5.6. Note on Decision threshold and Detection limit for linear fitting¶
At first, for the equation used in the Least squares fitting
\(Y\left( t_{k} \right) = a_{1} \cdot X_{1}\left( t_{k} \right) + \ a_{2} \cdot X_{2}\left( t_{k} \right) + \ a_{3} \bullet X_{3}\left( t_{k} \right)\)
it must be defined, which of the fitting parameters \(a_{i}\) corresponds to the actually valid output quantity. For the following notes let us assume that the parameter \(a_{1}\) is the one representing the output quantity.
The procedure for calculating the Decision threshold and the Detection limit follows ISO 11929:2010. The parameter \(a_{1}\) is modified by iteration. At first, the parameter \(a_{1}\) in the above equation (i.e. the value of the Y-90 counting rate in the case of a Y-90 decay-curve) is replaced by an iterated value \(a_{1}^{'}\) while all other values remain unchanged. From this, new measured net counting rates \(Y^{'}\left( t_{k} \right)\) are calculated as well as its new uncertainties \(u\left( Y'\left( t_{k} \right) \right)\) (the uncertainty function). With these new values, i.e. (\(a_{1}^{'}\),\(Y^{'}\left( t_{k} \right)\),\(\ u\left( Y'\left( t_{k} \right) \right)\)), the least squares analysis calculations are repeated yielding the uncertainty \(u\left( a_{1}^{'} \right)\) of the iterated parameter value \(a_{1}^{'}\). With this pair of values (\(a_{1}^{'}\),\(\ u\left( a_{1}^{'} \right)\)) it is then tested whether the termination condition of the iteration procedure is fulfilled; if not, the next iterated value \(a_{1}^{''}\) is determined and the above procedure is repeated in order to find its uncertainty \(u\left( a_{1}^{''} \right)\); and so on.
A more detailed description of these calculations while iterating can be found at the end of the following help topics:
