Section inverse models/modeling

In Section 12.3.2, the fundamental differences between direct and inverse modeling methods were discussed. As will be discussed here, this distinction is not just a convenient means for classifying quantitative regression methods, but has profound implications regarding calibration strategy and supporting infrastructure. 

In maiK- applications, the spectral residuals will not behave in the ideal manner as picted here. Some nonrandom behavior may be tolerable depending on the performance requirements of the model. If the residuals are lai e and fire model performance is not acceptable, an inverse model approach sudias PLS or PCR (Section 5.3) can be considered. 

The maiber of samples that need to be measured in the calibration phase is smiS as compared to inverse modeling (Section 5.3). The minimum number of mistures to measure is equal to the number of analytes in the system. 

A commonly perceived weakness of PCR/PLS is that it usually takes many samples to comstruct rhe model. This can be tnie if relying on natural designs (see Chapter 2 and Appendix A). It is tnie that, in general, more samples are required to build a PLS model than a CLS model (see Section 5-2). However, this is because the inverse models are typically correcting for effects that cannot be modeled using the classical methods. The perception that hundreds of samples are always required to build inverse models is simply not true. One rule of thumb is that there should be at least three times as many samples as factors. 

Experimental planning concepts are emphasized in Chapter 2 because it is ver> important to carefully consider the variables that affect the data before they are collected. The concepts taught in Chapter 2 are most appropriate for classical regression modeling. When using inverse modeling methods in chemometrics. a modification to the classical approach is required (see Section 5.5 for details on inverse models). To explain why this is necessary, the standard regression model shown in Equation A.1 is discussed. 

In this particular example, the salt and temperature information was recorded, however with the inverse modeling approach, the values are not used in the computation of the PLS model. One might be tempted to want to account for these variables by including them as additional columns of R. However, this is not necessary, because the effects of these variables are already captured by the spectra. Complimenting the R matrix with variables related to or correlated with the c vector may be helpful if that correlation is different from what is already in R. This is in contrast to a more classical approach for analyzing these same data, discussed in Section 5.2.2.2. 

This is an acceptable model performance based on the requirements of the application. The inverse model accounts for tlie eifects of salt and temperature on the water peak without explicitly using the salt or temperature infonnation in the calculations. This is in contrast to the ICLS analysis of these same data where the concentrations and temperatures of the calibration data are required to obtain a satisfactory model (Section 3.2.2.2) ... 

It is an inverse model with the associated implicit modeling capability as discussed in Section 5.3. 

Tracer behavior is different in each of the ideal chemical reactors, so the first step in developing a tracer kinetics model is to decide which reactor type best simulates the real situation and which kind of tracer dosing has occurred. If the amount and timing of tracer introduction is known, a forward model can be developed. If the amount and timing of tracer detected is known, an inverse model can be produced. The equations derived in the following section use concentration imits of mg/L because these units are typical of field studies. For laboratory experiments the models would use molal concentration units. The mass (M) variables would be replaced by mole quantities ( ). The volume variables (V) would be replaced by mass of water (M) and the flow rate (0 would have units of kg/sec. 

In subsequent sections, the notions direct model and inverse model will be used. 

So far, parameter sensitivities of transfer functions of direct models have been considered. This section presents an incremental bond graph-based procedure to the symbolic determination of parameter sensitivities of transfer functions of linear inverse models given that the latter exist. 

In the following, the results (4.74) and (4.75) obtained by symbolic differentiation will be derived from the incremental bond graph of the inverse model (Fig. 4.23) by building the matrices used in the approach presented in the previous section. 

One of the main characteristics of an inverse model is the presence of the output derivatives in the equations (vector y (t)). This will be discussed in more detail in the following sections. In particular, in structural analysis, the notion of essential orders enables the necessary minimal number of output time differentiations to be anticipated before the constmction of the inverse model. This notion will be translated into the bond graph language. 

The key principle for obtaining the inverse model from the direct one is to successively differentiate the outputs with respect to time until the inputs appear in the expression of the output derivatives. Then, from this transformation of the model, the aim is to express the inputs in terms of the outputs by inverting these equations if possible. The condition for the existence of this inversion will also be discussed in the following sections. 

The relative order indicates that the output ji will appear with a time derivative of order n[ at least in the inverse model. Depending on the model, this derivative order can be higher and then defined by the essential order of this output (see Section 6.2.2.3). It can be shown that the difference between the relative order and the essential order is related to the notion of the dynamic extension that must be introduced... 

Three definitions are given about the power line concepts. A power line characterizes the way energy flows between two points in a system. So, talking about inverse models (here implicitly between the inputs and the outputs), the input/output (I/O) power line concept is defined. Finally, the invertibility criteria presented in Section... 

This section defines the criteria that will be used in the bond graph sizing methodology based on model inversion. Then bicausality is presented as a tool for determining the inverse model directly from a bond graph representation. As seen in Procedure 4, bicausality also enables the essential orders to be determined. 

Criterion 4 (Specification differentiability) In order to simulate an inverse model each output specification must have a time differentiation order greater than or equal to nie, the essential order (Section 6.2.23 and Procedure 4) of the corresponding specified output yt in the system H. 

In the nemal network-based adaptive control scheme, a neurocontroller is trained to approximate an inverse model of the plant. We have introduced an adaptive activation function for increasing the training rate of the neural controller, and the proposed function is described in this section. 

Similar discrepancies as documented in the previous section for macroscopic data can be found for the spectroscopic approaches, which are now available for studying the structure of surface complexes in situ (i.e., wet samples). With respect to inverse modeling these studies would attempt to resolve the stmcture of the surface complexes in a certain system and to impose such structures in the surface complexation model. This would avoid extensive discussion about the mode of bonding (e.g., inner versus outer sphere monodentate versus polydentate). 

For this purpose, the inverse modeling approach described in Section 5.2 allows the crystal grower in a very comfortable way to obtain the required information. By inverse modeling the heater temperature-time profiles can be optimized in order... 

The general problems of forward and inverse modelling were discussed in Section 5 and the problem of integrating production data is, in principle, of the same t3rpe. However, in practice the large computer times needed to perform flow simulations and the strongly time dependent natme of the data to be processed makes the problem very difficult, compared to the inversion of small scale measurements. 

This paper is structured as follows in section 2, we recall the statement of the forward problem. We remind the numerical model which relates the contrast function with the observed data. Then, we compare the measurements performed with the experimental probe with predictive data which come from the model. This comparison is used, firstly, to validate the forward problem. In section 4, the solution of the associated inverse problem is described through a Bayesian approach. We derive, in particular, an appropriate criteria which must be optimized in order to reconstruct simulated flaws. Some results of flaw reconstructions from simulated data are presented. These results confirm the capability of the inversion method. The section 5 ends with giving some tasks we have already thought of. 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

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