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Lambda Selection Algorithms

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IterateRidge IterateRols StepItRols

Lambda is the regularization parameter.

IterateRidge

For a specified width, this algorithm optimizes the regularization parameter with respect to the GCV criterion (generalized cross-validation; see the discussion under GCV criterion).

The initial centers either are selected by one of the low-level center selection algorithms or the previous choice of centers is used (see discussion under the parameter Do not reselect centers). You can select an initial start value for by testing an initial number of values for lambda (set by the user) that are equally spaced on a logarithmic scale between 10^-10 and 10 and choosing the one with the best GCV score. This helps avoid falling into local minima on the GCV - curve. The parameter is then iterated to try to minimize GCV using the formulas given in the GCV criterion section. The iteration stops when either the maximum number of updates is reached or the log10(GCV) value changes by less than the tolerance.

Fit Parameters

Center selection algorithm — The center selection algorithm to use.

Maximum number of updates — Maximum number of times that the update of is made. The default is 10.

Minimum change in log10(GCV) — Tolerance. This defines the stopping criterion for iterating ; the update stops when the difference in the log10(GCV) value is less than the tolerance. The default is 0.005.

Number of initial test values for lambda — Number of test values of to determine a starting value for . Setting this parameter to 0 means that the best so far is used.

Do not reselect centers for new width — This check box determines whether the centers are reselected for the new width value, and after each lambda update, or if the best centers to date are to be used. It is cheaper to keep the best centers found so far, and often this is sufficient, but it can cause premature convergence to a particular set of centers.

Display — When you select this check box, this algorithm plots the results of the algorithm. The starting point for is marked with a black circle. As is updated, the new values are plotted as red crosses connected with red lines. The best found is marked with a green asterisk.

If too many graphs are likely to be produced, because of the Display check box being activated here, a warning is generated, and you have the option to stop execution.

A lower bound of 10^-12 is placed on , and an upper bound of 10.

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IterateRols

For a specified width, this algorithm optimizes the regularization parameter in the Rols algorithm with respect to the GCV criterion. An initial fit and the centers are selected by Rols using the user-supplied . As in IterateRidge, you select an initial start value for by testing an initial number of start values for lambda that are equally spaced on a logarithmic scale between 10^-10 and 10, and choosing the one with the best GCV score.

is then iterated to improve GCV. Each time that is updated, the center selection process is repeated. This means that IterateRols is much more computationally expensive than IterateRidge.

A lower bound of 10^-12 is placed on , and an upper bound of 10.

Fit Parameters

Center selection algorithm — The center selection algorithm to use. For IterateRols the only center selection algorithm available is Rols.

Maximum number of updates — The same as for IterateRidge.

Minimum change in log10(GCV) — The same as for IterateRidge.

Number of initial test values for lambda — The same as for IterateRidge.

Do not reselect centers for new width — This check box determines whether the centers are reselected for the new width value or if the best centers to date are to be used.

Display — When you select this check box, this algorithm plots the results of the algorithm. The starting point for is marked with a black circle.

As the above figure is updated, the new values are plotted as red crosses connected with red lines. The best found is marked with a green asterisk.

If too many graphs are likely to be produced, because of the Display check box being activated here, a warning is generated, and you have the option to stop execution.

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StepItRols

This algorithm combines the center-selection and lambda-selection processes. Rather than waiting until all centers are selected before is updated (as with the other lambda-selection algorithms), this algorithm offers the ability to update after each center is selected. It is a forward selection algorithm that, like Rols, selects centers on the basis of regularized error reduction. The stopping criterion for StepItRols is on the basis of log10(GCV) changing by less than the tolerance more than a specified number of times in a row (given in the parameter Maximum number of times log10(GCV) change is minimal). Once the addition of centers has stopped, the intermediate fit with the smallest log10(GCV) is selected. This can involve removing some of the centers that entered late in the algorithm.

Fit Parameters

Maximum number of centers — As in the Rols algorithm.

Percentage of data to candidate centers — As in the Rols algorithm.

Number of centers to add before updating — How many centers are selected before iterating begins.

Minimum change in log10(GCV) — Tolerance. It should be a positive number between 0 and 1. The default is 0.005.

Maximum number of times log10(GCV) change is minimal — Controls how many centers are selected before the algorithm stops. The default is 5. Left at the default, the center selection stops when the log10(GCV) values change by less than the tolerance five times in a row.

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Center Selection Algorithms

Width Selection Algorithms

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