Predictive of a model

Once a model has been fitted to the available data and parameter estimates have been obtained, two further possible questions that the experimenter may pose are How important is a single parameter in modifying the prediction of a model in a certain region of independent variable space, say at a certain point in time and, moreover. How important is the numerical value of a specific observation in determining the estimated value of a particular parameter Although both questions fall within the domain of sensitivity analysis, in the following we shall address the first. The second question is addressed in Section 3.6 on optimal design. 

Figure 2.24 Comparison of experimental results of bubble period with predictions of a model involving different mechanisms (a) nucleate boiling only (b) nucleate boiling and natural convection (c) nucleate boiling, natural convection, and microlayer evaporation. (From Judd, 1989. Copyright 1989 by American Society of Mechanical Engineers, New York. Reprinted with permission.)...

Fig. 12. The rheological functions G ((o) and G"(co) for an H-shaped PI of arm molecular weigh 20 kg mol and backbone 110 kg mol" [46]. The high-frequency arm-retraction modes can be seen as the shoulder from co 10 to co 10 together with a low-frequency peak due to the cross-bar dynamics at co 10. The smooth curves are the predictions of a model which takes Eq. (33) as the basis for the arm-retraction times and a Doi-Edwards reptation spectrum with fluctuations for the backbone. The reptation time is correctly predicted, as is the spectrum from the arm modes, though the low frequency form is more polydisperse than the simple theory predicts...

Fig. 6.20 Small angle scattering intensity (triangles log I) and effective diffusion DgfKQ) obtained from g=A carbosiloxane dendrimers with perfluorinated end groups in perfluo-rohexane. The dashed line is a fit to the prediction of a model for shape fluctuations of micro-emulsion droplets, the resulting bending modulus was 0.5 k T. (Reprinted with permission from [308]. Copyright 2003 Springer Berlin Heidelberg New York)...

Figure 5.3. Left. The gamma-ray emission from XX annihilation in a rich, Coma-like, nearby galaxy cluster is shown Mx = 70 — 500 GeV (from top down). The integral flux is compared to the sensitivity of ongoing and planned gamma-ray experiments, as labelled. Right. The diffuse synchrotron emission spectrum of secondary electrons produced in XX annihilation is shown to fit the Coma radio-halo spectrum the green area represent the prediction of a model in which the x annihilates predominantly into fermions, while the blue area represent the gauge-boson dominated x annihilation (from Colafrancesco Mele 2001).

Degree of agreement between average predictions of a model or the average of measurements and the true value of the quantity being predicted or measured. Accuracy is also a criterion used to evaluate the knowledge base uncertainty. It focuses on the identification of the most important bottlenecks in the available knowledge and the determination of their impact on the quality of the result. 

A measure of the reproducibility of the predictions of a model or repeated measurements, usually in terms of the standard deviation or other measures of variation among such predictions or measurements. 

One restriction of QSAR is that in most cases a good predictivity of a model is limited to a congeneric series or a specific class of compounds, which... 

Appropriate measures of goodness-of-fit, robustness and predictivity. This principle expresses the need to provide two types of information the internal performance of a model (as expressed by goodness-of-fit and robustness) and the predictivity of a model using an appropriate test set. 

In order to estimate the predictive capabilities of a model by validation techniques, the data set can be split into different parts the training set (or learning set), the set of objects used for modelling, a test set, the set of objects used to optimize the goodness of prediction of a model obtained from the training set, and the external evaluation set (or evaluation set), which is a new data set used to perform further external validation of the model obtained from the training set. 

The most difficult step in building QSAR models is getting access to relevant descriptors and the selection of adequate descriptors. TSAR provides one of the best source of descriptors for chemists. Often the statistical tool is of less importance than the descriptors in the predictivity of a model. In practice, consensus models using a variety of properties and multivariate methods can often maximize usefulness. 

More recently, Pena and Miller investigated solubilzation rates of mixtures of n-decane and squalane into 2.5 wt% solutions of pure C,2Eg at 23°C using the oil drop method described above. They first measured the rate of solubilization of pure decane, confirming that the rate was controlled by interfacial phenomena as in Carroll s work, and demonstrated that pure squalane was not solubilized to any significant extent under these conditions. Next they measured solubilization rates of decane from various mixtures of the two hydrocarbons. Figure 9.7 shows results from one of these experiments together with predictions of a model based on assuming that the rate of decane... 

Fig. 19. XPS spectra of pure SmS and Gd substituted SmS. Curves (a) Experiment, curves (b) prediction of a model similar to that of Jaccarino and Walker curve (c) prediction of a random model assuming additive effects.

Fig. 10.2. Oxidation states of Cut (A) and Cu2(B) in Ba2YCu30. The squares and crosses represent observed values from two different studies. The lines are the predictions of a model based on a bond valence analysis of the crystallographic constraints...

The division into a training and test set can be driven by rational criteria such as training set diversity or optimum coverage of the target range [85,88-90] or simply by random. A more objective way to assess the predictivity of a model is fhe applicafion on a test set which was not used for model generation, often... 

Fig. 1. Carbon abundances as a function of values for a suite of 16 Apollo 16 soils, compared with predictions of a model based primarily on solar-wind implantation and proton stripping. The curves represent predictions for three trial values of the model parameter which is a rate constant incorporating both production and diffusive loss of methane. The numbered data points are for samples known from other evidence to be anomalous. The remaining data appear to be loosely consistent with the model predictions. From Kerridge et al (1974).

Mechanical Impedance of Muscle Tissue. The (input) mechanical impedance is the complex ratio between the dynamic force applied to the body and the velocity at the interface where vibration enters the body. The real and imaginaty parts of the mechanical impedance of human muscle in vivo are shown as a function of frequency in Fig. 10.3 (von Gierke et al., 1952). In this diagram the measured resistance (open circles) and reactance (diamonds) are compared with the predictions of a model, from which some tissue properties may be derived (see Table 10.1). It should be noted that... 

How does experimental error impact the apparent predictivity of a model ... 

Validation This is the process of comparing the predictions of a model, which has been run... 

Reflection of X-rays and neutrons from a surface responds to the variation of electron density and scattering-length density, respectively. Penetration depths can be several thousand angstroms and resolution 10A. Hence, in principle these techniques should be ideal for providing information on the nature, composition and size of polymer surfaces and interfaces. However, a direct description of the surface is not provided the predictions of a model have to be compared with literature data, and in many cases finding a unique solution may be difficult. Only neutron reflectometry is dealt with here, since this has been applied to a wider range of polymer systems than X-ray reflectometry. 

Moreover, the combination of T) and T, (T), ) is carried out by means of a weighted sum. This approach is justified by the fact that the model should be kept relatively simple and directly understandable by the operative personnel. In fact, operators would possibly trust more the risk prediction of a model they can understand than a black-box model based on complex data elaboration. However, this choice should be demonstrated by appropriate on-field tests. 

Controlled strain is the preferred mode of operation for nonlinear studies. In step-strain experiments, an important source of experimental error is the deviation of the actual strain history from a perfect step. Laun [96] and Venerus and Kahvand [43] have discussed this problem and how it can be addressed. Gevgilili and Kalyon [ 100] found that the actual strain pattern generated by a popular coimnerdal rheometer in response to a command for a step was, in fact, a rather complex function of time. One approach that is of use in comparing data from any transient test with the predictions of a model is to record the actual, non-ideal, strain history and use this same history to calculate the model predictions. 

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