Page 411 - Numerical Methods for Chemical Engineering

400 8 Bayesian statistics and parameter estimation

Linear least-squares calculations in MATLAB

For linear models, we use the function regress,
[theta, theta CI, residuals, res CI, Stats] = regress(y, X, alpha);
N
y is the vector y ∈ of measured responses. X is the N × P design matrix of predictor
variables. alpha sets the width of the confidence intervals (α = 0.05 for 95% CIs). theta
P
is the vector θ M ∈ of fitted parameters. theta CI contains lower and upper bounds on
each parameter. residuals contains the values in each experiment of the residual errors
[k]
e [k] = y [k] − ˆ y (θ M ) and res CI contains lower and upper bounds on these errors. Stats
contains various goodness-of-fit measures.
We demonstrate the use of regress for the data set comparing the protein expression levels
of wild-type and mutant bacterial strains, (8.35). The following MATLAB code performs the
least-squares calculation:

y = [121.9; 113.4; 112.2; 106.1; 120.7; 119.5; 116.5; 124.0];
X=[10;10;10;10;11;11;11;11];
alpha = 0.05;
[theta, theta CI, residuals, res CI, Stats] = regress(y, X, alpha);
The results of these calculations are

theta =
113.4000
6.7750
theta CI =
107.1646 119.6354
−2.0432 15.5932
Thus, we have the 95% confidence intervals,

θ 1 = 113.40 ± 6.33 θ 2 = 6.78 ± 8.82 (8.145)

As the confidence interval for θ 2 includes θ 2 = 0, the difference in protein expression levels
between the two strains is not statistically significant.

Nonlinear least-squares calculations in MATLAB
For nonlinear models, the MATLAB statistics toolkit contains functions to compute the
least-squares estimates and generate confidence intervals using the approximate posterior
density. First, the least-squares estimate is found using the function nlinfit,
[theta, residuals, Jac] = nlinfit(X pred, y, fun, theta 0);
X pred is a matrix that contains in each row the set of predictors for the correspond-
ing experiment. y is the vector of measured response values. fun name is the name of a
function,