Construction of the fit covariance matrix
In the following we provide an explicit example of how the covariance matrix is built,
given two datasets having both uncorrelated and correlated systematics.
We consider 2 datasets having respectively 2 datapoints with 3 systematic uncertainties, and
3 datapoints with 2 systematic uncertainties.
The total statistic uncertainties for the two datasets are denoted as
\[\sigma_i\,, \,\,\, i = 1,2 \,\,\,\,\,\,\,\, \text{and} \,\,\,\,\,\,\,\, \bar{\sigma}_j \,,\,\,\, j = 1,2,3\]
while the systematic uncertainties as
\[\sigma^{\text{sys},\theta}_i\,, \,\,\,\, i = 1,2\,, \,\,\,\, \theta = 1,2,3\,,
\,\,\,\,\,\,\,\, \text{and} \,\,\,\,\,\,\,\,
\bar{\sigma}^{\text{sys},\theta}_i\,, \,\,\,\,\,\,\,\, i = 1,2,3\,, \,\,\,\, \theta = 1,2\,,\]
with the upper and lower indices labelling the systematic and the datapoint respectively.
We further assume that
the systematic \(\sigma^{\text{sys},1}_i\,,\,\,\, \sigma^{\text{sys},2}_i\) of the first dataset
are correlated within the dataset, and therefore named CORR according to our convention,
while \(\sigma^{\text{sys},3}_i\) is named as SPECIAL
regarding the second dataset, the systematic \(\bar{\sigma}^{\text{sys},1}_i\) is taken as uncorrelated,
and therefore denoted as UNCORR, while \(\bar{\sigma}^{\text{sys},2}_i\) is named SPECIAL, and
therefore will be correlated with the one of the other dataset having the same name
Considering a single dataset, the covariance matrix is given by
\[\text{cov}_{ij} = \sigma_i\,\sigma_j \, \delta_{ij} + \sum_{\theta}\sigma^{\text{sys},\theta}_i \sigma^{\text{sys},\theta}_j\]
where the sum on \(\theta\) runs on the correlated systematics between the point \(i\,,\,\,j\).
When considering the two datasets together, additional off diagonal contributions should be added to account for
the correlated systematics denoted as SPECIAL.
Specifically we have, from the statistical, CORR and UNCORR uncertainties
\[\begin{split}\begin{pmatrix}
\sigma_1^2 & 0 & 0 & 0 & 0 \\
0 & \sigma_2^2 & 0 & 0 & 0 \\
0 & 0 & \bar{\sigma}_1^2 & 0 & 0 \\
0 & 0 & 0 & \bar{\sigma}_2^2 & 0 \\
0 & 0 & 0 & 0 & \bar{\sigma}_3^2
\end{pmatrix}
+
\begin{pmatrix}
{\left(\sigma^{\text{sys},1}_1\right)}^2 + {\left(\sigma^{\text{sys},2}_1\right)}^2 & \sigma^{\text{sys},1}_1 \sigma^{\text{sys},1}_2 + \sigma^{\text{sys},2}_1 \sigma^{\text{sys},2}_2 & 0 & 0 & 0 \\
\sigma^{\text{sys},2}_1 \sigma^{\text{sys},1}_1 + \sigma^{\text{sys},2}_2 \sigma^{\text{sys},2}_1 & {\left(\sigma^{\text{sys},1}_2\right)}^2 + {\left(\sigma^{\text{sys},2}_2\right)}^2 & 0 & 0 & 0 \\
0 & 0 & {\left(\bar{\sigma}^{\text{sys},1}_1\right)}^2 & 0 & 0 \\
0 & 0 & 0 & {\left(\bar{\sigma}^{\text{sys},1}_2\right)}^2 & 0 \\
0 & 0 & 0 & 0 & {\left(\bar{\sigma}^{\text{sys},1}_3\right)}^2
\end{pmatrix}\end{split}\]
while from the cross correlated systematic
\[\begin{split}\begin{pmatrix}
\left(\sigma_1^{\text{sys},3}\right)^2 & \sigma_1^{\text{sys},3}\sigma_2^{\text{sys},3} & \sigma_1^{\text{sys},3}\bar{\sigma}_1^{\text{sys},2} & \sigma_1^{\text{sys},3}\bar{\sigma}_2^{\text{sys},2} & \sigma_1^{\text{sys},3}\bar{\sigma}_3^{\text{sys},2} \\
\sigma_2^{\text{sys},3}\sigma_1^{\text{sys},3} & \left(\sigma_2^{\text{sys},3}\right)^2 & \sigma_2^{\text{sys},3}\bar{\sigma}_1^{\text{sys},2} & \sigma_2^{\text{sys},3}\bar{\sigma}_2^{\text{sys},2} & \sigma_2^{\text{sys},3}\bar{\sigma}_3^{\text{sys},2} \\
\bar{\sigma}_1^{\text{sys},2}\sigma_1^{\text{sys},3} & \bar{\sigma}_1^{\text{sys},2}\sigma_2^{\text{sys},3} & \left(\bar{\sigma}_1^{\text{sys},2}\right)^2 & \bar{\sigma}_1^{\text{sys},2}\bar{\sigma}_2^{\text{sys},2} & \bar{\sigma}_1^{\text{sys},2}\bar{\sigma}_3^{\text{sys},2} \\
\bar{\sigma}_2^{\text{sys},2}\sigma_1^{\text{sys},3} & \bar{\sigma}_2^{\text{sys},2}\sigma_2^{\text{sys},3} & \bar{\sigma}_2^{\text{sys},2}\bar{\sigma}_1^{\text{sys},2} & \left(\bar{\sigma}_2^{\text{sys},2}\right)^2 & \bar{\sigma}_2^{\text{sys},2}\bar{\sigma}_3^{\text{sys},2} \\
\bar{\sigma}_3^{\text{sys},2}\sigma_1^{\text{sys},3} & \bar{\sigma}_3^{\text{sys},2}\sigma_2^{\text{sys},3} & \bar{\sigma}_3^{\text{sys},2}\bar{\sigma}_1^{\text{sys},2} & \bar{\sigma}_3^{\text{sys},2}\bar{\sigma}_2^{\text{sys},2} & \left(\bar{\sigma}_3^{\text{sys},2}\right)^2
\end{pmatrix}\end{split}\]