Estimating Error Bars on Model Predictions

In order to estimate the uncertainties in a model s predictions (the ut in Eq. (21)) for consistency checking, one must have estimates of the uncertainties in the model s input parameters. In the 20th century, model input uncertainties 

Knowing which of the Sy are large in magnitude can be very helpful to a modeler, but Eq. (24a) requires some care. First, in systems where T and P are not constant, one usually means the sensitivity with respect to the preexponential factor in the rate constant, i.e. a uniform scaling of kfT,P) at all T,P (Kee et al., 1989). 

More importantly, as written, the partial derivative implies that a single rate constant is to be varied, holding all the others constant, and indeed this is the way it is implemented in many sensitivity analysis routines. The index y runs out to 2Areactions because each reaction has two rate constants, one for the forward direction, and one for the reverse. However, in order for the model to remain consistent with the laws of thermodynamics, the rate constant for the reverse of reaction y must vary simultaneously with the forward reaction, since the two rate constants must maintain a detailed-balance ratio related to the AGreaction, Eq. (2). This can be assured by specifying that the partial derivative is taken only for the forward reaction, while holding the thermochemistry fixed. Note that this also cuts the number of partial derivatives to be computed in half. These sensitivities should then be supplemented with sensitivities to the individual species thermochemistry as in Eq. (25) overall the number of partial derivatives to be computed per model prediction M, is 

If one (optimistically) believes the boundary conditions, initial conditions, and other molecular properties (e.g. transport properties, heat capacities) have all been specified very precisely, and one thinks the uncertainties in the rate constants and the enthalpies are not highly correlated, one might then estimate the uncertainty in prediction M, by Eq. (26). 

If one believes that the uncertainties in rate constants and thermo are highly correlated, a more conservative (pessimistic) estimate would be 

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