Principal Component Analysis (PCA) employs an eigenvalue-eigenvector analysis to extract kinetic information from linear sensitivities calculated for species of a reacting system [4], [5]. It studies the effect on the calculated behavior of a reaction mechanism brought about by a variation in the rate coefficients. The effect is most sensitive to changes in the rate coefficients along the principle axis that corresponds to the largest eigenvalue of the sensitivity matrix and is least sensitive to changes along the axis that corresponds to the smallest eigenvalue. Therefore reactions in the principal components with small eigenvalues can be dropped.

Ansys Chemkin calculates the normalized linear sensitivities where is the mass fraction of the -th species and is the reaction rate constant of the j-th reaction. Let us denote as the sensitivity matrix of the reaction mechanism where . Let us assign and denote for the nominal case. The effect on the calculated behavior of a reaction mechanism brought about by a variation in the rate constants can be expressed as

(3–13)

Using the Taylor series expansion, we can get an approximation of as

(3–14)

where is the gradient vector of such that , and is the Hessian matrix of such that . Because and ,

(3–15)

The Hessian matrix can be expressed in terms of the sensitivity matrix as

(3–16)

where is the matrix of second derivatives of species concentrations regarding the rate constants, [5]. Using the Gauss approximation, we can ignore the second derivatives, , and the effect on the calculated behavior of a reaction mechanism by variations in the rate constants can be expressed in terms of the sensitivity matrix of the reaction mechanism as

(3–17)

We can perform an eigenvalue-eigenvector decomposition of the symmetric matrix ,

(3–18)

where is a diagonal matrix formed by the eigenvalues of and is the matrix of normed eigenvectors of such that for each . We can further denote principal components of the matrix as , then can be expressed in terms of the principal components as

(3–19)

where . This expression informs us that the effect on the calculated behavior of a reaction mechanism is approximately the sum of the variations in the rate constants along each principle axis denoted by . This shows that the behavior is most sensitive to variations in the rate constants along the principle axis corresponding to the largest eigenvalue of the sensitivity matrix and is least sensitive to variations along the axis corresponding to the smallest eigenvalue. Therefore reactions in the principal components with small eigenvalues can be dropped since they contribute very little to the overall effect. Once reactions have been removed, the Reaction Workbench implementation will automatically remove species that are no longer required due to absence of participation in any reaction.