System Identification Toolbox

Interpretation of the Loss Function

The value of the quadratic loss function is given as the field LossFcn in the EstimationInfo of the model.

• m.es.LossFcn
  

For multi-output systems, this is equal to the determinant of the estimated covariance matrix of the noise source e

For most models the estimated covariance matrix of the innovations is obtained by forming the corresponding sample mean of the prediction errors (squared), computed (using pe) from the model with the data for which the model was estimated.

Note the discrepancy between this value and the values shown during the minimization procedure (in pem, armax, bj, or oe), since these are the values of the robustified loss function (see under LimitError). Note also that it is the non-robustified residuals that are used to estimate the variance of e, as stored in Model.NoiseCovariance. It is also this value that is used to estimate the covariance matrix of the estimated parameters. Outliers may thus influence the estimate of NoiseVariance and the covariance matrix, while the parameter estimates are made robust against them.

Be careful when comparing loss function values between different structures that use very different disturbance models. An Output-Error model may have a better input-output fit, even though it displays a higher value of the loss function than, say, an ARX model.

Note that for ARX models computed using iv4, the covariance matrix of the innovations is estimated using the provisional disturbance model that is used to form the optimal instruments. The loss function therefore differs from what would be obtained if you computed the prediction errors using the model directly from the data. It is still the best available estimate of the innovations covariance. In particular, it is difficult to compare the loss function in an ARX model estimated using arx and one estimated using iv4.

Spectrum Normalization and the Sampling Interval

Enumeration of Estimated Parameters