correlation: Gnostic Correlation Metric

The correlation function computes the Gnostic correlation coefficient between two data samples. This metric provides a robust, assumption-free measure of association, leveraging Mathematical Gnostics to deliver reliable results even in the presence of noise, outliers, or non-Gaussian data.

---

Overview

The Gnostic correlation metric measures the association between two arrays using either estimation geometry (EGDF) or quantifying geometry (QGDF):

• Case 'i': Uses estimation geometry (EGDF) for correlation.
• Case 'j': Uses quantifying geometry (QGDF) for correlation.

Unlike classical correlation coefficients, the Gnostic approach is resilient to data uncertainty and is meaningful for both small and large datasets.

---

Parameters

Parameter	Type	Description
X	array-like	Feature data sample (1D array or single column from 2D array, no NaN/Inf).
y	array-like	Target data sample (1D array, no NaN/Inf).
case	str	'i' for estimation geometry, 'j' for quantifying geometry. Default: 'i'.
verbose	bool	If True, enables detailed logging for debugging. Default: False.

---

Returns

• float
  The Gnostic correlation coefficient between the two data samples.

---

Raises

• TypeError
  If X or y are not array-like.
• ValueError
  If input arrays are empty, contain NaN/Inf, are not 1D, have mismatched shapes, or if case is not 'i' or 'j'.

---

Example Usage

from machinegnostics.metrics import correlation
import numpy as np

# Example 1: Simple 1D arrays
X = np.array([1, 2, 3, 4, 5])
y = np.array([5, 4, 3, 2, 1])
corr = correlation(X, y, case='i')
print(corr)

# Example 2: Multi-column X
X = np.array([[1, 10], [2, 20], [3, 30], [4, 40], [5, 50]])
y = np.array([5, 4, 3, 2, 1])
for i in range(X.shape[1]):
    corr = correlation(X[:, i], y)
    print(f"Correlation for column {i}: {corr}")

---

Notes

• Both X and y must be 1D arrays of the same length, with no missing or infinite values.
• For multi-column X, pass each column separately (e.g., X[:, i]).
• The metric is robust to outliers and provides meaningful estimates even for small or noisy datasets.
• If data homogeneity is not met, the function will adjust parameters and issue a warning for best results.
• For optimal results, ensure your data is preprocessed and cleaned.

---

Gnostic vs. Classical Correlation

Note: Unlike classical correlation metrics that rely on statistical means and linear relationships, the Gnostic correlation uses irrelevance measures derived from gnostic theory. This approach is assumption-free and designed to reveal true data relationships, even in the presence of outliers or non-normal distributions.

---

Author: Nirmal Parmar
Date: 2025-09-24

---