Positioning Error Probability for Some Forms of Center-of-Gravity Algorithms Calculated with the Cumulative Distributions. Part I
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Positioning Error Probability for Some Forms ofCenter-of-Gravity Algorithms Calculated with theCumulative Distributions. Part I.
Gregorio Landi a ∗ , Giovanni E. Landi ba Dipartimento di Fisica e Astronomia, Universita’ di Firenze and INFNLargo E. Fermi 2 (Arcetri) 50125, Firenze, Italy b ArchonVR S.a.g.l.,Via Cisieri 3, 6900 Lugano, Switzerland.June 1, 2020
Abstract
To complete a previous paper, the probability density functions of the center-of-gravity as po-sitioning algorithm are derived with classical methods. These methods, as suggested by the text-book 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 cumula-tive probability distributions could be useful. The combinations of random variables are those es-sential for track fitting x = ξ / ( ξ + η ) , x = θ ( x − x )( − x ) / ( x + x ) + θ ( x − x ) x / ( x + x ) and x = ( x − x ) / ( x + x + x ) . 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 publica-tions. The standard assumption is that ξ , η , x , x and x are independent random variables. Contents ∗ Corresponding author. Gregorio.Landi@fi.infn.it Introduction
In reference [1], the probability density functions (PDFs) of positioning errors for some forms of Center-of-Gravity (COG) algorithms were reported. Their derivations were obtained with a straightforwardmethods, the Fermi golden rule condition sine qua non for the publication of ref. [5]. We rejected this pretense con-sidering that as substantial distortion of ref. [5], without no additional contribution to our results. In ourplans, ref. [5] was a phenomenological discussion of two of our unexpected results, the linear growth andthe lucky model, without resorting to long equations. In any case, refs. [3, 4] contain all the equationsmissed to ref. [5] (for the referee point of view). The PDFs, derived here and in ref. [1], are one of the keyelements of our approach of refs. [6, 7]. The other key element is the theorem of ref. [6], this is essentialto insert the functional dependence from the impact point in the PDFs. In fact, it is the estimation of thetrack impact points the aim of all these developments. Figures 15 and 16 of ref. [6] illustrate very wellthe ability of these methods to estimate the track impact points and the unexpected importance of theCauchy-(Agnesi) tails. A detailed discussion of the use of the theorem of ref. [6] to complete the PDFswith the functions of the track impact points will be the subject of a future paper. When we started thisstudy of the COG PDFs, we expected that a large part of these developments were well known, the COGalgorithms are in use by a long time. With our surprise, we discovered that nobody worried to calculatethem. The belief that all the probabilities are Gaussian functions is very hard to die. The perception tomove in an unexplored land obliged us to pay attention to any detail and to select the best allowed path.This strategy was very rewarding, producing good expected outcomes and excellent unexpected results.The following are our original
LateX -notes where we suppressed the parts reported in ref. [1]. The lastsubsection contains figures of the PDFs for three strip COG compared with the PDFs for two strip COG.
Now we calculate the cumulative probability of a simple two-strip COG, extending the compresseddiscussion of ref [1]. The parameters of the problem are the two energies collected by adjacent strips,each one affected by an additive random noise (probably Gaussian as our data look to support). We willproceeds along the lines of the application of ref. [2] to a ratio of two random variables. In our case wehave two random variables but the ratio is that of a COG ξ / ( ξ + η ) . As in ref. [2] we will consider theregions where: F ( x ) = P ( ξξ + η ≤ x ) (1)The derivative of Eq. 1 respect to x gives the PDF of this case. We have first two conditions: • ξ + η ≥ • ξ + η < ( η , ξ ) is divided in two parts by the line ξ = − η (figure 1). η ξ+η>0ξ+η<0 ξ+η=0 ξ Figure 1:
Sector of the plane ( η , ξ ) where ξ + η > or ξ + η < and its boundary ξ + η = F ( x ) . ξξ + η < x → ξ + η > ⇒ ξ < x ( ξ + η ) ⇒ ξ ( − x ) < x η x < ⇒ ξ ( − x ) x > η x > ⇒ ξ ( − x ) x < η (2)and similarly we have: ξξ + η < x → ξ + η < ⇒ ξ > x ( ξ + η ) ⇒ ξ ( − x ) > x η x < ⇒ ξ ( − x ) x < η x > ⇒ ξ ( − x ) x > η (3)From the definition of x , we see that the equation ξ = x =
0. For x <
0, the function ξ ( − x ) / x = η is the dashed line reported in fig. 2. All the lines for x < η -axis for x → x < ξ + η >
0, the integration region is that in the lower half plane within the twolines ξ + η = η is indicated in fig. 2 as a thick arrow. The contribution to F ( x ) of this part of the plane is F a ( x ) : F a ( x ) = Z − ∞ d ξ P ( ξ ) Z ξ ( − x ) / x − ξ P ( η ) d η . (4) P ( ξ ) and P ( η ) are the two distributions of the random variables ξ and η . The contribution to F ( x ) from the region with ( ξ + η ) < ξ ( − x ) / x = η and ξ = − η . A part of the integration path in η is indicated with a thick arrow in fig. 3.Its contribution to F ( x ) , we call it F b ( X ) , is given by: F b ( x ) = Z ∞ d ξ P ( ξ ) Z − ξξ ( − x ) / x P ( η ) d η . (5)3 ξ+η>0ξ+η<0 ξ+η=0 ξ η ξ(1− x)= η Figure 2:
Sector of the plane ( η , ξ ) where ξ + η > or ξ + η < and its boundary ξ + η = , the dashedline is the line ξ ( − x ) / x = η for negative x. The arrow indicates the integration path for ( ξ + η ) > x > x =
0, (the η -axis), and an integration must coverthe remaining part up to the line ξ ( − x ) / x = η for each ( ξ + η ) > ( ξ + η ) < F c ( x ) and F d ( x ) respectively for ( ξ + η ) > ( ξ + η ) < F c ( x ) = Z − ∞ d ξ P ( ξ ) Z + ∞ − ξ P ( η ) d η + Z + ∞ d ξ P ( ξ ) Z + ∞ξ ( − x ) / x P ( η ) d η (6) F d ( x ) = Z − ∞ d ξ P ( ξ ) Z ξ ( − x ) / x − ∞ P ( η ) d η + Z + ∞ d ξ P ( ξ ) Z − ξ − ∞ P ( η ) d η So for x ≤ F ( x ) = Z − ∞ d ξ P ( ξ ) Z ξ ( − x ) / x − ξ P ( η ) d η + Z ∞ d ξ P ( ξ ) Z − ξξ ( − x ) / x P ( η ) d η (7)and x > F ( x ) is: F ( x ) = Z − ∞ d ξ P ( ξ ) Z + ∞ − ξ P ( η ) d η + Z + ∞ d ξ P ( ξ ) Z + ∞ξ ( − x ) / x P ( η ) d η + Z − ∞ d ξ P ( ξ ) Z ξ ( − x ) / x − ∞ P ( η ) d η + Z + ∞ d ξ P ( ξ ) Z − ξ − ∞ P ( η ) d η (8)It is easy to verify the consistency of F ( x ) , in fact F ( x → − ∞ ) = F ( x → + ∞ ) =
1. When x → − ∞ it is ( − x ) / x → − F ( x → − ∞ ) = Z − ∞ d ξ P ( ξ ) Z − ξ − ξ P ( η ) d η + Z ∞ d ξ P ( ξ ) Z − ξ − ξ P ( η ) d η = ξ+η>0ξ+η<0 ξ+η=0 ξ η ξ(1− x)= η Figure 3:
Sector of the plane ( η , ξ ) where ξ + η > or ξ + η < and its boundary ξ + η = , thedashed line is the line ξ ( − x ) / x = η for negative x, the arrow indicates the η integration-path for ( ξ + η ) < x ξ+η>0ξ+η<0 ξ+η=0 ξ η ξ(1− x)= η Figure 4:
Sector of the plane ( η , ξ ) where ξ + η > or ξ + η < and its boundary ξ + η = , thedashed line is the line ξ ( − x ) / x = η for positive x. The integration regions are that with the arrowsand must cover positive and negative values of ξ the integration on η gives zero in the two integrals. For x → + ∞ and ( − x ) / x → − F ( x ) = Z − ∞ d ξ P ( ξ ) Z + ∞ − ξ P ( η ) d η + Z + ∞ d ξ P ( ξ ) Z + ∞ − ξ P ( η ) d η + Z − ∞ d ξ P ( ξ ) Z − ξ − ∞ P ( η ) d η + Z + ∞ d ξ P ( ξ ) Z − ξ − ∞ P ( η ) d η = Z + ∞ − ∞ P ( ξ ) d ξ Z + ∞ − ∞ P ( η ) d η = P and P . This check assures the absence of trivial errors. The probability density function (PDF) is extracted by the function F ( x ) with a derivative respect to x .Due to the complex dependence from x of the boundaries of integration, the derivative must be done withthe definition of an auxiliary variable y ( x ) and multiplying the derivative respect to y for d y / d x . We willindicate this PDF with P xg R ( x ) . The result is: P xg R ( x ) = d F ( x ) d x = x (cid:2) Z + ∞ d ξ P ( ξ ) ξ P ( ξ − xx ) − Z − ∞ d ξ P ( ξ ) ξ P ( ξ − xx ) (cid:3) . (11)An identical result is obtained differentiating F ( x ) for x ≥ F ( x ) for x <
0. Equation 11 is thePDF of the two strip COG assuming the independence of the two strip noise and the strip 1 is the rightstrip. The factor 1 / x and the factor ξ in the integral come from the derivative of ξ ( − x ) / x respect to x .To consider the left strip one can proceed as for the right strip obtaining different figures. The finalresult is equivalent to substitute to the probability P of the right strip the probability P of the left stripand change x in − y to obtain the PDF P xg L ( y ) of the two strip COG with the left strip. Even if the steps to obtain this distribution are identical to the previous ones, with a small modificationswe will reproduce the path because it will be used in the following. y = − ββ + η (12)As usual we calculate the function F L ( y ) defined as the region where − β / ( β + η ) < y . We have toseparate the two regions β + η ≥ β + η < y < y η β+η<0 β β+η>0β+η=0 η=−β −−−−−−(1+y) Figure 5:
Integration regions for y < for the two strip COG with the left strip The function F L ( y ) for y < F L ( y ) = Z ∞ d β P ( β ) Z β ( − − y ) / y − β P ( η ) d η + Z − ∞ d β P ( β ) Z − ββ ( − − y ) / y P ( η ) d η (13)6or y ≥ β = F L ( y ) for y ≥ F L ( y ) = Z + ∞ d β P ( β ) Z + ∞ − β P ( η ) d η + Z − ∞ d β P ( β ) Z + ∞β ( − − y ) / y P ( η ) d η + Z − ∞ d β P ( β ) Z − β − ∞ P ( η ) d η + Z + ∞ d β P ( β ) Z β ( − − y ) / y − ∞ P ( η ) d η (14) −−−−−−− η β+η<0 β β+η>0β+η=0 η=−β y(1+y) Figure 6:
Integration regions for y < for the two strip COG with the left strip To control the consistency of the equations, the limit of y → ∞ must give one. It is easy to verify thisas in the approach of F R ( x ) .Differentiating F L ( y ) respect to y gives the PDF for the COG distribution with the left strip. P xg L ( y ) = y (cid:2) Z + ∞ d β P ( β ) β P ( − − yy η ) − Z − ∞ d β P ( β ) β P ( − − yy β ) (cid:3) . (15)The variables β , ξ , η can be approximated as an average value and an additive Gaussian noise. In thisassumption the PDF P , P , P can be represented as Gaussian functions with averages corresponding tothe unperturbed energy values collected by the strips. With MATHEMATICA [9], it is possible to obtainan exact form of the above integrals with Gaussian PDF. This full form is very complex, but we have toconsider that we are interested in x or y values less than or equal to 0 . x → ∞ . It goes to ∞ as constant / x with anextremely small constant (10 − ). This convergence to ∞ is very slow compared to a Gaussian function.The PDF has infinite variance and no average. Thus, for these PDF the central limit theorem cannot beapplied. Numerical calculations does not reveal the divergence. The equations developed up to now consider only two strips. The full two strip algorithm selects thesecond strip as the greatest of two (left and right) nearby strips. For x g ≈
0, the noise can favor one or theother strip. So, the energy of the third strip can plays its role in generating the typical anomaly of the twostrip COG. The addition of the third strip requires to study the function F ( x ) in three dimensions. The7ombinations of the previous plots will be useful in this case. As in the previous sections, the three stripsare indicated with the index 1 for the right strip, with the index 2 the central strip, and the index 3 withthe left strip. The random variables are ξ , η and β for the right, central and left strip and their axis aredirected as y , x and z axis of a 3D reference system . If the energy of the right strip is much higher thanthat of left strip, the effect of the left strip is negligible. At a → − ∞ the calculation of F ( x ) is identicalto that of fig. 2, 3, 4. When ξ ≥ β one get eq. 7 and multiplies F by P ( β ) and limits the integrations tothe due regions ξ ≥ β as plotted in fig. 7. ξ=β ξ+η>0ξ+η<0 ξ+η=0 ξ η ξ(1− x)= η x ξ>ββ>ξ Figure 7:
This is fig. 3 with the dash-dotted line indicating the boundary of the region with ξ ≥ β . Nowthe integration region is above this line, here we have x < . When β > ξ the function F ( y ) is used with the substitution y → x and P ( ξ ) → P ( β ) and integratingover the regions β ≥ ξ . The result is: x < F ( x ) = Z − ∞ d β P ( β ) Z β d ξ P ( ξ ) Z ξ ( − x ) / x − ξ P ( η ) d η + Z ∞ d β P ( β ) Z ∞β d ξ P ( ξ ) Z − ξξ ( − x ) / x P ( η ) d η + Z − ∞ d β P ( β ) Z ∞ d ξ P ( ξ ) Z − ξξ ( − x ) / x P ( η ) d η + Z − ∞ d ξ P ( ξ ) Z ξ d β P ( β ) Z − ββ ( − − x ) / x P ( η ) d η + Z ∞ d ξ P ( ξ ) Z ∞ξ d β P ( β ) Z β ( − − x ) / x − β P ( η ) d η Z − ∞ d ξ P ( ξ ) Z ∞ d β P ( β ) Z β ( − − x ) / x − β P ( η ) d η (16)The condition lim x →− ∞ F ( x ) = x ≥ >β ξ+η>0ξ+η=0 ξ η ξ(1− x)= η x ξ+η<0 ξ<βξ=β Figure 8:
This is fig. 4 with the dash-dotted line indicating the boundary of the region with ξ ≥ β . Theintegration region is above this line, here we have x > .x ≥ F ( x ) = Z − ∞ d β P ( β ) Z β d ξ P ( ξ ) Z + ∞ − ξ P ( η ) d η + Z ∞ d β P ( β ) Z + ∞β d ξ P ( ξ ) Z + ∞ξ ( − x ) / x P ( η ) d η + Z − ∞ d β P ( β ) Z β d ξ P ( ξ ) Z ξ ( − x ) / x − ∞ P ( η ) d η + Z ∞ d β P ( β ) Z + ∞β P ( ξ ) d ξ Z − ξ − ∞ P ( η ) d η + Z ∞ d ξ P ( ξ ) Z + ∞ξ d β P ( β ) Z + ∞ − β d η P ( η ) + Z − ∞ d ξ P ( ξ ) Z ξ d β P ( β ) Z + ∞β ( − − x ) / x d η P ( η )+ Z − ∞ d ξ P ( ξ ) Z ξ d β P ( β ) Z − β − ∞ P ( η ) d η + Z ∞ d ξ P ( ξ ) Z + ∞ξ d β P ( β ) Z β ( − − x ) / x − ∞ P ( η ) d η + Z − ∞ d β P ( β ) Z ∞ d ξ P ( ξ ) Z ∞ξ ( − x ) / x d η P ( η ) + Z − ∞ d β P ( β ) Z ∞ d ξ P ( ξ ) Z − ξ − ∞ d η P ( η )+ Z − ∞ d ξ P ( ξ ) Z ∞ d β P ( β ) Z ∞ − β d η P ( η ) + Z − ∞ d ξ P ( ξ ) Z ∞ d β P ( β ) Z β ( − − x ) / x − ∞ d η P ( η ) (17)This complex set of integrals are necessary to complete the integration space of the variables for F ( x ) , and they give the correct limits for x → ± ∞ (in a first calculation the last four were lost and thelimit for x going to ∞ was wrong). It is required a transformation of the integrals on a triangular regionwith the Fubini’s theorem. 9he probability distribution is obtained with a derivative of eq. 17 in the x variable. P xg ( x ) = x h Z ∞ d β P ( β ) Z ∞β d ξ P ( ξ ) ξ P ( ξ − xx ) − Z − ∞ d β P ( β ) Z β d ξ P ( ξ ) ξ P ( ξ − xx )+ Z ∞ d ξ P ( ξ ) Z ∞ξ d β P ( β ) β P ( β − − xx ) − Z − ∞ d ξ P ( ξ ) Z ξ d β P ( β ) β P ( β − − xx )+ Z − ∞ d β P ( β ) Z ∞ d ξ P ( ξ ) ξ P ( ξ − xx ) + Z − ∞ d ξ P ( ξ ) Z ∞ d β P ( β ) β P ( β − − xx ) i (18)The first four integrals can be rearranged with the Fubini’s theorem and summed with the last twointegrals giving: P xg ( x ) = x h Z ∞ d ξ P ( ξ ) ξ P ( ξ − xx ) Z ξ − ∞ d β P ( β ) − Z − ∞ d ξ P ( ξ ) ξ P ( ξ − xx ) Z ξ − ∞ d β P ( β )+ Z ∞ d β P ( β ) β P ( β − − xx ) Z β − ∞ d ξ P ( ξ ) − Z − ∞ d β P ( β ) β P ( β − − xx ) Z β − ∞ d ξ P ( ξ ) i (19)that can be recast in: P xg ( x ) = x h Z ∞ − ∞ d ξ (cid:12)(cid:12) ξ (cid:12)(cid:12) P ( ξ ) P ( ξ − xx ) Z ξ − ∞ d β P ( β )+ Z ∞ − ∞ d β (cid:12)(cid:12) β (cid:12)(cid:12) P ( β ) P ( β − − xx ) Z β − ∞ d ξ P ( ξ ) i (20)This last form is easer to handle numerically or analytically. MATHEMATICA does not give ananalytical form of the integrals with Gaussian probability distributions, erf-functions do not allow ananalytical result. The numerical integrals have convergence problems around x ≈ x are perfect. The function requires three energiesand three standard deviations to produce the results. The three energies should be the unperturbed onesbut it is evident that these are impossible to have. Around x g = x g = x g = ± . x g it could be convenient totest this strategy in a best fit. For this task it is required the probability distribution.As for x g we have to work in a three dimensional space. a = ξ , a = η , a = β x g = ξ − βξ + η + β For β = ξ + η > ξ − βξ + η + β < x . (21)10 ξ+η>0ξ+η<0 ξ+η=0 ξ η ξ(1− x)= η Figure 9:
Sector of the plane ( η , ξ ) where ξ + η > or ξ + η < and its boundary ξ + η = , thedashed line is the line ξ ( − x ) / x = η for negative x, the arrow indicates the η integration-path for ( ξ + η ) < ξ + η + β > ξ + η + β <
0, in any case we have two limiting planes: ξ ( − x ) + β ( − − x ) − η x = ξ + η + β = x < β =
0, the traces of these two planes in the plane η , ξ are those of fig. 9. Let us see thecase with β > β = b . The traces of the two planes are those of fig. 10. The intersection point is {− b , b } and it changes with β . It is evident that the only change in moving from β > β < F xg ( x ) becomes ( x < F xg ( x ) = Z ∞ − ∞ d β P ( β ) Z ∞β d ξ P ( ξ ) Z − ξ − βξ ( − x ) x + β ( − − x ) x P ( η ) d η + Z ∞ − ∞ d β P ( β ) Z β − ∞ , d ξ P ( ξ ) Z ξ ( − x ) x + β ( − − x ) x − ξ − β P ( η ) d η (22)For x ≥ β = β = {− b , b } and the F xg ( x ) becomes: F xg ( x ) = Z ∞ − ∞ d β P ( β ) Z β − ∞ d ξ P ( ξ ) Z ∞ − ξ − β P ( η ) d η + Z ∞ − ∞ d β P ( β ) Z ∞β d ξ P ( ξ ) Z + ∞ξ ( − x ) x + β ( − − x ) x P ( η ) d η + Z ∞ − ∞ d β P ( β ) Z β − ∞ , d ξ P ( ξ ) Z ξ ( − x ) x + β ( − − x ) x − ∞ P ( η ) d η Z ∞ − ∞ d β P ( β ) Z + ∞β , d ξ P ( ξ ) Z − ξ − β − ∞ P ( η ) d η (23)11 +η=0β+η<0 β+η>0ξ+ ξ+ ξ b2b− ηξ+ Figure 10:
Sector of the plane ( η , ξ ) where ξ + η + β > or ξ + η + β < and its boundary ξ + η + β = , the dashed line has the equation ξ ( − x ) + b ( − − x ) − η x = for negative x, the arrows indicatethe η integration-paths for ( ξ + η + β ) < and for ( ξ + η + β ) > x ξ+η>0ξ+η<0 ξ+η=0 ξ η ξ(1− x)= η Figure 11:
Sector of the plane ( η , ξ ) where ξ + η > or ξ + η < and its boundary ξ + η = , thedashed line is the line ξ ( − x ) / x = η for positive x. The integration regions are that with the arrowsand must cover positive and negative values of ξ It is easy to prove that lim x →− ∞ F xg ( x ) = x → + ∞ F xg ( x ) =
1. In fact the first limit is easy giventhat lim x →± ∞ ( − x ) / x = −
1. With this position eq. 22 has the limits of the last integrals identical, and12 =0ξ+η>0β+ ηξξ+η<0β+
2b b β+ ξ+η=0ξ(1− β(−1 −x)−xx)+
Figure 12:
Sector of the plane ( η , ξ ) where ξ + η + β > or ξ + η + β < and its boundary ξ + η + β = , the dashed line is the line ξ ( − x ) + β ( − − x ) − x η = for positive x. The integration regions arethat with the arrows and must cover positive and negative values of ξ the integrals are zero. For x → + ∞ the integrals of eq. 23 become: F xg (+ ∞ ) = Z ∞ − ∞ d β P ( β ) Z β − ∞ d ξ P ( ξ ) Z ∞ − ξ − β P ( η ) d η + Z ∞ − ∞ d β P ( β ) Z ∞β d ξ P ( ξ ) Z + ∞ − ξ − β P ( η ) d η + Z ∞ − ∞ d β P ( β ) Z β − ∞ , d ξ P ( ξ ) Z − ξ − β − ∞ P ( η ) d η Z ∞ − ∞ d β P ( β ) Z + ∞β , d ξ P ( ξ ) Z − ξ − β − ∞ P ( η ) d η (24)The first and third integrals have identical integration limits in the variables β and ξ , and the sum of thelast integrals produces the normalization of the probability distribution P ( η ) . Identically for the secondand forth integrals. The two remaining integrals add completing the normalization of the probability P ( ξ ) that is multiplied by the normalization of P ( β ) giving 1.Now, after this consistency check, we can extract the probability P xg ( x ) differentiating F xg ( x ) and F xg ( x ) respect to x . The result is: P xg ( x ) = x h Z + ∞ − ∞ d β P ( β ) Z + ∞β d ξ P ( ξ ) P ( ξ − xx + β − − xx )( − β + ξ )+ Z + ∞ − ∞ d β P ( β ) Z β − ∞ d ξ P ( ξ ) P ( ξ − xx + β − − xx )( β − ξ ) i (25)With the transformation δ = ξ − β , eq. 25 becomes: P xg ( x ) = x h Z + ∞ − ∞ d β P ( β ) Z + ∞ d δ P ( δ + β ) P ( δ − xx − β ) δ − Z + ∞ − ∞ d β P ( β ) Z − ∞ d δ P ( δ + β ) P ( δ − xx − β ) δ i (26)13r better: P xg ( x ) = x h Z + ∞ − ∞ d β P ( β ) Z + ∞ − ∞ d δ P ( δ + β ) P ( δ − xx − β ) (cid:12)(cid:12) δ (cid:12)(cid:12)i (27)That can be recast in a form more appropriate for the coming developments: P xg ( x ) = x h Z + ∞ − ∞ d δ (cid:12)(cid:12) δ (cid:12)(cid:12) Z + ∞ − ∞ d β P ( β ) P ( δ + β ) P ( δ − xx − β ) i (28)A set of variable transformations reports this form to be identical to equation 21 of ref. [1]. With additive Gaussian noise, eq. 28 shows the possibility of an analytical expression of the integrals,in fact the integral on β is on product of Gaussian functions and it will give a Gaussian function. Theintegral on δ has itself an analytical expression. P xg ( x ) assumes the form (with a consistent help ofMATHEMATICA): P xg ( x ) = n exp (cid:2) − ( a − a a + a + a − x ) ( a + a + a ) [( − x ) σ + x σ + ( + x ) σ ] (cid:3)(cid:12)(cid:12)(cid:12) a [ x σ + ( + x ) σ ] + a [ ( − x ) σ − x σ ] + a [( − x ) σ + ( + x ) σ ] (cid:12)(cid:12)(cid:12) √ π [( − x ) σ + x σ + ( + x ) σ ] / o (29)This form is simplified, with the suppression of elements with negligible contributions to the final result(as far as for the x values of our needs). We have an expression of the type A erf ( A ) that is approximatedas | A | . The differences with the full integral are really extremely small. The normalization of the dis-tributions can be numerically verified in the range ≈ ±
3, beyond these values MATHEMATICA giveswrong a normalization due to the errors introduced by singularities in the integrands. These singularitiesare characteristic of the distributions of ratios of Gaussian random variables as shown in ref. [2].Another form to write eq. 29 consists in the observation that the factors each σ i i = , , i . Defining X = ( a − a ) / ( a + a + a ) , we have: a [ x σ + ( + x ) σ ] + a [ ( − x ) σ − x σ ] + a [( − x ) σ + ( + x ) σ ] =( a − a ) x σ + ( a + a )( + x ) σ + ( a + a )( − x ) σ =( a + a + a )[ X x σ + ( + X )( + x ) σ + ( X − )( x − ) σ ] (30)With the approximation X ≈ x , a further simplified form of eq. 29 can be obtained , this implies a smallincrease of the error (even in the normalization): P xg ( x ) = n exp (cid:2) − (cid:0) a − a a + a + a − x (cid:1) ( a + a + a ) ( σ ( − x ) + x σ + ( + x ) σ ) (cid:3) ( a + a + a ) √ π q ( σ ( − x ) + x σ + ( + x ) σ ) o (31)The following figure illustrates the probability reproduction of the data. With MATLAB [10], we gen-erate a set of 500000 events all at the identical noiseless energy. The normalized histograms of x g and x g at θ = o and at a noiseless energy of 150 ADC five-strip energy are reported with the calculatedprobability distributions. The calculated distributions exactly overlaps the histograms.The cumulative distributions for the complete three strip COG (with the gap at x ± / Figure 13:
Right plot. Simulation of the data distributions of x g in red and x g blue at the incidence angle θ = o , impact point ε = and noiseless energy 150 ADC counts (floating-strip detector). The blackcurve is the x g and the magenta one is the x g calculated probability distribution. Left plot. Simulationsat θ = o , ε = and noiseless energy 150 ADC for normal-strip detector with higher noise ( . ADC)
The probability distributions for the center-of-gravity as positioning algorithms are calculated with thetextbook method with the cumulative probability. This method is very cumbersome but it is reported forcompleteness. This method was the first, we used long time ago, for this type of calculations. Otherfaster methods are reported in another previous report. Other type of center-of-gravity algorithms will bediscussed in Part II.
References [1] Landi G.; Landi G. E.;
Probability Distributions of Positioning Errors for Some Forms of Center-of-Gravity Algorithms. arXiv:2004.08975 [physics.ins-det] [2] B. V. Gnedenko "The Theory of Probability and Elements of Statistics" (AMS Chelsea Publishing-Providence Rhode Island )[3] Landi G.; Landi G. E.
The Cramer-Rao inequality to go beyomd the √ N -limit of the stan-dard least-squares method in track fitting arXiv:1910.14494 [physics.ins-det] https://arxiv.org/abs/1910.14494.[4] Landi G.; Landi G. E. Proofs of non-optimality of the standard least-squares method for trackreconstructions arXiv:2003.10021 [physics.ins-det] [5] Landi G.; Landi G. E.
Beyond the √ N-limit of the least squares resolution and the lucky-model arXiv:1808.06708[physics.ins-det] https://arxiv.org/abs/1808.06708.[6] Landi G.; Landi G. E.
Improvement of track reconstruction with well tuned prob-ability distributions JINST arXiv:1404.1968[physics.ins-det] https://arxiv.org/abs/1404.1968 157] Landi, G.; Landi G. E.
Optimizing momentum resolution with a new fitting method for silicon-stripdetectors
INSTRUMENTS , 2, 22[8] Landi G.,
The center of gravity as an algorithm for position measurements
Nucl. In-str. and Meth.
A 485 (2002) 698 arXiv:1908.04447 [physics.ins-det]arXiv:1908.04447 [physics.ins-det]