%% Simulate States of Time-Varying State-Space Model Using Simulation Smoother 
% This example generates data from a known model, fits a state-space model
% to the data, and then simulates series from the fitted model using the
% simulation smoother.
%%
% Suppose that a latent process comprises an AR(2) and an MA(1) model.
% There are 50 periods, and the MA(1) process drops out of the model for
% the final 25 periods.  Subsequently, the state equation for the first 25
% periods is
%
% $$\begin{array}{l}
% {x_{1,t}} = 0.7{x_{1,t - 1}} - 0.2{x_{1,t - 2}} + {u_{1,t}}\\
% {x_{2,t}} = {u_{2,t}} + 0.6{u_{2,t - 1}},
% \end{array}$$
%
% and for the last 25 periods, it is 
%
% $${x_{1,t}} = 0.7{x_{1,t - 1}} - 0.2{x_{1,t - 2}} + {u_{1,t}},$$
%
% where $u_{1,t}$ and $u_{2,t}$ are Gaussian with mean 0 and standard
% deviation 1.
%%
% Assuming that the series starts at 1.5 and 1, respectively, generate a
% random series of 50 observations from $x_{1,t}$ and $x_{2,t}$.

% Copyright 2015 The MathWorks, Inc.

T = 50;
ARMdl = arima('AR',{0.7,-0.2},'Constant',0,'Variance',1);
MAMdl = arima('MA',0.6,'Constant',0,'Variance',1);
x0 = [1.5 1; 1.5 1];
rng(1);
x = [simulate(ARMdl,T,'Y0',x0(:,1)),...
    [simulate(MAMdl,T/2,'Y0',x0(:,2));nan(T/2,1)]];
%%
% The last 25 values for the simulated MA(1) data are |NaN| values.
%%
% Suppose further that the latent processes are measured using
%
% $${y_t} = 2\left( {{x_{1,t}} + {x_{2,t}}} \right) + {\varepsilon _t},$$
%
% for the first 25 periods, and
%
% $${y_t} = 2{x_{1,t}} + {\varepsilon _t}$$
%
% for the last 25 periods, where $\varepsilon_t$ is Gaussian with mean 0
% and standard deviation 1.
%
% Use the random latent state process (|x|) and the observation equation to
% generate observations.
y = 2*nansum(x')'+randn(T,1);
%%
% Together, the latent process and observation equations compose a
% state-space model.  Supposing that the coefficients are unknown
% parameters, the state-space model is
% 
% $$\begin{array}{l}
% \left[ {\begin{array}{*{20}{c}}
% {{x_{1,t}}}\\
% {{x_{2,t}}}\\
% {{x_{3,t}}}\\
% {{x_{4,t}}}
% \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
% {{\phi _1}}&{{\phi _2}}&0&0\\
% 1&0&0&0\\
% 0&0&0&{{\theta _1}}\\
% 0&0&0&0
% \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
% {{x_{1,t - 1}}}\\
% {{x_{2,t - 1}}}\\
% {{x_{3,t - 1}}}\\
% {{x_{4,t - 1}}}
% \end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
% 1&0\\
% 0&0\\
% 0&1\\
% 0&1
% \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
% {{u_{1,t}}}\\
% {{u_{2,t}}}
% \end{array}} \right]\\
% {y_t} = a({x_{1,t}} + {x_{3,t}}) + {\varepsilon _t}
% \end{array}$$
%
% for the first 25 periods,
%
% $$\begin{array}{l}
% \left[ {\begin{array}{*{20}{c}}
% {{x_{1,t}}}\\
% {{x_{2,t}}}
% \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
% {{\phi _1}}&{{\phi _2}}&0&0\\
% 1&0&0&0
% \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
% {{x_{1,t - 1}}}\\
% {{x_{2,t - 1}}}\\
% {{x_{3,t - 1}}}\\
% {{x_{4,t - 1}}}
% \end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
% 1\\
% 0
% \end{array}} \right]{u_{1,t}}\\
% {y_t} = b{x_{1,t}} + {\varepsilon _t}
% \end{array}$$
%
% for period 26, and
%
% $$\begin{array}{l}
% \left[ {\begin{array}{*{20}{c}}
% {{x_{1,t}}}\\
% {{x_{2,t}}}
% \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
% {{\phi _1}}&{{\phi _2}}\\
% 1&0
% \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
% {{x_{1,t - 1}}}\\
% {{x_{2,t - 1}}}
% \end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
% 1\\
% 0
% \end{array}} \right]{u_{1,t}}\\
% {y_t} = b{x_{1,t}} + {\varepsilon _t}
% \end{array}$$
%
% for the last 24 periods.
%%
% Write a function that specifies how the parameters in |params| map to the
% state-space model matrices, the initial state values, and the type of
% state.
%
% <include>AR2MAParamMap.m</include>
%
%%
% Save this code as a file named |AR2MAParamMap| on your MATLAB(R) path.
%%
% Create the state-space model by passing the function |AR2MAParamMap| as a
% function handle to |ssm|.
Mdl = ssm(@(params)AR2MAParamMap(params,T));
%%
% |ssm| implicitly creates the state-space model. Usually, you
% cannot verify an implicitly defined state-space model.
%%
% Simulate one path of states from Mdl using the simulation smoother.
% Specify that the parameter-to-matrix mapping function has seven output
% arguments. Also, specify the unknown values of the parameters.
simParams = [0.48 0.0081 0.56 1.63 1.9];
X = simsmooth(Mdl,y,'NumOut',7,'Params',simParams);
%%
% |X| is a |T|-by-1 cell vector of simulated states. Cells 1 through 25
% contain 4-by-1 vectors, and cells 26 through 50 contain 2-by-1 vectors.
%%
% Access a cell using cell indexing, for example, access cell 5 using
% |X{5}|.
simStatesPeriod5 = X{5}