Model refinement: Parameter reduction example

In this example, statistical analysis will be carried out on the surface reaction r.d.s.

model with product desorption (Eq. (3.6b)), using a methodology based on Quiney and Schuurman, (2007). The initial rate expressions, which describe the four reaction pathways detailed in Figure 3.1, are shown overleaf:

99 within k₁ - k₄) and four adsorption constants, one for each species. To begin analysis of this kinetic model, an initial parameter estimation pass is carried out using the multi-temperature data in a hexane solvent.

Results for this initial estimation routine are shown in Table 3.6. The R² for this model fit is 0.999. It is clear however from the estimated parameter values that the model is over parameterised, with evidence from the following observations:

 Two parameters, A2,373 and KCBL are indeterminate; this means the solver cannot reliable iterate towards a final value for that parameter.

 The fitted value of E_a4 is negative which is not feasible for an activation energy value.

 The critical t-value, tcrit for individual parameters is ±1.98. This is calculated assuming 108 degrees of freedom (120 observations – 12 parameters) at 95%

confidence (p = 0.05). Parameters A3,373, Ea4 and KPBL are below this threshold suggesting that they are statistically insignificant. In accordance with this, these three parameters all have 95% confidence intervals that are greater than their estimated value.

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Table 3.6: Estimated parameter values, confidence intervals and t-values from

non-linear least squares fitting of a 12 parameter surface reaction r.d.s model with product desorption

Parameter ^a Result Fitted Value 95%

Confidence Interval

t-value ^b

A1,373 Estimated 0.071 ± 0.0207 6.79

A2,373 Indeterminate 0.695 - -

A3,373 Estimated 0.084 ± 0.228 0.731

A_4,373 Estimated 0.044 ± 0.0352 2.50

Ea1 Estimated 33.6 ± 6.28 11.6

Ea₂ Estimated 47.0 ± 2.23 41.9

Ea3 Estimated 59.1 ± 30.1 3.89

Ea4 Estimated -6.7 ± 23.5 -0.564

KPBN Indeterminate 0.55 - -

K_PBL Estimated 8.23 ± 21.2 0.771

K_CBN Estimated 0.25 ± 0.199 2.52

KCBL Estimated 0.71 ± 0.0601 18.1

a Units are Ai,373 (mol L^-1 min^-1), Eai (kJ mol^-1), Ki (L mol^-1)

b tcrit (measurement for parameter significance) is ± 1.96 for this dataset.

Based on these observations, three parameters in particular (A_3,373, E_a4 and K_PBL) appear to have little influence on the model fitting. The same could be true for A_2,373 and K_PBN however the over parameterisation of this model may have resulted in the indeterminate result returned for these parameters; ergo, these parameters may still be significant at a later point in the model refinement process.

The final steps in quantifying the significance of all 12 model parameters is to address the sensitivity, B(t), of all model responses to model parameters as well as cross correlation between parameters themselves. This is achieved using the Jacobian Matrix for each experiment, which is a matrix of all first order partial derivatives of a vector or scalar function with respect to another vector (Caracotsios and Stewart, 1985):

101 denote correspondence to experiment i and parameter j respectively. This norm gives relative information on the influence of model parameters within the system (Quiney and Schuurman, 2007). By comparing these ‘lumped’ sensitivities, non-influential parameters can be removed systematically from the model.

Figure 3.7a shows the Jacobian norms for each parameter based over all four responses in the system. As Jacobian norms are relative values, a normalised form is shown here for clarity. Across the entire system, the most influential parameters appear to be A_2,373, E_a2 and K_PBN. This is not surprising as conversion of PBN to CBN (k₂ route) is the most selective and dominant route across the entire dataset and the parameters which predict this route will have strong influence. Parameter Ea4 appears to be most insignificant followed by the intermediate species hydrogenation pre-exponentials A_3,373 and A_4,373.

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Figure 3.7: A) Normalised Jacobian norms for all model responses with respect to model parameters and B) for individual responses (♦) PBN, (■) PBL, (▲) CBN, (●) CBL.

It is critical to view parameter significance in terms of individual model responses (Figure 3.7b). Model responses PBL and CBL, which are much smaller in magnitude across the dataset in comparison to PBN and CBN can therefore be examined with greater clarity.

In this case, A3,373 and A4,373 have a much greater impact on CBL response than the other species. Ea4 remains low in relative significance however.

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An extension of Jacobian Matrix analysis involves studying cross correlation of parameters; a further measure of over parameterisation in a model. This can be calculated for all parametric interactions using the following equation (Marquardt, 1963):





Where CC denotes cross correlation coefficient, which is always in the range -1 < CC

< +1, superscript T denotes matrix transpose and j1 and j2 are the two considered parameters respectively. CC values approaching -1 or +1 suggest a strong cross correlation between model parameters.

Table 3.7 shows the cross correlation coefficients for all 12 parameters during the first estimation routine. A2,373 and KPBN return a string of zero coefficient values as both are indeterminate. A3,373 and KPBL show a strong cross correlation suggesting their joint presence in the model may not be required, in particular in the r3 rate expression.

Table 3.7: Cross correlation matrix for 12 model parameters

A1,373 A2,373 A3,373 A4,373 Ea1 Ea2 Ea3 Ea4 KPBN KPBL KCBN KCBL

At this point of analysis, the kineticist may make any of the following four decisions in order to attempt to improve the model:

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 Fully or partially remove a parameter from the rate expression(s). This may involve setting an E_a value to zero or the removal of Kterms from some or all of the rate expressions.

 Equate two or more model parameters, for example: two reactions may share a common reaction mechanism and equating their Ea values may be a viable step.

 Fix the value of a parameter; for example, a model may contain n as a fitting parameter which subsequently tends towards logical mechanistic values during estimation (such as 0.5 or 1) and therefore can be fixed. Also, activation energies or heats of adsorption may be fixed based on previous literature study, a common practice in micro-kinetic analysis works such as those by Dumesic and co-workers (Dumesic and Trevino, 1989).

 Make no further changes.

Consideration of the real physical and chemical implications of model parameter reduction is critical during this process. From the initial estimation process, Ea4 appears to be a weak parameter showing low sensitivity values, an estimated value lacking physical meaning, a wide confidence interval and a low t value. Setting this value to zero would make the assumption that hydrogenation of the ketone group of CBN is a fast process with no temperature dependency. This is unlikely as ketone hydrogenation is not a selective reaction in the conditions tested and C=O bond hydrogenation will have an energy barrier that constitutes a rate determining step (Chang et al., 2000; Vargas et al., 2008). Equating Ea4 to Ea1 is more logical: both describe the activation energy for ketone hydrogenation of similar species PBN and CBN.

The next step is to rerun the parameter estimation routine with 11 fitting parameters, following equating Ea1 and Ea4. This process will generate a new set of fitted parameters, sensitivities, cross correlation coefficients and residuals. A procedure known as the F-test is then invoked which addresses whether the change in residuals of model responses in the 11 parameter model in comparison to the 12 parameter one is statistically significant. This is often defined as a ‘nested model’ problem and the F-statistic can be calculated as follows:

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Where F is the F-statistic, RSS1 and RSS2 are the residual sum of squares in the nested and original model respective, par₁ and par₂ are number of parameters and obs is the total number of observations. In this study (par₂ – par₁) is always equal to 1 as parameters are removed on an individual basis. The F generated is compared to F_crit (p = 0.05) under these constraints. If F is smaller than F_crit, the removal, equating or fixing of a parameter is deemed acceptable as a statistically significant increase in residuals has not been induced.

Figure 3.8a shows the values of F calculated for the successive removal of parameters in the model. In terms of the entire system response, seven parameters can be removed from the fitting procedure without a statistically significant effect. Individual model responses were also examined in Figure 3.8b and largely demonstrate the same result. The Fcrit value is exceeded once for CBL and PBL respectively, however the residuals of both responses improve in subsequent parameter removals, correcting this statistically significant shift. The F value response is significant on removal of the eighth parameter however, in particular for the CBL response. Further removals, which would reduce the entire model to 3 or less parameters, amplify this effect further.

Table 3.8 documents the parameter reduction process from 12 down to 4 parameters in the entire model. The reduction contains a number of steps where an activation energy or pre-exponential is equated with another. In each case this involved ‘pairing’ of reaction pathways which have the same mechanism: ketone hydrogenation (r₁ with r₄) and aromatic ring hydrogenation (r₂ with r₃). Adsorption parameters K_PBN, K_PBL and K_CBL were all removed at various stages leaving the parameter for the selective product, K_CBN.

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Figure 3.8: A) Calculated F-statistic for successive parameter removal across the entire system response and B) with respect to individual model responses (♦) PBN, (■) PBL,

(▲) CBN, (●) CBL. N.B.: Black line denotes Fcrit

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Table 3.8: Parameters removed in surface reaction limited model with product desorption

No. parameters in model

Most recent parameter removed

RSS (x 10⁴)

Reason for parameter removal ^a

12 - 6.94 -

11 Ea4 6.95  Negative, poor confidence

 Equated to Ea1: both are ketone hydrogenation steps

10 A_4,373 7.06  Equated to A1,373: both are available

sites for ketone hydrogenation

9 KPBL 7.20  Indeterminate

8 Ea3 7.24  Poor confidence, similar value to Ea2

 Equated to Ea2: both are aromatic ring hydrogenation steps

7 A_3,373 7.25  Equated to A_2,373: both are available

sites for aromatic ring hydrogenation

6 K_CBL 7.41  Lowest Jacobian norm only

5 K_PBN 7.38  Strong cross-correlation with A2,373

4 KCBN 8.38  Statistically significant, should not be removed.

a In addition to showing the lowest Jacobian norm