mean: Gnostic Mean Metric

The mean function computes the Gnostic mean (Local Estimate of Location) of a data sample. This metric uses gnostic theory to provide robust, assumption-free estimates of central tendency, leveraging irrelevance and fidelity measures for deeper insight into data structure and uncertainty.

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Overview

Gnostic mean generalizes classical mean by using irrelevance and fidelity measures:

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

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

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Parameters

Parameter	Type	Description	Default
data	np.ndarray	Input data array (1D, no NaN/Inf).	Required
S	float/str	Scaling parameter for ELDF/QLDF (float or 'auto').	1
case	str	'i' for estimating mean (ELDF), 'j' for quantifying mean (QLDF).	'i'
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

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Returns

• float The Gnostic mean of the data.

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Raises

• TypeErrorIf input is not a numpy array, or if S is not a float or 'auto'.
• ValueError If input is not 1D, is empty, contains NaN/Inf, or if case/data_form is invalid.

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Example Usage

import machinegnostics as mg
import numpy as np

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

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

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Notes

• The function uses ELDF or QLDF to compute irrelevance and fidelity values, which are then used to estimate the mean.
• 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 mean.
• Scaling (S), optimization (z0_optimize), and data form (data_form) parameters allow for flexible analysis.

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Gnostic vs. Classical Mean

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

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Authors: Nirmal Parmar
Date: 2025-09-24

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