BLUE: combining correlated estimates of physics observables within ROOT using the Best Linear Unbiased Estimate method
BBLUE : combining correlated estimates of physicsobservables within ROOT using the Best LinearUnbiased Estimate method
Richard Nisius
Max-Planck-Institut f¨ur Physik (Werner-Heisenberg-Institut), F¨ohringer Ring 6,D-80805 M¨unchen, Germany
Abstract
This software performs the combination of m correlated estimates of n physicsobservables ( m ≥ n ) using the Best Linear Unbiased Estimate ( BLUE )method. It is implemented as a
C++ class, to be used within the ROOT anal-ysis package. It features easy disabling of specific estimates or uncertaintysources, the investigation of different correlation assumptions, and allowsperforming combinations according to the importance of the estimates. Thisenables systematic investigations of the combination on details of the mea-surements from within the software, without touching the input.
Keywords: combination, correlation, estimate, uncertainty
Code metadata
Current code version 2.4.0Permanent link to code/repository used https://github.com/ElsevierSoftwareX/SOFTX 2020 16for this code version https://blue.hepforge.orgCode Ocean compute capsuleLegal Code License LGPLCode versioning system used noneSoftware code languages, tools, and ser-vices used
C++
Compilation requirements, operatingenvironments & dependencies depends on https://root.cern.chIf available Link to developer documen-tation/manual https://blue.hepforge.orgSupport email for questions [email protected] a r X i v : . [ phy s i c s . d a t a - a n ] A p r . Motivation and significance The combination of a number of correlated estimates for a single observ-able is discussed in Ref. [1]. Here, the term estimate denotes a particularoutcome (measurement) of an experiment based on an experimental esti-mator (an algorithm for a measurement) of the observable, which follows aprobability density function (PDF). The particular estimate obtained by theexperiment may be a likely or unlikely outcome for that PDF. Repeating themeasurement numerous times under identical conditions, the estimates willfollow the underlying PDF of the estimator.The metadata for the
BLUE software are listed in the Code metadatatable. The software uses the Gaussian approximation for the uncertaintiesand performs a χ minimization to obtain the combined value. In Ref. [1],this minimization is expressed using the mathematically equivalent BLUE ansatz. Provided the estimators are unbiased, when applying this formalismthe
Best Linear Unbiased Estimate of the observable is obtained with thefollowing meaning:
Best: the combined result for the observable obtainedthis way has the smallest variance;
Linear: the result is constructed as alinear combination of the individual estimates;
Unbiased Estimate: whenthe procedure is repeated for a large number of cases consistent with the un-derlying multi-dimensional PDF, the mean of all combined results equals thetrue value of the observable. The formulas for more than one observable [2]are implemented in the
BLUE software, which is programmed as a separateclass of the ROOT analysis framework [3].The easiest case of two correlated estimates of the same observable isbriefly illustrated here. Already for this case, the main features of the com-bination can easily be understood. For further information and the derivationof the formulas the reader is refereed to Ref. [4]. Let x and x with vari-ances σ and σ be two estimates from two unbiased estimators X and X of the true value x T of the observable and ρ the total correlation of the twoestimators. Without loss of generality it is assumed that the estimate x stems from an estimator X of x T that is at least as precise as the estimator X yielding the estimate x , such that z ≡ σ /σ ≥
1. In this situation the
BLUE x of x T is: x = (1 − β ) x + β x where β is the weight of the less precise estimate and the sum of weightsis unity by construction. The variable x is the combined result and σ x isits variance, i.e. the uncertainty assigned to the combined value is σ x . Fora number of z values the two quantities β and σ x / σ as functions of ρ are2 - - - - - b - - - x b + ) x b x = (1- - x x x - x = z + z r r b = s s z = 1.0 1.1 1.2 1.5 2.0 3.0 (a) β as a function of ρ r - - - - - s x s x b + ) x b x = (1- z + z r r (1 - z = s x s = s s z = 1.0 1.1 1.2 1.5 2.0 3.0 (b) σ x / σ as a function of ρ Figure 1: The variables β (a) and σ x / σ (b) shown as functions of ρ for a number of z values. shown in Fig. 1. Their functional forms are also written in the figures. Thefunctions are valid for − ≤ ρ ≤ z ≥
1, except for ρ = z = 1.Fig. 1 displays the strong dependence of the uncertainty in the combinedvalue on z and ρ . For the special situation of ρ = 1 /z the uncertainty σ x equals σ , i.e. the precision in the observable is not improved by adding thesecond measurement. For ρ = ± σ x = 0, a mere consequence of the conditional probability for X given the measured value of x , see Ref. [4] for details. It is worth noticingthat in most regions of the ( ρ , z )-plane the sensitivity of σ x / σ to ρ is strongerthan to z . This means, reducing the correlation of the estimates in most casesgives a larger improvement in precision in the combined value than reducing z .
2. Software description
To use the
BLUE combination software, a working installation of theROOT package [3] is needed. The software allows repeated combinationsinterleaved with an arbitrary number of changes to various aspects of theinput, achieved using
Set...() functions. In addition, it gives access tomany intermediate quantities used in the combination via
Get...() and
Print...() functions. Only at well-defined steps of the computation, willthe returned quantities be meaningful. To enable the above flexibility, the
BLUE object has several states like
IsFilledInp , IsFixed or IsSolved that can be false or true. The various functions are only enabled, if theobject has the appropriate state. 3 ree:
ReleaseInp(), ResetInp()
Create: myBlue = new Blue(..)
Fill:
FillEst(..), FillCor(..), FillSta(..), . . .
Change:
SetInActiveEst(..), SetInActiveUnc(..),SetRhoFacUnc(..), SetRelUnc(..), . . .
Fix:
FixInp()
Combine:
Solve(), SolveAccImp(..), . . .
Digest:
Get..(..), Print..(..), LatexResult(..), . . .
Anothercombination?yes noDelete: delete myBlue; myBlue = NULL
Figure 2: Flowchart of the
BLUE software. The gray boxes display the steps to beperformed for a single combination with unchanged inputs. The white boxes displaythe optional steps for further combinations with changed inputs and/or different solvingmethods. In each rectangular box, the first line indicates the performed action, while thefollowing line(s) list the functions to use.
BLUE object by means of the filling functions
FillEst() and
FillCor() . The end of filling the mandatory inputs is au-tomatically recognized by the software. At this point, the initial input issaved, auxiliary information is calculated,
IsFilledInp is set to true andall filling functions are disabled. Consequently, the optional additional inputlike names, print formats and a logo, all have to be filled before the end offilling the mandatory input is reached.The most important object states can be controlled by function calls. Forexample, a call to
FixInp() initiates the calculation of various quantities andsets
IsFixed to true, thereby enabling the use of a number of
Get...() and
Print...() functions. A subsequent call to any of the solving methods
Solve...() performs the desired combination, and finally sets
IsSolved to true. In this scheme, solving is only possible after a call to
FixInp() ,and changing the inputs only after a call to
ReleaseInp() or ResetInp() .Internally, the combination is always performed on a temporary copy of theinitial input, which can be successively altered using the sequence of calling
ReleaseInp() , Set...() and
FixInp() . Since the initial input was savedas described above, a call to
ResetInp() allows to return to this situation.
The flowchart of the software is shown in Fig. 2. After calling the con-structor of the class (
Create :), thereby defining the number of estimates,uncertainties and observables, the measurements together with their un-certainties and correlations are passed to the software (
Fill :). If wanted,the inputs can be adapted (
Change :). Combinations are performed follow-ing the loop of fixing (
Fix :), combining (
Combine :) and evaluating the re-sult (
Digest :). Before changing the inputs for the next combination, theyhave to be freed (
Free :). For this step, two options are available, Relea-seInp() means the change proceeds from the status at the last fix, while
ResetInp() reverts to the original inputs. If no further combinations arewanted, the object is deleted (
Delete :).
One main quality of this software is the built-in flexibility for easy andthorough investigations of the impact of details of the input measurementsand their correlation. With single function calls estimates or uncertaintysources can be removed from the combination, different uncertainty mod-els (e.g. absolute or relative uncertainties) and correlation assumptions can5 stimates Correlations Result M M M ρ ρ ρ M [GeV] [GeV] [GeV] [GeV]Value 174.86 172.63 173.25 173.92Stat 0.35 0.54 0.24 0.20Syst ± ± ± .
00 +1 .
00 +1 .
00 0.36Syst ± ± ± .
00 +1 .
00 +1 .
00 0.17Syst ± ± ± − . − . − .
00 0.09Syst ± ± ± − . − . − .
00 0.02Syst ± ± ± .
50 +0 . − .
30 0.25Syst 0.59 ± ± ± ± ± ± .
29 +0 .
29 +0 .
35 0.52 χ Table 1: Combination of one observable M from three correlated estimates M , M and M using the BLUE software. Counting from the left, columns2-4 show the individual results, their uncertainties and the statistical preci-sion in the systematic uncertainties. Columns 5-7 report the estimator cor-relations ρ ijk for the pair of estimates ( i, j ) for all sources k of uncertainty.The line denoted by Total lists the total uncertainties σ i and correlations ρ ij . The rightmost column shows the combined result. The lower part ofthe table reports the estimator consistencies expressed as pairwise χ values,the weights of the estimates within the combination and the pulls of the es-timates. Details on the calculation of χ , the weights and pulls are given inRef. [4].be investigated. Another strength is the large number of different solvingmethods implemented, ranging from only using measurements with positiveweights in the combination to a successive combination method in which theinput measurements are included one at a time according to their importance,allowing an in-depth investigation of their impact on the combination.
3. Illustrative Example
A compact example of three estimates of a single observable is listed inTable 1. The code reproducing the content of this table and all followingfigures is given in Appendix A. It contains detailed documentation relatingthe various function calls to the steps described in Fig. 2, and the obtainedresults to Table 1 and the respective figures. After installing the package,the only two commands to be executed from the shell prompt are:6 [GeV] (M D - - - ) [ G e V ] ( M D = +1 r = -1 r (a) M vs. M ) [GeV] (M D - - - ) [ G e V ] ( M D = +1 r = -1 r (b) M vs. M Figure 3: The sources of systematic uncertainties with estimator correlations of ρ ijk =+1 (full red points) or ρ ijk = − to Syst from Table 1. r o o t − b < B SoftExample . i n p > B SoftExample . o u tp d f l a t e x B SoftExample . t e xAlthough for this example the values chosen are similar to what is ob-tained in measurements of the top quark mass, this example is purely ar-tificial. The estimate M deliberately was chosen such as to have large χ values in the compatibility evaluations with the other two estimates. Sincethe three measurements should come from the same underlying x T , this indi-cates either an unlikely outcome or a potential systematic problem with thisresult. Only after a careful investigation of this measurement, resulting ina low probability for the second possibility, should this measurement be in-cluded in the combination. The systematic uncertainties are shown togetherwith the statistical precisions at which they are known. Those statistical pre-cisions allow evaluating whether two estimators have a significantly differentsensitivity for a source of uncertainty . In addition, they indicate whichsystematic effect should be evaluated with higher statistical precision.The sources of systematic uncertainties for which the estimator correla-tions are ρ ijk = ± ρ ijk = +1 corresponds tothe situation where simultaneously applying a systematic effect to both es-timates (e.g. increasing a component l of the jet energy scale uncertainty by Two quantities ( x ± σ , x ± σ ) are significantly different, if their difference (∆ = x − x ) with σ = σ + σ − ρ σ σ is significantly different from zero. In this case,the x i correspond to the systematic uncertainties and the σ i to their statistical precisions. - - - - - M [ G e V ] = 173.25 GeV M = 174.86 GeV MM = 173.92 GeV r M vs.
BLUE (a) Combined value r - - - - - ( M ) [ G e V ] s ) = 0.61 GeV (M s ) = 0.69 GeV (M s (M) = 0.52 GeV s r (M) vs. s BLUE (b) Uncertainty in the combined valueFigure 4: Results of the combination of the estimates M and M as functions of ρ , wherethe blue point corresponds to the actual correlation ρ ( M , M ). The combined value isshown in (a), the uncertainty in the combined value in (b). +1 σ JES , l as e.g. performed in Ref. [5]) leads to both measured values movinginto the same direction, either both get larger or both get smaller than theoriginal result. The case ρ ijk = − ρ ijk = +1 or ρ ijk = −
1. For example, thisis the case for Syst of M from Table 1, i.e. for the upper point in the leftquadrant in both subfigures of Fig. 3. This will be exploited in the stabilityevaluation discussed below.Without combining, the precision of the knowledge about the observableis defined by the most precise result, here M . The impact that an additionalestimate has can be digested by performing pairwise combinations with themost precise result. An example of such a pairwise combination of M and M is shown in Fig. 4. Apart from the range ρ > .
8, the combined value isalmost independent of ρ . In contrast, the uncertainty in the combined valuehas a very strong dependence on ρ .The combination of all estimates is shown in Fig. 5(a). The input mea-surements are listed in the first three lines, the combined result is listed in redin the last line. Fig. 5(b) reveals that not all results significantly contributeto the combined value. In this figure, the lines show the results of succes-sive combinations, always adding the estimate listed, to the previous list ofestimates. Also here, the suggested combined result is shown in red. At thequoted precision, the estimate M does not improve the already accumulatedresult obtained from combining M and M . This means M merely serves8 [GeV]170 175 180 185 - M – – M – – M – – M – – BLUE (stat) (syst) (a) Full combination
M [GeV]172 174 176 178 - +M – – +M – – M – – BLUE (stat) (syst) (b) Successive combinationFigure 5: The three estimates and the combined value of all estimates are shown in (a). Incontrast, (b) shows the combined results obtained in successive combinations adding oneestimate at a time, i.e. the second line is the result of combining M and M , the thirdline the result of the combination of all three measurements. as a cross-check measurement for the combination.Fig. 6 shows the stability of the combination of all three results, takinginto account the statistical precisions at which the systematic uncertaintiesare known, see Table 1. For this figure, all systematic uncertainties arealtered within their statistical precisions. For sources with ρ ijk = ± .
15 GeV and the related uncertainty by 0 .
04 GeV.These uncertainties exist on top of what is quoted in Table 1. Frequentlythose are not provided, or even not evaluated. This is only justified if they aremuch smaller than the quoted uncertainties. For the statistical uncertaintyin the combined result of 0 .
20 GeV, see Table 1, the situation is at the borderof being acceptable.
4. Impact
The software can be used for an in-depth analysis of the impact of variousassumptions made in the combination. In case the relevant input is provided,it also allows assessing the stability of the combination.Because of the large reduction in the uncertainty in the combined resultobtained by lowering the estimator correlations, see Fig. 4, it is advisable touse this software already in the design stage of the various analyses performed9 [GeV] C o m b i n a t i on s / . G e V Mean = 173.90 GeV RMS = 0.15 GeV
BLUE (a) Combined value (M)[GeV] s C o m b i n a t i on s / . G e V Mean = 0.53 GeV RMS = 0.04 GeV
BLUE (b) Uncertainty in combined valueFigure 6: The stability of the combined value (a) and the uncertainty in the combinedvalue (b), obtained in 500 combinations, while varying the input uncertainties and corre-lations according to their statistical precision. for obtaining the same observable within a single experiment. Usually, theuncertainties in the various systematic effects (e.g. the uncertainty in jetenergy scales for experiments at hadron collider) are determined by the actuallevel of understanding of the detector and have to be taken into account atface value. In contrast, the sensitivity of the estimators to those effects canbe influenced by the estimator design. This way their correlation can bereduced, thereby improving the gain obtained in the combination. Generallyspeaking, the strategy should not be to take over an aspect of the analysisthat has worked for one estimator to another estimator. Instead, alternativeapproaches should be pursuit, such as to potentially lower the estimatorcorrelations, even at the expense of a larger uncertainty. This is becauseachieving an anti-correlated pair of estimates with the same sensitivity to aspecific source of uncertainty, renders this a significantly smaller uncertaintyin the combined result. This can be seen for the sources Syst and Syst , forwhich ρ ij = ρ ij = −
1. Those sources exhibit the largest fractional gain inuncertainty when comparing M with M , e.g. σ x /σ = 0 . / .
10 = 1 / BLUE softwarewas used in a number of combinations, mostly in the context of high energyphysics, especially at the Large Hadron Collider (LHC). Examples from theALICE, ATLAS, CMS and LHCb collaborations are detailed in Refs. [6, 7,10, 9]. The first world combination of the top quark mass [10] has also beenperformed with this software. In addition to the LHC collaborations, thesoftware has been used by the PHENIX [11] and STAR [12] collaborations,and in a combination of the strong coupling constant α s from many results inRef. [13]. Further examples of the software usage are described in the manuallisted in the Code metadata table. To assist the users in developing theirown combination code, the corresponding C++ routines to reproduce thosepublished results are included in the software package. Although the aboveexamples are all particle physics applications, the use of this software is notconfined to a specific area of research. Any set of correlated measurementsof one or more observables can be combined.
5. Conclusions
The software performs the combination of m correlated estimates of n physics observables ( m ≥ n ) using the Best Linear Unbiased Estimate ( BLUE )method. The large flexibility, together with the several implemented correla-tion models and combination methods makes it a useful tool to assess detailson the combination in question. Exploring the combination of various esti-mators of the same observable within a single experiment allows a design ofestimators with low correlation. This enhances the gain achieved in combi-nations of estimates obtained from those estimators.
Declaration of competing interest
The author declares that he has no known competing financial interestsor personal relationships that could have appeared to influence the workreported in this paper. 11 ppendix A. Example code
Listing 1: B SoftExample.inp1 // Load Blue l i b r a r y − > Load ( ” l i b B l u e . s o ” ) ;34 // Compile and e x e c u t e code . This w i l l produce // t h e b e l o w ’ . c x x ’ f i l e s and a ’ . t e x ’ f i l e // F i g u r e 3a
10 . L B S o f t E x a m p l e x 0 x 1 C o r P a i . cxx++11 B S o f t E x a m p l e x 0 x 1 C o r P a i ( )1213 // F i g u r e 3 b
14 . L B S o f t E x a m p l e x 0 x 2 C o r P a i . cxx++15 B S o f t E x a m p l e x 0 x 2 C o r P a i ( )1617 // F i g u r e 4a , b
18 . L B S o f t E x a m p l e x 0 x 2 D i s P a i . cxx++19 B S o f t E x a m p l e x 0 x 2 D i s P a i ( )2021 // F i g u r e 5a
22 . L B SoftExample DisRes Obs 0 . cxx++23 B SoftExample DisRes Obs 0 ( )2425 // F i g u r e 5 b
26 . L B SoftExample AccImp Obs 0 . cxx++27 B SoftExample AccImp Obs 0 ( )28 . q isting 2: B SoftExample.cxx1 ” Blue . h”23 void B SoftExample ( ) { // −− S t a r t p r e p a r i n g t h e i n p u t s f o r l a t e r use i n BLUE // Numbers o f e s t i m a t e s , u n c e r t a i n t i e s and o b s e r v a b l e s s t a t i c const I n t t NumEst = 3 ;8 s t a t i c const
I n t t NumUnc = 6 ;9 s t a t i c const
I n t t NumObs = 1 ;1011 // Array o f names o f e s t i m a t e s ( i , j = 0 , 1 , 2) s t a t i c const T S t r i n g NamEst [ NumEst ] = { ”M { } ” , ”M { } ” , ”M { } ” } ;1314 // Array o f names o f u n c e r t a i n t i e s ( k = 0 , . . . , 5) s t a t i c const T S t r i n g NamUnc [ NumUnc ] = {
16 ” S t a t ” , ” S y s t { } ” , ” S y s t { } ” , ” S y s t { } ” , ” S y s t { } ” , ” S y s t { } ” } ;1718 // Array o f names o f o b s e r v a b l e s ( n = 0) s t a t i c const T S t r i n g NamObs [ NumObs ] = { ”M” } ;2021 // Array o f e s t i m a t e s and u n c e r t a i n t i e s , copy t o m a t r i x s t a t i c const I n t t LenXEst = NumEst ∗ (NumUnc+1);23 s t a t i c const D o u b l e t XEst [ LenXEst ] = { // S t a t Sys1 Sys2 Sys3 Sys4 Sys5
25 1 7 4 . 8 6 , 0 . 3 5 , 0 . 2 6 , 0 . 0 9 , 0 . 1 2 , 0 . 1 8 , 0 . 4 8 ,26 1 7 2 . 6 3 , 0 . 5 4 , 0 . 6 6 , 0 . 6 4 , 0 . 4 7 , 0 . 2 4 , 0 . 5 3 ,27 1 7 3 . 2 5 , 0 . 2 4 , 0 . 4 3 , 0 . 2 3 , 0 . 2 3 , 0 . 1 0 , 0 . 1 2 } ;28 TMatrixD ∗ InpEst = new
TMatrixD ( NumEst , NumUnc+1, &XEst [ 0 ] ) ;2930 // Array o f s t a t i s t i c a l p r e c i s i o n i n s y s t e m a t i c u n c e r t a i n t i e s , copy t o m a t r i x s t a t i c const I n t t LenSUnc = NumEst ∗ NumUnc ;32 s t a t i c const
D o u b l e t SUnc [ LenSUnc ] = { // S t a t Sys1 Sys2 Sys3 Sys4 Sys5
34 0 . 0 0 , 0 . 0 6 , 0 . 0 5 , 0 . 1 4 , 0 . 0 8 , 0 . 0 9 ,35 0 . 0 0 , 0 . 0 4 , 0 . 0 5 , 0 . 0 9 , 0 . 0 5 , 0 . 0 8 ,36 0 . 0 0 , 0 . 0 6 , 0 . 0 8 , 0 . 1 1 , 0 . 0 8 , 0 . 0 5 } ;37 TMatrixD ∗ InpSta = new
TMatrixD ( NumEst , NumUnc, &SUnc [ 0 ] ) ;3839 // Array o f r h o i j k == c o n s t c o r r e l a t i o n s f o r k < s t a t i c const D o u b l e t RhoVal [ NumUnc −
1] = {
41 0 . 0 , 1 . 0 , 1 . 0 , − − } ;4243 // Array f o r c o r r e l a t i o n m a t r i x f o r k == 5 w i t h r h o i j 5 != c o n s t s t a t i c const I n t t LenInd = NumEst ∗ ( NumEst −
1) / 2 ;45 s t a t i c const
D o u b l e t RhoInd [ LenInd ] = { − } ;4647 // Formats and f i l e name f o r f i g u r e s and L a t e x f i l e s t a t i c const T S t r i n g ForVal = ” %5.2 f ” ; s t a t i c const T S t r i n g ForUnc = ” %4.2 f ” ;50 s t a t i c const
T S t r i n g ForWei = ForUnc ;51 s t a t i c const
T S t r i n g ForRho = ”%+4.2 f ” ;52 s t a t i c const
T S t r i n g ForPul = ForUnc ;53 s t a t i c const
T S t r i n g ForChi = ForUnc ;54 s t a t i c const
T S t r i n g ForUni = ”GeV” ;55 s t a t i c const
T S t r i n g F i l B a s = ” B SoftExample ” ;5657 // Axis r a n g e s f o r F i g s . 3 ab and Fig . 6 ab s t a t i c const D o u b l e t XvaMax = 0 . 3 4 , YvaMin = − s t a t i c const D o u b l e t ValMin = 1 7 3 . 2 , ValMax = 1 7 4 . 6 ;60 s t a t i c const
D o u b l e t UncMin = 0 . 3 4 , UncMax = 0 . 7 6 ;61 // −− End o f p r e p a r a t i o n o f i n p u t s f o r BLUE // −− S t a r t u s i n g BLUE −−− no i n p u t s b e l o w t h i s l i n e −−− // −− Keywords : r e l a t e t o t h e f l o w c h a r t o f t h e s o f t w a r e // Create : c a l l c o n s t r u c t o r o f BLUE o b j e c t
66 Blue ∗ myBlue = new Blue ( NumEst , NumUnc ) ;6768 // F i l l : s e t d i s p l a y f o r m a t s and d e f i n e names
69 myBlue − > SetFormat ( ForVal , ForUnc , ForWei , ForRho , ForPul , ForChi , ForUni ) ;70 myBlue − > FillNamEst(&NamEst [ 0 ] ) ;71 myBlue − > FillNamUnc(&NamUnc [ 0 ] ) ;72 myBlue − > FillNamObs(&NamObs [ 0 ] ) ;7374 // F i l l : i n s e r t e s t i m a t e s w i t h t h e i r u n c e r t a i n t i e s
75 myBlue − > F i l l E s t ( InpEst ) ;7677 // F i l l : i n s e r t s t a t i s t i c a l p r e c i s i o n i n s y s t e m a t i c u n c e r t a i n t i e s
78 myBlue − > F i l l S t a ( InpSta ) ;7980 // F i l l : i n s e r t t h e e s t i m a t o r c o r r e l a t i o n s f o r a l l s o u r c e s o f u n c e r t a i n t y for ( I n t t k = 0 ; k < NumUnc ; k++) { i f ( k != 5 ) { myBlue − > F i l l C o r ( k , RhoVal [ k ] ) ;83 } e l s e { myBlue − > F i l l C o r ( − k , &RhoInd [ 0 ] ) ; } } // Change : // . . . a l t e r a t i o n s would go h e r e // Fix :
90 myBlue − > FixInp ( ) ;9192 // Combine :
93 myBlue − > S o l v e ( ) ;9495 // D i g e s t : show some e s t i m a t e q u a n t i t i e s , p r e p a r e code f o r F i g s . 3 and 4
96 myBlue − > P r i n t E s t ( ) ;97 myBlue − > C o r r e l P a i r ( 0 , 1 , FilBas , − XvaMax , XvaMax , YvaMin , YvaMax ) ; − > C o r r e l P a i r ( 0 , 2 , FilBas , − XvaMax , XvaMax , YvaMin , YvaMax ) ;99 myBlue − > D i s p l a y P a i r ( 0 , 2 , F i l B a s ) ;100101 // D i g e s t : show some o b s e r v a b l e q u a n t i t i e s , p r e p a r e code f o r Fig . 5a
102 myBlue − > PrintChiPro ( ) ;103 myBlue − > D i s p l a y R e s u l t ( 0 , F i l B a s ) ;104105 // D i g e s t : w r i t e L a t e x f i l e
106 myBlue − > L a t e x R e s u l t ( F i l B a s ) ;107108 // Free : , Fix : , Combine : s o l v e a c c o r d i n g t o im p or ta nc e
109 myBlue − > R e l e a s e I n p ( ) ;110 myBlue − > FixInp ( ) ;111 myBlue − > SolveAccImp ( 0 . 1 ) ;112113 // D i g e s t : p r e p a r e code f o r Fig . 5 b
114 myBlue − > DisplayAccImp ( 0 , F i l B a s ) ;115116 // Free : , Fix : , Combine : s o l v e v a r y i n g t h e s y s t e m a t i c u n c e r t a i n t i e s
117 myBlue − > R e l e a s e I n p ( ) ;118 myBlue − > FixInp ( ) ;119 myBlue − > S o l v e S c a S t a ( ) ;120121 // D i g e s t : produce Fig . 6
122 myBlue − > P r i n t S c a S t a ( FilBas , ValMin , ValMax , UncMin , UncMax ) ;123124 // D e l e t e : d e l e t e o b j e c t as w e l l as l o c a l m a t r i c e s and r e t u r n delete myBlue ; myBlue = NULL;126 InpEst − > D e l e t e ( ) ; InpEst = NULL;127 InpSta − > D e l e t e ( ) ; InpSta = NULL;128 return ;129 } ; References [1] L. Lyons, D. Gibaut and P. Clifford, How to combine correlated esti-mates of a single physical quantity, Nucl. Instr. and Meth. A 270 (1988)110. doi:10.1016/0168-9002(88)90018-6 .[2] A. Valassi, Combining correlated measurements of several differentquantities, Nucl. Instr. and Meth. A 500 (2003) 391. doi:10.1016/S0168-9002(03)00329-2 .[3] R. Brun and F. Rademakers, ROOT - An Object Oriented Data Anal-ysis Framework, Nucl. Instr. and Meth. A 389 (1997) 81, Proceed-ings of AIHENP’96 Workshop, Lausanne, Sep. 1996. doi:10.1016/S0168-9002(97)00048-X . 154] R. Nisius, On the combination of correlated estimates of a physics ob-servable, Eur. Phys. J. C 74 (2014) 3004. arXiv:1402.4016 , doi:10.1140/epjc/s10052-014-3004-2 .[5] ATLAS Collaboration, Measurement of the top quark mass in the t ¯ t → lepton+jets and t ¯ t → dilepton channels using √ s = 7 TeV ATLASdata, Eur. Phys. J. C 75 (2015) 330. arXiv:1503.05427 , doi:10.1140/epjc/s10052-015-3544-0 .[6] ALICE Collaboration, Neutral pion and η meson production in p-Pbcollisions at √ s NN = 5 .
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