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<title>Figure 7.1: Logistic regression</title>
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<h1>Figure 7.1: Logistic regression</h1>
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<pre class="codeinput">
<span class="comment">% Section 7.1.1</span>
<span class="comment">% Boyd &amp; Vandenberghe, "Convex Optimization"</span>
<span class="comment">% Original by Lieven Vandenberghe</span>
<span class="comment">% Adapted for CVX by Argyris Zymnis - 01/31/06</span>
<span class="comment">%</span>
<span class="comment">% We consider a binary random variable y with prob(y=1) = p and</span>
<span class="comment">% prob(y=0) = 1-p. We assume that that y depends on a vector of</span>
<span class="comment">% explanatory variables u in R^n. The logistic model has the form</span>
<span class="comment">% p = exp(a'*u+b)/(1+exp(a'*u+b)), where a and b are the model parameters.</span>
<span class="comment">% We have m data points (u_1,y_1),...,(u_m,y_m).</span>
<span class="comment">% We can reorder the data so that for u_1,..,u_q the outcome is y = 1</span>
<span class="comment">% and for u_(q+1),...,u_m the outcome is y = 0. Then it can be shown</span>
<span class="comment">% that the ML estimate of a and b can be found by solving</span>
<span class="comment">%</span>
<span class="comment">% minimize sum_{i=1,..,q}(a'*u_i+b) - sum_i(log(1+exp(a'*u_i+b)))</span>
<span class="comment">%</span>
<span class="comment">% In this case we have m = 100 and the u_i are just scalars.</span>
<span class="comment">% The figure shows the data as well as the function</span>
<span class="comment">% f(x) = exp(aml*x+bml)/(1+exp(aml*x+bml)) where aml and bml are the</span>
<span class="comment">% ML estimates of a and b.</span>
randn(<span class="string">'state'</span>,0);
rand(<span class="string">'state'</span>,0);
<span class="comment">% Generate data</span>
a = 1;
b = -5 ;
m= 100;
u = 10*rand(m,1);
y = (rand(m,1) &lt; exp(a*u+b)./(1+exp(a*u+b)));
plot(u,y,<span class="string">'o'</span>)
axis([-1,11,-0.1, 1.1]);
<span class="comment">% Solve problem</span>
<span class="comment">%</span>
<span class="comment">% minimize -(sum_(y_i=1) ui)*a - b + sum log (1+exp(a*ui+b)</span>
U = [ones(m,1) u];
cvx_expert <span class="string">true</span>
cvx_begin
variables <span class="string">x(2)</span>
maximize(y'*U*x-sum(log_sum_exp([zeros(1,m); x'*U'])))
cvx_end
<span class="comment">% Plot results and logistic function</span>
ind1 = find(y==1);
ind2 = find(y==0);
aml = x(2); bml = x(1);
us = linspace(-1,11,1000)';
ps = exp(aml*us + bml)./(1+exp(aml*us+bml));
dots = plot(us,ps,<span class="string">'-'</span>, u(ind1),y(ind1),<span class="string">'o'</span>,<span class="keyword">...</span>
u(ind2),y(ind2),<span class="string">'o'</span>);
axis([-1, 11,-0.1,1.1]);
xlabel(<span class="string">'x'</span>);
ylabel(<span class="string">'y'</span>);
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<pre class="codeoutput">
Successive approximation method to be employed.
SDPT3 will be called several times to refine the solution.
Original size: 602 variables, 400 equality constraints
200 exponentials add 1600 variables, 1000 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
200/200 | 5.422e+00 2.242e+00 0.000e+00 | Solved
200/200 | 5.049e-01 1.636e-02 0.000e+00 | Solved
194/199 | 2.442e-02 3.880e-05 0.000e+00 | Solved
0/148 | 3.819e-04 7.845e-09 0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -32.9797
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