Computing Model Uncertainty

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Why Analyze Model Uncertainty? What Is Model Covariance? Types of Model Uncertainty Information

Why Analyze Model Uncertainty?

In addition to estimating model parameters, the toolbox algorithms also estimate variability of the model parameters that result from random disturbances in the output.

Understanding model variability helps you to understand how different your model parameters would be if you repeated the estimation using a different data set (with the same input sequence as the original data set) and the same model structure.

When validating your parametric models, check the uncertainty values. Large uncertainties in the parameters might be caused by high model orders, inadequate excitation, and poor signal-to-noise ratio in the data.

Note   You can get model uncertainty data for linear parametric black-box models, and both linear and nonlinear grey-box models. Supported model objects include idproc, idpoly, idss, idarx, idgrey, idfrd, and idnlgrey.

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What Is Model Covariance?

Uncertainty in the model is called model covariance.

If you estimate model uncertainty data, this information is stored in the Model.CovarianceMatrix model property. The covariance matrix is used to compute all uncertainties in model output, Bode plots, residual plots, and pole-zero plots.

Computing the covariance matrix is based on the assumption that the model structure gives the correct description of the system dynamics. For models that include a disturbance model H, a correct uncertainty estimate assumes that the model produces white residuals. To determine whether you can trust the estimated model uncertainty values, perform residual analysis tests on your model, as described in Residual Analysis. If your model passes residual analysis tests, there is a good chance that the true system lies within the confidence interval and any parameter uncertainties results from random disturbances in the output.

In the case of output-error models, where the noise model H is fixed to 1, computing the covariance matrix does not assume that the residuals are white. Instead, the covariance is estimated based on the estimated color of the residual correlations. This estimation of the noise color is also performed for state-space models with K=0, which is equivalent to an output-error model.

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Types of Model Uncertainty Information

You can view the following uncertainty information from linear and nonlinear grey-box models:

• Uncertainties of estimated parameters.

  Type present(model) at the prompt, where model represents the name of a linear or nonlinear model.

• Confidence intervals on the linear model plots, including step-response, impulse-response, Bode, and pole-zero plots.

  Confidence intervals are computed based on the variability in the model parameters. For information about displaying confidence intervals, see the corresponding plot section.

• Covariance matrix of the estimated parameters in linear and nonlinear grey-box models.

  Type model.CovarianceMatrix at the prompt, where model represents the name of the model object.

• Estimated standard deviations of polynomial coefficients or state-space matrices

  Type model.dA at the prompt to access the estimated standard deviations of the model.A estimated property, where model represents the name of the model object, and A represents any estimated model property. In general, you prefix the name of the estimated property with a d to get the standard deviation estimate for that property. For example, to get the standard deviation value of the A polynomial in an estimated ARX model, type model.da.

  Note   State-space models, estimated with free parameterization, do not have well-defined standard deviations of the matrix elements. To display matrix parameter uncertainties in this case, first transform the model to a canonical parameterization by setting the ss model property to model.ss = 'canon'. For more information about free and canonical parameterizations, see Identifying State-Space Models.

• Simulated output values for linear models with standard deviations using the sim command.

  Call the sim command with output arguments, where the second output argument is the estimated standard deviation of each output value. For example, type [ysim,ysimsd]=sim(model,data), where ysim is the simulated output, ysimsd contains the standard deviations on the simulated output, and data is the simulation data.

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Akaike's Criteria for Model Validation

Troubleshooting Models

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