Figure 9.11:

Reducing parameter correlation by centering the data.

Code for Figure 9.11

Text of the GNU GPL.

main.m

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% Copyright (C) 2001, James B. Rawlings and John G. Ekerdt
%
% This program is free software; you can redistribute it and/or
% modify it under the terms of the GNU General Public License as
% published by the Free Software Foundation; either version 2, or (at
% your option) any later version.
%
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
% General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program; see the file COPYING.  If not, write to
% the Free Software Foundation, 59 Temple Place - Suite 330, Boston,
% MA 02111-1307, USA.

%
% arhenius_center.m
%
chisq = 5.99;  % chi square value for 95% confidence level with 2 parameters
lnk0  = 1;
E     = 100;
ndata = 10;
Tmin  = 300;
Tmax  = 500;
Tmeas = linspace(Tmin,Tmax,ndata)';
X     = [ones(ndata,1) -1./Tmeas];
lnk   = X*[lnk0; E];
k     = exp(lnk);
measvar = 1e-3;
measstddev = sqrt(measvar);

% set seed for "reproducible" random numbers
randn('seed',0);

% some noisy data points
nexpts = 500;
clear lnkmeas;

for i = 1:nexpts
    lnkmeas(:,i) = lnk + measstddev*randn(ndata,1);
end

theta = (inv(X'*X)*X' * lnkmeas)';

%center the data first
Tcenter = -1./Tmeas + 1/mean(Tmeas);
Xcenter = [ones(ndata,1) Tcenter];
thetacenter = (inv(Xcenter'*Xcenter)*Xcenter' * lnkmeas)';

% calculate the elliptical confidence interval and bounding box
npts = 181;
amat = X'*X/measvar;
level = chisq;
[x, y, major, minor, bbox] = ellipse(amat, level, npts);

% shift the ellipse's center to the optimal parameters
x = x + lnk0;
y = y + E;
minor(:,1) = minor(:,1) + lnk0;
minor(:,2) = minor(:,2) + E;
major(:,1) = major(:,1) + lnk0;
major(:,2) = major(:,2) + E;
bbox(:,1) = bbox(:,1) + lnk0;
bbox(:,2) = bbox(:,2) + E;
bbox1 = bbox;

outline1 = [x, y];

amat = Xcenter'*Xcenter/measvar;
[x, y, major, minor, bbox] = ellipse(amat, level, npts);

% shift the ellipse's center to the optimal parameters
lnkmean = lnk0 - E/mean(Tmeas);
x = x + lnkmean;
y = y + E;

minor(:,1) = minor(:,1) + lnkmean;
minor(:,2) = minor(:,2) + E;
major(:,1) = major(:,1) + lnkmean;
major(:,2) = major(:,2) + E;
bbox(:,1) = bbox(:,1) + lnkmean;
bbox(:,2) = bbox(:,2) + E;
bbox2 = bbox;

outline2 = [x, y];

save arrhenius_center.dat theta thetacenter bbox1 outline1 bbox2 outline2;

if (~ strcmp (getenv ('OMIT_PLOTS'), 'true')) % PLOTTING
    plot (theta(:,1), theta(:,2), '+', ...
           thetacenter(:,1), thetacenter(:,2), 'x', ...
           bbox1(:,1), bbox1(:,2), outline1(:,1), outline1(:,2), ...
           bbox2(:,1), bbox2(:,2), outline2(:,1), outline2(:,2));
    % TITLE
end % PLOTTING