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Hybrid Radial Basis Functions

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Introducing Hybrid Radial Basis Functions Width Selection Algorithm: TrialWidths Lambda and Term Selection Algorithms: Interlace Lambda and Term Selection Algorithms: Two-Step

Introducing Hybrid Radial Basis Functions

Hybrid RBFs combine a radial basis function model with more standard linear models such as polynomials or hybrid splines. The two parts are added together to form the overall model. This approach offers the ability to combine a priori knowledge, such as the expectation of quadratic behavior in one of the variables, with the nonparametric nature of RBFs.

The model setup GUI for hybrid RBFs has a top Set Up button, where you can set the fitting algorithm and options. The interface also has two tabs, one to specify the radial basis function part, and one for the linear model part.

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Width Selection Algorithm: TrialWidths

This is the same algorithm as is used in ordinary RBFs, that is, a guided search for the best width parameter.

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Lambda and Term Selection Algorithms: Interlace

This algorithm is a generalization of StepItRols for RBFs. The algorithm chooses radial basis functions and linear model terms in an interlaced way, rather than in two steps. At each step a forward search procedure is performed to select the radial basis function (with a center chosen from within the set of data points) or the linear model term (chosen from the ones specified in the linear model setup pane) that decreases the regularized error the most. This process continues until the maximum number of terms is chosen. The first few terms are added using the stored value of lambda, until the Number of terms to add before updating has been reached. Subsequently lambda is iterated after each center is added to improve GCV.

The fit options for this algorithm are as follows:

• Maximum number of terms: Maximum number of terms that will be chosen. The default is the number of data points.

• Maximum number of centers: Maximum number of terms that can be radial basis functions. The default is a quarter of the data points, or 25, whichever is smaller.

  Note   The maximum number of terms used is a combination of the maximum number of centers and the number of linear model terms. It is limited as follows: Maximum number of terms used = Minimum(Maximum number of terms, Maximum number of centers + number of linear model terms) As a result of this, the model may have more centers than specified in Maximum number of centers, but there will always be fewer terms than (Maximum number of centers + number of linear model terms). You can view the number of possible linear model terms on the Linear Part tab of the Global Model Setup dialog box (Total number of terms).

• Percentage of data to be candidate centers: Percentage of the data points that are available to be chosen as centers. The default is 100% when the number of data points is 200.

• Number of terms to add before updating: How many terms to add before updating lambda begins.

• Minimum change in log10(GCV): Tolerance.

• Maximum no. times log10(GCV) change is minimal: Number of steps in a row that the change in log10(GCV) can be less than the tolerance before the algorithm terminates.

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Lambda and Term Selection Algorithms: Two-Step

This algorithm starts by fitting the linear model specified in the linear model pane, and then fits a radial basis function network to the residual. You can specify the linear model terms to include in the usual way using the term selector. If desired, you can activate the stepwise options. In this case, after the linear model part is fitted, some of the terms are automatically added or removed before the RBF part is fitted. You can choose the algorithm and options that are used to fit the nonlinear parameters of the RBF by clicking the Set Up button in the RBF training options.

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Statistics

Tips for Modeling with Radial Basis Functions

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