A2 Orthogonalization of Parameters to Remove Correlation

If all else fails, serious partial correlation of parameters can be removed by orthogonalization of the parameter space. This orthogonalization method is general. In principle, it should be applicable to a wide variety

Table 4.3 Results of Model in Eq. (3.6) Fit onto Steady State Voltammetry Data with Background with Different Initial Parameters from Those in Table 3.7

Parameter/statistic

Initial value

True value"

Final value

Error

0.9

1.000

0.984

1.6%

b2

39

38.92

39.66

2.0%

bi

-0.2

-0.200

-0.1996

0.2%

b4

-0.4

-0.200

-0.264

32%

b5

0.005

0.005

0.0053

6%

SD

0.0033

Deviation plot

Random

" Data generated by using eq. (3.6) with absolute normally distributed noise at 0.5% of maximum y.

" Data generated by using eq. (3.6) with absolute normally distributed noise at 0.5% of maximum y.

of models. However, it is not successful for models with fully correlated parameters.

Recall that in nonlinear regression the error sum S is minimized by systematic iterative variation of the parameters by an appropriate algorithm (Section 2.B.2). The approach to minimum S starting from a set of initial guesses for m parameters can be thought of as the journey of the initial point toward the global minimum of an error surface in an (m + 1)-dimensional coordinate system called parameter space. For a two-parameter model, we have a three-dimensional coordinate system. The z axis of this parameter space corresponds to S. The x and y axes correspond to the two parameters, respectively. The minimum S is called the convergence point (cf. Figure 2.4).

If correlation between two parameters is strong, convergence of the regression analysis or unique estimates of each parameter can be difficult to achieve. Parameter correlation is equivalent to having nonorthogonal axes in the parameter space (Figure 4.1). As an example of orthogonaliza-tion, we discuss parameter correlation that arises in models for obtaining rate constants of chemical reactions coupled to electrode reactions using a technique called chronocoulometry.

In chronocoulometry, a step of potential is applied to a working electrode, and the accumulated charge passing through the electrochemical cell

is recorded vs. time. Models describing charge response in chronocoulome-try can be expressed as explicit functions of time. Here, we summarize the application of the Gram-Schmidt orthogonalization to chronocoulometry of an electrode reaction occurring by an electron transfer-chemical reaction-electron transfer, or ECE, pathway [2]. We will return to the chemistry of ECE reactions in Chapter 12. At present we need to focus on only the form of the model.

The initial potential (£■, ) of the working electrode is chosen so that no electrolysis occurs. At time t = 0, the potential is rapidly pulsed to a value where the first electron transfer is so fast that its rate does not influence the shape of the charge-time response. Under these conditions, the response of the experiment Q(t) for the ECE model is of the form

Q(t) = b2 + b,[2- (7T/4b0t)ia erf (V)172] + b3t. (4.5)

The b0, ...,b3 are regression parameters. The rate constant for the chemical reaction is k = b0. This kinetic constant is generally the required parameter in this type of analysis. Identities of the other parameters are not relevant to our present discussion.

Gram-Schmidt orthogonalization was used to transform eq. (4.5) to the form yj = BMtj) + B2h2(tj) + B3h,(tj; B0) (4.6)

where the h, (tj) are orthonormal functions, the B, are a new set of parameters that depend on the original b0, ..., b3, and B0 = k. The details of the mathematical manipulations follow.

Orthogonalization defines a new orthogonal parameter space. Here, the most desired parameter is the rate constant B0, which is determined by the regression analysis. The other original parameters b, are computed from the Bt estimated in the regression analysis.

Gram-Schmidt orthogonalization of an ECE reaction model in single potential step chronocoulometry removed correlations between parameters and greatly improved the speed of convergence when using the MarquardtLevenberg nonlinear regression algorithm [2]. The orthogonalized ECE model was used to estimate chemical rate constants for decomposition of unstable anion radicals produced during the reduction of aryl halides in organic solvents. The orthogonalized model converged three to four times faster than the nonorthogonal model.

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