theta, expanding 2-factor ratios to n-factors.

This repository provides implementations of a function, f_theta, that calculates the angle (theta) between each row vector of an input matrix X and a vector of all ones of the same dimension. The angle is returned in radians.

This can be appropriate as a normative model (hence the vector of ones), to detect feature deviations across any number dimensions.

The equivalent of the traditional ratio would be $\tan(\theta)$, but $\theta$ is visually more simple.

Taking $\cos{\theta}$ instead will return values where 1 is the closest to the controls, and 0 the furthest.

Figure below shows an example in 3D, and two of its projections to 2D. Theta adapts to the number of dimensions (2 to N).

Formula

For each row vector x in the input matrix X, and a normalized all-ones vector v1 (where v1 = [1, 1, ..., 1] / ||[1, 1, ..., 1]||), the angle theta is calculated as:

theta = arccos( (x ⋅ v1) / (||x|| ⋅ ||v1||) )

Since ||v1|| = 1, this simplifies to:

theta = arccos( (x ⋅ v1) / ||x|| )

Implementations

This repository provides implementations in:

• R: R/f_theta.R
• Python (NumPy): python/f_theta.py

Usage

Please see the example scripts in the examples/ directory for each language.

R Example

See R/f_theta.R and R/example.R

# Source the function
source("R/f_theta.R")

# Create a sample matrix
X_matrix <- matrix(c(1, 2, 3,
                     4, 5, 6,
                     1, 1, 1,
                     -1, -1, -1,
                     1, 0, 0), ncol = 3, byrow = TRUE)

# Calculate theta
thetas_r <- f_theta(X_matrix)
print(thetas_r)
# Expected output for [1,1,1] is approx 0 radians
# Expected output for [-1,-1,-1] is approx pi radians

Python Example

See python/f_theta.py and python/example.py

import numpy as np
from f_theta import f_theta # Assuming f_theta.py is in the same directory or PYTHONPATH

X_matrix_py = np.array([[1, 2, 3],
                        [4, 5, 6],
                        [1, 1, 1],
                        [-1, -1, -1],
                        [1, 0, 0]], dtype=float)

thetas_py = f_theta(X_matrix_py)
print(thetas_py)