Learn more about Model-Based Calibration

Width Selection Algorithms

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TrialWidths WidPerDim Tree Regression

TrialWidths

This routine tests several width values by trying different widths. A set of trial widths equally spaced between specified initial upper and lower bounds is selected. The width with the lowest value of log10(GCV) is selected. The area around the best width is then tested in more detail — this is referred to as a zoom. Specifically, the new range of trial widths is centered on the best width found at the previous range, and the length of the interval from which the widths are selected is reduced to 2/5 of the length of the interval at the previous zoom. Before the new set of trial widths is tested, the center selection is updated to reflect the best width and found so far. This can mean that the location of the optimum width changes between zooms because of the new center locations.

Fit Parameters

Lambda selection algorithm — Midlevel fit algorithm that you test with the various trial values of . The default is IterateRidge.

Number of trial widths in each zoom — Number of trials made at each zoom. The widths tested are equally spaced between the initial upper and lower bounds. Default is 10.

Number of zooms — Number of times you zoom in. Default is 5.

Initial lower bound on width — Lower bound on the width for the first zoom. Default is 0.01.

Initial upper bound on width — Upper bound on the width for the first zoom. Default is 20.

Display — If you select this check box, a stem plot of log10(GCV) against width is plotted. The best width is marked by a green asterisk.

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WidPerDim

In the WidPerDim algorithm (Width Per Dimension), the radial basis functions are generalized. Rather than having a single width parameter, a different width in each input factor can be used; that is, the level curves are elliptical rather than circular (or spherical, with more factors). The basis functions are no longer radially symmetric.

This can be especially helpful when the amount of variability varies considerably in each input direction. This algorithm offers more flexibility than TrialWidths, but is more computationally expensive.

You can set Initial width in the RBF controls on the Global Model Setup dialog box. For most algorithms the Initial width is a single value. However, for WidPerDim (available in the Width selection algorithm pull down), you can specify a vector of widths to use as starting widths.

If supplying a vector of widths, there should be the same number as the number of global variables, and they must be in the same order as specified in the test plan. If you provide a single width, then all dimensions start off from the same initial width, but are likely to move from there to a vector of widths during model fitting.

An estimation of the time for the width per dimension algorithm is computed. This is given as a number of time units (as it depends on the machine). A time estimate of over 10 but less than 100 generates a warning. A time estimate of over 100 might take a prohibitively long amount of time (probably over five minutes on most machines). You have the option to stop execution and change some of the parameters to reduce the run time.

Fit Parameters

As for the TrialWidths algorithm.

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Tree Regression

There are three parts to the tree regression algorithm for RBFs:

• Tree building

• Alpha selection

• Center selection.

Tree Building

The tree regression algorithm builds a regression tree from the data and uses the nodes (or panels) of this tree to infer candidate centers and widths for the RBF. The root panel of the tree corresponds to a hypercube that contains all of the data points. This panel is divided into two child panels such that each child contains the same amount of variation, as much as is possible. The child panel with the most variation is then split in a similar way. This process continues until there are no panels left to split, i.e., no childless panel has more than the minimum number of data points, or until the maximum number of panels has been reached. Each panel in the tree corresponds to a candidate center and the size of the panel determines the width that goes with that vector.

The size of the child panels can be based solely on the size of the parent panel or can be determined by shrinking the child panel onto the data that it contains.

Once you have selected Radial Basis Function in the Global Model Setup dialog box, you can choose Tree Regression from the Width Selection Algorithm drop-down menu.

Click Advanced to open the Radial Basis Functions Options dialog box to reach settings such as maximum number of panels and minimum number of data points per panel. To shrink child panels to fit the data, select the check box Shrink panels to data.

Alpha Selection Algorithm

The size for the candidate widths are not taken directly from the panel sizes: we need to scale the panel sizes to get the corresponding widths. This scaling factor is called alpha. The same scaling factor needs to be applied to every panel in the tree and to determine the optimal value of alpha we use an alpha selection algorithm.

You can choose the parameter Specify Alpha to specify the exact value of alpha to use, or you can select Trial Alpha. Trial Alphais very similar to the Trial Widths algorithm. The only difference is that the trial alpha algorithm can specify how to space the values to search. Linear is the same as used by trial widths but Logarithmic searches more values near the lower range.

Click Advanced to open the Radial Basis Functions Options dialog box to reach further settings such as bounds on alpha, number of zooms and number of trial alphas. Here you can select the Display check box to see the progress of the algorithm and the values of alpha trailed.

Center Selection Algorithm

The tree building generates candidate centers and the alpha selection generates candidate widths for these centers. The center selection chooses which of those centers to use.

Generic Center Selection is a center selection algorithm that knows nothing about the tree structure to be used. It uses Rols, which is very fast way to choose centers and works in this case as well as the usual RBF cases. However, in this case the candidates for centers are not the data by the centers from the regression tree.

Tree-based center selection uses the regression tree. It is natural to use the regression tree to select centers because of the way it is built. In particular, the panel corresponding to the root node should be considered for selection before any of its children as it captures coarse detail while nodes at the leaves of the tree capture fine detail. This is what the Tree-based center selection does. You can also set the maximum number of centers.

Click Advanced to open the Radial Basis Functions Options dialog box to reach the Model selection criteria setting. The Model selection criteria determines what function should be used as a measure of how good a model is. This can be BIC (Bayesian information criterion) or GCV (generalized cross- validation). BIC is usually less susceptible to over fitting.

The Stepwise menu is the same for all RBFs, see Global Model Class: Radial Basis Function.

Tree Regression and CentreExchange are the only algorithms that permit centers that are not located at the data points. This means that you do not see centers on model plots.

If you leave the Alpha selection algorithm at the default, Trial Alpha, you will see a progress dialog box when you click OK to begin modeling. An example is shown.

This is an example progress dialog box of a Tree Regression RBF model fitting in progress. Here you can see each trial value of alpha with its calculated cost and the best number of centers with that value of alpha. The alpha value in red is the best so far. Alpha values no longer red, but in bold, are previous best values. You can then refine your model by zooming in on the best values for alpha and number of centers.

Reference: M. Orr, J. Hallam, K. Takezawa, A. Murray, S. Ninomiya, M. Oide, T. Leonard, "Combining Regression Trees and Radial Basis Function Networks," International Journal of Neural Systems, Vol. 10, No. 6 (2000) 453-465.

http://www.anc.ed.ac.uk/rbf/rbf.html

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Lambda Selection Algorithms

Prune Functionality

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