Principal Component Analysis of the Sensitivity Matrix

The accuracy of the decomposition in equation (5.12) depends on the accuracy of the determination of the period sensitivity vector s. Several methods have been proposed for the accurate calculation of s, including the application of an approximate term (Edelson and Thomas 1981), a more accurate integral method (Larter 1983) and the singular value decomposition (SVD) of the original sensitivity matrix S (Zak et al. 2005). 

Elements of the local concentration sensitivity matrix show the effect of changing a single parameter on the calculated concentration of a species. However, we are frequently interested in the effect of parameter changes on the concentrations of a group of species. This effect is indicated by the overall sensitivity measure (Vajda et al. 1985)  

Quantity Bj shows the effect of changing parameter Xj on the concentration of all species present in the summation at time t (Whitehouse et al. 2004a, b). The utilisation of such overall sensitivity measures for identifying unimportant species or reactions as part of the model reduction process will be discussed further in Sect. 7.2.3. 

If a group of species is important for us in the time interval [fj, f2] (e.g. the concentrations of these species can be measured in this interval), we may be interested in which parameters or groups of parameters are highly influential on the measured concentrations. To answer this question, the following scalar valued function, called the objective function, will be used  

The principal component analysis of matrix S (PCAS) (Vajda et al. 1985) investigates the effect of the change in parameters on the value of the objective function. The objective function Q can be approximated (Vajda et al. 1985) using the local sensitivity matrix S  

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Practical identifiability is not the only problem that can be adressed by principal component analysis of the sensitivity matrix. In (refs. 29-30) several examples of model reduction based on this technique are discussed. 

From the redundant species analysis it is clear that all reactions which consume H2O2 and O3 are redundant and can be removed automatically from the mechanism. In order to identify other redundant reactions the techniques of rate sensitivity analysis coupled with a principal component analysis of the resulting matrix can be used. The principal component analysis of the rate sensitivity matrix containing only the remaining important and necessary species will reveal the important reactions leading to reduced mechanisms applicable at various ambient temperatures. In principle it may be possible to produce a reduced scheme which models non-isothermal behaviour from analysis carried out on an isothermal model. An isothermal system is easier to model since thermodynamic and heat-transfer properties can be excluded from the calculations. However,... 

As seen from Fig. 5.3, the substrate concentration is most sensitive to the parameters around t = 7 hours. It is therefore advantageous to select more observation points in this region when designing identification experiments (see Section 3.10.2). The sensitivity functions, especially with respect to Ks and Kd, seem to be proportional to each other, and the near—linear dependence of the columns in the Jacobian matrix may lead to ill-conditioned parameter estimation problem. Principal component analysis of the matrix STS is a powerful help in uncovering such parameter dependences. The approach will be discussed in Section 5.8.1. 

Principal component analysis of the normalized sensitivity matrix both concentrations observed... 

For a long time the main topic of research in the area of sensitivity analysis was to find an accurate and effective method for the calculation of local concentration sensitivities. This question now seems to be settled, and the decoupled direct method (ddm) is generally considered the best numerical method. All the main combustion simulation packages such as CHEMKIN, LSENS, RUNIDL and FACSIMILE calculate sensitivities as well as the simulation results and, therefore, many publications contain sensitivity calculations. However, usually very little information is actually deduced from the sensitivity results. It is surprising that the application of principal component analysis is not widespread, since it is a simple postprocessing method which can be used to extract a lot of information from the sensitivities about the structure of the kinetic mechanism. Also, methods for parameter estimation should always be preceded by the principal component analysis of the concentration sensitivity matrix. 

The method of the principal component analysis of the rate sensitivity matrix with a previous preselection of necessary species is a relatively simple and effective way for finding a subset of a large reaction mechanism that produces very similar simulation results for the important concentration profiles and reaction features. This method has an advantage over concentration sensitivity methods, in that the log-normalized rate sensitivity matrix depends algebraically on reaction rates and can be easily computed. For large mechanisms this could provide considerable time savings for the reduction process. This method has been applied for mechanism reduction to several reaction schemes [96-102]. 

Computing the sensitivities is time consuming. Fortunately the direct integral approximation of the sensitivity matrix and its principal component analysis can offer almost the same information whenever the direct integral method of parameter estimation applies. 

Another method for removing redundant reaction steps is the principal component analysis of matrix F PCAF), where V=[dfjdxk (Turmyi et al. 1989 Tomlin et al. 1992 Borger et al. 1992 Heard et al. 1998 Carslaw et al. 1999 Zsely and Turanyi 2001 Bahlouli et al. 2014). Here the sensitivity of the net rates of production of species to changes in the input parameters is investigated. Using the PCAF method, the objective function has the following form ... 

Principal component analysis is ideally suited for the analysis of bilinear data matrices produced by hyphenated chromatographic-spectroscopic techniques. The principle component models are easy to construct, even when large or complicated data sets are analyzed. The basis vectors so produced provide the fundamental starting point for subsequent computations. Additionally, PCA is well suited for determining the number of chromatographic and spectroscopically unique components in bilinear data matrices. For this task, it offers superior sensitivity because it makes use of all available data points in a data matrix. 

The sensitivity matrix shows the effect of individual parameter changes on the calculated concentrations. In most applications the parameters may change simultaneously. Principal component analysis is a mathematical method that assesses the effect of simultaneously changing parameters on several outputs of a model [74]. 

Generally, local concentration sensitivities are used for finding the parameters that have to be known with high precision, and for the identification of rate-limiting steps. An important achievement in the field of sensitivity analysis has been the introduction of principal component analysis as a method for the interpretation of the large amount of information contained in the sensitivity matrix. In the future, a more wide-spread application of principal component analysis is expected for the interpretation of concentration sensitivity results. This would help the extraction of further mechanistic details, or could be used for the detection of redundant reactions. The calculation of initial concentration sensitivities is very useful in some cases, but most simulation packages cannot calculate these sensitivities and there is no sign that such a feature will be incorporated. 

The singular value decomposition (SVD) method, and the similar principal component analysis method, are powerful computational tools for parametric sensitivity analysis of the collective effects of a group of model parameters on a group of simulated properties. The SVD method is based on an elegant theorem of linear algebra. The theorem states that one can represent an w X n matrix M by a product of three matrices ... 

Principal component analysis can also be carried out by using simulation data obtained at different conditions, such as at different temperatures, so that more observables can be used to construct a larger sensitivity matrix. This has been done for the evaluation of serine and threonine dipeptides in methano , but the key findings were essentially the same as those described above when the results from only two simulations were used in the analysis. 

A more systematic approach for reducing the reaction kinetic model using the sensitivity analysis method, known as the method of principal components analysis, was developed by T. Turanyi and co-workers [56,57]. This method is applicable after calculation of the local sensitivity array and the compiling of the approptiate matrix from the normalized sensitivities... 

Further simplification is possible by eliminating the redundant reactions, through principal components analysis (PCA) of the rate sensitivity matrix, F, which has elements... 

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