TomSym Maximum Likelihood Estimation
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TomSym implementation of GAMS Example (LIKE,SEQ=25)
This application from the biomedical area tests the hypothesis that a population of systolic blood pressure can be separated into three distinct groups.
Bracken, J, and McCormick, G P, Chapter 8.5. In Selected Applications of Nonlinear Programming. John Wiley and Sons, New York, 1968, pp. 90-92.
i: Observations (1-31)
g: Groups (one, two, three)
% Systolic blood pressure data
pressure = [95 105 110 115 120 125 130 135 140 145 150 155 160 165 170 ...
175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 260]';
frequency = [1 1 4 4 15 15 15 13 21 12 17 4 20 8 17 ...
8 6 6 7 4 3 3 8 1 6 0 5 1 7 1 2]';
y = pressure;
w = frequency;
% Constant
c = 1/sqrt(2*3.14159);
% p(g): proportion of population
% m(g): population mean
% s(g): population standard deviation
toms 3x1 p m s
% Maximum likelihood function
toms i
mlf = fsum(lookup(w,i)*log(c*sum(p./s.*exp(-.5*((lookup(y,i)-m)./s).^2))),...
i, 1:31);
eq1 = {sum(p) == 1};
eq2 = {m(2) >= m(1); m(3) >= m(2)};
eq3 = {p >= 0.1; s >= 0.1; m >= 0};
x0 = {p == 1/3; m == 100+30*(1:3)'; s == 15};
options = struct;
options.solver = 'conopt';
solution = ezsolve(-mlf,{eq1,eq2,eq3},x0,options);
Problem type appears to be: con Time for symbolic processing: 1.4168 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: f_k 1138.410564424085100000 f(x_0) 1204.299328240221300000 Solver: CONOPT. EXIT=0. INFORM=2. Feasible Path GRG, CONOPT 3.14R Normal completion : Locally optimal FuncEv 21 GradEv 16 HessEv 9 Iter 15 CPU time: 0.249602 sec. Elapsed time: 0.255000 sec.