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Computing Model Uncertainty

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, idtf, idgrey, idfrd, and idnlgrey.

What Is Model Covariance?

Uncertainty in the model is called model covariance.

When you estimate a model, the covariance matrix of the estimated parameters is stored with the model. Use getcov to fetch the covariance matrix. Use getpvec to fetch the list of parameters and their individual uncertainties that have been computed using the covariance matrix. 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. For more details about residual analysis, see the topics on the Residual Analysis page. 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.

For output-error models, such as transfer function models, state-space with K=0 and polynomial models of output-error form, with the noise model H fixed to 1, the covariance matrix computation does not assume white residuals. 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.

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, Nyquist, noise spectrum and pole-zero plots.

    Confidence intervals are computed based on the variability in the model parameters. For information about displaying confidence intervals, see Definition of Confidence Interval for Specific Model Plots.

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

  • Estimated standard deviations of polynomial coefficients, poles/zeros, or state-space matrices using idssdata, tfdata, zpkdata, and polydata.

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

    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.

  • Perform Monte-Carlo analysis using rsample to generate a random sampling of an identified model in a given confidence region. An array of identified systems of the same structure as the input system is returned. The parameters of the returned models are perturbed about their nominal values in a way that is consistent with the parameter covariance.

  • Simulate the effect of parameter uncertainties on a model's response using simsd.

Definition of Confidence Interval for Specific Model Plots

You can display the confidence interval on the following plot types:

Plot TypeConfidence Interval Corresponds to the Range of ...More Information on Displaying Confidence Interval
Simulated and Predicted OutputOutput values with a specific probability of being the actual output of the system.Model Output Plots
ResidualsResidual values with a specific probability of being statistically insignificant for the system. Residuals Plots
Impulse and StepResponse values with a specific probability of being the actual response of the system.Impulse and Step Plots
Frequency ResponseResponse values with a specific probability of being the actual response of the system.Frequency Response Plots
Noise SpectrumPower-spectrum values with a specific probability of being the actual noise spectrum of the system.Noise Spectrum Plots
Poles and ZerosPole or zero values with a specific probability of being the actual pole or zero of the system. Pole-Zero Plots

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