Heteroscedasticity and angle resolution in high-energy particle tracking: revisiting "Beyond the N − − √ limit of the least squares resolution and the lucky model", by G. Landi and G. E. Landi
HHeteroscedasticity and angle resolution in high-energy particletracking: revisiting “Beyond the √ N limit of the least squaresresolution and the lucky model”, by G. Landi and G. E. Landi
D. Bernard,LLR, Ecole Polytechnique, CNRS/IN2P3, 91128 Palaiseau, FranceOctober 8, 2020
Abstract
I re-examine a recent work by G. Landi and G. E. Landi. [arXiv:1808.06708 [physics.ins-det]],in which the authors claim that the resolution of a tracker can vary linearly with the number ofdetection layers, N , that is, faster than the commonly known √ N variation, for a tracker of fixedlength, in case the precision of the position measurement is allowed to vary from layer to layer, i.e.heteroscedasticity, and an appropriate analysis method, a weighted least squares fit, is used. keywords :Tracking, weighted least squares, homoscedasticity, heteroscedasticity, Cramer-Rao Bound The momentum of charged particles, including the magnitude and the direction, is one of the verybasic observables on which event reconstruction is built in particle physics. In the case of a detectorof fixed given length, L , containing a number N of detection layers, it is common wisdom that theprecision on the track angle improves as 1 / √ N , asymptotically at large N , so that the resolutionimproves as √ N (e.g. [1], and references therein).In a recent work, though, G. Landi and G. E. Landi. are claiming that “ A very simple Gaussianmodel is used to illustrate a new fitting result: a linear growth of the resolution with the number N ofdetecting layers. This rule is well beyond the well-known rule proportional to √ N for the resolutionof the usual fit ” [2] (and further developments in [3, 4]).As I didn’t find the graphical pieces of evidence that were presented in [2] to support the allegationquite convincing, I am trying here to re-examine the matter. As in [2], I consider a simple situation ofa tracker consisting of equally-spaced parallel layers, without magnetic field, and for which multiplescattering can be neglected. Let’s consider a tracker consisting of N layers, i = 1 · · · N , equally spaced at position x i along the x axis with spacing D . Each detector i measures the position in the transverse direction y of eachtrack traversing it, y i , with a Gaussian point spread function (PSF) with RMS σ i . We aim at fittingstraight tracks y = ax + b (1)1 a r X i v : . [ phy s i c s . d a t a - a n ] O c t N N Figure 1: Homoscedastic trackers: single-Gaussian-distributed measurement precision: variation ofthe inverse precision, 1 /σ a , as a function of the number of detectors, for a fixed total detector length L , and for σ/L = 1. Left plot: up to N − N − a and b are the slope and the intercept of the track. Minimization of the χ , χ = N (cid:88) i =1 (cid:18) y i − ( ax i + b ) σ i (cid:19) , (2)provides the values of a and b : a = s xy s − s x s y s x s − ( s x ) and b = s y s x − s xy s x s x s − ( s x ) (3)with precisions σ a = (cid:114) ss x s − ( s x ) and σ b = (cid:114) s x s x s − ( s x ) (4)and with s = N (cid:88) i =1 σ i , s x = N (cid:88) i =1 x i σ i , s y = N (cid:88) i =1 y i σ i , s xy = N (cid:88) i =1 x i y i σ i and s x = N (cid:88) i =1 x i σ i . (5)Given that x i = iD , we have σ a = 1 D (cid:80) Ni =1 σ i (cid:18)(cid:80) Ni =1 i σ i (cid:19) (cid:18)(cid:80) Ni =1 σ i (cid:19) − (cid:18)(cid:80) Ni =1 iσ i (cid:19) (6) In case the precisions σ i are the same for all layers (homoscedasticity) and equal to a common value σ , eq. (6) simplifies and the precision of the measurement of the track angles boils down to [1]2 a = 2 σL (cid:115) N − N ( N + 1) = 2 σD (cid:115) N − N ( N + 1) , (7)where L = ( N − D is the total length of the detector. For N = 1, σ a is undefined as wasexpected for an angle measurement. For trackers with a large number of detection layers, and for atotal length L being kept constant, the precision of the measurement of the angle varies asymptoticallyas (cid:112) /N σ/L , and the resolution as (cid:112) N/ L/σ .It seems clear from the variation of the inverse precision, 1 /σ a , as a function of the number ofdetectors, Fig. 1, that the linear variation with N alluded in [2] is an “impression” when focusingattention on the very smallest numbers of detectors (left plot), while the asymptotic √ N variation isclearly visible for larger numbers (right plot). I now turn to the two-Gaussian toy model that G. Landi and G. E. Landi [2] have used as being agood approximation of tracking with silicon strip detectors, and with which they say they observe alinear growth. The point spread function consists of two Gaussians with different standard deviations,the first one with σ = 0 .
18 with a probability of 80 % and the second one with σ = 0 .
018 with aprobability of 20 % [2].With this model, fitting each track with a weighted least squares provides values of a with aGaussian probability density function with standard deviation σ a given by eq. (6), (as demonstratedby the pull distribution, that is found to follow a perfect N (0 ,
1) distribution) but the value of σ a varies from track to track, depending on the distributions of the precisions of the measurements ( σ or σ ) along the track. As the a distribution of the whole event sample is not Gaussian-distributed, Iuse the same method as in [2] to obtain a samplewise estimate of 1 /σ a , that is, the maximum of the a distribution. N N Figure 2: Heteroscedastic trackers: double-Gaussian-distributed measurement precision: variation ofthe height of the a peak at maximum, as a function of the number of detectors, for a fixed totaldetector length L . Left plot: up to N − N − N , I obtain a √ N -like variation, something which is moreeasily observed on the variation with √ N (Fig. 3). √ N √ N Figure 3: Variation with √ N of the estimators shown in Figs. 1 and 2, up to N − /σ a . Right: Heteroscedastic trackers: double-Gaussian-distributed measurement precision:variation of the height of the a peak at maximum. The present work does confirm that for tracking detectors consisting of a small number of layers,the angle resolution seems to vary as N as was shown in Fig. 2 of [2]. Examination of detectorsconsisting of a large number of layers, though, shows a variation of the resolution as √ N , compatiblewith common wisdom. Neither homoscedasticity nor heteroscedasticity are found to play any role inthe matter, in contrast with what alleged in [2]. References [1] M. Regler and R. Fruhwirth, “Generalization of the Gluckstern formulas. I: Higher orders, alternativesand exact results,” Nucl. Instrum. Meth. A (2008) 109.[2] G. Landi and G. E. Landi, “Beyond the √ N limit of the least squares resolution and the lucky model,”[arXiv:1808.06708 [physics.ins-det]].[3] G. Landi and G. E. Landi, “The Cramer-Rao Inequality to Improve the Resolution of the Least-SquaresMethod in Track Fitting,” Instruments (2020) 2[4] G. Landi and G. E. Landi, “Proofs of non-optimality of the standard least-squares method for trackreconstructions,” [arXiv:2003.10021 [math.ST]].(2020) 2[4] G. Landi and G. E. Landi, “Proofs of non-optimality of the standard least-squares method for trackreconstructions,” [arXiv:2003.10021 [math.ST]].