r/generate_ndset: Add inverted-simplex and concave-simplex shapes, fix convex-simplex

r/NEWS.md

@@ -1,5 +1,6 @@
 # moocore 0.2.0.900
 
+ * `generate_ndset()`: New shapes `"inverted-simplex"` and `"concave-simplex"`. Shape `"convex-simplex"` is now equivalent to `generate_ndset(..., method="simplex")^2`, which is slightly more uniform than the previous approach.
  * `r2_exact()` implements the exact computation of the R2 indicator for bi-objective solution sets.
  * `hypervolume()` is significantly faster for more than four dimensions (Andreia P. Guerreiro).
  * `hypervolume()` now handles 1D inputs and provides a clear error for 0D inputs (#58).

r/R/generate.R

@@ -28,11 +28,21 @@
 #'   standard simplex.
 #' * `"concave-sphere"`, `"sphere"`, or `"C"`: Uniformly samples points
 #'   on the positive orthant of the unit hypersphere (concave for minimisation).
-#' * `"convex-sphere"` or `"X"`: Equivalent to `1 - generate_ndset(..., method="concave-sphere")`,
-#'   which is convex for minimisation.
-#' * `"convex-simplex"`: Equivalent to `generate_ndset(..., method="concave-sphere")^4`,
+#' * `"convex-simplex"`: Equivalent to `generate_ndset(..., method="simplex")^2`,
 #'   which is convex for minimisation. Such a set cannot be obtained by any
-#'   affine transformation of a subset of the hyper-sphere.
+#'   affine transformation of a subset of the hyper-sphere.  This sampling is
+#'   *not* uniform.
+#' * `"convex-sphere"` or `"X"`: Equivalent to `1 - generate_ndset(...,
+#'   method="concave-sphere")`, which is convex for minimisation.  This shape
+#'   has also been called *inverted convex* \citep{IshHeSha2019regular}. This
+#'   sampling is uniform.
+#' * `"inverted-simplex"` or `"inverted-linear"`: Equivalent to `1 -
+#'   generate_ndset(..., method="simplex")`. This sampling is uniform.
+#' * `"concave-simplex"`: Equivalent to `1 - generate_ndset(...,
+#'   method="convex-simplex")`, which is concave for minimisation.  This shape
+#'   has also been called *inverted concave* \citep{IshHeSha2019regular}.  This
+#'   sampling is *not* uniform because `method="convex-simplex"` is also not
+#'   uniform.
 #'
 #' Method `"simplex"` uniformly samples points on the standard
 #' \eqn{(d-1)}-simplex defined by \eqn{\{x \in R_+^d : \sum_i x_i = 1\}}. This
@@ -46,7 +56,7 @@
 #'
 #' Sampling from either the standard normal distribution
 #' \citep{GueFonPaq2021hv} or the uniform distribution \citep{LacKlaFon2017box}
-#' does not produce a uniform distribution when projected into the simplex.
+#' does not produce a uniform distribution when projected onto the simplex.
 #'
 #' Method `"concave-sphere"` uniformly samples points on the positive orthant
 #' of the hyper-sphere, which is concave when all objectives are minimised.
@@ -56,19 +66,26 @@
 #' then dividing each value by the l2-norm of the vector, \eqn{z_i =
 #' \frac{|x_i|}{\|\vec{x}\|_2}} \citep{Muller1959sphere}. The absolute value in
 #' the numerator ensures that points are sampled on the positive orthant of the
-#' hyper-sphere.  Sampling from the uniform distribution
-#' \citep{LacKlaFon2017box} does not result in a uniform sampling when
-#' projected onto the surface of the hyper-sphere.
+#' hyper-sphere.  Sampling from the uniform distribution \citep{LacKlaFon2017box} would
+#' not result in a uniform sampling when projected onto the surface of the
+#' hyper-sphere.
+#'
+#' Method `"convex-simplex"` is equivalent to `generate_ndset(...,
+#' method="simplex")^2`, which is convex for minimisation problems.  The
+#' corresponding surface is equivalent to a simplex curved towards the
+#' origin. Although the sampling on the simplex is uniform, the transformed
+#' points are not.
 #'
 #' Method `"convex-sphere"` is equivalent to `1 - generate_ndset(...,
 #' method="concave-sphere")`, which is convex for minimisation problems.  It
 #' corresponds to translating points from the negative orthant of the
-#' hyper-sphere to the positive orthant.
+#' hyper-sphere to the positive orthant. Thus, the sampling remains uniform.
 #'
-#' Method `"convex-simplex"` is equivalent to `generate_ndset(...,
-#' method="concave-sphere")^4`, which is convex for minimisation problems.
-#' The corresponding surface is equivalent to a simplex curved towards the
-#' origin.
+#' Methods `"inverted-simplex"` and `"concave-simplex"` are similar
+#' translations of `"simplex"` and `"convex-simplex"`, respectively.
+#' These translations have been called `inverted` shapes in the literature and
+#' have different properties than their `regular` counterparts
+#' \citep{IshHeSha2019regular}.
 #'
 #' @references
 #'
@@ -102,13 +119,17 @@ generate_ndset <- function(n, d, method, seed = NULL, integer = FALSE)
   }
 
   sample_convex_sphere <- function() 1. - sample_sphere()
-  sample_convex_simplex <- function() sample_sphere()^4L
+  sample_convex_simplex <- function() sample_simplex()^2
+  sample_inverted_simplex <- function() 1. - sample_simplex()
+  sample_concave_simplex <- function() 1. - sample_convex_simplex()
 
   sample_fun <- (
     if (method %in% c("simplex", "linear", "L")) sample_simplex
     else if (method %in% c("concave-sphere", "sphere", "C")) sample_sphere
     else if (method %in% c("convex-sphere", "X")) sample_convex_sphere
     else if (method == "convex-simplex") sample_convex_simplex
+    else if (method %in% c("inverted-simplex", "inverted-linear")) sample_inverted_simplex
+    else if (method == "concave-simplex") sample_concave_simplex
     else stop("Unknown method =", method)
   )
 

r/vignettes/articles/generate.Rmd

@@ -36,9 +36,10 @@ mutually nondominated points.
 
 ```{r generate_rnd_2d}
 n <- 100
-methods <- c("simplex", "concave-sphere", "convex-sphere", "convex-simplex")
-colors <- c("red", "blue", "green", "purple")
-shapes <- c(16, 15, 17, 18)  # ggplot2 shape codes
+methods <- c("simplex", "concave-sphere", "convex-sphere", "convex-simplex",
+             "inverted-simplex", "concave-simplex")
+colors <- c("red", "blue", "green", "purple", "orange", "yellow")
+shapes <- c(16, 15, 17, 18, 16, 15)  # ggplot2 shape codes
 
 df_all <- do.call(rbind.data.frame, lapply(methods, function(method) {
   points <- moocore::generate_ndset(n, 2, method = method, seed = 42)
@@ -86,9 +87,9 @@ ggplot(df_int, aes(x = x, y = y, color = method, shape = method)) +
 A popular way to generate a convex nondominated set is to translate the
 negative orthant of the hyper-sphere to the unit hyper-cube
 (`convex-sphere`).  However, there are other possible convex sets with
-different properties. For example the method `convex-simplex` applies a
-non-linear transformation to a `concave-sphere` set, resulting in a convex
-transformation of the standard simplex.
+different properties. For example the method `convex-simplex` squares the
+points generated by method `simplex`, resulting in a convex transformation
+of the standard simplex.
 
 First, let's define a few functions useful for plotting:
 <details><summary>(show plotting code)</summary>
@@ -179,11 +180,38 @@ plotly_side_by_side <- function(fig1, fig2, height="500px") {
 ```{r generate_3d_convex}
 n <- 2000
 
-points1 <- moocore::generate_ndset(n, 3, "convex-sphere", seed = 42)
-fig1 <- plot_3d("simplex", points1, title = "Convex-sphere")
+fig1 <- plot_3d("simplex", moocore::generate_ndset(n, 3, "convex-sphere", seed = 42), title = 'method="convex-sphere"')
+fig2 <- plot_3d("simplex", moocore::generate_ndset(n, 3, "convex-simplex", seed = 42), title = 'method="convex-simplex"')
 
-points2 <- moocore::generate_ndset(n, 3, "convex-simplex", seed = 42)
-fig2 <- plot_3d("simplex", points2, title = "Convex-simplex")
+plotly_side_by_side(fig1, fig2)
+```
+
+# Inverted shapes
+
+@IshHeSha2019regular analyze the differences between *regular* and
+*inverted* shapes, shown below on the left and right figures,
+respectively. The differences between `simplex` and `inverted-simplex`
+are not noticeable in 2D, but are significant in higher dimensions.
+
+```{r generate_3d_inverted_simplex}
+n <- 2000
+
+fig1 <- plot_3d("simplex", moocore::generate_ndset(n, 3, "simplex", seed = 42), title = 'method="simplex"')
+fig2 <- plot_3d("simplex", moocore::generate_ndset(n, 3, "inverted-simplex", seed = 42), title = 'method="inverted-simplex"')
+
+plotly_side_by_side(fig1, fig2)
+```
+
+```{r generate_3d_concave_convex_sphere}
+fig1 <- plot_3d("simplex", moocore::generate_ndset(n, 3, "concave-sphere", seed = 42), title = 'method="concave-sphere"')
+fig2 <- plot_3d("simplex", moocore::generate_ndset(n, 3, "convex-sphere", seed = 42), title = 'method="convex-sphere"')
+
+plotly_side_by_side(fig1, fig2)
+```
+
+```{r generate_3d_convex_concave_simplex}
+fig1 <- plot_3d("simplex", moocore::generate_ndset(n, 3, "convex-simplex", seed = 42), title = 'method="convex-simplex"')
+fig2 <- plot_3d("simplex", moocore::generate_ndset(n, 3, "concave-simplex", seed = 42), title = 'method="concave-simplex"')
 
 plotly_side_by_side(fig1, fig2)
 ```