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 $\eta$-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 $\eta$-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 $\sqrt{N}$.Comment: This article supersedes arXiv:1606.03051, 22 pages and 10 figure

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.Comment: 15 pages, 11 figures. Illustration of a random number generator derived from the algorithm for positioning correction

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.Comment: It completes the results of INSTRUMENTS 2020 4,2-- 13 pages, 2 figure

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

A very simple Gaussian model is used to illustrate an interesting 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 N for the resolution of the usual fits. The effect is obtained with the appropriate form of the variance for each hit (observation). 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 micro-strip 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. The maximum likelihood evaluations complete this exploration of the growth in resolution