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Computing Accurate Probabilistic Estimates of One-D Entropy from Equiprobable Random Samples.

Hoshin V Gupta1, Mohammad Reza Ehsani1, Tirthankar Roy2

  • 1Hydrology and Atmospheric Sciences, The University of Arizona, Tucson, AZ 85721, USA.

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Summary
This summary is machine-generated.

A new Quantile Spacing (QS) method accurately estimates entropy from data. QS outperforms Bin-Counting (BC) and Kernel Density (KD) methods, requiring no hyper-parameter tuning and showing low bias even with small sample sizes.

Keywords:
accuracybootstrapentropyestimationquantile spacingsmall-sample efficiencyuncertainty

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Area of Science:

  • Statistics
  • Information Theory
  • Data Analysis

Background:

  • Accurate estimation of entropy is crucial for understanding probability distributions.
  • Existing methods like Bin-Counting (BC) and Kernel Density (KD) have limitations in hyper-parameter tuning and applicability to diverse distributions.
  • There is a need for a robust and adaptable entropy estimation method.

Purpose of the Study:

  • To develop and validate a simple Quantile Spacing (QS) method for accurate one-dimensional entropy estimation.
  • To compare the performance of QS against established BC and KD methods.
  • To assess the effectiveness of QS across various probability density functions (pdfs) and sample sizes.

Main Methods:

  • Developed the Quantile Spacing (QS) method using equiprobable intervals based on quantile estimates.
  • Compared QS with Bin-Counting (BC) and Kernel Density (KD) methods using Gaussian, Log-Normal, Exponential, and Bimodal Gaussian Mixture distributions.
  • Employed bootstrapping to approximate the sampling variability of the entropy estimates.

Main Results:

  • The QS method demonstrated accurate probabilistic estimation of entropy with low bias (<1%) across tested distributions.
  • Optimal number of quantiles for QS was found to be a fixed fraction (~0.25-0.35) of the sample size, insensitive to distributional form.
  • QS significantly outperformed BC (bias up to -50%) and KD (bias up to -10%), especially for small sample sizes, and requires no hyper-parameter tuning or kernel selection.

Conclusions:

  • Quantile Spacing (QS) offers a superior, robust, and user-friendly method for entropy estimation compared to BC and KD.
  • QS efficiently utilizes data information by estimating quantile locations, leading to more accurate approximations of underlying pdfs.
  • The method's insensitivity to hyper-parameter tuning and broad applicability make it valuable for diverse scientific and data analysis applications.