Time series modeling prediction error method

Autocorrelation in data affects the accuracy of the charts developed based on the iid assumption. One way to reduce the impact of autocorrelation is to estimate the value of the observation from a model and compute the error between the measured and estimated values. The errors, also called residuals, are assumed to have a Normal distribution with zero mean. Consequently regular SPM charts such as Shewhart or CUSUM charts could be used on the residuals to monitor process behavior. This method relies on the existence of a process model that can predict the observations at each sampling time. Various techniques for empirical model development are presented in Chapter 4. The most popular modeling technique for SPM has been time series models [1, 202] outlined in Section 4.4, because they have been used extensively in the statistics community, but in reality any dynamic model could be used to estimate the observations. If a good process model is available, the prediction errors (residual) e k) = y k)—y k) can be used to monitor the process status. If the model provides accurate predictions, the residuals have a Normal distribution and are independently distributed with mean zero and constant variance (equal to the prediction error variance). 

For time-series data, the contiguous block method can provide a good assessment of the temporal stability of the model, whereas the Venetian blinds method can better assess nontemporal errors. For batch data, one can either specify custom subsets where each subset is assigned to a single batch (i.e., leave one batch out cross-validation), or use Venetian blinds or contiguous blocks to assess within-batch and between-batch prediction errors, respectively. For blocked data that contains replicates, one must be very careful with the Venetian blinds and contiguous block methods to select parameters such that the rephcate sample trap and the external subset traps, respectively, are avoided. 

Another method of predicting human pharmacokinetics is physiologically based pharmacokinetics (PB-PK). The normal pharmacokinetic approach is to try to fit the plasma concentration-time curve to a mathematical function with one, two or three compartments, which are really mathematical constructs necessary for curve fitting, and do not necessarily have any physiological correlates. In PB-PK, the model consists of a series of compartments that are taken to actually represent different tissues [75-77] (Fig. 6.3). In order to build the model it is necessary to know the size and perfusion rate of each tissue, the partition coefficient of the compound between each tissue and blood, and the rate of clearance of the compound in each tissue. Although different sources of errors in the models have been... 

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