Stochastic error spatial

The stochastic error is expressed in (9.23) by the variance Var [Aj (t)] and co-variance Cov [Nj (t) Nk (t)] that did not exist in the deterministic model. This error could also be named spatial stochastic error, since it describes the process uncertainty among compartments for the same t and it depends on the number of drug particles initially administered in the system. For the sake of simplicity, assume riQi = uq for each compartment i. From the previous relations, the coefficient of variation CVj (t) associated with a time curve Nj (t) in compartment 3 is... 

Besides the hypothesis of spatially homogeneous processes in this stochastic formulation, the particle model introduces a structural heterogeneity in the media through the scarcity of particles when their number is low. In fact, the number of differential equations in the stochastic formulation for the state probability keeps track of all of the particles in the system, and therefore it accounts for the particle scarcity. The presence of several differential equations in the stochastic formulation is at the origin of the uncertainty, or stochastic error, in the process. The deterministic version of the model is unable to deal with the stochastic error, but as stated in Section 9.3.4, that is reduced to zero when the number of particles is very large. Only in this last case can the set of Kolmogorov differential equations be adequately approximated by the deterministic formulation, involving a set of differential equations of fixed size for the states of the process. 

It is important to note that the approximations made in the derivation of Eqs. (2) and (4) are questionable for very short times. As a result, it is expected that short time dynamics from LD and BD simulations are likely to contain significant errors. For example, the equation of motion for BD simulations (Eq. (4)) neglects inertial terms. Because of this, all motions are damped by the same friction coefficient and must scale exactly with the viscosity. This is, of course, inconsistent with the experimental observation that high frequency motions such as bond vibrations are largely independent of the molecular environment. Although LD simulations retain the inertial term, spatial and temporal correlations in the stochastic forces at short times are neglected. Even GLE simulations, which contain some of these correlations, would not be expected to be quantitatively correct at short times. 

According to GUM, a measurement has imperfections which give rise to errors in the measurement result. A random error presumably arises from unpredictable or stochastic temporal and spatial variations of influence quantities. Although it is not possible to compensate for random error, it can usually be reduced by increasing the number of observations. Systematic error, like random error, cannot be eliminated but it too can often be reduced. Once the effect causing the systematic error has been recognized, the effect can be quantifled and a correction can be applied to compensate for the effect. The uncertainty of the result of a measurement reflects the lack of exact knowledge of the value of the measurand. 

---