Standard Reference Material 2881 - National Institute of Standards ...

11. Discussion of Type B Uncertainties11.1 Instrument contribution to Type B uncertaintyIn an earlier paper 2 a novel approach is described for the selection of optimal instrumentparameters that yield a mass spectrum which best replicates the molecular mass distribution of asynthet ic polymer. The application of stochastic gradient approximation algorithms was shown tobe a viable method to find the best instrument settings while simultaneously minimizing the totalnumber of experiments that need to be performed. This includes considerations of when to halt theiterative optimization process at a point when statistically-significant gains can no longer beexpected. An algorithm to determine the confidence intervals for each parameter is also given.Details on sample preparation and data analysis that ensure stability of the measurement over thetime scale of the optimization experiments are provided.We can write J(x) from equation [7.1] in terms of the calibration equation for the signal axis:2expexp⎛ ⎡G⎤ ⎞⎛ ⎡G⎤ ⎞02⎜TOB⎟02⎜TOC⎟⎛ ⎞ ⎢ exp ⎥ ⎛ ⎞ ⎢ expG⎥TOB( ) =⎜00 1⎟⎜ ⎢00 ⎥ −1⎟⎜⎟ ⎢GGTBATOC⎥ − +⎜⎟GJ xTBA[11.1]⎝ G ⎠⎜ ⎢ G ⎥ ⎟⎝ ⎠⎜ ⎢ ⎥ ⎟TBATOB GGTBATOC⎜0 ⎟⎜0 ⎟⎝⎢⎣ G ⎥⎦ ⎠⎝⎢⎣⎥TBAGTBA⎦ ⎠2Substituting equation [6.19] into equation [11.1] yields:J ( x)=( Q k )02⎡⎛G⎢⎜⎢⎣⎝ G0TOB0TBA20⎞ ⎛ ( MwOB− M⎟⎜⎠ ⎝1+( Q / k0)(M0wBA0wBA−)M0⎞)⎟⎠2+⎛ G⎜⎝ G0TOC0TBA20⎞ ⎛ ( MwOC− M⎟⎜⎠ ⎝1+( Q / k0)(M0wBA0wBA−)M2⎞ ⎤⎥)⎟⎠ ⎥⎦02[11.2]From this we obtain partial derivatives of Q/k 0 as:δ ln( J ( x))δ ln(Q/k0)≈ 2δ p δ p[11.3]32