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QLDF: Quantifying Local Distribution Function (Machine Gnostics)

The QLDF class provides robust, assumption-free local quantification of data distributions using the Machine Gnostics framework. QLDF is designed for inlier-resistant, detailed local analysis, making it ideal for heterogeneous, clustered, or uncertain datasets where dense regions may dominate.

Overview

QLDF is optimized for local quantification and density estimation, especially when data may contain dense clusters, inliers, or unknown distributions. It leverages gnostic algebra and error geometry to deliver resilient, interpretable results without requiring prior statistical assumptions.

  • Assumption-Free: No parametric forms or distributional assumptions.
  • Inlier-Resistant: Robust to dense clusters and inliers.
  • Flexible: Supports additive and multiplicative data forms.
  • Weighted Data: Incorporates sample weights for advanced analysis.
  • Automatic Z0 Identification: Finds local minima in probability density.
  • Advanced Interpolation: Precise estimation of critical points.
  • Memory-Efficient: Optimized for large datasets.
  • Visualization: Built-in plotting for QLDF and PDF.
  • Customizable: Multiple solver options, bounds, and precision settings.

Key Features

  • Fits a local quantifying distribution function to your data
  • Robust to inliers and dense clusters
  • Supports weighted and unweighted samples
  • Automatic or manual bounds and scale selection
  • Additive ('a') and multiplicative ('m') data forms
  • Advanced optimization with customizable tolerance and solver
  • Visualization of QLDF, PDF, and bounds
  • Memory-efficient for large datasets
  • Detailed results and diagnostics
  • Variable scale parameter option for heteroscedasticity

Parameters

Parameter Type Default Description
DLB float or None None Data Lower Bound (absolute minimum, optional)
DUB float or None None Data Upper Bound (absolute maximum, optional)
LB float or None None Lower Probable Bound (practical lower limit, optional)
UB float or None None Upper Probable Bound (practical upper limit, optional)
S float or 'auto' 1 Scale parameter (auto-estimated or fixed value)
varS bool False Use variable scale parameter during optimization
z0_optimize bool True Optimize location parameter Z0 during fitting
tolerance float 1e-9 Convergence tolerance for optimization
data_form str 'a' Data form: 'a' (additive), 'm' (multiplicative)
n_points int 500 Number of points for distribution curve
homogeneous bool True Assume data homogeneity
catch bool True Store intermediate results (memory usage)
weights np.ndarray or None None Prior weights for data points
wedf bool False Use Weighted Empirical Distribution Function
opt_method str 'L-BFGS-B' Optimization method (scipy.optimize)
verbose bool False Print progress and diagnostics
max_data_size int 1000 Max data size for smooth QLDF generation
flush bool True Flush large arrays (memory management)

Attributes

  • params: dict
    Fitted parameters and results after fitting.
  • DLB, DUB, LB, UB, S, varS, z0_optimize, tolerance, data_form, n_points, homogeneous, catch, weights, wedf, opt_method, verbose, max_data_size, flush:
    Configuration parameters as set at initialization.

Methods

fit(data, plot=False)

Fits the QLDF to your data, estimating all relevant parameters and generating the local quantifying distribution function.

  • data: np.ndarray, shape (n_samples,)
    Input data array.
  • plot: bool (optional)
    If True, automatically plots the fitted distribution.

Returns:
None (results stored in params)

plot(plot_smooth=True, plot='both', bounds=True, extra_df=True, figsize=(12,8))

Visualizes the fitted QLDF and related plots.

  • plot_smooth: bool
    Plot smooth interpolated curve.
  • plot: str
    'qldf', 'pdf', or 'both'.
  • bounds: bool
    Show bound lines.
  • extra_df: bool
    Include additional distribution functions.
  • figsize: tuple
    Figure size.

Returns:
None (displays plot)

results()

Returns a dictionary of all fitted parameters and results.

Returns:
dict (fitted parameters, bounds, scale, diagnostics, etc.)

Example Usage

import numpy as np
from machinegnostics.magcal import QLDF

# Example data
data = np.array([ -13.5, 0, 1., 2., 3., 4., 5., 6., 7., 8., 9., 10.])

# Initialize QLDF
qldf = QLDF()

# Fit the model
qldf.fit(data)

# Plot the results
qldf.plot()

# Access fitted parameters
results = qldf.results()
print("Local scale parameter:", results['S_opt'])
print("Distribution bounds:", results['LB'], results['UB'])

Notes

  • QLDF is robust to inliers and suitable for non-Gaussian, clustered, or uncertain data.
  • Supports both additive and multiplicative data forms.
  • Use weights for advanced analysis (e.g., clustering, risk).
  • For large datasets, set catch=False to save memory.
  • Visualization options allow in-depth analysis of local distribution structure.
  • For more information, see GDF documentation and Machine Gnostics.

Author: Nirmal Parmar
Date: 2025-09-24