median: Gnostic Median Metric

The median function computes the Gnostic median (Global 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.

---

Overview

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

• Case 'i': Estimates median using EGDF (Empirical Gnostics Distribution Function).
• Case 'j': Quantifies median using QGDF (Quantile Gnostics 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 median (EGDF), 'j' for quantifying median (QGDF).	'i'
S	float	Scaling parameter for EGDF/QGDF.	1
z0_optimize	bool	Whether to optimize z0 in EGDF/QGDF.	True
data_form	str	Data form for EGDF/QGDF:'a' for additive, 'm' for multiplicative.	'a'
tolerance	float	Tolerance for EGDF/QGDF fitting.	1e-6
verbose	bool	If True, enables detailed logging for debugging.	False

---

Returns

• float The Gnostic median of the data.

---

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.

---

Example Usage

import machinegnostics as mg
import numpy as np

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

# Example 2: Quantifying median with QGDF
median_j = mg.median(data, case='j')
print(median_j)

---

Notes

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

---

Gnostic vs. Classical Median

Note: Unlike classical median metrics that use statistical order statistics, the Gnostic median 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.

---

Author: Nirmal Parmar
Date: 2025-09-24

---