Linearization

(6.1) |

with

The symbol indicates that we are dealing with

(6.3) |

Let us indicate with
and
the best estimates and the standard uncertainty
of the raw values:

E | (6.4) | ||

Var | (6.5) |

For any possible configuration of conditioning hypotheses ,

The function which relates the corrected value to the raw value and to the systematic effects has been denoted by so as not to be confused with a probability density function. Expanding () in series around we finally arrive at the expression which will allow us to make the approximated evaluations of uncertainties:

(All derivatives are

Neglecting the terms of the expansion above the first order,
and taking the expected values, we get

The terms included within vanish if the unknown systematic errors are uncorrelated, and the formulae become simpler. Unfortunately, very often this is not the case, as when several calibration constants are simultaneously obtained from a fit (for example, in most linear fits slope and intercept have a correlation coefficient close to ).

Sometimes the expansion
() is not performed around the best values
of
but around their __nominal__
__values__, in the
sense that the correction for the known value of the systematic errors
has not yet been applied
(see Section ). In this case ()
should be replaced by

where the subscript stands for

() and () instead remain valid, with the condition that the derivative is calculated at . If

is the component of the standard uncertainty due to effect . is equal to the product of signs of the derivatives, which takes into account whether the uncertainties are positively or negatively correlated.

To summarize, when systematic effects are not correlated with each other, the following quantities are needed to evaluate the corrected result, the combined uncertainties and the correlations:

- the raw and ;
- the best estimates of the corrections for each systematic effect ;
- the best estimate of the standard deviation due to the imperfect knowledge of the systematic effect;
- for any pair the sign of the correlation due to the effect .

In High Energy Physics applications it is frequently the case that
the derivatives appearing in
()-() cannot be calculated directly,
as for example when are parameters of a simulation program,
or acceptance cuts. Then variations of
are
usually studied by varying a particular within
a *reasonable* interval, holding the other influence
quantities at the nominal value.
and are calculated from
the interval
of variation of the true value
for a given variation
of
and from the probabilistic meaning of the intervals (i.e.
from the assumed distribution of the true value).
This empirical procedure for determining
and has the advantage that it
can take into account
non-linear effects[45], since it
directly measures the difference
for a given difference
.

Some examples are given in Section , and two typical experimental applications will be discussed in more detail in Section .