Estimate state-space model parameters using Parameter Mapping function

Hi everyone,

I am currently having a set of differential equations, namely

dB = - mu*S,

dS = mu*S - beta*S*I + gamma*I - nu_s*S,

dI = beta*S*I - gamma*I - alpha*I - nu_i*I,

dP = nu_s*S + nu_i*I, and

dD = alpha*I,

as well as a dataset, containing observations of the states of B, S, I, P, and D.

I am trying to estimate the parameter [mu, beta, gamma, alpha, nu_s, nu_i], using the econometrics toolbox approach to estimate state space models with unknown parameters.

The state transition matrix A, with unknown parameter denoted as NaN is

 A = [0 NaN  0  0 0;...
      0 NaN NaN 0 0;...
      0 NaN NaN 0 0;...
      0 NaN NaN 0 0;...
      0  0  NaN 0 0]

In accordance with the documentation, I formulated the parameter mapping function:

function [A,B,C,D,Mean0,Cov0,StateType] = timeInvariantParamMap(params)
  A = [0 -params(1) 0 0 0;...
       0 params(1)-params(3) params(4)-params(2) 0 0;...
       0 0 params(2)-params(4)-params(5)-params(6) 0 0;...
       0 params(3) params(6) 0 0; 0 0 params(5) 0 0];
  B = [params(7) 0 0 0 0;...
       0 params(8) 0 0 0;...
       0 0 params(9) 0 0;...
       0 0 0 params(10) 0;...
       0 0 0 0 params(11)];
  C = eye(5);
  D = zeros(5,1);
  Mean0 = [30161, 3, 0, 0, 0]; % First observed values
  Cov0 = [];
  StateType = [0, 0, 0, 0, 0];
end

But I don't know, how to incorporate the term beta*S*I, so a state variable in the transition matrix. The current approach just ignores that fact

Since the econometrics toolbox is a quite new tool, I was not able to find any related Q&As.

I would really appreciate any hint on my problem, as well as additional information.

Best regards,

David

Answer 1

[Hang Qian]

If we included the beta*S*I term, the state transition would be non-linear with respect to past states B, S, I, P, and D. In that case, we could not apply the standard Kalman filter. We probably need some approximation filters such as extended or unscented Kalman filters. MATLAB central has some examples of them. Usually non-linear filter needs to be applied with care, since approximation could be crude.

http://www.mathworks.com/matlabcentral/fileexchange/18217

http://www.mathworks.com/matlabcentral/fileexchange/18189

Hang Qian