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

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".

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.

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.

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

# Load data
data("auto_mpg", package = "ascentTraining")

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

# Retain only selected variables and rename dataset as X
X <- auto_mpg[, selected_variables] # Select a subset of variables
rm(auto_mpg)
#> Warning: object 'auto_mpg' not found

# 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 <- geometry::pol2cart(theta, r)

# Select variable for exact estimates, and use it for coloring the embedded
# points
n <- nrow(V)
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
)
#> [1] 0