%Class for CO907.
%__________________________________________
%Author: Jiarui Cao
%Date: 19 Nov 2013
%Purpose: Problem Sheet 2. Q2.5c


close all;
clear all;
clc;
clf;

n=500;
rng('default');
%% generate data
sigma=0.5;
y=normrnd(0,sigma,n,1); %noise

x=zeros(n,1);

% %% AR(1)
% 
% c=1;
% phi=0.99;
% x(1)= 0;
% for t =2 :n
%     x(t) = c + phi*x(t-1) + y(t);
% end
% 
% %hold on
% 
% plot(x,'or');
% xlabel('t','fontsize',30);
% ylabel('X(t)', 'fontsize', 30);


% model = arima('Constant',c,'AR',phi,'Variance',sigma^2);
% 
% s=simulate(model,n);
% plot(s);

%% AR(2)
c = 0;
phi = [1.5, -0.5];
x(1)=0;
x(2)=0;
for t=3:n
   x(t) = c + phi(1)*x(t-1) + phi(2)*x(t-2) + y(t); 
end
plot(x, 's-');

xlabel('t','fontsize',30);
ylabel('X(t)', 'fontsize', 30);

%% sovle lambda
syms lambda;
vpasolve(lambda^2 - phi(1)*lambda - phi(2)==0,lambda) %compute 

%% use estimate to fit the parameters
model =arima(2,0,0); %guess this is a AR(2) model. 

estimate(model,x) %estimate the data