---
title: "Frontier shapes and near-ties"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Frontier shapes and near-ties}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 7, fig.height = 5)
```

```{r setup}
library(paretoinfer)
```

Identification with confidence boxes does not care whether the frontier is
convex: a certificate compares two boxes, objective by objective. This
vignette runs the procedure on the frontier shapes that trouble scalarised
methods (weighted sums find only the convex hull) and on near-ties, and shows
what the three sets look like in each case.

```{r helpers}
make_sim <- function(means, h = 0.08) {
  function(alternative, n) {
    mu <- means[alternative, ]
    data.frame(y1 = runif(n, mu[1] - h, mu[1] + h), y2 = runif(n, mu[2] - h, mu[2] + h))
  }
}
run <- function(means, epsilon = 0.04, seed = 1, ...) {
  d <- pareto_design(alternatives = rownames(means), objectives = c("y1", "y2"),
                     bounds = c(0, 1), alpha = 0.05, epsilon = epsilon, max_evaluations = 4000, ...)
  set.seed(seed)
  pareto_run(d, make_sim(means), batch = 5, initial = 5)
}
```

## Convex frontier

```{r convex}
convex <- rbind(A = c(0.10, 0.80), B = c(0.25, 0.50), C = c(0.45, 0.30), D = c(0.80, 0.15),
                E = c(0.50, 0.60), F = c(0.70, 0.45))
s <- run(convex)
plausible_pareto_set(s)
autoplot(s)
```

## Concave (non-convex) frontier

The middle alternative `C` is Pareto-optimal but lies above the segment that
joins its neighbours, so no weighted sum of the objectives would select it.
The boxes certify it without difficulty.

```{r concave}
concave <- rbind(A = c(0.10, 0.80), B = c(0.30, 0.65), C = c(0.45, 0.55), D = c(0.60, 0.30),
                 E = c(0.80, 0.10), F = c(0.55, 0.70))
s <- run(concave)
plausible_pareto_set(s)$table[, c("alternative", "status", "dominated_by", "n_evaluations")]
autoplot(s)
```

## Disconnected frontier

Two clusters of optimal alternatives with a dominated gap between them.

```{r disconnected}
disconnected <- rbind(A = c(0.10, 0.85), B = c(0.15, 0.75), C = c(0.70, 0.20), D = c(0.80, 0.12),
                      E = c(0.45, 0.55), F = c(0.40, 0.65))
s <- run(disconnected)
plausible_pareto_set(s)$table[, c("alternative", "status", "dominated_by", "n_evaluations")]
autoplot(s)
```

## Near-ties

Two alternatives within 0.02 of each other on both objectives, with a
tolerance of 0.04: they are an `epsilon`-tie. The procedure keeps the first
in design order as the representative and certifies the other as dominated by
it; the report names the tie. With a tolerance below the gap the two could
not be separated and the identification would run to its budget.

```{r ties}
ties <- rbind(A = c(0.20, 0.60), B = c(0.21, 0.61), C = c(0.60, 0.20), D = c(0.50, 0.50))
s <- run(ties, epsilon = 0.04)
plausible_pareto_set(s)$table[, c("alternative", "status", "dominated_by", "n_evaluations")]
s_tight <- run(ties, epsilon = 0.005)
glance(s_tight)[, c("n_evaluations", "n_uncertain", "stopping_reason")]
```

## Reading the plots

In every plot the boxes are the simultaneous confidence boxes, the colour is
the status, and the orange step line joins the point estimates of the
estimated frontier. A box that overlaps the region to the lower left of
another box's corner cannot be certified dominated; a box entirely to the
upper right of another one's upper-right corner (shifted by `epsilon`) is.