Help for package aramappings

Title:

Computes Adaptable Radial Axes Mappings

Version:

0.1.3

Description:

Computes low-dimensional point representations of high-dimensional numerical data according to the data visualization method Adaptable Radial Axes described in: Manuel Rubio-Sánchez, Alberto Sanchez, and Dirk J. Lehmann (2017) "Adaptable radial axes plots for improved multivariate data visualization" <doi:10.1111/cgf.13196>.

License:

MIT + file LICENSE

Encoding:

UTF-8

RoxygenNote:

7.3.3

Imports:

clarabel, CVXR, glpkAPI, Matrix, parallel, pracma, Rglpk, slam, ggplot2, grDevices, grid, stats

URL:

https://github.com/manuelrubio/aramappings, https://manuelrubio.github.io/aramappings/

BugReports:

https://github.com/manuelrubio/aramappings/issues

Suggests:

testthat (≥ 3.0.0), parallelly, knitr, rmarkdown

Config/testthat/edition:

VignetteBuilder:

knitr

Depends:

R (≥ 3.5)

LazyData:

true

NeedsCompilation:

Packaged:

2026-01-29 15:58:02 UTC; manuel

Author:

Manuel Rubio-Sánchez

[aut, cre, cph], Dirk J. Lehmann

[ctb], Miguel Ángel Muñoz Mohedano

[ctb], Alberto Sánchez Campos

[ctb], Cristina Soguero-Ruiz

[ctb]

Maintainer:

Manuel Rubio-Sánchez <manuel.rubio@urjc.es>

Repository:

CRAN

Date/Publication:

2026-01-29 16:20:02 UTC

Exact Adaptable Radial Axes (ARA) mappings using the L1 norm

Description

ara_exact_l1() computes exact Adaptable Radial Axes (ARA) mappings for the L1 norm

Usage

ara_exact_l1(
  X,
  V,
  variable = 1,
  solver = "glpkAPI",
  use_glpkAPI_simplex = TRUE,
  cluster = NULL
)

Arguments

X

Numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

V

Numeric matrix defining the axes or "axis vectors". Its dimensions are n x m, where 1<=m<=3 is the dimension of the visualization space. Each row of V defines an axis vector.

variable

Integer that indicates the variable (in [1,n]) for which the estimates of high-dimensional data will be exact. Default: variable = 1.

solver

String indicating a package for solving the linear problem(s). It can be "clarabel" (default), "glpkAPI", "Rglpk", or "CVXR".

use_glpkAPI_simplex

Boolean parameter that indicates whether to use the simplex algorithm (if TRUE) or an interior point method (if FALSE), when using the glpkAPI solver. The default is TRUE.

cluster

Optional cluster object related to the parallel package. If supplied, and n_LP_problems is N, the method computes the mappings using parallel processing.

Details

ara_exact_l1() computes low-dimensional point representations of high-dimensional numerical data (X) according to the data visualization method "Adaptable Radial Axes" (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196), which describes a collection of convex norm optimization problems aimed at minimizing estimates of original values in X through dot products of the mapped points with the axis vectors (rows of V). This particular function solves the constrained optimization problem in Eq. (13), for the L1 vector norm. Its equality constraint forces estimates to be exact for one of the data variables.

Value

A list with the three following entries:

P: A numeric N x m matrix containing the mapped points. Each row is the low-dimensional representation of a data observation in X.
status: A vector of length N where the i-th element contains the status of the chosen solver when calculating the mapping of the i-th data observation. The type of the elements depends on the particular chosen solver.
objval: The numeric objective value associated with the solution to the optimization problem, considering matrix norms.

If the chosen solver fails to map one or more data observations (i.e., fails to solve the related optimization problems), their rows in P will contain NA (not available) values. In that case, objval will also be NA.

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables of the auto_mpg dataset
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Select variable for exact estimates, and use it for coloring the embedded
# points
variable <- sample(1:n, 1)

# Set the number of CPU cores/workers
# NCORES <- parallelly::availableCores(omit = 1)
# NCORES <- max(1,parallel::detectCores() - 1)
NCORES <- 2L

# Create a cluster for parallel processing
cl <- parallel::makeCluster(NCORES)

# Compute the mapping
mapping <- ara_exact_l1(
  Z,
  V,
  variable = variable,
  solver = "glpkAPI",
  use_glpkAPI_simplex = TRUE,
  cluster = cl
)

# Stop cluster
parallel::stopCluster(cl)

# Select variables with labeled axis lines on ARA plot
axis_lines <- variable

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  axis_lines = axis_lines,
  color_variable = variable
)

Exact Adaptable Radial Axes (ARA) mappings using the L2 norm

Description

ara_exact_l2() computes exact Adaptable Radial Axes (ARA) mappings for the L2 norm

Usage

ara_exact_l2(X, V, variable = 1, solver = "formula")

Arguments

X

Numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

V

Numeric matrix defining the axes or "axis vectors". Its dimensions are n x m, where 1<=m<=3 is the dimension of the visualization space. Each row of V defines an axis vector.

variable

Integer that indicates the variable (in [1,n]) for which the estimates of high-dimensional data will be exact. Default: variable = 1.

solver

String indicating a package or method for solving the optimization problem. It can be "formula" (default), where the solution is obtained through a closed-form formula, or "CVXR".

Details

ara_exact_l2() computes low-dimensional point representations of high-dimensional numerical data (X) according to the data visualization method "Adaptable Radial Axes" (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196), which describes a collection of convex norm optimization problems aimed at minimizing estimates of original values in X through dot products of the mapped points with the axis vectors (rows of V). This particular function solves the constrained optimization problem in Eq. (13), for the squared-Euclidean norm. Its equality constraint forces estimates to be exact for one of the data variables. The problem admits closed-form solutions.

Value

A list with the three following entries:

P: A numeric N x m matrix containing the mapped points. Each row is the low-dimensional representation of a data observation in X.
status: A vector of length N where the i-th element contains the status of the chosen solver when calculating the mapping of the i-th data observation. The type of the elements depends on the particular chosen solver.
objval: The numeric objective value associated with the solution to the optimization problem, considering matrix norms.

The output status vector returns the 2-norm condition number of V. If the chosen solver fails to map the data (i.e., fails to solve the related optimization problem), P will contain NA (not available) values. In that case, objval will also be NA.

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Select variable for exact estimates, and use it for coloring the embedded
# points
variable <- sample(1:n, 1)

# Compute the mapping
mapping <- ara_exact_l2(
  Z,
  V,
  variable = variable,
  solver = "formula"
)

# Select variables with labeled axis lines on ARA plot
axis_lines <- variable

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  axis_lines = axis_lines,
  color_variable = variable
)

Exact Adaptable Radial Axes (ARA) mappings using the L-infinity norm

Description

ara_exact_linf() computes exact Adaptable Radial Axes (ARA) mappings for the L-infinity norm

Usage

ara_exact_linf(
  X,
  V,
  variable = 1,
  solver = "glpkAPI",
  use_glpkAPI_simplex = TRUE,
  cluster = NULL
)

Arguments

X

Numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

V

Numeric matrix defining the axes or "axis vectors". Its dimensions are n x m, where 1<=m<=3 is the dimension of the visualization space. Each row of V defines an axis vector.

variable

Integer that indicates the variable (in [1,n]) for which the estimates of high-dimensional data will be exact. Default: variable = 1.

solver

String indicating a package for solving the linear problem(s). It can be "clarabel" (default), "glpkAPI", "Rglpk", or "CVXR".

use_glpkAPI_simplex

Boolean parameter that indicates whether to use the simplex algorithm (if TRUE) or an interior point method (if FALSE), when using the glpkAPI solver. The default is TRUE.

cluster

Optional cluster object related to the parallel package. If supplied, and n_LP_problems is N, the method computes the mappings using parallel processing.

Details

ara_exact_linf() computes low-dimensional point representations of high-dimensional numerical data (X) according to the data visualization method "Adaptable Radial Axes" (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196), which describes a collection of convex norm optimization problems aimed at minimizing estimates of original values in X through dot products of the mapped points with the axis vectors (rows of V). This particular function solves the constrained optimization problem in Eq. (13), for the L-infinity vector norm. Its equality constraint forces estimates to be exact for one of the data variables.

Value

A list with the three following entries:

P: A numeric N x m matrix containing the mapped points. Each row is the low-dimensional representation of a data observation in X.
status: A vector of length N where the i-th element contains the status of the chosen solver when calculating the mapping of the i-th data observation. The type of the elements depends on the particular chosen solver.
objval: The numeric objective value associated with the solution to the optimization problem, considering matrix norms.

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Select variable for exact estimates, and use it for coloring the embedded
# points
variable <- sample(1:n, 1)

# Set the number of CPU cores/workers
# NCORES <- parallelly::availableCores(omit = 1)
# NCORES <- max(1,parallel::detectCores() - 1)
NCORES <- 2L

# Create a cluster for parallel processing
cl <- parallel::makeCluster(NCORES)

# Compute the mapping
mapping <- ara_exact_linf(
  Z,
  V,
  variable = variable,
  solver = "glpkAPI",
  use_glpkAPI_simplex = TRUE,
  cluster = cl
)

# Stop cluster
parallel::stopCluster(cl)

# Select variables with labeled axis lines on ARA plot
axis_lines <- variable

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  axis_lines = axis_lines,
  color_variable = variable
)

Ordered Adaptable Radial Axes (ARA) mappings using the L1 norm

Description

ara_ordered_l1() computes ordered Adaptable Radial Axes (ARA) mappings for the L1 norm

Usage

ara_ordered_l1(X, V, variable = 1, solver = "clarabel")

Arguments

X

Numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

V

Numeric matrix defining the axes or "axis vectors". Its dimensions are n x m, where 1<=m<=3 is the dimension of the visualization space. Each row of V defines an axis vector.

variable

Integer that indicates the variable (in [1,n]) for which the estimates of high-dimensional data will be exact. Default: variable = 1.

solver

String indicating a package for solving the linear problem(s). It can be "clarabel" (default), "glpkAPI", "Rglpk", or "CVXR".

Details

ara_ordered_l1() computes low-dimensional point representations of high-dimensional numerical data (X) according to the data visualization method "Adaptable Radial Axes" (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196), which describes a collection of convex norm optimization problems aimed at minimizing estimates of original values in X through dot products of the mapped points with the axis vectors (rows of V). This particular function solves the constrained optimization problem in Eq. (14), for the L1 norm. The inequality constraint ensures that the estimates for a selected variable are ordered in accordance with its original values. In other words, ignoring any ties, the estimate for the data observation with the i-th smallest value will correspond to the i-th smallest estimate.

Value

A list with the three following entries:

P: A numeric N x m matrix containing the mapped points. Each row is the low-dimensional representation of a data observation in X.
status: A vector of length N where the i-th element contains the status of the chosen solver when calculating the mapping of the i-th data observation. The type of the elements depends on the particular chosen solver.
objval: The numeric objective value associated with the solution to the optimization problem, considering matrix norms.

If the chosen solver fails to map the data (i.e., fails to solve the related optimization problem), P will contain NA (not available) values. In that case, objval will also be NA.

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Select variable for exact estimates, and use it for coloring the embedded
# points
variable <- sample(1:n, 1)

# Compute the mapping
mapping <- ara_ordered_l1(
  Z,
  V,
  variable = variable,
  solver = "clarabel"
)

# Select variables with labeled axis lines on ARA plot
axis_lines <- variable

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  axis_lines = axis_lines,
  color_variable = variable
)

Ordered Adaptable Radial Axes (ARA) mappings using the L2 norm

Description

ara_ordered_l2() computes ordered Adaptable Radial Axes (ARA) mappings for the L2 norm

Usage

ara_ordered_l2(X, V, variable = 1, solver = "clarabel")

Arguments

X

Numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

V

Numeric matrix defining the axes or "axis vectors". Its dimensions are n x m, where 1<=m<=3 is the dimension of the visualization space. Each row of V defines an axis vector.

variable

Integer that indicates the variable (in [1,n]) for which the estimates of high-dimensional data will be exact. Default: variable = 1.

solver

String indicating a package for solving the linear problem(s). It can be "clarabel" (default), "glpkAPI", "Rglpk", or "CVXR".

Details

ara_ordered_l2() computes low-dimensional point representations of high-dimensional numerical data (X) according to the data visualization method "Adaptable Radial Axes" (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196), which describes a collection of convex norm optimization problems aimed at minimizing estimates of original values in X through dot products of the mapped points with the axis vectors (rows of V). This particular function solves the constrained optimization problem in Eq. (14), for the squared-Euclidean norm. The inequality constraint ensures that the estimates for a selected variable are ordered in accordance with its original values. In other words, ignoring any ties, the estimate for the data observation with the i-th smallest value will correspond to the i-th smallest estimate.

Value

A list with the three following entries:

P: A numeric N x m matrix containing the mapped points. Each row is the low-dimensional representation of a data observation in X.
status: A vector of length N where the i-th element contains the status of the chosen solver when calculating the mapping of the i-th data observation. The type of the elements depends on the particular chosen solver.
objval: The numeric objective value associated with the solution to the optimization problem, considering matrix norms.

If the chosen solver fails to map the data (i.e., fails to solve the related optimization problem), P will contain NA (not available) values. In that case, objval will also be NA.

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Select variable for exact estimates, and use it for coloring the embedded
# points
variable <- sample(1:n, 1)

# Compute the mapping
mapping <- ara_ordered_l2(
  Z,
  V,
  variable = variable,
  solver = "clarabel"
)

# Select variables with labeled axis lines on ARA plot
axis_lines <- variable

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  axis_lines = axis_lines,
  color_variable = variable
)

Ordered Adaptable Radial Axes (ARA) mappings using the L-infinity norm

Description

ara_ordered_linf() computes ordered Adaptable Radial Axes (ARA) mappings for the Linf norm

Usage

ara_ordered_linf(X, V, variable = 1, solver = "clarabel")

Arguments

X

Numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

V

Numeric matrix defining the axes or "axis vectors". Its dimensions are n x m, where 1<=m<=3 is the dimension of the visualization space. Each row of V defines an axis vector.

variable

Integer that indicates the variable (in [1,n]) for which the estimates of high-dimensional data will be exact. Default: variable = 1.

solver

String indicating a package for solving the linear problem(s). It can be "clarabel" (default), "glpkAPI", "Rglpk", or "CVXR".

Details

ara_ordered_linf() computes low-dimensional point representations of high-dimensional numerical data (X) according to the data visualization method "Adaptable Radial Axes" (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196), which describes a collection of convex norm optimization problems aimed at minimizing estimates of original values in X through dot products of the mapped points with the axis vectors (rows of V). This particular function solves the constrained optimization problem in Eq. (14), for the L-infinity norm. The inequality constraint ensures that the estimates for a selected variable are ordered in accordance with its original values. In other words, ignoring any ties, the estimate for the data observation with the i-th smallest value will correspond to the i-th smallest estimate.

Value

A list with the three following entries:

P: A numeric N x m matrix containing the mapped points. Each row is the low-dimensional representation of a data observation in X.
status: A vector of length N where the i-th element contains the status of the chosen solver when calculating the mapping of the i-th data observation. The type of the elements depends on the particular chosen solver.
objval: The numeric objective value associated with the solution to the optimization problem, considering matrix norms.

If the chosen solver fails to map the data (i.e., fails to solve the related optimization problem), P will contain NA (not available) values. In that case, objval will also be NA.

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Select variable for exact estimates, and use it for coloring the embedded
# points
variable <- sample(1:n, 1)

# Compute the mapping
mapping <- ara_ordered_linf(
  Z,
  V,
  variable = variable,
  solver = "clarabel"
)

# Select variables with labeled axis lines on ARA plot
axis_lines <- variable

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  axis_lines = axis_lines,
  color_variable = variable
)

Unconstrained Adaptable Radial Axes (ARA) mappings using the L1 norm

Description

ara_unconstrained_l1() computes unconstrained Adaptable Radial Axes (ARA) mappings for the L1 norm

Usage

ara_unconstrained_l1(
  X,
  V,
  weights = rep(1, ncol(X)),
  solver = "glpkAPI",
  use_glpkAPI_simplex = TRUE,
  cluster = NULL
)

Arguments

X

Numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

V

Numeric matrix defining the axes or "axis vectors". Its dimensions are n x m, where 1<=m<=3 is the dimension of the visualization space. Each row of V defines an axis vector.

weights

Numeric array specifying optional non-negative weights associated with each variable. The function only considers them if they do not share the same value. Default: array of n ones.

solver

String indicating a package for solving the linear problem(s). It can be "clarabel" (default), "glpkAPI", "Rglpk", or "CVXR".

use_glpkAPI_simplex

Boolean parameter that indicates whether to use the simplex algorithm (if TRUE) or an interior point method (if FALSE), when using the glpkAPI solver. The default is TRUE.

cluster

Optional cluster object related to the parallel package. If supplied, and n_LP_problems is N, the method computes the mappings using parallel processing.

Details

ara_unconstrained_l1() computes low-dimensional point representations of high-dimensional numerical data (X) according to the data visualization method "Adaptable Radial Axes" (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196), which describes a collection of convex norm optimization problems aimed at minimizing estimates of original values in X through dot products of the mapped points with the axis vectors (rows of V). This particular function solves the unconstrained optimization problem in Eq. (10), for the L1 vector norm. Specifically, it solves equivalent linear problems as described in (11). Optional non-negative weights (weights) associated with each data variable can be supplied to solve the problem in Eq. (15).

Value

A list with the three following entries:

P: A numeric N x m matrix containing the mapped points. Each row is the low-dimensional representation of a data observation in X.
status: A vector of length N where the i-th element contains the status of the chosen solver when calculating the mapping of the i-th data observation. The type of the elements depends on the particular chosen solver.
objval: The numeric objective value associated with the solution to the optimization problem, considering matrix norms, and ignoring weights.

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Define weights
weights <- c(1, 0.75, 0.75, 1)

# Set the number of CPU cores/workers
# NCORES <- parallelly::availableCores(omit = 1)
# NCORES <- max(1,parallel::detectCores() - 1)
NCORES <- 2L

# Create a cluster for parallel processing
cl <- parallel::makeCluster(NCORES)

# Compute the mapping
mapping <- ara_unconstrained_l1(
  Z,
  V,
  weights = weights,
  solver = "glpkAPI",
  use_glpkAPI_simplex = TRUE,
  cluster = cl
)

# Stop cluster
parallel::stopCluster(cl)

# Select variables with labeled axis lines on ARA plot
axis_lines <- c(1, 4) # 1:"mpg", 4:"acceleration")

# Select variable used for coloring embedded points
color_variable <- 1 # "mpg"

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  weights = weights,
  axis_lines = axis_lines,
  color_variable = color_variable
)

Unconstrained Adaptable Radial Axes (ARA) mappings using the L2 norm

Description

ara_unconstrained_l2() computes unconstrained Adaptable Radial Axes (ARA) mappings for the L2 norm

Usage

ara_unconstrained_l2(X, V, weights = rep(1, ncol(X)), solver = "formula")

Arguments

X

Numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

V

Numeric matrix defining the axes or "axis vectors". Its dimensions are n x m, where 1<=m<=3 is the dimension of the visualization space. Each row of V defines an axis vector.

weights

Numeric array specifying optional non-negative weights associated with each variable. The function only considers them if they do not share the same value. Default: array of n ones.

solver

String indicating a package or method for solving the optimization problem. It can be "formula" (default), where the solution is obtained through a closed-form formula, or "CVXR".

Details

ara_unconstrained_l2() computes low-dimensional point representations of high-dimensional numerical data (X) according to the data visualization method "Adaptable Radial Axes" (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196), which describes a collection of convex norm optimization problems aimed at minimizing estimates of original values in X through dot products of the mapped points with the axis vectors (rows of V). This particular function solves the unconstrained optimization problem in Eq. (10), for the squared-Euclidean norm. Optional non-negative weights (weights) associated with each data variable can be supplied to solve the problem in Eq. (15).

Value

A list with the three following entries:

P: A numeric N x m matrix containing the mapped points. Each row is the low-dimensional representation of a data observation in X.
status: A vector of length N where the i-th element contains the status of the chosen solver when calculating the mapping of the i-th data observation. The type of the elements depends on the particular chosen solver.
objval: The numeric objective value associated with the solution to the optimization problem, considering matrix norms, and ignoring weights.

When solver is "formula" this function always produces valid solutions (P), since the pseudo-inverse matrix always exists. Thus, the output status vector is not relevant, but is returned in consonance with other adaptable radial axes functions in the package. If CVRX were used and failed to map the data observations (i.e., failed to solve the related optimization problem), P would be a matrix containing NA (not available) values, and objval would be also be NA.

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Define weights
weights <- c(1, 0.75, 0.75, 1)

# Compute the mapping
mapping <- ara_unconstrained_l2(
  Z,
  V,
  weights = weights,
  solver = "formula"
)

# Select variables with labeled axis lines on ARA plot
axis_lines <- c(1, 4) # 1:"mpg", 4:"acceleration")

# Select variable used for coloring embedded points
color_variable <- 1 # "mpg"

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  weights = weights,
  axis_lines = axis_lines,
  color_variable = color_variable
)

Unconstrained Adaptable Radial Axes (ARA) mappings using the L-infinity norm

Description

ara_unconstrained_linf() computes unconstrained Adaptable Radial Axes (ARA) mappings for the L-infinity norm

Usage

ara_unconstrained_linf(
  X,
  V,
  weights = rep(1, ncol(X)),
  solver = "glpkAPI",
  use_glpkAPI_simplex = TRUE,
  cluster = NULL
)

Arguments

X

Numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

V

Numeric matrix defining the axes or "axis vectors". Its dimensions are n x m, where 1<=m<=3 is the dimension of the visualization space. Each row of V defines an axis vector.

weights

Numeric array specifying optional non-negative weights associated with each variable. The function only considers them if they do not share the same value. Default: array of n ones.

solver

String indicating a package for solving the linear problem(s). It can be "clarabel" (default), "glpkAPI", "Rglpk", or "CVXR".

use_glpkAPI_simplex

Boolean parameter that indicates whether to use the simplex algorithm (if TRUE) or an interior point method (if FALSE), when using the glpkAPI solver. The default is TRUE.

cluster

Optional cluster object related to the parallel package. If supplied, and n_LP_problems is N, the method computes the mappings using parallel processing.

Details

ara_unconstrained_linf() computes low-dimensional point representations of high-dimensional numerical data (X) according to the data visualization method "Adaptable Radial Axes" (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196), which describes a collection of convex norm optimization problems aimed at minimizing estimates of original values in X through dot products of the mapped points with the axis vectors (rows of V). This particular function solves the unconstrained optimization problem in Eq. (10), for the L-infinity vector norm. Specifically, it solves equivalent linear problems as described in (12). Optional non-negative weights (weights) associated with each data variable can be supplied to solve the problem in Eq. (15).

Value

A list with the three following entries:

P: A numeric N x m matrix containing the mapped points. Each row is the low-dimensional representation of a data observation in X.
status: A vector of length N where the i-th element contains the status of the chosen solver when calculating the mapping of the i-th data observation. The type of the elements depends on the particular chosen solver.
objval: The numeric objective value associated with the solution to the optimization problem, considering matrix norms, and ignoring weights.

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Define weights
weights <- c(1, 0.75, 0.75, 1)

# Set the number of CPU cores/workers
# NCORES <- parallelly::availableCores(omit = 1)
# NCORES <- max(1,parallel::detectCores() - 1)
NCORES <- 2L

# Create a cluster for parallel processing
cl <- parallel::makeCluster(NCORES)

# Compute the mapping
mapping <- ara_unconstrained_linf(
  Z,
  V,
  weights = weights,
  solver = "glpkAPI",
  use_glpkAPI_simplex = TRUE,
  cluster = cl
)

# Stop cluster
parallel::stopCluster(cl)

# Select variables with labeled axis lines on ARA plot
axis_lines <- c(1, 4) # 1:"mpg", 4:"acceleration")

# Select variable used for coloring embedded points
color_variable <- 1 # "mpg"

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  weights = weights,
  axis_lines = axis_lines,
  color_variable = color_variable
)

Auto MPG Data Set

Description

Data concerns city-cycle fuel consumption - revised from CMU StatLib library.

Usage

auto_mpg

Format

A matrix containing 398 observations and 10 attributes.

mpg: Miles per gallon of the engine. Predictor attribute
cylinders: Number of cylinders in the engine
displacement: Engine displacement
horsepower: Horsepower of the car
weight: Weight of the car (lbs)
acceleration: Acceleration of the car (seconds taken for 0-60mph)
model_year: Model year of the car in the 1900s
origin: Car origin
make: Car manufacturer
car_name: Name of the car

Source

http://archive.ics.uci.edu/ml/datasets/Auto+MPG

References

Quinlan,R. (1993). Combining Instance-Based and Model-Based Learning. In Proceedings on the Tenth International Conference of Machine Learning, 236-243, University of Massachusetts, Amherst. Morgan Kaufmann.

Examples

data(auto_mpg)    # Lazy loading

Breakfast Cereal Data Set

Description

Nutritional information and manufacturer data for 70+ popular US breakfast cereals

Usage

cereal

Format

A matrix containing 77 observations and 16 attributes.

name: name of cereal
manuf: manufacturer of cereal, coded into seven categories: "A" for American Home Food Products, "G" for General Mills, "K" for Kelloggs, "N" for Nabisco, "P" for Post, "Q" for Quaker Oats, and "R" for Ralston Purina
type: cold or hot
calories: calories per serving
protein: grams of protein
fat: grams of fat
sodium: milligrams of sodium
fiber: grams of dietary fiber
carbo: grams of complex carbohydrates
sugars: grams of sugars
potass: milligrams of potassium
vitamins: vitamins and minerals - 0, 25, or 100, indicating the typical percentage of FDA recommended
shelf: display shelf (1, 2, or 3, counting from the floor)
weight: weight in ounces of one serving
cups: number of cups in one serving
rating: a rating of the cereals

Source

https://lib.stat.cmu.edu/datasets/1993.expo/

References

Reza Mohammadi (2025). Data Science Foundations and Machine Learning with R: From Data to Decisions. https://book-data-science-r.netlify.app.

Examples

data(cereal)
str(cereal)

Draws a 2D Adaptable Radial Axes (ARA) plot for standardized data

Description

Creates a plot associated with an Adaptable Radial Axes (ARA) mapping

Usage

draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  P,
  weights = rep(1, ncol(Z)),
  axis_lines = NULL,
  color_variable = NULL
)

Arguments

Z

Standardized numeric data matrix of dimensions N x n, where N is the number of observations, and n is the number of variables.

X

Original numeric data matrix (before standardizing) of dimensions N x n

V

Numeric matrix of "axis vectors" of dimensions n x 2, where each row of V defines an axis vector.

P

Numeric data matrix of dimensions N x 2 containing the N 2-dimensional representations of the data observations (i.e., the embedded points).

weights

Numeric array specifying non-negative weights associated with each variable. Can also be a 1D matrix. Default: array of n ones.

axis_lines

Array of integer variable indices (in [1,n]) indicating which calibrated axis lines are to be displayed. Default: NULL.

color_variable

Integer (in [1,n]) that indicates the variable used to color the embedded points. Default: NULL.

Details

The function draw_ara_plot_2d_standardized() generates a basic two-dimensional plot related to an "Adaptable Radial Axes" (ARA) mapping (M. Rubio-Sánchez, A. Sanchez, and D. J. Lehmann (2017), doi: 10.1111/cgf.13196) for high-dimensional numerical data (X) that has been previously standardized (Z). The plot displays a set of 2D points (P), each representing an observation from the high-dimensional dataset. It also includes a collection of axis vectors (V), each corresponding to a specific data variable. If the ARA mapping incorporates weights (weights), these axis vectors are colored accordingly to reflect the weighting. For a user-specified subset of variables (axis_lines), the function additionally draws axis lines with tick marks that represent values of the selected variables. Users can estimate the values of the high-dimensional data by visually projecting the plotted points orthogonally onto these axes. The plotted points can also be colored according to the values of the variable color_variable.

Value

Returns 0 if the function terminates without errors

References

M. Rubio-Sánchez, A. Sanchez, D. J. Lehmann: Adaptable radial axes plots for improved multivariate data visualization. Computer Graphics Forum 36, 3 (2017), 389–399. doi:10.1111/cgf.13196

Examples

# Define subset of (numerical) variables
# 1:"mpg", 4:"horsepower", 5:"weight", 6:"acceleration"
selected_variables <- c(1, 4, 5, 6)
n <- length(selected_variables)

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables

# Remove rows with missing values from X
N <- nrow(X)
rows_to_delete <- NULL
for (i in 1:N) {
  if (sum(is.na(X[i, ])) > 0) {
    rows_to_delete <- c(rows_to_delete, -i)
  }
}
X <- X[rows_to_delete, ]

# Convert X to matrix
X <- apply(as.matrix.noquote(X), 2, as.numeric)

# Standardize data
Z <- scale(X)

# Define axis vectors (2-dimensional in this example)
r <- c(0.8, 1, 1.2, 1)
theta <- c(225, 100, 315, 80) * 2 * pi / 360
V <- pracma::zeros(n, 2)
for (i in 1:n) {
  V[i,1] <- r[i] * cos(theta[i])
  V[i,2] <- r[i] * sin(theta[i])
}

# Define weights
weights <- c(1, 0.75, 0.75, 1)

# Compute the mapping
mapping <- ara_unconstrained_l2(Z, V, weights = weights, solver = "formula")

# Select variables with labeled axis lines on ARA plot
axis_lines <- c(1, 4) # 1:"mpg", 4:"acceleration")

# Select variable used for coloring embedded points
color_variable <- 1 # "mpg"

# Draw the ARA plot
draw_ara_plot_2d_standardized(
  Z,
  X,
  V,
  mapping$P,
  weights = weights,
  axis_lines = axis_lines,
  color_variable = color_variable
)

Wine Data Set

Description

Chemical analysis to determine the origin of wines.

Usage

wine

Format

A matrix containing 178 observations and 14 attributes (including 1 classification attribute).

Class: Class of cultivar
Alcohol: Alcohol
Malic acid: Malic acid
Ash: Ash
Alcalinity of ash: Alcalinity of ash
Magnesium: Magnesium
Total phenols: Total phenols
Flavanoids: Flavanoids
Nonflavanoid phenols: Nonflavanoid phenols
Proanthocyanins: Proanthocyanins
Color intensity: Color intensity
Hue: Hue
OD280/OD315 of diluted wines: OD280/OD315 of diluted wines
Proline: Proline

Source

http://archive.ics.uci.edu/dataset/109/wine

References

Dua, D., Graff, C.: UCI Machine Learning Repository. University of California, School of Information and Computer Science, Irvine, CA (2019)

Examples

data(wine)
X <- wine[,-1]
class <- wine[,1]