n4sid 'prediction' focus vs. n4sid 'simulation' focus

Why are there 'prediction' or 'simulation' settings in n4sidOptions? I thought that subspace identification finds a state space realization directly from input/output data.

I tried toggling this option on the n4sid algorithm for my dataset and found that the state space matrices were identical between the 'prediction' focus and 'simulation' focus options.

Is there something that I am missing here?

Answer 1

[Chidvi Modala]

As specified in the n4sid documentation, it estimates state space model using measured input-output data. The output represents the following system

xdot(t)=Ax(t)+Bu(t)+Ke(t)

y(t)=Cx(t)+Du(t)+e(t)

where e(t) is the disturbance

The model parameters are estimated by minimizing the error between the model output and measured response. The noise component is not trivial for n4sid. So ep(t) and es(t) are not equivalent. You can refer to https://www.mathworks.com/help/ident/ug/model-quality-metrics.html

Answer 2

[Chidvi Modala]

n4sid estimates state-space realization from input/output data. ‘prediction’ and ‘simulation’ settings in n4sidOptions determines the error to be minimized in the loss function during estimation.

‘prediction’ option focuses on minimizing error between measured and predicted outputs during estimation. ‘simulation’ option focuses on minimizing error between measured and simulation outputs during estimation.

[DemandQModeling]

Can you point me to a reference explaining how the loss function is different between n4sid with a "prediction" focus and n4sid with a "simulation" focus?

I can understand that these options would yield different results for the prediction error method, where an iterative model parameter optimization routine is involved. But because n4sid uses projections on I/O data to realize the model's state space matrices instead of parameter optimization, I cannot understand how a "prediction" and "simulation" focus makes any difference in the final model.