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, for example, 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 is 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 Equation (9.42) in Ljung (1999).

Initial States for Frequency Domain Data

No Covariance