Beyond the N-Limit of the Least Squares Resolution and the Lucky Model

Landi, G.; Landi, G.

doi:10.3390/instruments6010010

Cited by 2 publications

(14 citation statements)

References 20 publications

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“…The construction of the weights for the fits is very simple. Simulations of track fitting with this upgraded tool show resolution improvements also for combination of two types of very different detectors, near to the resolutions of the schematic model of reference [1].…”

mentioning

confidence: 75%

“…The availability of the analytical equations for the COG PDFs allows a direct discussion of the lucky model [1]. The structure of [1] had to be heavily deviated from its aims to complete this discussion.…”

Section: A Demonstration Of the Lucky Modelmentioning

confidence: 99%

“…The signals released by a minimum ionizing particle (MIP), if carefully analyzed, allow to extract sub-optimal effective standard deviations defined in ref. [1] as lucky-model. This model was illustrated without a detailed discussion.…”

Section: Introductionmentioning

confidence: 99%

“…A more accurate discussion of the lucky model, promised in ref. [1], will be developed here. A brief recall to the equations of ref.…”

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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Sub-optimal Approaches to Heteroscedasticity in Silicon Strip Detectors: the Lucky Model and the Super-Lucky Model

Landi¹,

Landi²

2022

Preprint

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The approach to heteroscedasticity of ref.[1] contains a sketchy application of a sub-optimal method of very easy implementation: the lucky model. The supporting proof of this method could not be inserted in [1]. The proof requires the analytical forms of the probability of ref. [2] for the two strip center of gravity. However, those analytical forms suggest also a completion of the lucky-model for the absence of a scaling constant, relevant for combinations of different detector types. The advanced lucky-model (the super-lucky model) can be directly used for track fitting in trackers composed of non-identical detectors. The construction of the weights for the fits is very simple. Simulations of track fitting with this upgraded tool show resolution improvements also for combination of two types of very different detectors, near to the resolutions of the schematic model of reference [1].

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mentioning

confidence: 75%

Section: A Demonstration Of the Lucky Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…A more accurate discussion of the lucky model, promised in ref. [1], will be developed here. A brief recall to the equations of ref.…”

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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Sub-optimal Approaches to Heteroscedasticity in Silicon Strip Detectors: the Lucky Model and the Super-Lucky Model

Landi¹,

Landi²

2022

Preprint

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Probability Distributions for Track Fitting and the Advanced Lucky Model

Landi¹,

Landi²

2022

Physics

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Probability distributions for the center of gravity are fundamental tools for track fitting. The center of gravity is a widespread algorithm for position reconstruction in tracker detectors for particle physics. Its standard use is always accompanied by an easy guess (Gaussian) for the probability distribution of the positioning errors. This incorrect assumption degrades the results of the fit. The explicit error forms evident Cauchy–Agnesi tails that render the use of variance minimizations problematic. Therefore, it is important to report probability distributions for some combinations of random variables essential for track fitting: x=ξ/(ξ+μ), y=(ξ−μ)/[2(ξ+μ)], w=ξ/μ, x=θ(x3−x1)(−x3)/(x3+x2)+θ(x1−x3)x1/(x1+x2) and x=(x1−x3)/(x1+x2+x3). The first two are partial forms of the two strip center of gravity. The fourth is the complete two strip center of gravity, and the fifth is a partial form of the three strip center of gravity. For the complexity of the forth equation, only approximate expressions of the probability are allowed. Analytical expressions are calculated assuming ξ, μ, x1, x2 and x3 independent Gaussian random variables. The analytical form of the probability for the two strip center of gravity allows one to construct an approximate proof for the lucky model of our previous paper. This proof also suggests how to complete the lucky model by its absence of a scaling constant, relevant to combine different detector types. This advanced lucky model (the super-lucky model) can be directly used in trackers composed of non-identical detectors. The construction of the super-lucky model is very simple. Simulations with this upgraded tool also show resolution improvements for a combination of two types of very different detectors, near to the resolutions of the schematic model.

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Beyond the N-Limit of the Least Squares Resolution and the Lucky Model

Cited by 2 publications

References 20 publications

Sub-optimal Approaches to Heteroscedasticity in Silicon Strip Detectors: the Lucky Model and the Super-Lucky Model

Sub-optimal Approaches to Heteroscedasticity in Silicon Strip Detectors: the Lucky Model and the Super-Lucky Model

Probability Distributions for Track Fitting and the Advanced Lucky Model

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