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Validating the results of online estimation can indicate low confidence in the estimation results. Try the following tips to improve the quality of the fit.
Check that you have chosen a model structure that is appropriate for the system to be estimated. Ideally, you want the simplest model structure that adequately captures the system dynamics.
Recursive least squares (RLS) estimation — Use the Recursive Least Squares Estimator block to estimate a system that is linear in the parameters to be estimated.
Suppose the inputs to the RLS block are simply the time-shifted versions of some fundamental input/output variables. You can estimate this system using an ARX model structure instead. ARX models can express the time-shifted regressors using the A and B parameters. The autoregressive term, A(q), allows representation of dynamics using fewer coefficients than an RLS model. Also, configuring an ARX structure is simpler, because you provide fewer inputs.
For example, the a and b parameters of the system y(t) = b1u(t)+b2u(t-1)-a1y(t-1) can be estimated using either recursive least squares (RLS) or ARX models. The RLS estimation requires you to provide u(t), u(t-1) and y(t-1) as regressors. An ARX model eliminates this requirement because you can express these time-shifted parameters using the A and B parameters. Therefore, you provide the Recursive Polynomial Model Estimator block only u and y. For more information regarding ARX models, see What Are Polynomial Models?.
ARX and ARMAX models — Use the Recursive Polynomial Model Estimator block to estimate ARX models (SISO and MISO) and ARMAX models (SISO). For information about these models, see What Are Polynomial Models?
The ARMAX model has more dynamic elements (C parameters) than the ARX model to express noise. However, ARMAX models are more sensitive to initial guess values than ARX models, and therefore require more careful initialization.
You can underfit (model order is too low) or overfit (model order is too high) data by choosing an incorrect model order. For example, suppose you estimate the model parameters using the Recursive Polynomial Model Estimator block and the estimated parameters underfit the data. Increasing the number of parameters to be estimated can improve the goodness of the fit. Ideally, you want the lowest-order model that adequately captures the system dynamics.
Estimation data that contains deficiencies can lead to poor estimation results. Data deficiencies include drift, offset, missing samples, equilibrium behavior, seasonalities, and outliers. It is recommended that you preprocess the estimation data as needed. For information on how to preprocess estimation data, see Preprocess Online Estimation Data.
ARMAX models are especially sensitive to the initial guess of the parameter values. Poor guesses can result in the algorithm finding a local minima of the objective function in the parameter space, which can lead to a poor fit. You can also change the initial parameter covariance matrix values. For uncertain initial guesses, use large values in the initial parameter covariance matrix.
Check that you have specified appropriate settings for the estimation algorithm. For example, for the Forgetting Factor algorithm, you must choose the forgetting factor (λ) carefully. If λ is too small, the estimation algorithm assumes that the parameter value is varying quickly with respect to time. Conversely, if λ is too large, the estimation algorithm assumes that the parameter value does not vary much with respect to time. For more information regarding the estimation algorithms, see Recursive Algorithms for Online Estimation.