how to estimate and optimize the parameters of coupled nonlinear ODE's - matlab

Question

I have a set of coupled nonlinear ODE's, i need to solve these ODE's, then estimate and optimize the parameters using genetic algorithm or any function so as to minimize the difference between experimental and simulated data.
i have all the initial values for all state variables and the range(1-10) for the parameters to be estimated. I am new to MATLAB please forgive if i had made any syntax errors.
following is the code:
function dx = reaction( t,x,k )
dx = zeros(6,1);
dx(1)=0;
dx(2)= k(1)*x(1) - k(2)*x(2) - k(3)*x(5) + k(4)*x(6);
dx(3)=0;
dx(4) = -k(2)*x(3)*x(4) + k(3)*x(5);
dx(5) = k(8)*x(3)*x(4) - k(7)*x(5) - k(3)*x(2)*x(5) + k(5)*x(6);
dx(6) = k(6)*x(2)*x(5) - k(3)*x(6) - k(4)*x(2)*x(6);
end
function objective
k01=1:10;
k02=1:10;
k03=1:10;
k04=1:10;
k05=1:10;
k06=1:10;
k07=1:10;
k08=1:10; %//range of parameters to lie in
exp=[8;0;0.8;12;0;0]; %//experimental data
time=[0;5;10;15;20;25]; %//time span
x01=8;
x02=0;
x03=0.8;
x04=12;
x05=0;
x06=0; %// initial values of state variables
tspan = [min(time),max(time)];
k_opt = fminsearch(#minimize, [k1,k2,k3,k4,k5,k6,k7,k8])
function e = minimize(k1,k2,k3,k4,k5,k6,k7,k8)
sol = ode45(#reaction, tspan, [x01,x02,x03,x04,x05,x06],[], [k1,k2,k3,k4,k5,k6,k7,k8]);
y_hat = deval(sol, time); % // evaluate solution at given times
e = sum((y_hat' - exp).^2); % // compute squarederror
end
% // plot with optimal parameter
[T,X] = ode45(#reaction, tspan, [x01,x02,x03,x04,x05,x06], [], k_opt);
figure
plot(time, exp,'ko', 'markersize',10,'markerfacecolor','black')
hold on
plot(T,Y, 'r--', 'linewidth', 2)
end

Answer 1

See this as a start (still needs some work). It is the best that i can come up with.
Your plot plot(time, exp,'ko', 'markersize',10,'markerfacecolor','black') doesn't make sense to me as it plots the data over the time, which (referring to your latest comment) is not what exp should be. Thats why i was asking in the comments
The optimization stops with an error of about 139.3828 for me, also raising some warnings about problems with the numerical integration. The optimal parameterset is
k_opt =
-0.2258 6.2298 -1.2666 3.6352 -1.8671 0.0001 2.5592 -2.2011
Here comes the code:
function main
k0 = ones(1,8);
exp=[8;0;0.8;12;0;0]; %//experimental data
time=[0;5;10;15;20;25]; %//time span
%// initial values of state variables
x01=8;
x02=0;
x03=0.8;
x04=12;
x05=0;
x06=0;
x0 = [x01,x02,x03,x04,x05,x06];
tspan = [min(time),max(time)];
k_opt = fminsearch(#minimize, k0)
function e = minimize(k0)
[t,y_hat] = ode45(#reaction, tspan, x0,[], k0);
% y_hat = deval(sol, time(end)); % // evaluate solution at given times
e = sum((y_hat(end,:)' - exp).^2) % // compute squarederror '
end
% // plot with optimal parameter
[T,Y] = ode45(#reaction, tspan, [x01,x02,x03,x04,x05,x06], [], k_opt);
figure
subplot(1,2,1)
plot(time(end), exp, '*', 'markersize',15)
hold on
plot(T,Y, 'linewidth', 2)
% // Scatter: data vs model
subplot(1,2,2)
scatter(Y(end,:), exp)
hold on
axis equal
grid on
xlabel('model')
ylabel('data')
disp('yay')
end
function dx = reaction(t, x, k)
dx = zeros(6,1);
dx(1)=0;
dx(2)= k(1)*x(1) - k(2)*x(2) - k(3)*x(5) + k(4)*x(6);
dx(3)=0;
dx(4) = -k(2)*x(3)*x(4) + k(3)*x(5);
dx(5) = k(8)*x(3)*x(4) - k(7)*x(5) - k(3)*x(2)*x(5) + k(5)*x(6);
dx(6) = k(6)*x(2)*x(5) - k(3)*x(6) - k(4)*x(2)*x(6);
end
And two plots of the result. Blue and red are fit quite well, however, they do not change while integrating. I do not know about the others i.e. if your equations are correct and actually can reproduce the result.