Standard Reference Material 2881 - National Institute of Standards ...
Standard Reference Material 2881 - National Institute of Standards ...
Standard Reference Material 2881 - National Institute of Standards ...
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11. Discussion <strong>of</strong> Type B Uncertainties11.1 Instrument contribution to Type B uncertaintyIn an earlier paper 2 a novel approach is described for the selection <strong>of</strong> optimal instrumentparameters that yield a mass spectrum which best replicates the molecular mass distribution <strong>of</strong> asynthet ic polymer. The application <strong>of</strong> stochastic gradient approximation algorithms was shown tobe a viable method to find the best instrument settings while simultaneously minimizing the totalnumber <strong>of</strong> experiments that need to be performed. This includes considerations <strong>of</strong> 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 <strong>of</strong> the measurement over thetime scale <strong>of</strong> the optimization experiments are provided.We can write J(x) from equation [7.1] in terms <strong>of</strong> 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 <strong>of</strong> Q/k 0 as:δ ln( J ( x))δ ln(Q/k0)≈ 2δ p δ p[11.3]32