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.

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)