Fast Optimal Space-Filling Design Generators: ISODEN, ISODIS

Quick and easy generation of equally-distributed samples in the unit hypersphere (ISODEN) or hypercube (ISODIS), in multiple dimensions

ISODEN

Generate n-uniformly distributed points within the unit diameter hypersphere [-1,1] in p-dimensions using energy minimization to equalize point density (minimum-entropy).

Syntax:

[x, k] = isoden(n, p)

[x, k] = isoden(n, p, maxIter)

[x, k] = isoden(n, p, maxIter, step)

Inputs:

• n (integer): Number of points to generate.
• p (integer): Dimensionality of the space.
• maxIter (optional, integer): Maximum number of iterations. Default is based on n and p.
• step (optional, double): Energy-step constant for energy minimization. Default is based on n and p.

Outputs:

• x (n-by-p matrix): Generated points.
• k (double): Final energy density of the sample.

Examples:

% Generate 200 points in 3D space
x = isoden(200, 3);
plotPoints(x)

% Generate 1000 points in 2D space (settings)
[x, e] = isoden(1000, 2, 100, 5e-4);
plotPoints(x)

ISODIS

Generate n uniformly distributed points within the unit length hypercube [0,1] in p-dimensions by maximizing minimum distances between points.

Syntax:

[x, d] = isodis(n, p)

[x, d] = isodis(n, p, maxIter)

[x, d] = isodis(n, p, maxIter, step)

Inputs:

• n (integer): Number of points to generate.
• p (integer): Dimensionality of the space.
• maxIter (optional, integer): Maximum number of iterations. Default is 1000.
• step (optional, double): Distance-step constant for distance maximization. Default is based on n, maxIter, and p.

Outputs:

• x (n-by-p matrix): Generated points.
• d (double): Final average neighbor distance.

Customization:

• maxIter(2) (optional, integer): Iteration step to update the connectivity matrix.
• step(2) (optional, integer): Number of changes (factor of n) before updating the connectivity matrix.

Examples:

% Generate 500 points in 2D space
x = isodis(500, 2);
plotPoints(x)

% Generate 100 points in 3D space (settings)
[x, d] = isodis(100, 3, 2e3, 1e-2);
plotPoints(x, 'scatter')

% Update connectivity every 50 iterations
x = isodis(200, 2, [1e3, 50]);
plotPoints(x)

% Update connectivity every 10*n changes
x = isodis(200, 2, [], [NaN, 10]);          % Default step(1)
plotPoints(x)

PLOTPOINTS

Visualize scattered point clouds.

Syntax:

plotPoints(X)

plotPoints(X1, X2, ..., Xp)

plotPoints(..., 'scatter')

Inputs:

• X (n-by-p matrix): Point cloud data.
• X1, X2, ..., Xp (vectors): Separate coordinate components.
• 'scatter' (optional, string): Use scatter plot for 3D data instead of spheres.

Example:

% Load data
load seamount x y z

% Plot sample as matrix
plotPoints([x, y, z])

% Plot sample as vectors
plotPoints(x, y, z, 'scatter')