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Moment-based Piecewise Polynomial Probability Density Estimation with Quantile-Based Binning

Meltem Turan, Joakim Munkhammar

Abstract

Accurate reconstruction of probability density functions (PDFs) from data is essential in engineering applications. Classical global moment-based polynomial approximations often suffer from oscillations, instability in the tails, and sensitivity to the choice of support. This work proposes a quantile-based piecewise polynomial density reconstruction approach that combines equal-probability binning with local moment-matched polynomials within each bin. Two variants are considered: piecewise monomial and piecewise Lagrange polynomials with Chebyshev nodes. The numbers of bins and polynomial degrees are selected by a proposed grid search approach guided by the Kolmogorov-Smirnov (K-S) test statistic under non-negativity constraints. Across several benchmark distributions, the proposed methods reduce K-S errors by about $80$-$96\%$ relative to standard monomial and Lagrange polynomial approaches, and by about $83$-$97\%$ compared with spline density estimation. For real-world household electricity consumption and solar irradiance data, the piecewise approaches achieve K-S test statistic performance comparable to kernel density estimation while offering improved control over tail behavior and oscillations. Overall, the results demonstrate that quantile-based localization substantially enhances the robustness and fidelity of moment-based polynomial PDF reconstruction.

Moment-based Piecewise Polynomial Probability Density Estimation with Quantile-Based Binning

Abstract

Accurate reconstruction of probability density functions (PDFs) from data is essential in engineering applications. Classical global moment-based polynomial approximations often suffer from oscillations, instability in the tails, and sensitivity to the choice of support. This work proposes a quantile-based piecewise polynomial density reconstruction approach that combines equal-probability binning with local moment-matched polynomials within each bin. Two variants are considered: piecewise monomial and piecewise Lagrange polynomials with Chebyshev nodes. The numbers of bins and polynomial degrees are selected by a proposed grid search approach guided by the Kolmogorov-Smirnov (K-S) test statistic under non-negativity constraints. Across several benchmark distributions, the proposed methods reduce K-S errors by about - relative to standard monomial and Lagrange polynomial approaches, and by about - compared with spline density estimation. For real-world household electricity consumption and solar irradiance data, the piecewise approaches achieve K-S test statistic performance comparable to kernel density estimation while offering improved control over tail behavior and oscillations. Overall, the results demonstrate that quantile-based localization substantially enhances the robustness and fidelity of moment-based polynomial PDF reconstruction.
Paper Structure (14 sections, 20 equations, 8 figures, 15 tables, 2 algorithms)

This paper contains 14 sections, 20 equations, 8 figures, 15 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Moment-based piecewise polynomial reconstruction via quantile binning
  3. Quantile-based binning and piecewise polynomial density estimation
  4. Piecewise monomial formulation
  5. Piecewise Lagrange polynomial formulation
  6. Applications
  7. Estimation of well-known distribution functions
  8. Estimation of empirical distribution functions
  9. Household electricity consumption.
  10. Solar irradiance and clear-sky index.
  11. Sensitivity analysis
  12. Conclusion
  13. Disclosure statement
  14. Data Availability Statement

Figures (8)

  • Figure 1: Normal distribution: Bandwidth $h=0.05$ for KDE; $NumBasis=14$ and $Degree=3$ for SDE; $N_M=11$ for standard monomial and Lagrange polnomial approaches; $N_M=5$, $N_B=18$ for piecewise monomial; and $N_M=4$, $N_B=14$ for piecewise Lagrange polynomial.
  • Figure 2: Weibull distribution: Bandwidth $h=0.05$ for KDE; $NumBasis=20$ and $Degree=3$ for SDE; $N_M=11$ for standard monomial and Lagrange polynomial approaches; $N_M=7$, $N_B=19$ for piecewise monomial; and $N_M=5$, $N_B=16$ for piecewise Lagrange polynomial.
  • Figure 3: Bimodal Normal distribution: Bandwidth $h=0.05$ for KDE; $NumBasis=19$ and $Degree=3$ for SDE; $N_M=11$ for standard monomial and Lagrange polynomial approaches; $N_M=8$, $N_B=16$ for piecewise monomial; and $N_M=5$, $N_B=14$ for piecewise Lagrange polynomial.
  • Figure 4: Trimodal Weibull distribution: Bandwidth $h=0.05$ for KDE; $NumBasis=18$ and $Degree=7$ for SDE; $N_M=11$ for standard monomial and Lagrange polnomial approaches; $N_M=5$, $N_B=19$ for piecewise monomial; and $N_M=4$, $N_B=19$ for piecewise Lagrange polynomial.
  • Figure 5: Household electricity consumption data set: Bandwidth $h=0.05$ for KDE; $NumBasis=17$ and $Degree=6$ for SDE; $N_M=11$ for standard monomial and Lagrange polynomial approaches; $N_M=4$, $N_B=19$ for piecewise monomial; and $N_M=4$, $N_B=19$ for piecewise Lagrange polynomial.
  • ...and 3 more figures