arXiv subjects

Gregorio Landi

Publications and source records attributed to Gregorio Landi.

17 recordsLinked to original sources

arXiv2023

Unifying positioning corrections and random number generations in silicon micro-strip trackers

The optimizations of the track fittings require complex simulations of silicon strip detectors to be compliant with the fundamental properties of the hit heteroscedasticity. Many different generations of random numbers must be available with distributions as similar as possible to the test-beam data. A fast way to solve this problem is an extension of an algorithm of frequent use for the center of gravity positioning corrections. Such extension gives a single method to generate the required types of random numbers. Actually, the starting algorithm is a random number generator, useful in a reverse mode: from non uniform sets of data to uniform ones. The inversion of this operation produces random numbers of given distributions. Many methods have been developed to generate random numbers, but none of those methods is directly connected with this positioning corrections. Hence, the adaptation of the correction algorithm to operate in both mode is illustrated. A sample distribution is generated and its consistency is verified with the Kolmogorov-Smirnov test. As final step, the elimination of the noise is explored, in fact, simulations require noiseless distributions to be modified by given noise models.

physics.ins-det↗

arXiv2022

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

The approach to heteroscedasticity of ref.1(Instruments 2022, 6(1), 10) 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 ref.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 ref.1.

physics.ins-det↗

arXiv2021

Problems of Position Reconstruction in Silicon Microstrip Detects

The algorithms for position reconstruction in silicon micro-strip detectors are studied, and the signals of a minimum ionizing particle are simulated. The center-of-gravity distributions of the data events allow the fine tuning of the signal forms and of the strip response functions. In the sensors with floating strips the response function turns out to roughly approximate the response of a triangular function for the size of the signal involved. The simulations are extended to non orthogonal incidence. In these directions, and in general for all the asymmetric signal distributions, the standard application of the $η$ algorithm introduces a systematic error. A signal reconstruction theorem gives the way to implement the corrections of this error.

physics.ins-det↗

arXiv2021

Positioning Error Probabilities for Some Forms of Center-of-Gravity Algorithm Calculated with the Cumulative Distributions. Part II

To complete a previous work, the probability density functions for the errors in the center-of-gravity as positioning algorithm are derived with the usual methods of the cumulative distribution functions. These methods introduce substantial complications compared to the approaches used in a previous publication on similar problems. The combinations of random variables considered are: $X_{g3}=θ(x_2-x_1) (x_1-x_3)/(x_1+x_2+x_3) + θ(x_1-x_2)(x_1+2x_4)/(x_1+x_2+x_4)$ and $X_{g4}=(θ(x_4-x_5)(2x_4+x_1-x_3)/(x_1+x_2+x_3+x_4)+ θ(x_5-x_4)(x_1-x_3-2x_5)/(x_1+x_2+x_3+x_5)$ The complete and partial forms of the probability density functions of these expressions of the center-of-gravity algorithms are calculated for general probability density functions of the observation noise. The cumulative probability distributions are the essential steps in this study, never calculated elsewhere.

physics.ins-det↗

arXiv2021

Properties of the Center of Gravity as an Algorithm for Position Measurements: Two-Dimensional Geometry

The center of gravity as an algorithm for position measurements is analyzed for a two-dimensional geometry. Several mathematical consequences of discretization for various types of detector arrays are extracted. Arrays with rectangular, hexagonal, and triangular detectors are analytically studied, and tools are given to simulate their discretization properties. Special signal distributions free of discretized error are isolated. It is proved that some crosstalk spreads are able to eliminate the center of gravity discretization error for any signal distribution (ideal detectors). Simulations, adapted to the CMS em-calorimeter and to a triangular detector array, are provided for energy and position reconstruction algorithms with a finite number of detectors.

physics.ins-det↗

arXiv2020

Probability Distributions of Positioning Errors for Some Forms of Center-of-Gravity Algorithms. Part II

The center of gravity is one of the most frequently used algorithm for position reconstruction with different analytical forms for the noise optimization. The error distributions of the different forms are essential instruments to improve the track fitting in particle physics. Their Cauchy-(Agnesi) tails have a beneficial effects to attenuate the outliers disturbance in the maximum likelihood search. The probability distributions are calculated for some combinations of random variables, impossible to find in literature, but relevant for track fitting: $x_{g3}=θ(x_2-x_1)[ (x_1-x_3)/(x_1+x_2+x_3)] + θ(x_1-x_2)[(x_1+2x_4)/(x_1+x_2+x_4)]$ and $x_{g4}=θ(x_4-x_5)[(2x_4+x_1-x_3)/(x_1+x_2+x_3+x_4)]+ θ(x_5-x_4)[(x_1-x_3-2x_5)/(x_1+x_2+x_3+x_5)]$ and $x_{g5}=(2x_4+x_1-x_3-2x_5)/(x_1+x_2+x_3+x_4+x_5)$. The probability density functions of $x_{g3}$, $x_{g4}$ and $x_{g5}$ have complex structures with regions of reduced probability. These regions must be handled with care to avoid false maximums in the likelihood function. General integral equations and detailed analytical expressions are calculated assuming the set $\{x_i\}$ as independent random variables with Gaussian probability distributions. %

physics.ins-det↗

arXiv2020

Positioning Error Probability for Some Forms of Center-of-Gravity Algorithms Calculated with the Cumulative Distributions. Part I

To complete a previous paper, the probability density functions of the center-of-gravity as positioning algorithm are derived with classical methods. These methods, as suggested by the textbook of Probability, require the preliminary calculation of the cumulative distribution functions. They are more complicated than those previously used for these tasks. In any case, the cumulative probability distributions could be useful. The combinations of random variables are those essential for track fitting $x=ξ/{(ξ+η)}$, $x=θ(x_3-x_1) (-x_3)/(x_3+x_2) +θ(x_1-x_3)x_1/(x_1+x_2)$ and $x=(x_1-x_3)/(x_1+x_2+x_3)$. The first combination is a partial form of the two strip center-of-gravity. The second is the complete form, and the third is a simplified form of the three strip center-of-gravity. The cumulative probability distribution of the first expression was reported in the previous publications. The standard assumption is that $ξ$, $η$, $x_1$, $x_2$ and $x_3$ are independent random variables.

cond-mat.stat-mech↗

arXiv2020

Probability Distributions of Positioning Errors for Some Forms of Center-of-Gravity Algorithms

The center of gravity is a widespread algorithm for position reconstruction in particle physics. For track fitting, its standard use is always accompanied by an easy guess for the probability distribution of the positioning errors. This is an incorrect assumption that degrades the results of the fit. The explicit error forms show evident Cauchy-(Agnesi) tails that render problematic the use of variance minimizations. Here, we report the probability distributions for some combinations of random variables, impossible to find in literature, but essential for track fitting: $x=ξ/{(ξ+η)}$, $y={(ξ-η)}/[2{(ξ+η)}]$, $w=ξ/η$, $x=θ(x_3-x_1) (-x_3)/(x_3+x_2) +θ(x_1-x_3)x_1/(x_1+x_2)$ and $x=(x_1-x_3)/(x_1+x_2+x_3)$. The first three are directly connected to each other and are partial forms of the two-strip center of gravity. The fourth is the complete two-strip center of gravity. For its very complex form, it allows only approximate expressions of the probability. The last expression is a simplified form of the three-strip center of gravity. General integral forms are obtained for all of them. Detailed analytical expressions are calculated assuming $ξ$, $η$, $x_1$, $x_2$ and $x_3$ independent random variables with Gaussian probability distributions (the standard assumption for the strip noise).

physics.ins-det↗

arXiv2020

Proofs of non-optimality of the standard least-squares method for track reconstructions

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.

math.ST↗

arXiv2019

The Cramer-Rao inequality to go beyond the $\mathbf{\sqrt{N}}$-limit of the standard least-squares method in track fitting

The Cramer-Rao-Frechet inequality is reviewed specializing it to track fitting. A diffused opinion attributes to this inequality the limitation of the resolution of the track fits with the number N of observations. It turns out that this opinion is incorrect, weighted least squares method is not subjected to that N-limitation. In a previous publication, simulations with a simple Gaussian model produced interesting results: a linear growth of the peaks of the distributions with the number $N$ of observations, much faster than the $\sqrt{N}$ of the standard least squares. These results could be considered a violation of a well known $1/N$-rule for the variance of an unbiased estimator, frequently reported as the Cramer-Rao-Frechet bound. To clarify this point beyond any doubt, it would be essential a direct proof of the consistency of those results with this inequality. Unfortunately, such proof is lacking or very difficult to find. Hence, the Cramer-Rao-Frechet developments are applied to prove the efficiency (optimality) of the simple Gaussian model and the consistency of its results. The inequality remains valid even for irregular models supporting the results of realistic models with similar growths.

physics.ins-det↗

arXiv2019

Properties of the Center of Gravity as an Algorithm for Position Measurements

The center of gravity $x_{g}= \sum_{i}E_{i} x_{i}/\sum_{i} E_{i}$ as an algorithm for position measurements is carefully analyzed. Many mathematical consequences of discretization are extracted. The origin of the systematic error of the algorithm is shown to be connected to the absence of band limits in the Fourier Transform of the signal distributions, which, owing to the intrinsic properties of the measuring devices, must have a finite supports. However, special signal distributions exist among the finite support functions which are free from the discretized error. In the presence of crosstalk, it is proved that some crosstalk spreads are able to eliminate the discretization error for any shape ({\em ideal detector}). For all other cases, analytical expressions and prescriptions are given to correct the error and to efficiently simulate various experimental situations.

physics.ins-det↗

arXiv2019

Beyond the $\sqrt{\mathrm{N}}$ limit of the least squares resolution and the lucky model

A very simple Gaussian model is used to illustrate a new fitting result: a linear growth of the resolution with the number N of detecting layers. This rule is well beyond the well-known rule proportional to $\sqrt{N}$ for the resolution of the usual fit. The effect is obtained with the appropriate form of the variance for each hit (measurement). The model reconstructs straight tracks with N parallel detecting layers, the track direction is the selected parameter to test the resolution. The results of the Gaussian model are compared with realistic simulations of silicon microstrip detectors. These realistic simulations suggest an easy method to select the essential weights for the fit: the lucky model. Preliminary results of the lucky model show an excellent reproduction of the linear growth of the resolution, very similar to that given by realistic simulations.

physics.ins-det↗

arXiv2018

Optimizing momentum resolution with a new fitting method for silicon-strip detectors

A new fitting method is explored for momentum reconstruction of tracks in a constant magnetic field for a silicon-strip tracker. Substantial increases of momentum resolution respect to standard fit is obtained. The key point is the use of a realistic probability distribution for each hit (heteroscedasticity). Two different methods are used for the fits, the first method introduces an effective variance for each hit, the second method implements the maximum likelihood search. The tracker model is similar to the PAMELA tracker. Each side, of the two sided of the PAMELA detectors, is simulated as momentum reconstruction device. One of the two is similar to silicon micro-strip detectors of large use in running experiments. Two different position reconstructions are used for the standard fits, the $η$-algorithm (the best one) and the two-strip center of gravity. The gain obtained in momentum resolution is measured as the virtual magnetic field and the virtual signal-to-noise ratio required by the two standard fits to reach an overlap with the best of two new methods. For the best side, the virtual magnetic field must be increased 1.5 times respect to the real field to reach the overlap and 1.8 for the other. For the high noise side, the increases must be 1.8 and 2.0. The signal-to-noise ratio has similar increases but only for the $η$-algorithm. The signal-to-noise ratio has no effect on the fits with the center of gravity. Very important results are obtained if the number N of detecting layers is increased, our methods provide a momentum resolution growing linearly with N, much higher than standard fits that grow as the $\sqrt{N}$.

physics.ins-det↗

arXiv2017

Nonlinear Volterra model of a loudspeaker behavior based on Laser Doppler Vibrometry

We demonstrate the capabilities of nonlinear Volterra models to simulate the behavior of an audio system and compare them to linear filters. In this paper a nonlinear model of an audio system based on Volterra series is presented and Normalized Least Mean Square algorithm is used to determine the Volterra series to third order. Training data for the models were collected measuring a physical speaker using a laser interferometer. We explore several training signals and filter's parameters. Results indicate a decrease in Mean Squared Error compared to the linear model with a dependency on the particular test signal, the order and the parameters of the model.

cs.SD↗

arXiv2016

Augmenting momentum resolution with well tuned probability distributions

The realistic probability distributions of a previous article are applied to the reconstruction of tracks in constant magnetic field. The complete forms and their schematic approximations produce excellent momentum estimations, drastically better than standard fits. A simplified derivation of one of our probability distributions is illustrated. The momentum reconstructions are compared with standard fits (least squares) with two different position algorithms: the eta-algorithm and the two-strip center of gravity. The quality of our results are expressed as the increase of the magnetic field and signal-to-noise ratio that overlap the standard fit reconstructions with ours best distributions. The data and the simulations are tuned on the tracker of a running experiment and its double sided microstrip detectors, here each detector side is simulated to measure the magnetic bending. To overlap with our best distributions, the magnetic field must be increased by a factor 1.5 for the least squares based on the eta-algorithm and 1.8 for the two-strip center of gravity for the low noise side, and 1.8 and 2.0 for the high noise side. The signal-to-noise ratio must be increased by 1.6 for the low noise side and 2.2 for the high noise side (eta-algorithms). The fits, built on the positioning with the center of gravity, are not modified by a reduction of the signal-to-noise ratio.}

physics.ins-det↗

arXiv2014

Improvements of Track Fitting with Well Tuned Probability Distributions for Silicon Strip Detectors

The construction of a well tuned probability distributions is illustrated in synthetic way, these probability distributions produce faithful realizations of the impact point distributions for particles in silicon strip detector. Their use for track fitting shows a drastic improvements of a factor two, for the low noise case, and a factor three, for the high noise case, respect to the standard approach. The tracks are well reconstructed even in presence of hits with large errors, with a surprising effect of hit discarding. The applications illustrated are simulations of the PAMELA tracker, but other type of trackers can be handled similarly. The probability distributions are calculated for the center of gravity algorithms, and they are very different from gaussian probabilities. These differences are crucial to accurately reconstruct tracks with high error hits and to produce the effective discarding of the too noisy hits (outliers). The similarity of our distributions with the Cauchy distribution forced us to abandon the standard deviation for our comparisons and instead use the full width at half maximum. A set of mathematical approaches must be developed for these applications, some of them are standard in wide sense, even if very complex. One is essential and, in its absence, all the others are useless. Therefore, in this paper, we report the details of this critical approach. It extracts physical properties of the detectors, and allows the insertion of the functional dependence from the impact point in the probability distributions. Other papers will be dedicated to the remaining parts.

physics.ins-det↗

arXiv2014

Asymmetries in Silicon Microstrip Response Function and Lorentz Angle

An experimental set up, dedicated to isolate an error present in the $η$-algorithm, gave an unexpected result. The average of a center of gravity algorithm at orthogonal particle incidence turns out to be non zero. This non zero average signals an asymmetry in the response function of the strips, and introduces a further parameter in the corrections: the shift of the strip response center of gravity respect its geometrical position. A strategy to extract this parameter from a standard data set is discussed. Some simulations with various asymmetric response functions are explored for this test. The method is able to detect easily the asymmetry parameters introduced in the simulations. Its robustness is tested against angular rotations, and we see an almost linear variation with the angle. This simple property is used to simulate a determination of a Lorentz angle with and without the asymmetry of the response function.

physics.ins-det↗