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:
x1 +- dx1s (statistical) +- dx1u (uncorrelated systematic) +- dx1c (correlated systematic)
x2 +- dx2s (statistical) +- dx2u (uncorrelated systematic) +- dx2c (correlated systematic)
resulting in mean m +- dmstat +- dmsyst. 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
dx1s2 + dx1u2 |
0 |
0 |
dx2s2 + dx2u2 |
and a non-diagonal covariance matrix for the correlated uncertainties with correlation coefficient rho=1
dx1c2 |
dx1c * dx2c |
dx1c * dx2c |
dx2c2 |
. From the form of the covariance matrix C
dx12 |
rho * dx1 * dx2 |
rho * dx1 * dx2 |
dx22 |
we can identify dxi2 = dxis2 + dxiu2 + dxic2 and rho = dx1c * dx2c / (dx1 * dx2).
Minimizing the generalized chi-squared XTC-1X, where X is a column vector
x1 - m |
x2 - m |
, we get for the minimum variance estimate of m
m = (x1/dx12 + x2/dx22 - rho * (x1 + x2) /(dx1 * dx2)) / (1/dx12 + 1/dx22 - 2 * rho/(dx1 * dx2))
and the variance of m
dm2 = (1-rho2) / ( 1/dx12 + 1/dx22 - 2 * rho/(dx1 * dx2)). We decompose the variance into statistical and systematic components by subtraction in quadrature of the statistical uncertainty dmstat2 = 1/(1/dx1s2 + 1/dx2s2), dmsyst2 = dm2 - dmstat 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 < δi > = 0 and < δi * δj > = δij
and where is the kroneker delta function (1 for and zero for ).
Pseudo-measurements of (x1,x2) can be generated from the expressions (x1,x2) = (x10 + α * δ1 + β * δ2,x20 + γ * δ3 + λ * δ2).
Expanding the covariance matrix one finds
.Using the properties of C12 = C21 = β * λ. Similar substitution gives for the diagonal matrix elements
and
so that
C=
α2 + β2 | β * λ |