Page 390 - Numerical Methods for Chemical Engineering

Linear least-squares regression 379

T
parameter values. X X is a real, symmetric matrix that is at least positive-semidefinite. For
a well-designed set of experiments that provides sufficient data to estimate each parameter
T
to at least some finite accuracy, X X is positive-definite.

Solving the least-squares linear system
T
Often, X X may be ill conditioned, i.e., it has one or more eigenvalues near zero, when the
experimental design does not provide sufficient information to measure well one or more
linear combinations of parameters. Therefore, QR decomposition is the favored solution
method,

T
X X = QR
(8.25)
T
−1
Q = Q R is upper-triangular
T
T
As Q is orthogonal, we write (X X)θ LS = X y as
T T T
QRθ LS = X y ⇒ Rθ LS = Q X y (8.26)
The solution of the latter system requires only backward substitution.
A more expensive approach, but one that provides further insight into experimental
T
design, is to diagonalize the real, symmetric matrix X X:
T
X X = V V T (8.27)
T
V is an orthogonal matrix, V = V −1 , whose columns are the normalized eigenvectors of
T
T
X X and is a diagonal matrix whose principal diagonal contains the eigenvalues of X X.
The least-squares estimate θ LS is
T
T
−1
θ LS = [V V ]X y (8.28)
T
T
Note also that since X Xθ LS = X y, we have an overdetermined system,
Xθ LS = y (8.29)
that can be solved by SVD (Chapter 3). The SVD approach is popular; however, here we
use the familiar eigenvalue decomposition (8.27) in lieu of SVD to analyze experimental
designs.

Example. Least-squares fitting of rate law parameters to transformed
batch data

We now apply the least-squares method to the transformed batch reactor data of Table 8.2
for the kinetics of A + B → C, using the model

(8.30)
log r R1 = log k 1 + ν a log c A + ν b log c B
10 10 10 10