inline_desirability: Desirability functions for in-line computations

Description

Desirability functions map some input to a [0, 1] scale where zero is unacceptable and one is most desirable. The mapping depends on the situation. For example, d_max() increases desirability with the input while d_min() does the opposite. See the plots in the examples to see more examples.

Currently, only the desirability functions defined by Derringer and Suich (1980) are implemented.

Usage

d_max(x, low, high, scale = 1, missing = NA_real_, use_data = FALSE)
d_min(x, low, high, scale = 1, missing = NA_real_, use_data = FALSE)
d_target(
  x,
  low,
  target,
  high,
  scale_low = 1,
  scale_high = 1,
  missing = NA_real_,
  use_data = FALSE
)
d_box(x, low, high, missing = NA_real_, use_data = FALSE)
d_custom(x, x_vals, desirability, missing = NA_real_)
d_category(x, categories, missing = NA_real_)

Value

A numeric vector on [0, 1] where larger values are more desirable.

Arguments

x: A vector of data to compute the desirability function
low, high, target: Single numeric values that define the active ranges of desirability.
scale, scale_low, scale_high: A single numeric value to rescale the desirability function (each should be great than 0.0). Values >1.0 make the desirability more difficult to satisfy while smaller values make it easier (see the examples below). scale_low and scale_high do the same for target functions with scale_low affecting the range below the target value and scale_high affecting values greater than target.
missing: A single numeric value on [0, 1] (or NA_real_) that defines how missing values in x are mapped to the desirability score.
use_data: Should the low, middle, and/or high values be derived from the data (x) using the minimum, maximum, or median (respectively)?
x_vals, desirability: Numeric vectors of the same length that define the desirability results at specific values of x. Values below and above the data in x_vals are given values of zero and one, respectively.
categories: A named vector of desirability values that match all possible categories to specific desirability values. Data that are not included in categories are given the value in missing.

Details

Each function translates the values to desirability on [0, 1].

Equations

Maximization

data > high: d = 1.0
data < low: d = 0.0
low <= data <= high: \(d = \left(\frac{data-low}{high-low}\right)^{scale}\)

Minimization

data > high: d = 0.0
data < low: d = 1.0
low <= data <= high: \(d = \left(\frac{data = low}{low - high}\right)^{scale}\)

Target

data > high: d = 0.0
data < low: d = 0.0
low <= data <= target: \(d = \left(\frac{data - low}{target - low}\right)^{scale\_low}\)
target <= data <= high: \(d = \left(\frac{data - high}{target - high}\right)^{scale\_high}\)

Box

data > high: d = 0.0
data < low: d = 0.0
low <= data <= high: d = 1.0

Custom

For the sequence of values given to the function, d_custom() will return the desirability values that correspond to data matching values in x_vals. Otherwise, linear interpolation is used for values in-between.

Data-Based Values

By default, most of the d_*() functions require specific user inputs for arguments such as low, target and high. When use_data = TRUE, the functions can use the minimum, median, and maximum values of the existing data to estimate those values (respectively) but only when users do not specify them.

References

Derringer, G. and Suich, R. (1980), Simultaneous Optimization of Several Response Variables. Journal of Quality Technology, 12, 214-219.

Examples

Run this code

library(dplyr)
library(ggplot2)

set.seed(1)
dat <- tibble(x = sort(runif(30)), y = sort(runif(30)))
d_max(dat$x[1:10], 0.1, 0.75)

dat |>
  mutate(d_x = d_max(x, 0.1, 0.75))

set.seed(2)
tibble(z = sort(runif(100))) |>
  mutate(
    no_scale = d_max(z, 0.1, 0.75),
    easier   = d_max(z, 0.1, 0.75, scale = 1/2)
  ) |>
  ggplot(aes(x = z)) +
  geom_point(aes(y = no_scale)) +
  geom_line(aes(y = no_scale), alpha = .5) +
  geom_point(aes(y = easier), col = "blue") +
  geom_line(aes(y = easier), col = "blue", alpha = .5) +
  lims(x = 0:1, y = 0:1) +
  coord_fixed() +
  ylab("Desirability")

# ------------------------------------------------------------------------------
# Target example

dat |>
  mutate(
    triangle = d_target(x, 0.1, 0.5, 0.9, scale_low = 2, scale_high = 1/2)
  ) |>
  ggplot(aes(x = x, y = triangle)) +
  geom_point() +
  geom_line(alpha = .5) +
  lims(x = 0:1, y = 0:1) +
  coord_fixed() +
  ylab("Desirability")

# ------------------------------------------------------------------------------
# Box constraints

dat |>
  mutate(box = d_box(x, 1/4, 3/4)) |>
  ggplot(aes(x = x, y = box)) +
  geom_point() +
  geom_line(alpha = .5) +
  lims(x = 0:1, y = 0:1) +
  coord_fixed() +
  ylab("Desirability")

# ------------------------------------------------------------------------------
# Custom function

v_x <- seq(0, 1, length.out = 20)
v_d <- 1 - exp(-10 * abs(v_x - .5))

dat |>
  mutate(v = d_custom(x, v_x, v_d)) |>
  ggplot(aes(x = x, y = v)) +
  geom_point() +
  geom_line(alpha = .5) +
  lims(x = 0:1, y = 0:1) +
  coord_fixed() +
  ylab("Desirability")

# ------------------------------------------------------------------------------
# Qualitative data

set.seed(3)
groups <- sort(runif(10))
names(groups) <- letters[1:10]

tibble(x = letters[1:7]) |>
  mutate(d = d_category(x, groups)) |>
  ggplot(aes(x = x, y = d)) +
  geom_bar(stat = "identity") +
  lims(y = 0:1) +
  ylab("Desirability")

# ------------------------------------------------------------------------------
# Apply the same function to many columns at once (dplyr > 1.0)

dat |>
  mutate(across(c(everything()), ~ d_min(., .2, .6), .names = "d_{col}"))

# ------------------------------------------------------------------------------
# Using current data

set.seed(9015)
tibble(z = c(0, sort(runif(20)), 1)) |>
  mutate(
    user_specified = d_max(z, 0.1, 0.75),
    data_driven   = d_max(z, use_data = TRUE)
  ) |>
  ggplot(aes(x = z)) +
  geom_point(aes(y = user_specified)) +
  geom_line(aes(y = user_specified), alpha = .5) +
  geom_point(aes(y = data_driven), col = "blue") +
  geom_line(aes(y = data_driven), col = "blue", alpha = .5) +
  lims(x = 0:1, y = 0:1) +
  coord_fixed() +
  ylab("Desirability")

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