Mean or average of two partially correlated measurements
See this article for background material or a similar article with an example application from particle physics.
Use the root macro to perform an average of two measurements:
x_{1} +- dx_{1s} (statistical) +- dx_{1u} (uncorrelated systematic) +- dx_{1c} (correlated systematic)
x_{2} +- dx_{2s} (statistical) +- dx_{2u} (uncorrelated systematic) +- dx_{}_{2c} (correlated systematic)
resulting in mean m +- dm_{stat} +- dm_{syst}. The systematic errors in each channel are decomposed in uncorrelated and (100%) correlated components. The correlation matrix which appears in the solution is composed of the sum of a diagonal covariance matrix with elements given by the uncorrelated uncertainties summed in quadrature
dx_{1s}^{2} + dx_{1u}^{2} |
0 |
0 |
dx_{2s}^{2 }+ dx_{2u}^{2} |
and a non-diagonal covariance matrix for the correlated uncertainties with correlation coefficient rho=1
dx_{1c}^{2} |
dx_{1c} * dx_{2c} |
dx_{1c} * dx_{2c} |
dx_{2c}^{2} |
. From the form of the covariance matrix C
dx_{1}^{2} |
rho * dx_{1} * dx_{2} |
rho * dx_{1} * dx_{2} |
dx_{2}^{2} |
we can identify dx_{i}^{2} = dx_{i}_{s}^{2} + dx_{iu}^{2} + dx_{ic}^{2} and rho = dx_{1c} * dx_{2c} / (dx_{1} * dx_{2}).
Minimizing the generalized chi-squared X^{T}C^{-1}X, where X is a column vector
x_{1} - m |
x_{2} - m |
, we get for the minimum variance estimate of m
m = (x_{1}/dx_{1}^{2} + x_{2}/dx_{2}^{2} - rho * (x_{1} + x_{2}) /(dx_{1} * dx_{2})_{}) / (1/dx_{1}^{2} + 1/dx_{2}^{2} - 2 * rho/(dx_{1} * dx_{2}))
and the variance of m
dm^{2} = (1-rho^{2}) / ( 1/dx_{1}^{2} + 1/dx_{2}^{2} - 2 * rho/(dx_{1} * dx_{2})). We decompose the variance into statistical and systematic components by subtraction in quadrature of the statistical uncertainty dm_{stat}^{2} = 1/(1/dx_{1}_{s}^{2} + 1/dx_{2s}^{2}), dm_{syst}^{2} = dm^{2} - dm_{stat} ^{2}.
There is protection in the code against the very special case that there is only a 100% correlated uncertainty. If this is truely the case then the 2 measurements must be equal by construction and the uncertainty may be taken as the smaller of the 2.
There is also code to show the results of a popular approximation that does not properly take into account the correlation between the 2 measurements.
Here is a sample output from the macro for some (almost) randomly chosen measurement results:
******************************************* * * * W E L C O M E to R O O T * * * * Version 5.28/00 14 December 2010 * * * * You are welcome to visit our Web site * * http://root.cern.ch * * * ******************************************* ROOT 5.28/00 (trunk@37585, Dec 14 2010, 15:20:27 on linuxx8664gcc) CINT/ROOT C/C++ Interpreter version 5.18.00, July 2, 2010 Type ? for help. Commands must be C++ statements. Enclose multiple statements between { }. root [0] Processing AverageMeasurements.C... The measurements being averaged: ------------------------------- x1 = 58.9 +- 3.4 (stat) +- 1.5 (uncorr syst) +- 2.4 (corr syst) = 58.9 +- 4.4238 (total) x2 = 68.7 +- 2.8 (stat) +- 0.3 (uncorr syst) +- 3.9 (corr syst) = 68.7 +- 4.81041 (total) Results for the generalized weighted average -------------------------------------------- Correlation coefficient (rho) = 0.439844 m = 63.0708 +- 2.1614 (stat) +- 3.24854 (syst) = 63.0708 +- 3.90188 (total) Generalized chi-squared = 4.00333 Approximate, simple formulae ---------------------------- m = 65.1253 +- 2.1614 (stat) +- 3.20753 (syst) = 65.1253 +- 3.8678 (total) Generalized chi-squared = 4.28057 (for the approximate minimum) Delta chi-squared with respect to exact minimum = 0.277241
The derivation of the terms in the correlation matrix
The covariance matrix is
.
Suppose are three independent sources of normally-distributed unit fluctuations (with < δ_{i} > = 0 and < δ_{i} * δ_{j} > = δ_{ij}
and where is the kroneker delta function (1 for and zero for ).
Pseudo-measurements of (x_{1},x_{2}) can be generated from the expressions (x_{1},x_{2}) = (x_{10} + α * δ_{1} + β * δ_{2},x_{20} + γ * δ_{3} + λ * δ_{2}).
Expanding the covariance matrix one finds
.Using the properties of C_{12} = C_{21} = β * λ. Similar substitution gives for the diagonal matrix elements
and
so that
C=
α^{2} + β^{2} | β * λ |