load 'unemployment.data';
Y=unemployment(:,3);

t=[1:252]'
X=[ones(252,1),cos(pi*t/6),cos(2*pi*t/6)]
>>a=inv(X'*X)*X'*Y
a =
    6.6913
    0.0181
    0.0258

>>mean(Y)
ans =
    6.6913

plot(t,Y);
mu=6.6913 + 0.0181*cos(pi*t/6) + 0.0258*cos(2*pi*t/6);
hold
plot(t,mu,'-')

X=[ones(252,1),cos(0.1047*t),cos(2*0.1047*t)]
>>a=inv(X'*X)*X'*Y
a =
    6.7060
   -0.4324
    0.0252
plot(t,Y);
mu=6.7060 - 0.4324*cos(0.1047*t) + 0.0252*cos(2*0.1047*t);
hold
plot(t,mu,'-')

%We need to increase the smoothness

>> 2*pi/252
ans =
    0.0249
    
X=[ones(252,1),cos(0.0249*t),cos(2*0.0249*t),cos(3*0.0249*t)]
>> a=inv(X'*X)*X'*Y
a =
    6.6898
   -1.2512
   -0.5112
    0.6323

plot(t,Y);
mu=6.6898 - 1.2512*cos(0.0249*t) - 0.5112*cos(2*0.0249*t) + 0.6323*cos(3*0.0249*t);
hold
plot(t,mu,'-')