Maximum Likelihood Estimates

Today, I looked into the application of MLE.

Summary:

function [theta] = LG_RML(u,y,oa,ob,od)
na = oa - 1;nb = ob - 1;nc = od - 1;d = 1;
L = length(u);
nn = max(na,nc);
uk = zeros(d + nb,1);
yk = zeros(na,1);
vk = zeros(nc,1);
yfk = zeros(nn,1);ufk = zeros(nn,1);vfk = zeros(nc,1);
thetae_1 = zeros(na+nb+1+nc,1);
thetae = zeros(na+nb+1+nc,L);
P = eye(na+nb+1+nc);
for k = 1:L
    hk = [-yk;uk(d:d+nb);vk];
    v = y(k) - hk'*thetae_1;
    hkf=[-yfk(1:na);ufk(d:d+nb);vfk];
    K=P*hkf/(1+hkf'*P*hkf);
    thetae(:,k)=thetae_1+K*v;
    P=( eye(na+nb+1+nc)-K*hkf' )*P;
    yf=y(k)-thetae(na+nb+2:na+nb+1+nc,k)*yfk(1:nc);%yf(k)
    uf=u(k)-thetae(na+nb+2:na+nb+1+nc,k)*ufk(1:nc);%uf(k)
    vf=v-thetae(na+nb+2:na+nb+1+nc,k)*vfk(1:nc);%vf(k)
    thetae_1=thetae(:,k);      
    for i=d+nb:-1:2
        uk(i)=uk(i-1);
    end
    uk(1)=u(k);    
    for i=na:-1:2
        yk(i)=yk(i-1);
    end
    yk(1)=y(k);
    for i=nc:-1:2
        vk(i)=vk(i-1);
        vfk(i)=vfk(i-1);
    end
    vk(1)=v;
    vfk(1)=vf;
    for i=nn:-1:2
        yfk(i)=yfk(i-1);
        ufk(i)=ufk(i-1);
    end
    yfk(1)=yf;
    ufk(1)=uf;
end
theta = thetae;
end

a=[1 -1.5 0.7];b=[1 0.5];c=[1 -0.5];d=1;
L=1000;
na=length(a)-1;nb=length(b)-1;nc=length(c)-1;
uk=zeros(d+nb,1);
yk=zeros(na,1);
u=randn(L,1);
xi=randn(L,1);
xik=zeros(nc,1);
for k=1:L
    y(k)=-a(2:na+1)*yk+b *uk(d:d+nb)+c *[xi(k);xik];
    for i=d+nb:-1:2
        uk(i)=uk(i-1);
    end
    uk(1)=u(k);
    for i=na:-1:2
        yk(i)=yk(i-1);
    end
    yk(1)=y(k);
    xik(1)=xi(k);
end
thetae = LG_RML(u,y,3,2,2);
figure(1)
plot([1:L],thetae(1:na,:),[1:L],thetae(na+nb+2:na+nb+1+nc,:));
xlabel(k);ylabel('parameter estimation a&d');
legend('a_1' ,'a_2' ,'d_1' );axis([0 L -2 2]);
figure(2)
plot([1:L],thetae(na+1:na+nb+1,:));
xlabel(k);ylabel('parameter estimation b' );
legend('b_0' ,'b_1' );axis([0 L 0 1.5]);

There are many applications of MLE. I will try to find one that is the most similar to my research.

Tomorrow, I will look more into MLE.