System Identification Toolbox | ![]() ![]() |
The Estimated Parameter Covariance Matrix
The estimated parameters are uncertain. The amount of uncertainty is measured and described by the covariance matrix of the estimated parameter vector, (this vector is a random variable, since it depends on the random noise that has affected the output). This covariance (uncertainty) can also be estimated from data, as described, e.g. in Chapter 9 of Ljung (1999). The estimated covariance matrix is contained in the estimated model as the property Model
.CovarianceMatrix
. It is used to compute all relevant uncertainty measures of various model input-output properties (Bode plots, uncertain model output, zeros and poles, etc.)
The estimate of the covariance matrix is based on the assumption that the model structure is capable of giving a correct description of the system. For models that contain a disturbance model (H is estimated) it, thus, assumed that the model will produce white residuals, for the uncertainty estimate to be correct.
However, for output-error models (H fixed to 1, corresponding to K = 0 for state space models, and C = D = A = 1 for polynomial models), it is not assumed that the residuals are white. Instead, their color is estimated and a correct estimate of the covariance estimate is used. This corresponds to eq (9.42) in Ljung (1999).
![]() | Initial State | No Covariance | ![]() |