std: Gnostic Standard Deviation Metric

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

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Overview

Gnostic standard deviation generalizes classical standard deviation by using irrelevance and fidelity measures: - Case 'i': Estimates standard deviation using the estimating geometry. - Case 'j': Quantifies standard deviation using the quantifying geometry.

Both approaches are robust to outliers and non-normal data, providing reliable diagnostics in challenging scenarios. The function returns lower and upper bounds for the standard deviation.

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Parameters

Parameter	Type	Description	Default
data	np.ndarray	Input data array (1D, no NaN/Inf).	Required
case	str	'i' for estimating standard deviation, 'j' for quantifying standard deviation.	'i'
S	float/str	Scaling parameter for ELDF. Can be 'auto' to optimize using EGDF. Suggested range: [0.01, 2].	'auto'
z0_optimize	bool	Whether to optimize z0 in ELDF.	True
data_form	str	Data form for ELDF: '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

• tuple
  Lower and upper bounds of the Gnostic standard deviation.

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Raises

• TypeError
  If 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 standard deviation (default case)
data = np.array([1, 2, 3, 4, 5])
std_lb, std_ub = mg.std(data)
print(std_lb, std_ub)

# Example 2: Quantifying standard deviation
std_lb_j, std_ub_j = mg.std(data, case='j')
print(std_lb_j, std_ub_j)

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Notes

• The function uses mean and variance from gnostic theory to compute lower and upper bounds for standard deviation.
• 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 standard deviation.
• Scaling (S), optimization (z0_optimize), and data form (data_form) parameters allow for flexible analysis.
• If S='auto', the function optimizes the scaling parameter using EGDF.

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Gnostic vs. Classical Standard Deviation

Note:
Unlike classical standard deviation metrics that use statistical means and variances, the Gnostic standard deviation 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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