TomSym Maximum Likelihood Estimation

This page is part of the TomSym Manual. See TomSym Manual.

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: 1.4495 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: 0.327602 sec. Elapsed time: 0.422000 sec.