[Paper Review] Proofs of non-optimality of the standard least-squares method for track reconstructions

Gregorio Landi, Giovanni Landi|arXiv (Cornell University)|Mar 22, 2020

Soil Geostatistics and Mapping9 references4 citations

TL;DR

This paper proves that the standard least-squares method is suboptimal for track reconstruction in particle detectors, even for non-Gaussian, irregular probability models—common in real silicon microstrip detectors—by extending Cramér-Rao-Fréchet inequalities beyond regular models. It demonstrates that weighted (heteroscedastic) least squares yield significantly better resolution, with gains up to 2.4× over standard least squares in simulations with rectangular hit distributions.

ABSTRACT

It is a standard criterium in statistics to define an optimal estimator the one with the minimum variance. Thus, the optimality is proved with inequality among variances of competing estimators. The inequalities, demonstrated here, disfavor the standard least squares estimators. Inequalities among estimators are connected to names of Cramer, Rao and Frechet. The standard demonstrations of these inequalities require very special analytical properties of the probability functions, globally indicated as regular models. These limiting conditions are too restrictive to handle realistic problems in track fitting. A previous extension to heteroscedastic models of the Cramer-Rao-Frechet inequalities was performed with Gaussian distributions. These demonstrations proved beyond any possible doubts the superiority of the heteroscedastic models compared to the standard least squares method. However, the Gaussian distributions are typical members of the required regular models. Instead, the realistic probability distributions, encountered in tracker detectors, are very different from Gaussian distributions. Therefore, to have well grounded set of inequalities, the limitations to regular models must be overtaken. The aim of this paper is to demonstrate the inequalities for least squares estimators for irregular models of probabilities, explicitly excluded by the Cramer-Rao-Frechet demonstrations. Estimators for straight and parabolic tracks will be considered. The final part deals with the form of the distributions of simplified heteroscedastic track models reconstructed with optimal estimators and the standard (non-optimal) estimators. A comparison among the distributions of these different estimators shows the large loss in resolution of the standard least-squares estimators.

Motivation & Objective

• To establish that the standard least-squares method is non-optimal for track reconstruction in realistic detector conditions, where hit distributions are non-Gaussian and irregular.
• To extend Cramér-Rao-Fréchet variance inequalities beyond regular models to include irregular probability distributions, such as rectangular and Cauchy-like forms, which are common in real tracker systems.
• To demonstrate that heteroscedastic least-squares estimators outperform standard least squares in resolution, even when the underlying hit PDFs are non-Gaussian.
• To quantify the resolution improvement through simulations using a rectangular hit distribution model, which is analytically intractable under classical Cramér-Rao theory.
• To show that the maximum-likelihood estimator can further improve resolution beyond weighted least squares, especially in non-Gaussian models.

Proposed method

• Derives variance inequalities for least-squares estimators under irregular probability models, using formal identities and weighted mean formulations of inverse variances.
• Applies the law of large numbers to reconstruct the maximums of empirical probability distributions of fitted track parameters from Monte Carlo simulations.
• Uses a toy-model with rectangular hit probability density functions (PDFs) as a prototype for irregular models, contrasting it with Gaussian and triangular PDFs.
• Compares the line-shapes of direction estimators from standard (homoscedastic) and weighted (heteroscedastic) least-squares fits across different numbers of detector layers (N = 2 to 13).
• Computes effective variances and maximums of empirical distributions, using the form $1/\sqrt{2\pi\Sigma_l}$ to compare resolution across models.
• Correlates resolution improvements with physical parameters such as magnetic field strength and signal-to-noise ratio, as in prior work [4].

Experimental results

Research questions

• RQ1 Can the Cramér-Rao-Fréchet inequality framework be extended to irregular probability models, such as rectangular or Cauchy-like distributions, which are common in real silicon tracker detectors?
• RQ2 Does the standard least-squares method remain suboptimal when applied to non-Gaussian, heteroscedastic hit distributions typical of real detector systems?
• RQ3 What is the quantitative resolution gain of heteroscedastic over standard least-squares estimators in a realistic, non-Gaussian toy-model with rectangular hit PDFs?
• RQ4 How do the maximums of the empirical parameter distributions (e.g., track direction) scale with the number of detector layers in different fitting methods?
• RQ5 Can maximum-likelihood estimation further improve resolution beyond weighted least squares in non-Gaussian models, and how does this compare to Gaussian-based models?

Key findings

• The standard least-squares method is proven non-optimal even for irregular models, as it fails to achieve the minimum variance among competing estimators.
• For a rectangular hit PDF model, the heteroscedastic least-squares estimator achieves a resolution gain of up to 2.4 times that of the standard least-squares method at N = 7 layers.
• The maximums of the empirical distributions for direction estimators grow linearly with the number of layers N in heteroscedastic fits, in contrast to the $\sqrt{N}$ growth of standard least squares.
• The line-shapes of the fitted track direction distributions are nearly identical across Gaussian, rectangular, and triangular PDFs, but their maximums differ significantly, with Gaussian yielding the highest peak.
• The effective variance $\Sigma_{\text{eff}}$ in heteroscedastic fits varies significantly across N, unlike in standard fits where variances are nearly constant beyond N = 3.
• Maximum-likelihood estimation can further enhance resolution in non-Gaussian models, suggesting that Gaussian-based models may be suboptimal even among optimal methods.