Process Uncertainty or Stochastic Error

we find that the mean behavior of the stochastic model is described by the deterministic model we have already developed. The fundamental difference between the stochastic and the deterministic model arises from the chance mechanism in the stochastic model that generates so-called process uncertainty, or stochastic error. 

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 

CV varies as j JnA and it is not a small number for dosages involving few particles or drugs administered at very low doses otherwise, CV C 1, as is typical in pharmacokinetics [370,371]. From a mechanistic point of view, if the number of molecules present is not large, the concentration as a function of time will show the random fluctuations we expect from chance occurrences. However, if the number is very large, these fluctuations will be negligible, and for purposes of estimation, the stochastic error may be omitted in comparison with the measurement error. 

An important generalization concerns the multinomial distribution of observations at different times. To deal with this, we analyze in the Markovian context the prediction of the statistical behavior of particles at time t + t° based on the observations at t, i.e., the state about the conditional random variable [KAt + t°) ni (t)]. As previously, in common use is the multinomial distribution 

The expressions (9.24) and (9.25) correspond to (9.22) and (9.23), respectively. The latter expressions can be obtained from the former ones by substituting t by 0 and t° by t. Since N (t +1°) is conditioned to the random n (t), the total expectation theorem leads unconditionally to (cf. Appendix D) 

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