Linearization

We have seen in the above examples how to use the general formula () for practical applications. Unfortunately, when the problem becomes more complicated one starts facing integration problems. For this reason approximate methods are generally used. We will derive the approximation rules consistently with the approach followed in these notes and then the resulting formulae will be compared with the ISO recommendations. To do this, let us neglect for a while all quantities of influence which could produce unknown systematic errors. In this case () can be replaced by (), which can be further simplified if we remember that correlations between the results are originated by unknown systematic errors. In the absence of these, the joint distribution of all quantities $\underline{\mu}$ is simply the product of marginal ones:

$$f_{R_i}(\underline{\mu_i}) = \prod_i f_{R_i}(\mu_i)\,,$$ (6.1)

with

$$f_{R_i}(\mu_i) = f_{R_i}(\mu_i\,\vert\,x_i,\underline{h}_\circ) =... ...t f(x_i\,\vert\,\mu_i, \underline{h}_\circ) \,f_\circ(\mu_i) \,\rm {d}\mu_i}\,.$$ (6.2)

The symbol $f_{R_i}(\mu_i)$ indicates that we are dealing with raw values^6.1 evaluated at $\underline{h}=\underline{h}_\circ$. Since for any variation of $\underline{h}$ the inferred values of $\mu_i$ will change, it is convenient to name with the same subscript $R$ the quantity obtained for $\underline{h}_\circ$:

$$f_{R_i}(\mu_i) \longrightarrow f_{R_i}(\mu_{R_i})\,.$$ (6.3)

Let us indicate with $\widehat{\mu}_{R_i}$ and $\sigma_{R_i}$ the best estimates and the standard uncertainty of the raw values:

$$\widehat{\mu}_{R_i} = [\mu_{R_i}]$$ (6.4)

$$\sigma_{R_i}^2 = (\mu_{R_i})\,.$$ (6.5)

For any possible configuration of conditioning hypotheses $\underline{h}$, corrected values $\mu_i$ are obtained:

$$\mu_i=\mu_{R_i} + g_i(\underline{h})\,.$$ (6.6)

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

$$\boxed{ \mu_i= \mu_{R_i} + \sum_l \frac{\partial g_i}{\partial h_l} \,(h_l-h_{\circ_l}) + \ldots\, }$$ (6.7)

(All derivatives are evaluated at $\{\widehat{\mu}_{R_i},\underline{h}_\circ\}$. To simplify the notation a similar convention will be used in the following formulae.)

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

$$\widehat{\mu}_i = [\mu_i]$$

$$\approx \widehat{\mu}_{R_i}\,;$$ (6.8)

$$\sigma_{\mu_i}^2 = \left[(\mu_i-\mbox{E}[\mu_i])^2\right]$$

$$\approx \sigma_{R_i}^2 + \sum_l\left(\frac{\partial g_i}{\partial h_l}\right)^2 \sigma_{h_l}^2$$

$$\left\{ + \,2\,\sum_{l< m} \left(\frac{\partial g_i}{\partial h_l... ...al g_i}{\partial h_m}\right) \rho_{lm}\,\sigma_{h_l}\,\sigma_{h_m} \right\} \,;$$ (6.9)

$$(\mu_i,\mu_j) = \left[\,(\mu_i-\mbox{E}[\mu_i])(\mu_j- \mbox{E}[\mu_j])\,\right]$$

$$\approx \sum_l\left(\frac{\partial g_i}{\partial h_l}\right) \left(\frac{\partial g_j}{\partial h_l}\right)\sigma_{h_l}^2$$

$$\left\{ + \,2\,\sum_{l< m} \left(\frac{\partial g_i}{\partial h_l... ...ial g_j}{\partial h_m}\right) \rho_{lm}\,\sigma_{h_l}\,\sigma_{h_m} \right\}\,.$$ (6.10)

The terms included within $\{\cdot\}$ 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 $-0.9$).

Sometimes the expansion () is not performed around the best values of $\underline{h}$ 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

$$\mu_i=\mu_{R_i} + \sum_l \frac{\partial g_i}{\partial h_l} \,(h_l-h_{N_l}) + \ldots\,,$$ (6.11)

where the subscript $N$ stands for nominal. The best value of $\mu_i$ is then

$$\widehat{\mu}_i = [\mu_i]$$

$$\approx \widehat{\mu}_{R_i} + \left[\sum_l \frac{\partial g_i}{\partial h_l}\,(h_l-h_{N_l})\right]$$

$$= \widehat{\mu}_{R_i} + \sum_l\delta \mu_{i_l}\,.$$ (6.12)

() and () instead remain valid, with the condition that the derivative is calculated at $\underline{h}_N$. If $\rho_{lm}=0$, it is possible to rewrite () and () in the following way, which is very convenient for practical applications:

$$\sigma_{\mu_i}^2 \approx \sigma_{R_i}^2 + \sum_l\left(\frac{\partial g_i}{\partial h_l}\right)^2 \sigma_{h_l}^2$$ (6.13)

$$= \sigma_{R_i}^2 + \sum_l u_{i_l}^2 \,;$$ (6.14)

$$(\mu_i,\mu_j) \approx \sum_l\left(\frac{\partial g_i}{\partial h_l}\right) \left(\frac{\partial g_j}{\partial h_l}\right)\sigma_{h_l}^2$$ (6.15)

$$= \sum_l s_{ij_{l}} \,\left\vert\frac{\partial g_i}{\partial h_l}\r... ...ma_{h_l} \,\left\vert\frac{\partial g_j}{\partial h_l}\right\vert\,\sigma_{h_l}$$ (6.16)

$$= \sum_l s_{ij_{l}}\, u_{i_l}\,u_{j_l}$$ (6.17)

$$= \sum_l _l(\mu_i,\mu_j) \,.$$ (6.18)

$u_{i_l}$ is the component of the standard uncertainty due to effect $h_l$. $s_{ij_{l}}$ 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 $\widehat{\mu}_{R_i}$ and $\sigma_{R_i}$;
• the best estimates of the corrections $\delta \mu_{i_l}$ for each systematic effect $h_l$;
• the best estimate of the standard deviation $u_{i_l}$ due to the imperfect knowledge of the systematic effect;
• for any pair $\{\mu_i,\mu_j\}$ the sign of the correlation $s_{ij_{l}}$ due to the effect $h_l$.

In High Energy Physics applications it is frequently the case that the derivatives appearing in ()-() cannot be calculated directly, as for example when $h_l$ are parameters of a simulation program, or acceptance cuts. Then variations of $\underline{\mu}_i$ are usually studied by varying a particular $h_l$ within a reasonable interval, holding the other influence quantities at the nominal value. $\delta \mu_{i_l}$ and $u_{i_l}$ are calculated from the interval $\pm\Delta_i^\pm$ of variation of the true value for a given variation $\pm\Delta_{h_l}^\pm$ of $h_l$ and from the probabilistic meaning of the intervals (i.e. from the assumed distribution of the true value). This empirical procedure for determining $\delta \mu_{i_l}$ and $u_{i_l}$ has the advantage that it can take into account non-linear effects[45], since it directly measures the difference $\widehat{\mu}_i - \widehat{\mu}_{R_i}$ for a given difference $h_l-h_{N_l}$.

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