaussian(), data = data,
file = tempfile(fileext = ".jags"), weights = NULL,
na.action = na.action, offset = NULL, knots = knots,
drop.unused.levels = drop.unused.levels, diagonalize = TRUE)
y <- jd$jags.data$y
}
# there is no offset allowed by gamm4::gamm4
offset <- validate_offset(as.vector(model.offset(jd$pregam$model)), y = y)
X <- jd$jags.data$X
mark <- which(colnames(X) != "")
colnames(X) <- colnames(jd$pregam$X) <- jd$pregam$term.names
S <- lapply(jd$pregam$smooth, FUN = function(s) {
ranks <- s$rank
start <- s$first.para
out <- list()
for (r in seq_along(ranks)) {
end <- start + ranks[r] - 1L
out[[r]] <- X[,start:end, drop = FALSE]
start <- end + 1L
}
return(out)
})
if (any(sapply(S, length) > 1)) S <- unlist(S, recursive = FALSE)
names(S) <- names(jd$pregam$sp)
X <- X[,mark, drop = FALSE]
for (s in seq_along(S)) {
# sometimes elements of S are lists themselves that need to be unpacked
# before passing to stan_glm.fit (https://github.com/stan-dev/rstanarm/issues/362)
if (is.list(S[[s]]))
S[[s]] <- do.call(cbind, S[[s]])
}
X <- c(list(X), S)
if (is.null(prior)) prior <- list()
if (is.null(prior_intercept)) prior_intercept <- list()
if (is.null(prior_aux)) prior_aux <- list()
if (is.null(prior_smooth)) prior_smooth <- list()
if (is.null(random)) {
group <- list()
prior_covariance <- list()
}
else {
group <- glmod$reTrms
group$decov <- prior_covariance
}
algorithm <- match.arg(algorithm)
stanfit <- stan_glm.fit(x = X, y = y, weights = weights,
offset = offset, family = family,
prior = prior, prior_intercept = prior_intercept,
prior_aux = prior_aux, prior_smooth = prior_smooth,
prior_PD = prior_PD, algorithm = algorithm,
adapt_delta = adapt_delta, group = group, QR = QR, ...)
if (algorithm != "optimizing" && !is(stanfit, "stanfit")) return(stanfit)
if (family$family == "Beta regression") family$family <- "beta"
X <- do.call(cbind, args = X)
if (is.null(random)) Z <- Matrix::Matrix(nrow = NROW(y), ncol = 0, sparse = TRUE)
else {
Z <- pad_reTrms(Ztlist = group$Ztlist, cnms = group$cnms,
flist = group$flist)$Z
colnames(Z) <- b_names(names(stanfit), value = TRUE)
}
XZ <- cbind(X, Z)
# make jam object with point estimates, see ?mgcv::sim2jam
mat <- as.matrix(stanfit)
mark <- 1:ncol(X)
jd$pregam$Vp <- cov(mat[,mark, drop = FALSE])
jd$pregam$coefficients <- colMeans(mat[,mark, drop = FALSE])
jd$pregam$sig2 <- if ("sigma" %in% colnames(mat)) mean(mat[,"sigma"]) else 1
eta <- X %*% t(mat[,mark,drop = FALSE])
mu <- rowMeans(family$linkinv(eta))
eta <- rowMeans(eta)
w <- as.numeric(jd$pregam$w * family$mu.eta(eta) ^ 2 / family$variance(mu))
XWX <- t(X) %*% (w * X)
jd$pregam$edf <- rowSums(jd$pregam$Vp * t(XWX)) / jd$pregam$sig2
class(jd$pregam) <- c("jam", "gam")
fit <- nlist(stanfit, family, formula, offset, weights,
x = XZ, y = y, data, terms = jd$pregam$terms,
model = if (is.null(random)) jd$pregam$model else glmod$fr,
call = match.call(expand.dots = TRUE),
algorithm, glmod = glmod,
stan_function = "stan_gamm4")
out <- stanreg(fit)
out$jam <- jd$pregam
class(out) <- c(class(out), "gamm4", if (!is.null(glmod)) "lmerMod")
return(out)
}
#' @rdname stan_gamm4
#' @export
#' @param x An object produced by \code{stan_gamm4}.
#' @param smooths An optional character vector specifying a subset of the smooth
#' functions specified in the call to \code{stan_gamm4}. The default is
#' include all smooth terms.
#' @param prob For univarite smooths, a scalar between 0 and 1 governing the
#' width of the uncertainty interval.
#' @param facet_args An optional named list of arguments passed to
#' \code{\link[ggplot2]{facet_wrap}} (other than the \code{facets} argument).
#' @param alpha,size For univariate smooths, passed to
#' \code{\link[ggplot2]{geom_ribbon}}. For bivariate smooths, \code{size/2} is
#' passed to \code{\link[ggplot2]{geom_contour}}.
#'
#' @return \code{plot_nonlinear} returns a ggplot object.
#'
#' @importFrom ggplot2 aes_ aes_string facet_wrap ggplot geom_contour geom_line geom_ribbon labs scale_color_gradient2
#'
plot_nonlinear <- function(x, smooths, ...,
prob = 0.9, facet_args = list(),
alpha = 1, size = 0.75) {
validate_stanreg_object(x)
if (!is(x, "gamm4"))
stop("Plot only available for models fit using the stan_gamm4 function.")
on.exit(message("try plot(x$jam) instead"))
scheme <- bayesplot::color_scheme_get()
XZ <- x$x
XZ <- XZ[,!grepl("_NEW_", colnames(XZ), fixed = TRUE)]
labels <- sapply(x$jam$smooth, "[[", "label")
xnames <- sapply(x$jam$smooth, "[[", "vn")
names(x$jam$smooth) <- labels
names(xnames) <- labels
fs <- sapply(x$jam$smooth, FUN = "inherits", what = "fs.interaction")
if (!missing(smooths)) {
found <- smooths %in% labels
if (all(!found)) {
stop("All specified terms are invalid. Valid terms are: ",
paste(grep(",", labels, fixed = TRUE, value = TRUE, invert = TRUE),
collapse = ", "))
} else if (any(!found)) {
warning("The following specified terms were not found and ignored: ",
paste(smooths[!found], collapse = ", "))
}
labels <- smooths[found]
fs <- fs[found]
if (!is.matrix(xnames)) xnames <- xnames[found]
}
else smooths <- 1:length(labels)
B <- as.matrix(x)[, colnames(XZ), drop = FALSE]
original <- x$jam$model
bivariate <- any(grepl(",", labels, fixed = TRUE))
if (bivariate && !any(fs)) {
if (length(labels) > 1) {
on.exit(NULL)
stop("Multivariate functions can only be plotted one at a time; specify 'smooths'.")
}
if (length(xnames) > 2)
stop("Only univariate and bivariate functions can be plotted currently.")
xrange <- range(original[, xnames[1]])
yrange <- range(original[, xnames[2]])
xz <- expand.grid(seq(from = xrange[1], to = xrange[2], length.out = 100),
seq(from = yrange[1], to = yrange[2], length.out = 100))
colnames(xz) <- xnames[1:2]
plot_data <- data.frame(x = xz[, 1], y = xz[, 2])
nd <- original
nd <- nd[sample(nrow(xz), size = nrow(xz), replace = TRUE), ]
nd[[xnames[1]]] <- xz[[xnames[1]]]
nd[[xnames[2]]] <- xz[[xnames[2]]]
requireNamespace("mgcv", quietly = TRUE)
XZ <- predict(x$jam, newdata = nd, type = "lpmatrix")
incl <- grepl(labels, colnames(B), fixed = TRUE)
b <- B[, incl, drop = FALSE]
xz <- XZ[, grepl(labels, colnames(XZ), fixed = TRUE), drop = FALSE]
plot_data$z <- apply(linear_predictor.matrix(b, xz), 2, FUN = median)
return(
ggplot(plot_data, aes_(x = ~x, y = ~y, z = ~z)) +
geom_contour(aes_string(color = "..level.."), size = size/2) +
labs(x = xnames[1], y = xnames[2]) +
scale_color_gradient2(low = scheme[[1]],
mid = scheme[[3]],
high = scheme[[6]]) +
bayesplot::theme_default()
)
}
df_list <- lapply(x$jam$smooth[smooths], FUN = function(s) {
incl <- s$first.para:s$last.para
b <- B[, incl, drop = FALSE]
if (inherits(s, "fs.interaction")) { # see mgcv:::plot.fs.interaction
xx <- original[,s$base$term]
fac <- original[,s$fterm]
out <- by(data.frame(fac, xx), list(fac), FUN = function(df) {
df <- df[order(df[,2]),]
names(df) <- c(s$fterm, s$base$term)
xz <- mgcv::PredictMat(s, df)
f <- linear_predictor.matrix(b, xz)
data.frame(
predictor = df[,2],
lower = apply(f, 2, quantile, probs = (1 - prob) / 2),
upper = apply(f, 2, quantile, probs = prob + (1 - prob) / 2),
middle = apply(f, 2, median),
term = paste(s$label, df[,1], sep = ".")
)
})
do.call(rbind, args = out)
}
else {
xz <- XZ[, incl, drop = FALSE]
x <- original[, s$term]
ord <- order(x)
x <- x[ord]
xz <- xz[ord, , drop=FALSE]
if (!is.null(s$by.level)) {
fac <- original[,s$by][ord]
mark <- fac == s$by.level
x <- x[mark]
xz <- xz[mark, , drop = FALSE]
}
f <- linear_predictor.matrix(b, xz)
data.frame(
predictor = x,
lower = apply(f, 2, quantile, probs = (1 - prob) / 2),
upper = apply(f, 2, quantile, probs = prob + (1 - prob) / 2),
middle = apply(f, 2, median),
term = s$label
)
}
})
plot_data <- do.call(rbind, df_list)
facet_args[["facets"]] <- ~ term
if (is.null(facet_args[["scales"]]))
facet_args[["scales"]] <- "free"
if (is.null(facet_args[["strip.position"]]))
facet_args[["strip.position"]] <- "left"
on.exit(NULL)
ggplot(plot_data, aes_(x = ~ predictor)) +
geom_ribbon(aes_(ymin = ~ lower, ymax = ~ upper),
fill = scheme[[1]], color = scheme[[2]],
alpha = alpha, size = size) +
geom_line(aes_(y = ~ middle), color = scheme[[5]],
size = 0.75 * size, lineend = "round") +
labs(y = NULL) +
do.call(facet_wrap, facet_args) +
bayesplot::theme_default()
}
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