On the Combination Procedure of Correlated Errors

When averages of different experimental determinations of the same quantity are computed, each with statistical and systematic error components, then frequently the statistical and systematic components of the combined error are quoted explicitly. These are important pieces of information since statistical errors scale differently and often more favorably with the sample size than most systematical or theoretical errors. In this communication we describe a transparent procedure by which the statistical and systematic error components of the combination uncertainty can be obtained. We develop a general method and derive a general formula for the case of Gaussian errors with or without correlations. The method can easily be applied to other error distributions, as well. For the case of two measurements, we also define disparity and misalignment angles, and discuss their relation to the combination weight factors.


Introduction
Error propagation as well as the averaging of results of individual measurements -at least in the context of strictly Gaussian errors, possibly with statistical or systematic correlations -are straightforward, they are covered in many textbooks 1 , and there seems to be no open issue, because all that is required is multi-variate analysis applied to normal distributions. It is the more surprising, that -to the best of the authors knowledge -no explicit analytical expression is available that serves to compute for a given set of measurements of some quantity with individual, generally correlated, errors of statistical and systematic nature, the statistical and systematic components of the uncertainty of the average.
That is to say, while in the Gaussian context it is clear how to obtain the average including its uncertainty, and that the total error ought to be the quadratic sum of the statistical and systematic (and perhaps other such as theoretical) error components, formulae of these individual components and their derivation have not received much attention. But the proper disclosure of the statistical (random) error component compared to the systematic uncertainty can be of importance. For example, in the context of the design of future experimental facilities it is crucial to know how much precision (say, over the world average) can be gained by simply generating larger data samples, in contrast to possible technological or scientific breakthroughs. The systematic error is, of course, more troublesome as it cannot be reduced 2 as straightforwardly by increasing the sample size N .
In the next Section, we review the standard procedure to average a number of experimental determinations of some observable quantity. We also mention an approximate method to obtain the statistical and systematic error of an average of similar experiments where the ratio of the systematic to statistical components are comparable, or where the statistical error is dominant.
Then we turn to the main point, the exact determination of the individual error components of an average. We show that in the absence of correlations the square of the statistical error or any other type of uncertainty is weighted by the fourth power of the total errors.
Then we turn to correlations, starting with the simplest case of two measurements for which we introduce the concepts of disparity and misalignment angles. Finally, we present exact relations for the case of more than two measurements, and address some problems that arise when new measurements are added to an existing average iteratively.

Simplified Procedures
Suppose one is given a set of measurements of some quantity v, with central values v i , statistical (random) errors r i and total systematic errors s i . For simplicity, we are going to assume that the r i and s i are Gaussian distributed (the generalization to other error distributions is straightforward) in which case the total errors of the individual measurements are given by If we furthermore temporarily assume that the measurements are uncorrelated, then the central valuev of their combination is given by the precision weighted average, with total errort = 1 Similarly, the statistical componentr oft can often be approximated bȳ The systematic components oft is then obtained from  Table 1: Central value, v i , random error, r i , systematic uncertainty, s i , total error, t i , and uncorrelated error component, u i , for measurements of the quantity A τ by each of the four LEP collaborations, as well as the corresponding numbers for the combined result.
Notice, that while the individual errors in Eq. (6) are symmetric under the simultaneous exchange of the statistical and systematic errors (we recall that all r i and s i are assumed Gaussian) and the labels of the two measurements, the result (7) does not exhibit the corresponding symmetry which would implyr =s exactly. The exact procedure introduced in the next section would indeed yield r =s in this example. Now consider the case where one of the systematic errors, say s 1 , is larger and eventually s 1 → ∞. Then the weight of the first measurement approaches zero, andt → t 2 , as expected. However, one would also expect thatr → r 2 ands → s 2 , while insteadr < r 2 remains constant ands → √ 1924 ≈ 44 > s 2 . Thus, one would face the unreasonable result that averaging some measurement with an irrelevant constraint (with infinite uncertainty) will decrease (increase) the statistical (systematic) error component, leaving only the total error invariant. In other words, if in a set of measurements there is one with negligible statistical error, then the average would also have vanishing statistical error, regardless of how unimportant that one measurement is compared to the others. Clearly, Eq. (4) is then unsuitable even as an approximation.
One can easily extend these consideration to the case where the individual measurements have a common contribution c entering the systematic error. The precision weighted average (2) and total error (3) are then to be replaced bȳ where the uncorrelated error components are given by and where t 2  Table 2: Same as Table 1, but for the weak mixing angle determinations by ATLAS.
The general case of correlated errors will be dealt with later, but we note that the case of two measurements with Pearson's correlation coefficient ρ can always be brought to this form with c 2 given by A proper (normalizable) probability distribution requires |ρ| ≤ 1, so that from Eq. (10), guaranteeing thatt is real. On the other hand, u 1 or u 2 , as well asū, in Eq. (9) may become imaginary provided that in which case the first or second measurement, respectively, contributes with negative weight, and v lays no longer between v 1 and v 2 . In this situation, one rather (but equivalently) regards the measurement with a negative weight as a measurement of some nuisance parameter related to c. Replacing the inequalities (12) by equalities, gives rise to an infinite weight (one of the u i = 0) as well asū = 0 andt = c. As a concrete example, each of the four experimental collaborations at the Z boson factory LEP 1 [3,4,5,6] has measured some quantity A τ (related to the polarization of final-state τ leptons produced in Z decays) with the results shown in the Table below. A number of uncertainties affected the four measurements in a similar way, leading to a relatively weak correlation matrix [7] which, while not quite corresponding to the form (8), (9), can be well approximated by it when using the average of the square root of the off-diagonal entries of the covariance matrix c ≈ 0.0016.
The values in the last line arev,r,s,t andū as calculated from Eqs. (4), (5), (8) and (9).v,r ands agree with Table 4.3 andt agrees with Eq. (4.9) of the LEP combination in Ref. [7]. Table 2 shows the more recent example of the determination of the weak mixing angle [8] which is based on purely central (CC) electron events, events with a forward electron (CF), as well as muon pairs. Here the average of the off-diagonal entries of the covariance matrix amounts to c ≈ 0.0010. This is an example where the dominant uncertainty is from common systematics, namely from the imperfectly known parton distribution functions affecting the three channels in very similar ways.
We will return to these examples after deriving exact alternatives to formula (4).

Derivatively Weighted Errors
Our starting point is the basic property of a statistical error to scale as N −1/2 with the sample size.
To implement this, we rewrite Eq. (1) as Thus, the statistical error satisfies the relation, In the absence of correlations we can use Eq. (3), and demand that analogously, Notice, that Eq. (4) can be recovered from Eq. (15) upon substituting t i → r i andt →r. Eq. (15) means that the relative statistical error of the combination,x, is given by the precision weighted averagex where Furthermore, giving the systematic components a similar treatment, we find so that the expected symmetry between the two types of uncertainty becomes manifest, and moreover, Eq. (5) now follows directly from Eqs. (15) and (18), rather than being enforced. The central result is that for uncorrelated errors, the squares of the statistical and systematic components (or those of any other type) of an average are the corresponding individual squares weighted by the inverse of the fourth power of the individual total errors, or equivalently, weighted by the square of the individual precisions t −2 i . Returning to the case where the only source of correlation is a common contribution c = 0 equally affecting all measurements, we find from Eq. (8), where Applied to the case of A τ measurements we now find r = 0.0035,s = t 2 −r 2 = 0.0026, which agree not exactly, but within round-off precision with the approximate numbers in Table 1.

Bivariate Error Distributions
As a preparation for the most general case of N measurements with arbitrary correlation coefficients, we first discuss in some detail the case N = 2. Recall that the covariance matrix in this case reads The precision weighted average is given by the expression, where obtained by minimizing the likelihood following a bivariate Gaussian distribution, where The one standard deviation total errort is defined by which results int or conversely, Eq.(29) is useful in practice if one needs to recover the correlation between a pair of measurement uncertainties and their combination error. We now turn to the generalization of Eq. (15) in the presence of a systematic correlation. When applying our method of derivatively weighted errors to Eq. (28) it is important to keep c 2 = ρt 1 t 2 fixed (this would be different in the presence of a statistical correlation). Doing this, we obtain For the systematic component we find and we also note thatū More generally, one can compute the error contributionq of any individual source of uncertainty q to the total error asq = q 2 1 + 2ωc 2 q + ω 2 q 2 where c 2 q is the contribution of q to c 2 with the constraint If the two uncertainties q i are fully correlated or anti-correlated between the two measurements, then where the minus sign corresponds to anti-correlation. The formalism is now general enough to allow statistical correlations, as well. As we will illustrate later, knowing all theq is particularly useful if one wishes to successively include additional measurements to a combination -one by one -rather than having to deal with a multi-dimensional covariance matrix. This situation frequently arises in historical contexts when new measurements add information to a set of older ones, rather than superseding them. But there is a problematic issue with this, which apparently is not widely appreciated.

Disparity and Misalignment Angles
Continuing with the case of two measurements, we can relate ρ to the rotation angle necessary to diagonalize the matrix T . If we define an angle β quantifying the disparity of the total errors of two measurements through and The angle α may be interpreted as a measure of the misalignment of the two measurements with respect to the primary observable of interest v. Uncorrelated measurements of v are aligned (ρ = α = 0), while the case |ρ| ≫ | tan β| is reflective of a high degree of misalignment. Indeed, in the extreme case where β = 0 (|α| = 90 • ) two correlated measurements (ρ = 0) of the same quantity v are equivalent to two uncorrelated measurements, only one of which having any sensitivity to v at all. To reach the decorrelated configuration involves subtle cancellations between correlations and anti-correlations of the statistical and systematic error components of the original measurements.
We can now express the weight factor ω in terms of the disparity and misalignment angles β and α, In the case ρ = α = 0 this reduces to and Eq. (23) now readsv One can write equations of the form (41) and (42) for ρ = 0, as well, with shifted anglesβ related to β by cscβ = csc β − tan α .
However, this ceases to work out in the presence of a negative weight (ω < 0), in which case one would need to replace the trigonometric by the hyperbolic functions.

Multivariate Error Distributions
To treat cases of more than two measurements with generic correlations, one can choose one of two strategies. Either one effectively reduces the procedure to cases of just two measurements (in general at the price of some precision loss) by iteratively including additional measurements, or one deals with a multi-dimensional covariance matrix. We first discuss the latter approach, starting with the trivariate case where and for its statistical and systematic components we find (in the absence of statistical correlations), respectively. The generalization of Eq. (33) is now also straightforward. E.g., in the case of 100% correlation between the three measurements we have,  Table 3: Central values and breakdown of uncertainties (×10 4 ) of the weak mixing angle determinations by ATLAS. E e and E µ refer to the e ± and µ ± energy scales, respectively, while ∆E e denotes the electron energy resolution. The last three uncertainties from PDFs, missing higher order corrections, and other sources are taken as fully correlated, whereas the other uncertainties are assumed uncorrelated. The fourth column is the average of the two electron channels displayed in the second and third columns. The 6 th column adds the muon channel (5 th column) to them. The 7 th column shows the exact combination result of the three channels. The 8 th column are the corresponding numbers quoted by ATLAS. The interpretation of the last column is explained in the text.
impossible to compute them beforehand. In fact, they depend on the new measurement to be added (here the muon channel) and not just the initial measurements (here the two electron channels). Moreover, the ∆c 2 q necessary to enforce the correct average central valuev differs strongly from the ∆c 2 q necessary to enforce the total errort. This observation is a reflection of the fact that the combination principle can be violated [11], which we state as the requirement that the combination of a number of measurements must not depend on the order in which they added to the average. Thus, the iterative procedure generally suffers from a loss of precision. In this example the procedure gives nevertheless an excellent approximation because the uncertainty from higher order corrections (the origin of the asymmetric uncertainty) is itself very small. But there are cases in which the iterative procedure does not provide even a crude approximation and where one should use -if possible -the exact method based on the full covariance matrix. Unfortunately, its construction is not always possible, e.g., due to incomplete documentation of past results. Recent discussions of related aspects of this conundrum can be found in Refs. [12,13].

Summary and Conclusions
In summary, we have introduced a formalism (derivatively weighted errors) to derive formulas for random errors or any error type of uncorrelated Gaussian nature. We introduced what we call the disparity and misalignment angles to describe the case of two measurements, and showed their relation to the statistical weight factors. For the case of more than two measurements with known covariance matrix, we derived some explicit formulas in a form which (as far as we are aware) did not appear before.
It is remarkable, that even in the context of purely Gaussian errors and perfectly known correla-tions there are intractable problems at the most fundamental statistical level. Specifically, they may arise even when a number of observations of the same quantity is combined and the error sources are recorded and the assumptions regarding their correlations are spelled out carefully. In statistical terms, one can conclude that such a combination -despite of all its recorded details -represents an insufficient statistics of the available information. The inclusion of further observations of the same quantity is then in general ambiguous. On the other hand, there is no ambiguity in the absence of correlations or when any correlation is common to the set of observations to be combined. The fact that the ambiguities disappear in certain limits then reopens the possibility of useful approximations. For example, if an iterative procedure has to be chosen, one should first combine measurements where the dominant correlation is given approximately by a common contribution. Similarly, the measurements with small or no correlation with the other ones, are ideally kept for last.