Response surface correlation

Note, however, there are two critical limitations to these "predicting" procedures. First, the mathematical models must adequately fit the data. Correlation coefficients (R ), adjusted for degrees of freedom, of 0.8 or better are considered necessary for reliable prediction when using factorial designs. Second, no predictions outside the design space can be made confidently, because no data are available to warn of unexpectedly abrupt changes in direction of the response surface. The areas covered by Figures 8 and 9 officially violate this latter limitation, but because more detailed... 

The discrete factor solvent number is recognized as a simple bookkeeping designation. We can replace it with the continuous factor dipole moment expressed on a ratio scale and obtain, finally, the response surface shown in Figure 2.13. A special note of caution is in order. Even when data such as that shown in Figure 2.13 is obtained, the suspected property might not be responsible for the observed effect it may well be that a different, correlated property is the true cause (see Section 1.2 on masquerading factors). 

The covariance of the partition coefficients can be estimated by the correlation between the tangent planes of the response surfaces in a given mixture composition. This is explained in the next part of this paragraph. 

The correlation of the slopes of the tangent planes of the response surfaces of P, in P2 in the investigated mixture composition M is represented by r. 

If the tangent planes in M are more or less parallel then P, and P, are approximately equally affected by a variation in the mixture composition and the correlation of the response surfaces in M is high and the variance in the selectivity is small. If the tangent planes in M have opposite slopes then P, and Pj are affected in an opposite way by a variation in the mixture composition and the correlation of the response surfaces in Mwill be low and the variance in the selectivity high. The calculation of the correlation r (i.e. the parallelism of the response surfaces) is outlined below. 

In Figure 7.3, the partition coefficients, their ratio, the correlation of the response surfaces and the variances of the partition coefficients and the ratio are plotted for extraction of two compounds i and j into a binary extraction liquid. The regression coefficients / , P i for P and P2... 

As with scale-up, two levels of implementation are possible. The first level only entails the ability to sense, and a directional characterization of the effect of variables. PAT methods can be extremely effective for this purpose by generating large datasets of process inputs and outputs that can then be correlated to generate statistical or polynomial control models. Provided that (i) deviations from desired set-points are small, (ii) interactions between inputs are weak, and (Hi) the response surface does not depart too much from linearity, such systems can provide the basis of an initial effort to control a system. 

The research objective has been to define the durability of a coating depending on mixture composition Ni-Cr-B. Besides, one had to determine the optimal composition of the given three-component mixture. Since there is a linear correlation between resistance on wear-out and hardness of coating, Rockwell hardness (HRC) has been chosen as the system response. Based on preliminary information, it is known that the response surface is smooth and continuous. Hence, it may be... 

CONTENTS 1. Chemometrics and the Analytical Process. 2. Precision and Accuracy. 3. Evaluation of Precision and Accuracy. Comparison of Two Procedures. 4. Evaluation of Sources of Variation in Data. Analysis of Variance. 5. Calibration. 6. Reliability and Drift. 7. Sensitivity and Limit of Detection. 8. Selectivity and Specificity. 9. Information. 10. Costs. 11. The Time Constant. 12. Signals and Data. 13. Regression Methods. 14. Correlation Methods. 15. Signal Processing. 16. Response Surfaces and Models. 17. Exploration of Response Surfaces. 18. Optimization of Analytical Chemical Methods. 19. Optimization of Chromatographic Methods. 20. The Multivariate Approach. 21. Principal Components and Factor Analysis. 22. Clustering Techniques. 23. Supervised Pattern Recognition. 24. Decisions in the Analytical Laboratory. 

Examples of mathematical methods include nominal range sensitivity analysis (Cullen Frey, 1999) and differential sensitivity analysis (Hwang et al., 1997 Isukapalli et al., 2000). Examples of statistical sensitivity analysis methods include sample (Pearson) and rank (Spearman) correlation analysis (Edwards, 1976), sample and rank regression analysis (Iman Conover, 1979), analysis of variance (Neter et al., 1996), classification and regression tree (Breiman et al., 1984), response surface method (Khuri Cornell, 1987), Fourier amplitude sensitivity test (FAST) (Saltelli et al., 2000), mutual information index (Jelinek, 1970) and Sobol s indices (Sobol, 1993). Examples of graphical sensitivity analysis methods include scatter plots (Kleijnen Helton, 1999) and conditional sensitivity analysis (Frey et al., 2003). Further discussion of these methods is provided in Frey Patil (2002) and Frey et al. (2003, 2004). 

Disadvantages are that these response surface models are not available in standard software packages. Like all nonlinear statistical methods, the methodology is still subject to research, which has 2 important consequences. First, correlation structure of the parameters in these nonlinear models is usually not addressed. Second, the assessment of the test statistic is based on approximate statistical procedures. The statistical analyses can probably be improved through bootstrap analysis or permutation tests. 

Figure 5.9 shows the response surface that gives the correlation between the dependent variable y (or t ) and two independent factors (with values Zj (or t) and Z2 (or c) respectively). The problem of this example concerns a chemical reaction... 

Since OH(A Z) is efficiently quenched by N2 via a strongly attractive interaction [174], there is clearly at least one adiabatic surface correlating to OH(A Z) + N2(X i ) that does not have a barrier associated with the final separation of the OHIA Z) and N2(X Z) fragments. Also, there can be only one adiabatic surface for H( S) + N20(X Z ), and this correlates in C, symmetry to OH(X n) + N2(X S), albeit via a substantial barrier that can be traced to an avoided crossing [40]. Thus, access to OH(A i ) + N2(X S) is via a nonadiabatic process, and surface-hopping may be responsible for the observed A Z - X Z chemiluminescence. With complexes, the lack of OH chemiluminescence can be attributed to factors such as (i) specific approach... 

When the number of components or reactions is too large, or the mechanism is too complex to deduce with statistical certainty, then response surface models can be used instead. Methods for the statistical design of experiments can be applied, reducing the amount of experimental data that must be collected to form a statistically meaningful correlation of selectivity and yield to the main process parameters. See Montgomery (2001) for a good introduction to the statistical design of experiments. 

Hall, L.H. (1995). Experimental Design in Synthesis Planning and Structure-Property Correlations. Total Response Surface Optimization. In Chemometrics Methods in Molecular Design -Vol 2 (van de Waterbeemd, H., ed.), VCH Publishers, New York (NY), pp. 91-102. 

When a reaction has many reactive species (which may be the case even for apparently simple processes such as pyrolysis of ethane or synthesis of methanol), a factorial or sequential experimental design should be developed and the data can be subjected to a response surface analysis (Box, Hunter, and Hunter, Statistics for Experimenters, 2d ed., Wiley Interscience, 2005 Davies, Design ana Analysis of Industrial Experiments, Oliver Boyd, 1954). This can result in a black box correlation or statistical model, such as a quadratic (limited to first- and second-order effects) for the variables Xi, x2, and X3 ... 

When there are many response variables to consider, simultaneous evaluation of response surface models from each response becomes cumbersome. In such cases, a considerable simplification can often be achieved by multivariate analysis of the response matrix. For such purposes, principal components analysis and/or multivariate correlation by PLS are useful. These methods are discussed in Chapters 15 and 17. 

Another disadvantage of evaluating each response separately is that such an analysis ignores the fact that the different responses have been obtained from the same experiment. Correlations among the responses over the set of experiments can be assumed to reflect a similar dependency of the experimental variables. A problem is that colinearities among the responses are not easily detected when they are evaluated separately. A joint evaluation of several responses can be accomplished by overlaying the contour plots of their response surfaces. This is a useful technique when there are few responses to consider. With many responses this can be very difficult. 

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