Statistics basic error model

At a later stage, the basic model was extended to comprise several organic substrates. An example of the data fitting is provided by Figure 8.11, which shows a very good description of the data. The parameter estimation statistics (errors of the parameters and correlations of the parameters) were on an acceptable level. The model gave a logical description of aU the experimentally recorded phenomena. 

All regression methods aim at the minimization of residuals, for instance minimization of the sum of the squared residuals. It is essential to focus on minimal prediction errors for new cases—the test set—but not (only) for the calibration set from which the model has been created. It is relatively easy to create a model— especially with many variables and eventually nonlinear features—that very well fits the calibration data however, it may be useless for new cases. This effect of overfitting is a crucial topic in model creation. Definition of appropriate criteria for the performance of regression models is not trivial. About a dozen different criteria— sometimes under different names—are used in chemometrics, and some others are waiting in the statistical literature for being detected by chemometricians a basic treatment of the criteria and the methods how to estimate them is given in Section 4.2. 

In such statistically designed experiments one wants to exclude the random effects of a limited number of features by varying them systematically, i.e. by variation of the so-called factors. At the same time the order in which the experiments are performed should be randomized to avoid systematic errors in experimentation. In another basic type of experiment, sequential experiments, the set-up of an experiment depends on the results obtained from previous experiments. For help in deciding which design is preferable, see Section 3.6. In principle, statistical design is one recommendation of how to perform the experiments. The design should always be based on an exact question or on a working hypothesis. These in turn are often based on models. 

Step 8 Measuring Results and Monitoring Performance The evaluation of MPC system performance is not easy, and widely accepted metrics and monitoring strategies are not available, ffow-ever, useful diagnostic information is provided by basic statistics such as the means and standard deviations for both measured variables and calculated quantities, such as control errors and model residuals. Another useful statistic is the relative amount of time that an input is saturated or a constraint is violated, expressed as a percentage of the total time the MPC system is in service. 

An estimator (or more specifically an optimal state estimator ) in this usage is an algorithm for obtaining approximate values of process variables which cannot be directly measured. It does this by using knowledge of the system and measurement dynamics, assumed statistics of measurement noise, and initial condition information to deduce a minimum error state estimate. The basic algorithm is usually some version of the Kalman filter.14 In extremely simple terms, a stochastic process model is compared to known process measurements, the difference is minimized in a least-squares sense, and then the model values are used for unmeasurable quantities. Estimators have been tested on a variety of processes, including mycelial fermentation and fed-batch penicillin production,13 and baker s yeast fermentation.15 The... 

The basic principle of experimental design is to vary all factors concomitantly according to a randomised and balanced design, and to evaluate the results by multivariate analysis techniques, such as multiple linear regression or partial least squares. It is essential to check by diagnostic methods that the applied statistical model appropriately describes the experimental data. Unacceptably poor fit indicates experimental errors or that another model should be applied. If a more complicated model is needed, it is often necessary to add further experimental runs to correctly resolve such a model. 

Even though many different covariates may be collected in an experiment, it may not be desirable to enter all these in a multiple regression model. First, not all covariates may be statistically significant—they have no predictive power. Second, a model with too many covariates produces models that have variances, e.g., standard errors, residual errors, etc., that are larger than simpler models. On the other hand, too few covariates lead to models with biased parameter estimates, mean square error, and predictive capabilities. As previously stated, model selection should follow Occam s razor, which basically states the simpler model is always chosen over more complex models. ... 

RMC is a variation of the standard Metropolis Monte Carlo (MMC) method (Metropolis et al., 1953 see also Chapters l and 5). The principle is that we wish to generate an ensemble of atoms, i.e. a structural model, which corresponds to a total structure factor (set of experimental data) within its errors. These are assumed to be purely statistical and to have a normal distribution. Usually the level and distribution of statistical errors in the data is not a problem, but systematic errors can be. We shall initially consider materials that are macro-scopically isotropic and that have no long range order, i.e. glasses, liquids and gases. The basic algorithm, as applied to a monatomic system with a single set of experimental data, is as follows ... 

Experimental design is a large topic and we can only mention several of the important issues here. To keep this discussion focused on parameter estimation for reactor models, we must assume the. reader has had exposure to a course in basic statistics [4]. We assume the reader understands the source of experimental error or noise, and knows the difference between correlation and causation. The process of estimating parameters in reactor models is part of the classic, iterative scientific method hypothesize, collect experimental data, compare data and model predictions, modify hypothesis, and repeat. The goal of experimental design is to make this iterative learning process efficient. 

Among the mathematical models of industrial chemical processes, the system of balance equations is basic. If some measured data do not satisfy the balance equations, this fact is attributed to measurement errors, and not perhaps to an inadequate description of the process by the model. In practice, measurement errors are always present. Hence before using the measured data, they are adjusted to obey the balance constraints. The adjustment by methods using statistical theory of errors is called reconciliation. 

Some basic statistical requirements were also overlooked when quantitative models based on coefficients were developed. The most important assumption underlying the linear regression model is that the dependent variable contains all the errors in each data pair (Maes 1984 Irvin and Quickenden 1983). The range for 1-octanol/water partition coefficients obtained for each molecule clearly demonstrates that this basic assumption is not met. It has been demonstrated (York 1966) that if this basic assumption is violated, the fitted slopes can deviate by as much as 40% from the correct value. Thus, the validity and applicability of published quantitative models describing the relationship between the Kq coefficients and soil sorption coefficients are highly questionable. 

The validity of most of the assumptions involved in this model has not been tested rigorously. Furthermore, as with other micellar kinetic models, the success of this model is basically claimed on the basis of fairly low residual errors between experimentally determined rate constants (ko,s) and calculated rate constants (kcaicd) in terms of this model. Such a satisfactory statistical fit of observed data to such a micellar kinetic model is necessary for the apparent success of the model but is not sufficient to guarantee the reliability of the values of the calculated parameters from this model. Such a problem has been encountered in both PIE and MA-SB models. ... 

---