Smooth Time-Varying State-Space Model

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. The state equation for the first 25 periods is

and for the last 25 periods, it is

where and 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 and .

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

for the first 25 periods, and

for the last 25 periods, where 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*sum(x','omitnan')'+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

for the first 25 periods,

for period 26, and

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.

% Copyright 2015 The MathWorks, Inc.

function [A,B,C,D,Mean0,Cov0,StateType] = AR2MAParamMap(params,T)
%AR2MAParamMap Time-variant state-space model parameter mapping function
%
% This function maps the vector params to the state-space matrices (A, B,
% C, and D), the initial state value and the initial state variance (Mean0
% and Cov0), and the type of state (StateType). From periods 1 to T/2, the
% state model is an AR(2) and an MA(1) model, and the observation model is
% the sum of the two states. From periods T/2 + 1 to T, the state model is
% just the AR(2) model.
    A1 = {[params(1) params(2) 0 0; 1 0 0 0; 0 0 0 params(3); 0 0 0 0]};
    B1 = {[1 0; 0 0; 0 1; 0 1]}; 
    C1 = {params(4)*[1 0 1 0]};
    Mean0 = ones(4,1);
    Cov0 = 10*eye(4);
    StateType = [0 0 0 0];
    A2 = {[params(1) params(2) 0 0; 1 0 0 0]};
    B2 = {[1; 0]};
    A3 = {[params(1) params(2); 1 0]};
    B3 = {[1; 0]}; 
    C3 = {params(5)*[1 0]};
    A = [repmat(A1,T/2,1);A2;repmat(A3,(T-2)/2,1)];
    B = [repmat(B1,T/2,1);B2;repmat(B3,(T-2)/2,1)];
    C = [repmat(C1,T/2,1);repmat(C3,T/2,1)];
    D = 1;
end

Save this code as a file named AR2MAParamMap on your MATLAB® 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.

Pass the observed responses (y) to estimate to estimate the parameters. Specify an arbitrary set of positive initial values for the unknown parameters.

params0 = 0.1*ones(5,1);
EstMdl = estimate(Mdl,y,params0);

Method: Maximum likelihood (fminunc)
Sample size: 50
Logarithmic  likelihood:     -114.957
Akaike   info criterion:      239.913
Bayesian info criterion:      249.473
      |     Coeff       Std Err   t Stat     Prob  
---------------------------------------------------
 c(1) |  0.47870       0.26634    1.79733  0.07229 
 c(2) |  0.00809       0.27179    0.02975  0.97626 
 c(3) |  0.55735       0.80958    0.68844  0.49118 
 c(4) |  1.62679       0.41622    3.90848  0.00009 
 c(5) |  1.90021       0.49563    3.83391  0.00013 
      |                                            
      |   Final State   Std Dev    t Stat    Prob  
 x(1) | -0.81229       0.46815   -1.73511  0.08272 
 x(2) | -0.31449       0.45918   -0.68490  0.49341 

EstMdl is an ssm model containing the estimated coefficients. Likelihood surfaces of state-space models might contain local maxima. Therefore, it is good practice to try several initial parameter values, or consider using refine.

Smooth the states and estimate the variance-covariance matrices of the smoothed states by passing EstMdl and the observed responses to smooth.

[~,~,Output]= smooth(EstMdl,y);

Output is a T-by-1 structure array containing the smoothed states and their variance-covariance matrices, among other things.

Extract the smoothed states and their variances from the cell arrays. Recall that the two, different states are in positions 1 and 3. The states in positions 2 and 4 help specify the processes of interest.

stateIndx = [1 3]; % State Indices of interest
SmoothedStates = NaN(T,numel(stateIndx));
SmoothedStatesCov = NaN(T,numel(stateIndx));

for t = 1:T
    maxInd1 = size(Output(t).SmoothedStates,1);
		maxInd2 = size(Output(t).SmoothedStatesCov,1);
    mask1 = stateIndx <= maxInd1;
		mask2 = stateIndx <= maxInd2;
    SmoothedStates(t,mask1) = ...
        Output(t).SmoothedStates(stateIndx(mask1),1);
		SmoothedStatesCov(t,mask2) = ...
        diag(Output(t).SmoothedStatesCov(stateIndx(mask2),...
        stateIndx(mask2)));
end

Plot the true state values, the smoothed state values, and their individual 95% Wald-type confidence intervals for each model.

AR2SSCIlb = SmoothedStates(:,1) - 1.95*sqrt(SmoothedStatesCov(:,1));
AR2SSCIub = SmoothedStates(:,1) + 1.95*sqrt(SmoothedStatesCov(:,1));
AR2SSIntervals = [AR2SSCIlb AR2SSCIub];

MA1SSCIlb = SmoothedStates(:,2) - 1.95*sqrt(SmoothedStatesCov(:,2));
MA1SSCIub = SmoothedStates(:,2) + 1.95*sqrt(SmoothedStatesCov(:,2));
MA1SSIntervals = [MA1SSCIlb MA1SSCIub];

figure
plot(1:T,x(:,1),'-k',1:T,SmoothedStates(:,1),':r',...
		1:T,AR2SSIntervals,'--b','LineWidth',2);
title('AR(2) State Values')
xlabel('Period')
ylabel('State Value')
legend({'True state values','Smoothed state values',...
    '95% Confidence Intervals'});

figure
plot(1:T,x(:,2),'-k',1:T,SmoothedStates(:,2),':r',...
		1:T,MA1SSIntervals,'--b','LineWidth',2);
title('MA(1) State Values')
xlabel('Period')
ylabel('State Value')
legend({'True state values','Smoothed state values',...
    '95% Confidence Intervals'});