BLUE: combining correlated estimates of physics observables within ROOT using the Best Linear Unbiased Estimate method

This software performs the combination ofm correlated estimates of n physics observables (m ≥ n) using the Best Linear Unbiased Estimate (BLUE) method. It is implemented as a C++ class, to be used within the ROOT analysis package. It features easy disabling of specific estimates or uncertainty sources, the investigation of different correlation assumptions, and allows performing combinations according to the importance of the estimates. This enables systematic investigations of the combination on details of the measurements from within the software, without touching the input.


Motivation and significance
The combination of a number of correlated estimates for a single observable is discussed in Ref. [1]. Here, the term estimate denotes a particular outcome (measurement) of an experiment based on an experimental estimator (an algorithm for a measurement) of the observable, which follows a probability density function (pdf). The particular estimate obtained by the experiment may be a likely or unlikely outcome for that pdf. Repeating the measurement numerous times under identical conditions, the estimates will follow the underlying pdf of the estimator. The BLUE software listed in Table 1 makes use of a χ 2 minimisation to obtain the combined value, i.e. it uses the Gaussian approximation for the uncertainties. In Ref. [1], this minimisation is expressed in the mathematically equivalent BLUE language.
Provided the estimators are unbiased, when applying this formalism the Best Linear Unbiased Estimate of the observable is obtained with the following meaning: Best: the combined result for the observable obtained this way has the smallest variance; Linear: the result is constructed as a linear combination of the individual estimates; Unbiased Estimate: when the procedure is repeated for a large number of cases consistent with the underlying multi-dimensional pdf, the mean of all combined results equals the true value of the observable. The formulas for more than one observable [2] are implemented in the BLUE software, which is programmed as a separate class of the ROOT analysis framework [3].
The easiest case of two correlated estimates of the same observable is briefly illustrated here. Already for this case, the main features of the combination can easily be understood. For further information and the derivation of the formulas the reader is refereed to Ref. [4]. Let x 1 and x 2 with variances σ 2 1 and σ 2 2 be two estimates from two unbiased estimators X 1 and X 2 of the true value x T of the observable and ρ the total correlation of the two estimators. Without loss of generality it is assumed that the estimate x 1 stems from an estimator X 1 of x T that is as least as precise as the estimator X 2 yielding the estimate x 2 , such that z ≡ σ 2 /σ 1 ≥ 1. In this situation the BLUE x of x T is: where β is the weight of the less precise estimate and the sum of weights is unity by construction. The variable x is the combined result and σ 2 x is its variance, i.e. the uncertainty assigned to the combined value is σ x . For a number of z values the two quantities β and σ x /σ 1 as functions of ρ are shown in Figure 1. Their functional forms are also written in the figures. The functions are valid for −1 ≤ ρ ≤ 1 and z ≥ 1, except for ρ = z = 1.   Figure 1 displays the strong dependence of the uncertainty in the combined value on z and ρ. For the special situation of ρ = 1/z the uncertainty σ x equals σ 1 , i.e. the precision in the observable is not improved by adding the second measurement. For ρ = ±1 the uncertainty in the combined result vanishes, i.e. σ x = 0, a mere consequence of the conditional probability for X 2 given the measured value of x 1 , see Ref. [4] for details. It is worth noticing that in most regions of the (ρ, z)-plane the sensitivity of σ x /σ 1 to ρ is stronger than to z. This means, reducing the correlation of the estimates in most cases gives a larger improvement in precision in the combined value than reducing z.

Software description
To use the BLUE software, a working installation of the ROOT package [3] is needed. The mandatory input to the software are the measured values, their uncertainties for the various statistical and systematic effects relevant to the measurements, and the estimator correlations for each of those uncertainties.

Software Architecture
The flowchart of the software is shown in Figure 2. The relations of the (Keywords) to the various steps are as follows. After creating the object (New), the measurements together with their uncertainties and correlations are passed to the software (Fill). Afterwards, combinations can be performed by fixing (Fix) and solving (Solve), i.e. combining the potentially Set. . .  altered (Set...) measurements, as often as wanted. Before altering the measurements for the next combination, they have to be Released (the alteration continues from the status at the last fix) or Reset (the alteration starts from the original measurements). Finally, the object is deleted (Delete).

Software Functionalities
One main quality of this software is the built-in flexibility for easy and thorough investigations of the impact of details of the input measurements and their correlation. With single function calls estimates or uncertainty sources can be removed from the combination, different uncertainty models (e.g. absolute or relative uncertainties) and correlation assumptions can be investigated. Another strength is the large number of different solving methods implemented, ranging from only using measurements with positive weights in the combination to a successive combination method in which the input measurements are included one at a time according to their importance, allowing an in-depth investigation of their impact on the combination.

Illustrative Examples
A compact example of three estimates of a single observable is listed in   compatibility evaluations with the other two estimates. Since the three measurements should come from the same underlying x T , this indicates either an unlikely outcome or a potential systematic problem with this result. Only after a careful investigation of this measurement, resulting in a low probability for the second possibility, should this measurement be included in the combination. The systematic uncertainties are shown together with the statistical precisions at which they are known. Those statistical precisions allow evaluating whether two estimators have a significantly different sensitivity for a source of uncertainty. In addition, they indicate which systematic effect should be evaluated with higher statistical precision. The sources of systematic uncertainties with estimator correlations ρ = ±1 are shown in Figure 3. The case ρ = +1 corresponds to the situation where simultaneously applying a systematic effect to both estimates (e.g. in- creasing a jet energy scale by +1σ JES ) leads to both measured values moving into the same direction, either both get larger or both get smaller than the original result. The case ρ = −1 means the two measurements move in opposite directions. The points for which the bars cross one of the coordinate axis indicate sources for which, within uncertainties, the correlation may be ρ = +1 or ρ = −1. This will be exploited in the stability evaluation discussed below.
Without combining, the precision of the knowledge about the observable is defined by the most precise result, here M 2 . The impact that an additional estimate has can be digested by performing pairwise combinations with the most precise result. An example of such a pairwise combination of M 0 and M 2 is shown in Figure 4. Apart from the range ρ > 0.8, the combined value is almost independent of ρ. In contrast, the uncertainty in the combined value has a very strong dependence on ρ.
The combination of all estimates is shown in Figure 5(a). The input measurements are listed in the first three lines, the combined result is listed in red in the last line. Figure 5(b) reveals that not all results significantly contribute to the combined value. In this figure, the lines show the results of successive combinations, always adding the estimate listed to the previous list of estimates. Also here the suggested combined result is shown in red. The estimate M 1 does not improve the already accumulated result obtained from combining M 2 and M 0 . Figure 6 shows the stability of the combination of all three results, taking into account the statistical precisions at which the systematic uncertainties  are known, see Table 2. For this figure all systematic uncertainties are altered within their statistical precisions, the correlations are re-evaluated and they also may change sign, see  Table 2. Frequently those are not provided, or even not evaluated. This is only justified if they are much smaller than the quoted uncertainties.

Impact
The software can be used for an in-depth analysis of the impact of various assumptions made in the combination. Provided that appropriate input is provided, it also allows assessing the stability of the combination.
Because of the large reduction in the uncertainty in the combined result obtained by lowering the estimator correlations, it is advisable to use this software already in the design stage of the various analyses performed for obtaining the same observable within a single experiment. Usually, the uncertainties in the various systematic effects (e.g. the uncertainty in jet energy scales for experiments at hadron collider) are determined by the actual level of understanding of the detector and have to be taken into account at face value. In contrast, the sensitivity of the estimators to those effects can be  influenced by the estimator design. This way their correlation can be reduced, thereby improving the gain obtained in the combination. Generally speaking, the strategy should not be to take over an aspect of the analysis that has worked for one estimator to another estimator. Instead, alternative approaches should be pursuit, such as to potentially lower the estimator correlations, even at the expense of a larger uncertainty. This is because achieving an anti-correlated pair of estimates with the same sensitivity to a specific source of uncertainty, renders this a significantly smaller uncertainty in the combined result. Some indications can be seen for the sources Syst 4 and Syst 8 , which exhibit the largest fractional gain in uncertainty when comparing M with M 0 . An example of such an optimisation is explained in Ref. [5]. This software can be of significant help in this process. According to the knowledge of the author, by now the BLUE software was used in a number of combinations, mostly in the context of high energy physics, especially at the Large Hadron Collider (LHC). Examples from the ALICE, ATLAS, CMS and LHCb collaborations are detailed in Refs. [6,7,8,9]. The first world combination of the top quark mass [10] has also been performed with this software. In addition to the LHC collaborations, the software 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 in Ref. [13]. Further examples of the software usage are described in the manual listed in Table 1. To assist the users in developing their own combination code, the corresponding C++ routines to reproduce those published results are included in the software package. Although the above examples are all particle physics applications, the use of this software is not confined to a specific area of research. Any set of correlated measurements of one or more observables can be combined.

Conclusions
The BLUE 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 correlation models and combination methods makes it a useful tool to assess details on the combination in question. Exploring the combination of various estimators of the same observable within a single experiment allows a design of estimators with low correlation. This enhances the gain achieved in combinations of estimates obtained from those estimators.

Conflict of Interest
I wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.   [9] LHCb Collaboration, Measurement of the branching fractions of the decays D + → K − K + K + , D + → π − π + K + and D + s → π − K + K + , JHEP 03 (2019) 176. arXiv:1810.03138, doi:10.1007/JHEP03(2019) 176.