Learn more about Model-Based Calibration

Tips for Modeling with Radial Basis Functions

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Plan of Attack How Many RBFs to Use Width Selection Algorithms Which RBF to Use Lambda Selection Algorithms Center Selection Algorithms General Parameter Fine-Tuning Hybrid RBFs How to Find RBF Model Formula

Plan of Attack

Determine which parameters have the most impact on the fit by following these steps:

1. Fit the default RBF. Remove any obvious outliers.

2. Get a rough idea of how many RBFs are going to be needed. If a center coincides with a data point, it is marked with a magenta asterisk on the Predicted/Observed plot. You can view the location of the centers in graphical and table format by using the View Centers toolbar button . If you remove an outlier which coincided with a center (marked with an asterisk), refit by clicking Update Fit in the toolbar.

3. Try with more than one kernel. You can alter the parameters in the fit by clicking the Set Up button in the Model Selection dialog box.

4. Decide on the main width selection algorithm. Try with both TrialWidths and WidPerDim algorithms.

5. Determine which types of kernel look most hopeful.

6. Narrow the corresponding width range to search over.

7. Decide on the center selection algorithm.

8. Decide on the lambda-selection algorithm.

9. Try changing the parameters in the algorithms.

10. If any points appear to be possible outliers, try fitting the model both with and without those points.

If at any stage you decide on a change that has a big impact (such as removal of an outlier), then you should repeat the previous steps to determine if this would affect the path you have chosen.

See Fitting Routines for details on all the fit parameters.

The Model Browser has a quick option for comparing all the different RBF kernels and trying a variety of numbers of centers.

1. After fitting the default RBF, select the RBF global model in the model tree.

2. Click the Build Models toolbar icon .

3. Select the RBF icon in the Build Models dialog box that appears and click OK.

4. The Model Building Options dialog box appears. You can specify a range of values for the maximum number of centers, and click Model settings to change any other model settings. The defaults used are the same as the parent RBF model type.

5. You can select the check box to Build all kernels to create models with the specified range of centers for each kernel type as a selection of child nodes of the current RBF model.

   Note this can take a long time for local models as you will create alternative models with a range of centers for each kernel type for each response feature; once model building is started you can always click Stop to abort if the process is taking too long.

6. Click Build to create the specified models.

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How Many RBFs to Use

• The main parameter that you must adjust in order to get a good fit with an RBF is the maximum number of centers. This is a parameter of the center selection algorithm, and is the maximum number of centers/RBFs that is chosen.

• Usually the maximum number of centers is the number of RBFs that are actually selected. However, sometimes fewer RBFs are chosen because the (regularized) error has fallen below the tolerance before the maximum was reached.

• You should use a number of RBFs that is significantly less than the number of data points, otherwise there are not enough degrees of freedom in the error to estimate the predictive quality of the model. That is, you cannot tell if the model is useful if you use too many RBFs. We would recommend an upper bound of 60% on the ratio of number of RBFs to number of data points. Having 80 centers when there are only 100 data points might seem to give a good value of PRESS, but when it comes to validation, it can sometimes become clear that the data has been overfitted, and the predictive capability is not as good as PRESS would suggest.

• One strategy for choosing the number of RBFs is to fit more centers than you think is needed (say 70 out of 100), then use the Prune toolbar button to reduce the number of centers in the model. After pruning the network, make a note of the reduced number of RBFs. Try fitting the model again with the maximum number of centers set to this reduced number. This recalculates the values of the nonlinear parameters (width and lambda) to be optimal for the reduced number of RBFs.

• One strategy for the use of Stepwise is to use it to minimize PRESS as a final fine-tuning for the network, once pruning has been done. Whereas Prune only allows the last RBF introduced to be removed, Stepwise allows any RBF to be taken out.

• Do not focus solely on PRESS as a measure of goodness of fit, especially at large ratios of RBFs to data points. Take log10(GCV) into account also.

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Width Selection Algorithms

• Try both TrialWidths and WidPerDim. The second algorithm offers more flexibility, but is more computationally expensive. View the width values in each direction to see if there is significant difference, to see whether it is worth focusing effort on elliptical basis functions (use the View Model toolbar button ).

  

• If with a variety of basis functions the widths do not vary significantly between the dimensions, and the PRESS/GCV values are not significantly improved using WidPerDim over TrialWidths, then focus on TrialWidths, and just return to WidPerDim to fine-tune in the final stages.

• Turn the Display option on in TrialWidths to see the progress of the algorithm. Watch for alternative regions within the width range that have been prematurely neglected. The output log10(GCV) in the final zoom should be similar for each of the widths tried; that is, the output should be approximately flat. If this is not the case, try increasing the number of zooms.

• In TrialWidths, for each type of RBF, try to narrow the initial range of widths to search over. This might allow the number of zooms to be reduced.

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Which RBF to Use

• It is hard to give rules of thumb on how to select the best RBF, as the best choice is highly data-dependent. The best guideline is to try all of them with both top-level algorithms (TrialWidths and WidPerDim) and with a sensible number of centers, compare the PRESS/GCV values, then focus on the ones that look most hopeful.

• If multiquadrics and thin-plate splines give poor results, it is worth trying them in combination with low-order polynomials as a hybrid spline. Try supplementing multiquadrics with a constant term and thin-plate splines with linear (order 1) terms. See Hybrid Radial Basis Functions.

• Watch out for conditioning problems with Gaussian kernels (say condition number > 10^8).

• Watch out for strange results with Wendland's functions when the ratio of the number of parameters to the number of observations is high. When these functions have a very small width, each basis function only contributes to the fit at one data point. This is because its support only encompasses the one basis function that is its center. The residuals will be zero at each of the data points chosen as a center, and large at the other data points. This scenario can indicate good RMSE values, but the predictive quality of the network will be poor.

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Lambda Selection Algorithms

Lambda is the regularization parameter.

• IterateRols updates the centers after each update of lambda. This makes it more computationally intensive, but potentially leads to a better combination of lambda and centers.

• StepItRols is sensitive to the setting of Number of centers to add before updating. Switch the Display option on to view how log10(GCV) reduces as the number of centers builds up.

• Examine the plots produced from the lambda selection algorithm, ignoring the warning "An excessive number of plots will be produced." Would increasing the tolerance or the number of initial test values for lambda lead to a better choice of lambda?

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Center Selection Algorithms

• On most problems, Rols seems to be the most effective.

• If less than the maximum number of centers are being chosen, and you want to force the selection of the maximum number, reduce the tolerance to epsilon (eps).

• CenterExchange is very expensive, and you should not use this on large problems. In this case, the other center selection algorithms that restrict the centers to be a subset of the data points might not offer sufficient flexibility.

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General Parameter Fine-Tuning

• Try Stepwise after pruning, then update the model fit with the new maximum number of centers set to the number of terms left after Stepwise.

• Update the model fit after removal of outliers; use the toolbar button.

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Hybrid RBFs

• Go to the linear part pane and specify the polynomial or spline terms that you expect to see in the model.

Fitting too many non-RBF terms is made evident by a large value of lambda, indicating that the underlying trends are being taken care of by the linear part. In this case, you should reset the starting value of lambda (to say 0.001) before the next fit.

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How to Find RBF Model Formula

With any model you can use the View Model toolbar button or View > Model Definition (or keyboard shortcut CTRL+V) to see the details of the current model. The Model Viewer dialog box appears. Here for any RBF model you can see the kernel type, number of centers, width and regularization parameter.

However to specify the formula of an RBF model completely, you also need to give the locations of the centers, and the height of each basis function. The center location information is available in the "View Centers" dialog box and the coefficients can be found in the "Stepwise" window. Note these values are all in coded units.

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Hybrid Radial Basis Functions

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