[Paper Review] Performance of Non-Parametric Reconstruction Techniques in the Late-Time Universe

Celia Escamilla‐Rivera, Jackson Levi Said+1 more|arXiv (Cornell University)|May 29, 2021

Cosmology and Gravitation TheoriesPhysics and Astronomy104 references33 citations

TL;DR

This paper compares Gaussian Process (GP) and LOESS-Simex non-parametric reconstruction techniques for cosmological data in the late-time universe, using cosmic chronometers, BAO, and Pantheon SNe Ia data. It finds GP produces significantly smaller uncertainties (σGP−min = 1.1) than LOESS-Simex (σLS−min = 50.8), with GP offering tighter, more precise reconstructions despite higher overfitting risk, while LOESS-Simex yields broader, more conservative error regions independent of H₀ priors.

ABSTRACT

In the context of a Hubble tension problem that is growing in its statistical significance, we reconsider the effectiveness of non-parametric reconstruction techniques which are independent of prescriptive cosmological models. By taking cosmic chronometers, Type Ia Supernovae and baryonic acoustic oscillation data, we compare and contrast two important reconstruction approaches, namely Gaussian processes (GP) and the extbf{Lo}cally w extbf{e}ighted extbf{S}catterplot extbf{S}moothing together with extbf{Sim}ulation and extbf{ex}trapolation method (LOESS-Simex or LS). In the context of these methods, besides not requiring a cosmological model, they also do not require physical parameters in their approach to their reconstruction of data (but they do depend on statistical hyperparameters). We firstly show how both GP and LOESS-Simex can be used to successively reconstruct various data sets to a high level of precision. We then directly compare both approaches in a quantitative manner by considering several factors, such as how well the reconstructions approximate the data sets themselves to how their respective uncertainties evolve. In light of the puzzling Hubble tension, it is important to consider how the uncertain regions evolve over redshift and the methods compare for estimating cosmological parameters at current times. For cosmic chronometers and baryonic acoustic oscillation compiled data sets, we find that GP generically produce smaller variances for the reconstructed data with a minimum value of $\sigma_{ m GP-min} = 1.1$, while the situation for LS is totally different with a minimum of $\sigma_{ m LS-min} = 50.8$. Moreover, some of these characteristics can be alleviate at low $z$, where LS presents less underestimation in comparison to GP.

Motivation & Objective

• To evaluate and compare the performance of non-parametric reconstruction techniques—Gaussian Processes (GP) and LOESS-Simex (LS)—in the context of late-time cosmology.
• To assess how these methods reconstruct Hubble parameter H(z) and distance modulus μ(z) from cosmic chronometers, BAO, and Pantheon SNe Ia data without assuming a cosmological model.
• To quantify the accuracy and uncertainty evolution of both methods, especially in light of the growing Hubble tension.
• To investigate whether the choice of reconstruction method affects cosmological parameter estimation, particularly at low redshift.
• To examine the robustness of both methods under different H₀ priors (TRGB, HW, R19) and redshift scaling (linear vs. logarithmic).

Proposed method

• Employs Gaussian Process (GP) regression to non-parametrically reconstruct H(z) and μ(z) using covariance functions that model data correlations and hyperparameters.
• Applies LOESS-Simex (Locally weighted Scatterplot Smoothing with Simulation and Extrapolation) to smooth data and estimate uncertainties without assuming a cosmological model.
• Uses observational data sets: cosmic chronometers (CC), baryonic acoustic oscillations (BAO), and the Pantheon SNe Ia sample, with and without H₀ priors.
• Applies statistical metrics such as the d(z) indicator and Υ = Σd²(z) to quantify reconstruction quality and deviation from original data.
• Tests sensitivity of GP to redshift axis scaling, finding that logarithmic scaling yields more stable and accurate hyperparameters than linear scaling.
• Compares reconstruction performance using metrics like variance, uncertainty propagation, and consistency with ΛCDM predictions across different priors and redshift regimes.

Experimental results

Research questions

• RQ1 How do GP and LOESS-Simex perform in reconstructing H(z) and μ(z) from CC, BAO, and Pantheon SNe Ia data without assuming a cosmological model?
• RQ2 What are the differences in uncertainty propagation and variance between GP and LOESS-Simex reconstructions across redshift, especially at low z?
• RQ3 How does the choice of redshift axis (linear vs. logarithmic) affect the stability and accuracy of GP reconstructions?
• RQ4 To what extent do the reconstruction methods' results depend on the adopted H₀ prior (TRGB, HW, R19)?
• RQ5 Which method provides a more conservative or reliable uncertainty estimate in the context of the Hubble tension?

Key findings

• GP reconstructions achieve a minimum uncertainty of σGP−min = 1.1, significantly lower than the σLS−min = 50.8 minimum for LOESS-Simex, indicating GP produces tighter, more precise reconstructions.
• LOESS-Simex yields broader and more conservative uncertainty regions compared to GP, with less fluctuation in variance across redshift, suggesting greater robustness to noise.
• At low redshift (z ≲ 0.5), LOESS-Simex shows reduced underestimation of uncertainties compared to GP, indicating better performance in data-sparse regions.
• The GP method is sensitive to the choice of redshift axis scale: logarithmic scaling leads to stable, well-fitted hyperparameters, while linear scaling results in poor high-redshift reconstruction due to dominance of low-z data points.
• The Υ statistic, measuring reconstruction fidelity, shows GP consistently produces smaller values (better fit) than LOESS-Simex, though GP's small variances may indicate overfitting.
• Both methods remain robust across different H₀ priors (TRGB, HW, R19), with reconstruction quality and uncertainty characteristics independent of the prior used.