Correlation coefficient, predictive model

As discussed in other chapters of this book, two-phase flows of gas and particles occur with different flow regimes. The mechanisms for heat transfer and the resulting heat transfer coefficients are strongly affected by the different flow characteristics, resulting in different design correlations and predictive models for each flow regime. This chapter will deal with the two most often encountered flow regimes ... 

FIGURE 4.2 Linear combinations of the x-variables (data set in Table 4.2) are useful for the prediction of property y. For the left plot, xh x2, and x3 have been used to create an OLS-model for y, Equation 4.2 for the right plot x and x2 have been used for the model, Equation 4.4. R2 is the squared Pearson correlation coefficient. Both models are very similar the noise variable x3 does not deteriorate the model. 

The idea behind this approach is simple. First, we compose the characteristic vector from all the descriptors we can compute. Then, we define the maximum length of the optimal subset, i.e., the input vector we shall actually use during modeling. As is mentioned in Section 9.7, there is always some threshold beyond which an inaease in the dimensionality of the input vector decreases the predictive power of the model. Note that the correlation coefficient will always be improved with an increase in the input vector dimensionality. 

Multiple linear regression analysis is a widely used method, in this case assuming that a linear relationship exists between solubility and the 18 input variables. The multilinear regression analy.si.s was performed by the SPSS program [30]. The training set was used to build a model, and the test set was used for the prediction of solubility. The MLRA model provided, for the training set, a correlation coefficient r = 0.92 and a standard deviation of, s = 0,78, and for the test set, r = 0.94 and s = 0.68. 

Example 8 Calculation of Rate-Based Distillation The separation of 655 lb mol/h of a bubble-point mixture of 16 mol % toluene, 9.5 mol % methanol, 53.3 mol % styrene, and 21.2 mol % ethylbenzene is to be earned out in a 9.84-ft diameter sieve-tray column having 40 sieve trays with 2-inch high weirs and on 24-inch tray spacing. The column is equipped with a total condenser and a partial reboiler. The feed wiU enter the column on the 21st tray from the top, where the column pressure will be 93 kPa, The bottom-tray pressure is 101 kPa and the top-tray pressure is 86 kPa. The distillate rate wiU be set at 167 lb mol/h in an attempt to obtain a sharp separation between toluene-methanol, which will tend to accumulate in the distillate, and styrene and ethylbenzene. A reflux ratio of 4.8 wiU be used. Plug flow of vapor and complete mixing of liquid wiU be assumed on each tray. K values will be computed from the UNIFAC activity-coefficient method and the Chan-Fair correlation will be used to estimate mass-transfer coefficients. Predict, with a rate-based model, the separation that will be achieved and back-calciilate from the computed tray compositions, the component vapor-phase Miirphree-tray efficiencies. 

Bubble size in the circulating beds increases with Ug, but decreases with Ul or solid circulation rate (Gs) bubble rising velocity increases with Ug or Ul but decreases with Gs the ffequeney of bubbles increases with Ug, Ul or Gs. The axial or radial dispersion coefficient of liquid phase (Dz or Dr) has been determined by using steady or unsteady state dispersion model. The values of Dz and D, increase with increasing Ug or Gs, but decrease (slightly) with increasing Ul- The values of Dz and Dr can be predicted by Eqs.(9) and (10) with a correlation coefficient of 0.93 and 0.95, respectively[10]. 

In QSAR equations, n is the number of data points, r is the correlation coefficient between observed values of the dependent and the values predicted from the equation, is the square of the correlation coefficient and represents the goodness of fit, is the cross-validated (a measure of the quality of the QSAR model), and s is the standard deviation. The cross-validated (q ) is obtained by using leave-one-out (LOO) procedure [33]. Q is the quality factor (quality ratio), where Q = r/s. Chance correlation, due to the excessive number of parameters (which increases the r and s values also), can. 

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 mechanistic simulation ACAT model was modified to account automatically for the change in small intestinal and colon k as a function of the local (pH-dependent) log D of the drug molecule. The rank order of %HIA from GastroPlus was directly compared with rank order experimental %HIA with this correction for the log D of each molecule in each of the pH environments of the small intestine. A significant Spearman rank correlation coefficient for the mechanistic simulation-based method of 0.58 (p < 0.001) was found. The mechanistic simulation produced 71% of %HIA predictions within 25% of the experimental values. 

Figure 4.14. Predictions of the multi-variate SR model for Re, = 90 and Sc = (1, 1/8) with collinear mean scalar gradients and no backscatter (cb = 0). For these initial conditions, the scalars are uncorrelated pap(0) = gap(0) = 0. The correlation coefficient for the dissipation range, pD, is included for comparison with pap. 

In order to illustrate how the multi-variate SR model works, we consider a case with constant Re>. = 90 and Schmidt number pair Sc = (1, 1/8). If we assume that the scalar fields are initially uncorrelated (i.e., pup 0) = 0), then the model can be used to predict the transient behavior of the correlation coefficients (e.g., pap(i)). Plots of the correlation coefficients without (cb = 0) and with backscatter (Cb = 1) are shown in Figs. 4.14 and 4.15, respectively. As expected from (3.183), the scalar-gradient correlation coefficient gap(t) approaches l/yap = 0.629 for large t in both figures. On the other hand, the steady-state value of scalar correlation pap depends on the value of Cb. For the case with no backscatter, the effects of differential diffusion are confined to the small scales (i.e., (), / h and s)d) and, because these scales contain a relatively small amount of the scalar energy, the steady-state value of pap is close to unity. In contrast, for the case with backscatter, de-correlation is transported back to the large scales, resulting in a lower steady-state value for p p. 

Typical model predictions without and with backscatter are shown in Figs. 4.16 and 4.17, respectively. It can be noted that for decaying scalars the effect of backscatter on de-correlation is dramatic. For the case without backscatter (Fig. 4.16), after a short transient period the correlation coefficients all approach steady-state values. In contrast, when backscatter is included (Fig. 4.17), the correlation coefficients slowly approach zero. The rate of long-time de-correlation in the multi-variate SR model is thus proportional to the backscatter constant Cb-... 

Note Heating value in kJ/kg, others in mass %. The squared Pearson correlation coefficients, between experimental values and predicted values from leave-one-out CV and the standard error of prediction from leave-one-out CV (SEPCV, see Section 4.2.3) are given for a joint PLS2 model, and for separate PLS models developed for each variable seperately using the optimal number of components opt f°r each model. 

Roberts et al. criticized the attempts to predict permeabilities since permeability is the result of two processes, partitioning and diffusion [40], Therefore, instead of following the approach of Potts and Guy, Roberts et al. tried to find a predictive model for each of these processes separately. For the partitioning step they found a Collander-type linear relationship (Eq. 11) between the logarithms of the stratum corneum-water and the octanol-water partition coefficients with a high correlation coefficient (r2 = 0.839) ... 

Figure 4.1 Correlation of predicted and observed retention times in reversed-phase chromatography. The predicted retention times for 58 peptides of 2 to 16 residues in length were obtained by summation of retention coefficients for each residue in the peptide. Retention coefficients were determined from the retention of model synthetic peptides with the structure Ac-Gly-XX-(Leu)3-(Lys)2-amide, where X was substituted by the 20 protein amino acids. (Reproduced from D. Guo, C.T. Mant, A.K. Taneja, and R.S. Hodges, J. Chromatogr., 359 519 [1986]. With permission from Elsevier Science.)...

Much of the variation in absolute quantity of an element was shown to be due to variation in dust concentration. The correlation coefficients of linear regression models predicting the concentration of calcium and silicon from the vertical elutriator dust concentration were 0.89 and 0.79, respectively. 

Gifford and Hanna tested their simple box model for particulate matter and sulfur dioxide predictions for annual or seasonal averages against diffusion-model predictions. Their conclusions are summarized in Table 5-3. The correlation coefficient of observed concentrations versus calculated concentrations is generally higher for the simple model than for the detailed model. Hanna calculated reactions over a 6-h period on September 30, 1%9, with his chemically reactive adaptation of the simple dispersion model. He obtained correlation coefficients of observed and calculated concentrations as follows nitric oxide, 0.97 nitrogen dioxide, 0.05 and rhc, 0.55. He found a correlation coefficient of 0.48 of observed ozone concentration with an ozone predictor derived from a simple model, but he pointed out that the local inverse wind speed had a correlation of 0.66 with ozone concentration. He derived a critical wind speed formula to define a speed below which ozone prediction will be a problem with the simple model. Further performance of the simple box model compared with more detailed models is discussed later. 

In addition to the temporal correlation coefficient, the spatial correlation coefficient was calculated approximately for fixed values of time. Except for one of the mathematical models, all techniques showed a better temporal correlation than spatial correlation. The two correlation coefficients are cross plotted in Figure 5-6. Nappo stressed that correlation coefficients express fidelity in predicting tends, rather than accuracy in absolute concentration predictions. Another measure is used for assessing accuracy in predicting concentrations the ratio of predicted to observed concentration. Nappo averaged this ratio over space and over time and extracted the standard deviation of the data sample for each. The standard deviation expresses consistency of accuracy for each model. For example, a model might have a predicted observed ratio near unity,... 

Model Ref. Average Temporal Correlation Coefficient Computer Time for 24-h Prediction, min Computer Cost for 24-h Prediction, ... 

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