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hstats.R
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hstats.R
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#' Calculate Interaction Statistics
#'
#' @description
#' This is the main function of the package. It does the expensive calculations behind
#' the following H-statistics:
#' - Total interaction strength \eqn{H^2}, a statistic measuring the proportion of
#' prediction variability unexplained by main effects of `v`, see [h2()] for details.
#' - Friedman and Popescu's statistic \eqn{H^2_j} of overall interaction strength per
#' feature, see [h2_overall()] for details.
#' - Friedman and Popescu's statistic \eqn{H^2_{jk}} of pairwise interaction strength,
#' see [h2_pairwise()] for details.
#' - Friedman and Popescu's statistic \eqn{H^2_{jkl}} of three-way interaction strength,
#' see [h2_threeway()] for details. To save time, this statistic is not calculated
#' by default. Set `threeway_m` to a value above 2 to get three-way statistics of the
#' `threeway_m` variables with strongest overall interaction.
#'
#' Furthermore, it allows to calculate an experimental partial dependence based
#' measure of feature importance, \eqn{\textrm{PDI}_j^2}. It equals the proportion of
#' prediction variability unexplained by other features, see [pd_importance()]
#' for details. This statistic is not shown by `summary()` or `plot()`.
#'
#' Instead of using `summary()`, interaction statistics can also be obtained via the
#' more flexible functions [h2()], [h2_overall()], [h2_pairwise()], and
#' [h2_threeway()].
#'
#' @param object Fitted model object.
#' @param X A data.frame or matrix serving as background dataset.
#' @param v Vector of feature names. The default (`NULL`) will use all column names of
#' `X` except the column name of the optional case weight `w` (if specified as name).
#' @param pred_fun Prediction function of the form `function(object, X, ...)`,
#' providing \eqn{K \ge 1} predictions per row. Its first argument represents the
#' model `object`, its second argument a data structure like `X`. Additional arguments
#' (such as `type = "response"` in a GLM, or `reshape = TRUE` in a multiclass XGBoost
#' model) can be passed via `...`. The default, [stats::predict()], will work in
#' most cases.
#' @param pairwise_m Number of features for which pairwise statistics are to be
#' calculated. The features are selected based on Friedman and Popescu's overall
#' interaction strength \eqn{H^2_j}. Set to to 0 to avoid pairwise calculations.
#' For multivariate predictions, the union of the `pairwise_m` column-wise
#' strongest variable names is taken. This can lead to very long run-times.
#' @param threeway_m Like `pairwise_m`, but controls the feature count for
#' three-way interactions. Cannot be larger than `pairwise_m`.
#' To save computation time, the default is 0.
#' @param approx Should quantile approximation be applied to dense numeric features?
#' The default is `FALSE`. Setting this option to `TRUE` brings a massive speed-up
#' for one-way calculations. It can, e.g., be used when the number of features is
#' very large.
#' @param grid_size Integer controlling the number of quantile midpoints used to
#' approximate dense numerics. The quantile midpoints are calculated after
#' subampling via `n_max`. Only relevant if `approx = TRUE`.
#' @param n_max If `X` has more than `n_max` rows, a random sample of `n_max` rows is
#' selected from `X`. In this case, set a random seed for reproducibility.
#' @param eps Threshold below which numerator values are set to 0. Default is 1e-10.
#' @param w Optional vector of case weights. Can also be a column name of `X`.
#' @param verbose Should a progress bar be shown? The default is `TRUE`.
#' @param ... Additional arguments passed to `pred_fun(object, X, ...)`,
#' for instance `type = "response"` in a [glm()] model, or `reshape = TRUE` in a
#' multiclass XGBoost model.
#' @returns
#' An object of class "hstats" containing these elements:
#' - `X`: Input `X` (sampled to `n_max` rows, after optional quantile approximation).
#' - `w`: Case weight vector `w` (sampled to `n_max` values), or `NULL`.
#' - `v`: Vector of column names in `X` for which overall
#' H statistics have been calculated.
#' - `f`: Matrix with (centered) predictions \eqn{F}.
#' - `mean_f2`: (Weighted) column means of `f`. Used to normalize \eqn{H^2} and
#' \eqn{H^2_j}.
#' - `F_j`: List of matrices, each representing (centered)
#' partial dependence functions \eqn{F_j}.
#' - `F_not_j`: List of matrices with (centered) partial dependence
#' functions \eqn{F_{\setminus j}} of other features.
#' - `K`: Number of columns of prediction matrix.
#' - `pred_names`: Column names of prediction matrix.
#' - `pairwise_m`: Like input `pairwise_m`, but capped at `length(v)`.
#' - `threeway_m`: Like input `threeway_m`, but capped at the smaller of
#' `length(v)` and `pairwise_m`.
#' - `eps`: Like input `eps`.
#' - `pd_importance`: List with numerator and denominator of \eqn{\textrm{PDI}_j}.
#' - `h2`: List with numerator and denominator of \eqn{H^2}.
#' - `h2_overall`: List with numerator and denominator of \eqn{H^2_j}.
#' - `v_pairwise`: Subset of `v` with largest \eqn{H^2_j} used for pairwise
#' calculations. Only if pairwise calculations have been done.
#' - `combs2`: Named list of variable pairs for which pairwise partial
#' dependence functions are available. Only if pairwise calculations have been done.
#' - `F_jk`: List of matrices, each representing (centered) bivariate
#' partial dependence functions \eqn{F_{jk}}.
#' Only if pairwise calculations have been done.
#' - `h2_pairwise`: List with numerator and denominator of \eqn{H^2_{jk}}.
#' Only if pairwise calculations have been done.
#' - `v_threeway`: Subset of `v` with largest `h2_overall()` used for three-way
#' calculations. Only if three-way calculations have been done.
#' - `combs3`: Named list of variable triples for which three-way partial
#' dependence functions are available. Only if three-way calculations have been done.
#' - `F_jkl`: List of matrices, each representing (centered) three-way
#' partial dependence functions \eqn{F_{jkl}}.
#' Only if three-way calculations have been done.
#' - `h2_threeway`: List with numerator and denominator of \eqn{H^2_{jkl}}.
#' Only if three-way calculations have been done.
#' @references
#' Friedman, Jerome H., and Bogdan E. Popescu. *"Predictive Learning via Rule Ensembles."*
#' The Annals of Applied Statistics 2, no. 3 (2008): 916-54.
#' @export
#' @seealso [h2()], [h2_overall()], [h2_pairwise()], [h2_threeway()],
#' and [pd_importance()] for specific statistics calculated from the resulting object.
#' @examples
#' # MODEL 1: Linear regression
#' fit <- lm(Sepal.Length ~ . + Petal.Width:Species, data = iris)
#' s <- hstats(fit, X = iris[, -1])
#' s
#' plot(s)
#' plot(s, zero = FALSE) # Drop 0
#' summary(s)
#'
#' # Absolute pairwise interaction strengths
#' h2_pairwise(s, normalize = FALSE, squared = FALSE, zero = FALSE)
#'
#' # MODEL 2: Multi-response linear regression
#' fit <- lm(as.matrix(iris[, 1:2]) ~ Petal.Length + Petal.Width * Species, data = iris)
#' s <- hstats(fit, X = iris[, 3:5], verbose = FALSE)
#' plot(s)
#' summary(s)
#'
#' # MODEL 3: Gamma GLM with log link
#' fit <- glm(Sepal.Length ~ ., data = iris, family = Gamma(link = log))
#'
#' # No interactions for additive features, at least on link scale
#' s <- hstats(fit, X = iris[, -1], verbose = FALSE)
#' summary(s)
#'
#' # On original scale, we have interactions everywhere.
#' # To see three-way interactions, we set threeway_m to a value above 2.
#' s <- hstats(fit, X = iris[, -1], type = "response", threeway_m = 5)
#' plot(s, ncol = 1) # All three types use different denominators
#'
#' # All statistics on same scale (of predictions)
#' plot(s, squared = FALSE, normalize = FALSE, facet_scale = "free_y")
hstats <- function(object, ...) {
UseMethod("hstats")
}
#' @describeIn hstats Default hstats method.
#' @export
hstats.default <- function(object, X, v = NULL,
pred_fun = stats::predict,
pairwise_m = 5L, threeway_m = 0L,
approx = FALSE, grid_size = 50L,
n_max = 500L, eps = 1e-10,
w = NULL, verbose = TRUE, ...) {
stopifnot(
is.matrix(X) || is.data.frame(X),
is.function(pred_fun)
)
# Is w a column name or a vector?
if (!is.null(w)) {
w2 <- prepare_w(w = w, X = X)
w <- w2[["w"]]
w_name <- w2[["w_name"]]
}
# Determine missing v or check consistency with X
if (is.null(v)) {
v <- colnames(X)
if (!is.null(w) && !is.null(w_name)) {
v <- setdiff(v, w_name)
}
} else {
stopifnot(all(v %in% colnames(X)))
}
p <- length(v)
stopifnot(p >= 2L)
pairwise_m <- min(pairwise_m, p)
threeway_m <- min(threeway_m, pairwise_m, p)
# Reduce size of X (and w)
if (nrow(X) > n_max) {
ix <- sample(nrow(X), n_max)
X <- X[ix, , drop = FALSE]
if (!is.null(w)) {
w <- w[ix]
}
}
# Quantile approximation to speedup things for dense features
if (isTRUE(approx)) {
X <- approx_matrix_or_df(X = X, v = v, m = grid_size)
}
# Predictions ("F" in Friedman and Popescu) always calculated (cheap)
f <- wcenter(prepare_pred(pred_fun(object, X, ...), ohe = TRUE), w = w)
mean_f2 <- wcolMeans(f^2, w = w) # A vector
# Initialize first progress bar
if (verbose) {
cat("1-way calculations...\n")
pb <- utils::txtProgressBar(max = p, style = 3)
}
F_j <- F_not_j <- stats::setNames(vector("list", length = p), v)
for (j in seq_len(p)) {
# Main effect of x_j
z <- v[j]
g <- if (is.data.frame(X)) X[[z]] else X[, z]
F_j[[z]] <- wcenter(
pd_raw(object = object, v = z, X = X, grid = g, pred_fun = pred_fun, w = w, ...),
w = w
)
# Total effect of all other features
not_z <- setdiff(colnames(X), z)
F_not_j[[z]] <- wcenter(
pd_raw(
object,
v = not_z,
X = X,
grid = X[, not_z],
pred_fun = pred_fun,
w = w,
compress_grid = FALSE, # grid has too many columns (saves a very quick check)
...
),
w = w
)
if (verbose) {
utils::setTxtProgressBar(pb, j)
}
}
if (verbose) {
cat("\n")
}
# Initialize output
out <- list(
X = X,
w = w,
v = v,
f = f,
mean_f2 = mean_f2,
F_j = F_j,
F_not_j = F_not_j,
K = ncol(f),
pred_names = colnames(f),
pairwise_m = pairwise_m,
threeway_m = threeway_m,
eps = eps
)
# 0-way and 1-way stats
out[["pd_importance"]] <- pd_importance_raw(out)
out[["h2"]] <- h2_raw(out)
out[["h2_overall"]] <- h2_overall_raw(out)
h2_ov <- out$h2_overall$num
if (pairwise_m >= 2L) {
out[["v_pairwise"]] <- v2 <- get_v(h2_ov, m = pairwise_m)
if (length(v2) >= 2L) {
out[c("combs2", "F_jk")] <- mway(
object, v = v2, X = X, pred_fun = pred_fun, w = w, way = 2L, verb = verbose, ...
)
}
out[["h2_pairwise"]] <- h2_pairwise_raw(out)
}
if (threeway_m >= 3L) {
out[["v_threeway"]] <- v3 <- get_v(h2_ov, m = threeway_m)
if (length(v3) >= 3L) {
out[c("combs3", "F_jkl")] <- mway(
object, v = v3, X = X, pred_fun = pred_fun, w = w, way = 3L, verb = verbose, ...
)
}
out[["h2_threeway"]] <- h2_threeway_raw(out)
}
structure(out, class = "hstats")
}
#' @describeIn hstats Method for "ranger" models.
#' @export
hstats.ranger <- function(object, X, v = NULL,
pred_fun = function(m, X, ...) stats::predict(m, X, ...)$predictions,
pairwise_m = 5L, threeway_m = 0L,
approx = FALSE, grid_size = 50L,
n_max = 500L, eps = 1e-10,
w = NULL, verbose = TRUE, ...) {
hstats.default(
object = object,
X = X,
v = v,
pred_fun = pred_fun,
pairwise_m = pairwise_m,
threeway_m = threeway_m,
approx = approx,
grid_size = grid_size,
n_max = n_max,
eps = eps,
w = w,
verbose = verbose,
...
)
}
#' @describeIn hstats Method for DALEX "explainer".
#' @export
hstats.explainer <- function(object, X = object[["data"]],
v = NULL,
pred_fun = object[["predict_function"]],
pairwise_m = 5L, threeway_m = 0L,
approx = FALSE, grid_size = 50L,
n_max = 500L, eps = 1e-10,
w = object[["weights"]], verbose = TRUE, ...) {
hstats.default(
object = object[["model"]],
X = X,
v = v,
pred_fun = pred_fun,
pairwise_m = pairwise_m,
threeway_m = threeway_m,
approx = approx,
grid_size = grid_size,
n_max = n_max,
eps = eps,
w = w,
verbose = verbose,
...
)
}
#' Print Method
#'
#' Print method for object of class "hstats". Shows \eqn{H^2}.
#'
#' @param x An object of class "hstats".
#' @param ... Further arguments passed from other methods.
#' @returns Invisibly, the input is returned.
#' @export
#' @seealso See [hstats()] for examples.
print.hstats <- function(x, ...) {
cat("'hstats' object. Use plot() or summary() for details.\n\n")
print(h2(x))
invisible(x)
}
#' Summary Method
#'
#' Summary method for "hstats" object. Note that only the top 4 overall, the top 3
#' pairwise and the top 1 three-way statistics are shown.
#'
#' @inheritParams h2_overall
#' @param ... Currently not used.
#' @returns
#' An object of class "summary_hstats" representing a named list with statistics
#' "h2", "h2_overall", "h2_pairwise", "h2_threeway", all of class "hstats_matrix".
#' @export
#' @seealso See [hstats()] for examples.
summary.hstats <- function(object, normalize = TRUE, squared = TRUE,
sort = TRUE, zero = TRUE, ...) {
args <- list(
object = object,
normalize = normalize,
squared = squared,
sort = sort,
zero = zero
)
out <- list(
h2 = do.call(h2, args),
h2_overall = do.call(h2_overall, args),
h2_pairwise = do.call(h2_pairwise, args),
h2_threeway = do.call(h2_threeway, args)
)
structure(out, class = "hstats_summary")
}
#' Print Method
#'
#' Print method for object of class "hstats_summary".
#'
#' @param x An object of class "hstats_summary".
#' @param ... Further arguments passed from other methods.
#' @returns Invisibly, the input is returned.
#' @export
#' @seealso See [hstats()] for examples.
print.hstats_summary <- function(x, ...) {
m <- c(1L, 4L, 3L, 1L)
for (i in seq_along(x)) {
if (is.null(x[[i]]$M))
next
cat("*")
if (i >= 2L)
cat("Largest ")
print(x[[i]], top_m = m[i])
cat("\n")
}
invisible(x)
}
#' Plot Method for "hstats" Object
#'
#' Plot method for object of class "hstats".
#'
#' @param x Object of class "hstats".
#' @param which Which statistic(s) to be shown? Default is `1:3`, i.e.,
#' show \eqn{H^2_j} (1), \eqn{H^2_{jk}} (2), and \eqn{H^2_{jkl}} (3).
#' @inheritParams plot.hstats_matrix
#' @inheritParams h2_overall
#' @returns An object of class "ggplot".
#' @export
#' @seealso See [hstats()] for examples.
plot.hstats <- function(x, which = 1:3, normalize = TRUE, squared = TRUE,
sort = TRUE, top_m = 15L, zero = TRUE,
fill = getOption("hstats.fill"),
viridis_args = getOption("hstats.viridis_args"),
facet_scales = "free", ncol = 2L, rotate_x = FALSE, ...) {
if (is.null(viridis_args)) {
viridis_args <- list()
}
su <- summary(x, normalize = normalize, squared = squared, sort = sort, zero = zero)
su <- su[sapply(su, FUN = function(z) !is.null(z[["M"]]))]
# This part could be simplified, especially the "match()"
stat_names <- c("h2_overall", "h2_pairwise", "h2_threeway")[which]
stat_labs <- c("Overall", "Pairwise", "Three-way")[which]
ok <- stat_names[stat_names %in% names(su)]
if (length(ok) == 0L) {
message("Nothing to plot!")
return(NULL)
}
dat <- lapply(
ok,
FUN = function(nm)
mat2df(utils::head(su[[nm]]$M, top_m), id = stat_labs[match(nm, stat_names)])
)
dat <- do.call(rbind, dat)
dat <- barplot_reverter(dat)
p <- ggplot2::ggplot(dat, ggplot2::aes(x = value_, y = variable_)) +
ggplot2::ylab(ggplot2::element_blank()) +
ggplot2::xlab(su$h2$description) # Generic enough?
if (x[["K"]] == 1L) {
p <- p + ggplot2::geom_bar(fill = fill, stat = "identity", ...)
} else {
p <- p +
ggplot2::geom_bar(
ggplot2::aes(fill = varying_), stat = "identity", position = "dodge", ...
) +
ggplot2::theme(legend.title = ggplot2::element_blank()) +
do.call(ggplot2::scale_fill_viridis_d, viridis_args) +
ggplot2::guides(fill = ggplot2::guide_legend(reverse = TRUE))
}
if (length(ok) > 1L) {
p <- p + ggplot2::facet_wrap(~ id_, ncol = ncol, scales = facet_scales)
}
if (rotate_x) {
p <- p + rotate_x_labs()
}
p
}
# Helper functions used only by hstats()
#' Pairwise or 3-Way Partial Dependencies
#'
#' Calculates centered partial dependence functions for selected pairwise or three-way
#' situations.
#'
#' @noRd
#' @keywords internal
#'
#' @param v Vector of column names to calculate `way` order interactions.
#' @inheritParams hstats
#' @param way Pairwise (`way = 2`) or three-way (`way = 3`) interactions.
#' @param verb Verbose (`TRUE`/`FALSE`).
#'
#' @returns
#' A list with a named list of feature combinations (pairs or triples), and
#' corresponding centered partial dependencies.
mway <- function(object, v, X, pred_fun = stats::predict, w = NULL,
way = 2L, verb = TRUE, ...) {
combs <- utils::combn(v, way, simplify = FALSE)
n_combs <- length(combs)
F_way <- vector("list", length = n_combs)
names(F_way) <- names(combs) <- sapply(combs, paste, collapse = ":")
if (verb) {
cat(way, "way calculations...\n", sep = "-")
pb <- utils::txtProgressBar(max = n_combs, style = 3)
}
for (i in seq_len(n_combs)) {
z <- combs[[i]]
F_way[[i]] <- wcenter(
pd_raw(object, v = z, X = X, grid = X[, z], pred_fun = pred_fun, w = w, ...),
w = w
)
if (verb) {
utils::setTxtProgressBar(pb, i)
}
}
if (verb) {
cat("\n")
}
list(combs, F_way)
}
#' Get Feature Names
#'
#' This function takes the unsorted and unnormalized H2_j matrix and extracts the top
#' m feature names (unsorted). If H2_j has multiple columns, this is done per column and
#' then the union is returned.
#'
#' @noRd
#' @keywords internal
#'
#' @param H Unnormalized, unsorted H2_j values.
#' @param m Number of features to pick per column.
#'
#' @returns A vector of the union of the m column-wise most important features.
get_v <- function(H, m) {
v <- rownames(H)
selector <- function(vv) names(utils::head(sort(-vv[vv > 0]), m))
if (NCOL(H) == 1L) {
v_cand <- selector(drop(H))
} else {
v_cand <- Reduce(union, lapply(asplit(H, MARGIN = 2L), FUN = selector))
}
v[v %in% v_cand]
}