Page 432 - Numerical Methods for Chemical Engineering

Analysis of composite data sets 421

Bayes MCMC 1Dmarginal MR( X pred, Y, . . .
fun yhat, j plot 1D, val lo, val hi, . . .
N bins, k1 0, MCOPTS);
% compute 95% HPD
alpha = 0.05;
[HPD lo,HPD hi] = Bayes 1D HPD MR( . . .
bin 1Dc, bin 1Dp, j plot 1D, alpha),
About the most probable estimate of k 1 = 0.0024, this analysis of the multiresponse data
of Table 8.3 yields a 95% HPD region for k 1 of
0.0022 ≤ k 1 ≤ 0.0026 (8.202)

2-D marginal posterior densities and HPD regions are computed from MCMC simulation
for multiresponse data using the routines

[bin 2Dic, bin 2Djc, bin 2Dp] = . . .
Bayes MCMC 2Dmarginal MR( X pred, Y, . . .
fun yhat, i plot 2D, j plot 2D, val lo, val hi, . . .
N bins, theta 0, MCOPTS);
and

HPD 2D = Bayes 2D HPD MR(bin 2Dic, bin 2Djc, bin 2Dp, . . .
i plot 2D, j plot 2D, alpha);

Again, the arguments in these routines carry the same definitions as the corresponding ones
in the routines for single-response data.

Analysis of composite data sets

Above, we have assumed that we measure the same set of responses in each experiment;
however, often we estimate parameters from composite data sets that mix different types of
data. Here, we treat composite data sets using the sequential learning aspects of Bayesian
analysis; hence, the routines take the suffix MRSL for multiresponse sequential-learning.
Let us say that we have some sets of response data Y 1 , Y 2 ,..., Y D that provide
P
information about the same set of parameters θ ∈ but that are dimensioned differently
and have different error properties. We want to compute the marginal posterior density,
taking into account the information provided by all sets of data,

p θ Y 1 , Y 2 , ..., Y D (8.203)

For each data set, we propose a model that predicts the responses Y ˆ j (θ), and compute the
L j × L j “sum of squared errors” matrix
T

j
S (θ) = Y j − Y ˆ j (θ) Y j − Y ˆ j (θ) (8.204)