variance: Gnostic Variance Metric

The variance function computes the Gnostic variance of a data sample. This metric uses gnostic theory to provide robust, assumption-free estimates of data variability, leveraging irrelevance measures for deeper insight into uncertainty and structure.

---

Overview

Gnostic variance generalizes classical variance by using irrelevance measures:

• Case 'i': Estimates variance using ELDF (Empirical Likelihood Distribution Function).
• Case 'j': Quantifies variance using QLDF (Quantile Likelihood Distribution Function).

Both approaches are robust to outliers and non-normal data, providing reliable diagnostics in challenging scenarios.

---

Parameters

Parameter	Type	Description	Default
data	np.ndarray	Input data array (1D, no NaN/Inf).	Required
case	str	'i' for estimating variance (ELDF), 'j' for quantifying variance (QLDF).	'i'
S	float	Scaling parameter for ELDF/QLDF.	1
z0_optimize	bool	Whether to optimize z0 in ELDF/QLDF.	True
data_form	str	Data form for ELDF/QLDF:'a' for additive, 'm' for multiplicative.	'a'
tolerance	float	Tolerance for ELDF fitting.	1e-6
verbose	bool	If True, enables detailed logging for debugging.	False

---

Returns

• float The Gnostic variance of the data.

---

Raises

• TypeErrorIf input is not a numpy array.
• ValueError If input is not 1D, is empty, contains NaN/Inf, or if case is not 'i' or 'j'.

---

Example Usage

import machinegnostics as mg
import numpy as np

# Example 1: Compute gnostic variance (default case)
data = np.array([1, 2, 3, 4, 5])
var = mg.variance(data)
print(var)

# Example 2: Quantifying variance with QLDF
var_j = mg.variance(data, case='j')
print(var_j)

---

Notes

• The function uses ELDF or QLDF to compute irrelevance values, which are then squared and averaged.
• Input data must be 1D, cleaned, and free of NaN/Inf.
• The metric is robust to outliers and non-normal data, providing more reliable diagnostics than classical variance.
• Scaling (S), optimization (z0_optimize), and data form (data_form) parameters allow for flexible analysis.

---

Gnostic vs. Classical Variance

Note: Unlike classical variance metrics that use statistical means, the Gnostic variance is computed using irrelevance measures from gnostic theory. This approach is assumption-free and designed to reveal the true diagnostic properties of your data.

---

Authors: Nirmal Parmar
Date: 2025-09-24

---