objectivefcn.m

function f = objectivefcn(x)
%% Input Generation
N = 200;
u0 = 0;
v_1 = 0;

for i=1:N
v = 1*randn(1);
u0 = 0.5*u0+v+0.5*v_1;
U0(i) = u0;
v_1 = v;
end

%% https://medium.com/auquan/time-series-analysis-for-finance-arma-models-21695e14c999
%% Output Generation
y_1 = 0;
y_2 = 0;
u_1 = 0;
u_2 = 0;

for i = 1:N
  y = 1.5*y_1-0.7*y_2+2*U0(i)-u_1+0.5*u_2;
  y_2 = y_1;
  y_1 = y;
  u_2 = u_1;
  u_1 = U0(i);
  Y(i) = y+0.0001*randn(1);;
end

%% FFT Calculation
for k = 0:N-1
 wk = 2*pi*k/N;
 Uw(k+1) = 0;
 Yw(k+1) = 0;
 
 for t = 1:N-1
   Uw(k+1) = U0(t+1)*exp(-1i*wk*t)+Uw(k+1);
   Yw(k+1) = Y(t+1)*exp(-1i*wk*t)+Yw(k+1);   
 end
Uw(k+1) = 1/sqrt(N)*Uw(k+1);
Yw(k+1) = 1/sqrt(N)*Yw(k+1);    
end

%%
t0 = 1;
t1 = 1;
teta1 = x(1);
teta2 = x(2);
teta3 = x(3);
teta4 = x(4);
teta5 = x(5);

for k = 0:N-1
 Phi_k = exp(-1i*2*pi*k/N);
 Ak(k+1) = 1+teta1*Phi_k+teta2*Phi_k^2;
 Bk(k+1) = teta3+teta4*Phi_k+teta5*Phi_k^2;
 Gk(k+1) = Bk(k+1)/Ak(k+1);
 Ik(k+1) = t0+t1*Phi_k;
 Tk(k+1) = Ik(k+1)/Ak(k+1);
end
%% 
m = 1;
for k = 0:N-1
for j = N-1:-1:0
K(k+1,m) = 1/Ak(k+1)*exp(-1i*(2*pi*k/N)*j);     
end
m = m+1;
end
%%
Va = 0;
Vb = 0;
Vc = 0;
Vd = 0;
rho = 0.01;
% 
g =0;
for i = 1:N
Va = Va+(norm(Yw(i)-Gk(i)))^2/((norm(Gk(i)))^2+rho);
Vb = Vb+conj((Yw(i)-Gk(i)))/((norm(Gk(i)))^2+rho)*K(i,:);
Vc = Vc+1/((norm(Gk(i)))^2+rho)*(conj(K(i,:)))'*K(i,:);
Vd = Vd+(conj(K(i,:)))'*((Yw(i)-Gk(i)))/((norm(Gk(i)))^2+rho);
g = g+log10((norm(Gk(i)))^2+rho);
end
f = abs(log10((1/N*(Va-Vb*Vc^-1*Vd))))+g/N;
end