10 - H1 - Desy

8.3 Error matrix evaluation 111
expansion series:
U i,j = ∑ k,l
∣
∂η i ∂η j
Vk,l ∂θ k ∂θ l
∣∣∣∣ˆθ
θ (8.24)
with ˆθ being the estimator of θ and V θ k,l = cov|θ k, θ l |. In the case of this analysis ⃗η = ⃗x true ,
while m × n A i,j elements of migration matrix A (m×n) are used as variables θ. In such a
case, using the equation 8.11 one can calculate derivatives:
∂η i
∂θ k
≡ ∂xi true
∂A p,q
= (E u ) i,p × (V(⃗y obs − A⃗x true )) q − (E u A T V) i,q × (⃗x true ) p , (8.25)
where indices i and p run over unfolding output bins and q runs over input bins.
The covariance matrix U takes then the form:
U i,j = ∑ p,q
∑
r,s
∣
∂x i true ∂x j ∣∣∣∣Â
true
V(p,q),(r,s) A ∂A p,q ∂A , (8.26)
r,s
where indices i, j, p and r run over unfolding output bins, while q and s are input bin
indices.
8.3.1 MC statistics error propagation
The migration matrix as used in this analysis needs to be sufficiently populated to avoid
large statistical fluctuations in the determination of the matrix itself. Special care was
taken to include the MC statistical fluctuations in the final errors. The error originating
from the limited statistics of the MC is calculated with each migration matrix element
treated as independent source of uncorrelated error. In such a case the covariance matrix
V A (p,q),(r,s) takes the form V A (p,q),(r,s) = {
0 if (p, q) ≠ (r, s)
σ 2 p,q if (p, q) = (r, s)
(8.27)
where σ 2 p,q is the statistical error of the element A p,q . The equation 8.26 simplifies to
U i,j = ∑ p,q
∂η i
∂A p,q
∣
∂η ∣∣∣∣Â
j
σp,q 2 ≡ E mc (8.28)
∂A p,q
In practice, since migration matrix is normalised, two errors are propagated. One coming
from the matrix binomial error and the second, originating from the limited statistics of
the normalisation factor.
The MC statistical error contributes to at most 10% of the final error in the low η γ -high
E γ T
region. Typically it is consistently below 5% of the final error and thus considered to
be reduced to negligible level over the whole studied phase space.